diff options
Diffstat (limited to 'mathcomp/ssreflect')
22 files changed, 31030 insertions, 0 deletions
diff --git a/mathcomp/ssreflect/Make b/mathcomp/ssreflect/Make new file mode 100644 index 0000000..9e7c5db --- /dev/null +++ b/mathcomp/ssreflect/Make @@ -0,0 +1,8 @@ +all.v +eqtype.v +seq.v +ssrbool.v +ssreflect.v +ssrfun.v +ssrmatching.v +ssrnat.v diff --git a/mathcomp/ssreflect/Makefile b/mathcomp/ssreflect/Makefile new file mode 100644 index 0000000..46a2d69 --- /dev/null +++ b/mathcomp/ssreflect/Makefile @@ -0,0 +1,33 @@ +ifeq "$(COQBIN)" "" +COQBIN=$(dir $(shell which coqtop))/ +endif + + +ifeq "$(shell $(COQBIN)/coqtop -v | head -1 | grep trunk | wc -l)" "1" +V=trunk +else +V=$(shell $(COQBIN)/coqtop -v | head -1 | \ + sed 's/.*version \([0-9]\.[0-9]\)[^ ]* .*/v\1/') +endif + +OLD_MAKEFLAGS:=$(MAKEFLAGS) +MAKEFLAGS+=-B + +%: + @[ -e Makefile.coq ] || $(call coqmakefile) + @[ Make -ot Makefile.coq ] || $(call coqmakefile) + @MAKEFLAGS=$(OLD_MAKEFLAGS) $(MAKE) --no-print-directory \ + -f Makefile.coq $* + +define coqmakefile + (echo "Generating Makefile.coq for Coq $(V) with COQBIN=$(COQBIN)";\ + cp Make Make.coq;\ + echo -I plugin/$(V)/src >> Make.coq;\ + echo plugin/$(V)/src/ssreflect.mllib >> Make.coq;\ + echo plugin/$(V)/src/ssrmatching.mli >> Make.coq;\ + echo plugin/$(V)/src/ssrmatching.ml4 >> Make.coq;\ + echo plugin/$(V)/src/ssreflect.ml4 >> Make.coq;\ + $(COQBIN)/coq_makefile -f Make.coq -o Makefile.coq) +endef + + diff --git a/mathcomp/ssreflect/all.v b/mathcomp/ssreflect/all.v new file mode 100644 index 0000000..c441810 --- /dev/null +++ b/mathcomp/ssreflect/all.v @@ -0,0 +1,7 @@ +Require Export eqtype. +Require Export seq. +Require Export ssrbool. +Require Export ssreflect. +Require Export ssrfun. +Require Export ssrmatching. +Require Export ssrnat. diff --git a/mathcomp/ssreflect/eqtype.v b/mathcomp/ssreflect/eqtype.v new file mode 100644 index 0000000..01b0f5e --- /dev/null +++ b/mathcomp/ssreflect/eqtype.v @@ -0,0 +1,860 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Require Import ssreflect ssrfun ssrbool. + +(******************************************************************************) +(* This file defines two "base" combinatorial interfaces: *) +(* eqType == the structure for types with a decidable equality. *) +(* Equality mixins can be made Canonical to allow generic *) +(* folding of equality predicates. *) +(* subType p == the structure for types isomorphic to {x : T | p x} with *) +(* p : pred T for some type T. *) +(* The eqType interface supports the following operations: *) +(* x == y <=> x compares equal to y (this is a boolean test). *) +(* x == y :> T <=> x == y at type T. *) +(* x != y <=> x and y compare unequal. *) +(* x != y :> T <=> " " " " at type T. *) +(* x =P y :: a proof of reflect (x = y) (x == y); this coerces *) +(* to x == y -> x = y. *) +(* comparable T <-> equality on T is decidable *) +(* := forall x y : T, decidable (x = y) *) +(* comparableClass compT == eqType mixin/class for compT : comparable T. *) +(* pred1 a == the singleton predicate [pred x | x == a]. *) +(* pred2, pred3, pred4 == pair, triple, quad predicates. *) +(* predC1 a == [pred x | x != a]. *) +(* [predU1 a & A] == [pred x | (x == a) || (x \in A)]. *) +(* [predD1 A & a] == [pred x | x != a & x \in A]. *) +(* predU1 a P, predD1 P a == applicative versions of the above. *) +(* frel f == the relation associated with f : T -> T. *) +(* := [rel x y | f x == y]. *) +(* invariant k f == elements of T whose k-class is f-invariant. *) +(* := [pred x | k (f x) == k x] with f : T -> T. *) +(* [fun x : T => e0 with a1 |-> e1, .., a_n |-> e_n] *) +(* [eta f with a1 |-> e1, .., a_n |-> e_n] == *) +(* the auto-expanding function that maps x = a_i to e_i, and other values *) +(* of x to e0 (resp. f x). In the first form the `: T' is optional and x *) +(* can occur in a_i or e_i. *) +(* Equality on an eqType is proof-irrelevant (lemma eq_irrelevance). *) +(* The eqType interface is implemented for most standard datatypes: *) +(* bool, unit, void, option, prod (denoted A * B), sum (denoted A + B), *) +(* sig (denoted {x | P}), sigT (denoted {i : I & T}). We also define *) +(* tagged_as u v == v cast as T_(tag u) if tag v == tag u, else u. *) +(* -> We have u == v <=> (tag u == tag v) && (tagged u == tagged_as u v). *) +(* The subType interface supports the following operations: *) +(* val == the generic injection from a subType S of T into T. *) +(* For example, if u : {x : T | P}, then val u : T. *) +(* val is injective because P is proof-irrelevant (P is in bool, *) +(* and the is_true coercion expands to P = true). *) +(* valP == the generic proof of P (val u) for u : subType P. *) +(* Sub x Px == the generic constructor for a subType P; Px is a proof of P x *) +(* and P should be inferred from the expected return type. *) +(* insub x == the generic partial projection of T into a subType S of T. *) +(* This returns an option S; if S : subType P then *) +(* insub x = Some u with val u = x if P x, *) +(* None if ~~ P x *) +(* The insubP lemma encapsulates this dichotomy. *) +(* P should be infered from the expected return type. *) +(* innew x == total (non-option) variant of insub when P = predT. *) +(* {? x | P} == option {x | P} (syntax for casting insub x). *) +(* insubd u0 x == the generic projection with default value u0. *) +(* := odflt u0 (insub x). *) +(* insigd A0 x == special case of insubd for S == {x | x \in A}, where A0 is *) +(* a proof of x0 \in A. *) +(* insub_eq x == transparent version of insub x that expands to Some/None *) +(* when P x can evaluate. *) +(* The subType P interface is most often implemented using one of: *) +(* [subType for S_val] *) +(* where S_val : S -> T is the first projection of a type S isomorphic to *) +(* {x : T | P}. *) +(* [newType for S_val] *) +(* where S_val : S -> T is the projection of a type S isomorphic to *) +(* wrapped T; in this case P must be predT. *) +(* [subType for S_val by Srect], [newType for S_val by Srect] *) +(* variants of the above where the eliminator is explicitly provided. *) +(* Here S no longer needs to be syntactically identical to {x | P x} or *) +(* wrapped T, but it must have a derived constructor S_Sub statisfying an *) +(* eliminator Srect identical to the one the Coq Inductive command would *) +(* have generated, and S_val (S_Sub x Px) (resp. S_val (S_sub x) for the *) +(* newType form) must be convertible to x. *) +(* variant of the above when S is a wrapper type for T (so P = predT). *) +(* [subType of S], [subType of S for S_val] *) +(* clones the canonical subType structure for S; if S_val is specified, *) +(* then it replaces the inferred projector. *) +(* Subtypes inherit the eqType structure of their base types; the generic *) +(* structure should be explicitly instantiated using the *) +(* [eqMixin of S by <:] *) +(* construct to declare the Equality mixin; this pattern is repeated for all *) +(* the combinatorial interfaces (Choice, Countable, Finite). *) +(* More generally, the eqType structure can be transfered by (partial) *) +(* injections, using: *) +(* InjEqMixin injf == an Equality mixin for T, using an f : T -> eT where *) +(* eT has an eqType structure and injf : injective f. *) +(* PcanEqMixin fK == an Equality mixin similarly derived from f and a left *) +(* inverse partial function g and fK : pcancel f g. *) +(* CanEqMixin fK == an Equality mixin similarly derived from f and a left *) +(* inverse function g and fK : cancel f g. *) +(* We add the following to the standard suffixes documented in ssrbool.v: *) +(* 1, 2, 3, 4 -- explicit enumeration predicate for 1 (singleton), 2, 3, or *) +(* 4 values. *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Module Equality. + +Definition axiom T (e : rel T) := forall x y, reflect (x = y) (e x y). + +Structure mixin_of T := Mixin {op : rel T; _ : axiom op}. +Notation class_of := mixin_of (only parsing). + +Section ClassDef. + +Structure type := Pack {sort; _ : class_of sort; _ : Type}. +Local Coercion sort : type >-> Sortclass. +Variables (T : Type) (cT : type). + +Definition class := let: Pack _ c _ := cT return class_of cT in c. + +Definition pack c := @Pack T c T. +Definition clone := fun c & cT -> T & phant_id (pack c) cT => pack c. + +End ClassDef. + +Module Exports. +Coercion sort : type >-> Sortclass. +Notation eqType := type. +Notation EqMixin := Mixin. +Notation EqType T m := (@pack T m). +Notation "[ 'eqMixin' 'of' T ]" := (class _ : mixin_of T) + (at level 0, format "[ 'eqMixin' 'of' T ]") : form_scope. +Notation "[ 'eqType' 'of' T 'for' C ]" := (@clone T C _ idfun id) + (at level 0, format "[ 'eqType' 'of' T 'for' C ]") : form_scope. +Notation "[ 'eqType' 'of' T ]" := (@clone T _ _ id id) + (at level 0, format "[ 'eqType' 'of' T ]") : form_scope. +End Exports. + +End Equality. +Export Equality.Exports. + +Definition eq_op T := Equality.op (Equality.class T). + +Lemma eqE T x : eq_op x = Equality.op (Equality.class T) x. +Proof. by []. Qed. + +Lemma eqP T : Equality.axiom (@eq_op T). +Proof. by case: T => ? []. Qed. +Implicit Arguments eqP [T x y]. + +Delimit Scope eq_scope with EQ. +Open Scope eq_scope. + +Notation "x == y" := (eq_op x y) + (at level 70, no associativity) : bool_scope. +Notation "x == y :> T" := ((x : T) == (y : T)) + (at level 70, y at next level) : bool_scope. +Notation "x != y" := (~~ (x == y)) + (at level 70, no associativity) : bool_scope. +Notation "x != y :> T" := (~~ (x == y :> T)) + (at level 70, y at next level) : bool_scope. +Notation "x =P y" := (eqP : reflect (x = y) (x == y)) + (at level 70, no associativity) : eq_scope. +Notation "x =P y :> T" := (eqP : reflect (x = y :> T) (x == y :> T)) + (at level 70, y at next level, no associativity) : eq_scope. + +Prenex Implicits eq_op eqP. + +Lemma eq_refl (T : eqType) (x : T) : x == x. Proof. exact/eqP. Qed. +Notation eqxx := eq_refl. + +Lemma eq_sym (T : eqType) (x y : T) : (x == y) = (y == x). +Proof. exact/eqP/eqP. Qed. + +Hint Resolve eq_refl eq_sym. + +Section Contrapositives. + +Variable T : eqType. +Implicit Types (A : pred T) (b : bool) (x : T). + +Lemma contraTeq b x y : (x != y -> ~~ b) -> b -> x = y. +Proof. by move=> imp hyp; apply/eqP; apply: contraTT hyp. Qed. + +Lemma contraNeq b x y : (x != y -> b) -> ~~ b -> x = y. +Proof. by move=> imp hyp; apply/eqP; apply: contraNT hyp. Qed. + +Lemma contraFeq b x y : (x != y -> b) -> b = false -> x = y. +Proof. by move=> imp /negbT; apply: contraNeq. Qed. + +Lemma contraTneq b x y : (x = y -> ~~ b) -> b -> x != y. +Proof. by move=> imp; apply: contraTN => /eqP. Qed. + +Lemma contraNneq b x y : (x = y -> b) -> ~~ b -> x != y. +Proof. by move=> imp; apply: contraNN => /eqP. Qed. + +Lemma contraFneq b x y : (x = y -> b) -> b = false -> x != y. +Proof. by move=> imp /negbT; apply: contraNneq. Qed. + +Lemma contra_eqN b x y : (b -> x != y) -> x = y -> ~~ b. +Proof. by move=> imp /eqP; apply: contraL. Qed. + +Lemma contra_eqF b x y : (b -> x != y) -> x = y -> b = false. +Proof. by move=> imp /eqP; apply: contraTF. Qed. + +Lemma contra_eqT b x y : (~~ b -> x != y) -> x = y -> b. +Proof. by move=> imp /eqP; apply: contraLR. Qed. + +Lemma contra_eq x1 y1 x2 y2 : (x2 != y2 -> x1 != y1) -> x1 = y1 -> x2 = y2. +Proof. by move=> imp /eqP; apply: contraTeq. Qed. + +Lemma contra_neq x1 y1 x2 y2 : (x2 = y2 -> x1 = y1) -> x1 != y1 -> x2 != y2. +Proof. by move=> imp; apply: contraNneq => /imp->. Qed. + +Lemma memPn A x : reflect {in A, forall y, y != x} (x \notin A). +Proof. +apply: (iffP idP) => [notDx y | notDx]; first by apply: contraTneq => ->. +exact: contraL (notDx x) _. +Qed. + +Lemma memPnC A x : reflect {in A, forall y, x != y} (x \notin A). +Proof. by apply: (iffP (memPn A x)) => A'x y /A'x; rewrite eq_sym. Qed. + +Lemma ifN_eq R x y vT vF : x != y -> (if x == y then vT else vF) = vF :> R. +Proof. exact: ifN. Qed. + +Lemma ifN_eqC R x y vT vF : x != y -> (if y == x then vT else vF) = vF :> R. +Proof. by rewrite eq_sym; apply: ifN. Qed. + +End Contrapositives. + +Implicit Arguments memPn [T A x]. +Implicit Arguments memPnC [T A x]. + +Theorem eq_irrelevance (T : eqType) x y : forall e1 e2 : x = y :> T, e1 = e2. +Proof. +pose proj z e := if x =P z is ReflectT e0 then e0 else e. +suff: injective (proj y) by rewrite /proj => injp e e'; apply: injp; case: eqP. +pose join (e : x = _) := etrans (esym e). +apply: can_inj (join x y (proj x (erefl x))) _. +by case: y /; case: _ / (proj x _). +Qed. + +Corollary eq_axiomK (T : eqType) (x : T) : all_equal_to (erefl x). +Proof. move=> eq_x_x; exact: eq_irrelevance. Qed. + +(* We use the module system to circumvent a silly limitation that *) +(* forbids using the same constant to coerce to different targets. *) +Module Type EqTypePredSig. +Parameter sort : eqType -> predArgType. +End EqTypePredSig. +Module MakeEqTypePred (eqmod : EqTypePredSig). +Coercion eqmod.sort : eqType >-> predArgType. +End MakeEqTypePred. +Module Export EqTypePred := MakeEqTypePred Equality. + +Lemma unit_eqP : Equality.axiom (fun _ _ : unit => true). +Proof. by do 2!case; left. Qed. + +Definition unit_eqMixin := EqMixin unit_eqP. +Canonical unit_eqType := Eval hnf in EqType unit unit_eqMixin. + +(* Comparison for booleans. *) + +(* This is extensionally equal, but not convertible to Bool.eqb. *) +Definition eqb b := addb (~~ b). + +Lemma eqbP : Equality.axiom eqb. +Proof. by do 2!case; constructor. Qed. + +Canonical bool_eqMixin := EqMixin eqbP. +Canonical bool_eqType := Eval hnf in EqType bool bool_eqMixin. + +Lemma eqbE : eqb = eq_op. Proof. by []. Qed. + +Lemma bool_irrelevance (x y : bool) (E E' : x = y) : E = E'. +Proof. exact: eq_irrelevance. Qed. + +Lemma negb_add b1 b2 : ~~ (b1 (+) b2) = (b1 == b2). +Proof. by rewrite -addNb. Qed. + +Lemma negb_eqb b1 b2 : (b1 != b2) = b1 (+) b2. +Proof. by rewrite -addNb negbK. Qed. + +Lemma eqb_id b : (b == true) = b. +Proof. by case: b. Qed. + +Lemma eqbF_neg b : (b == false) = ~~ b. +Proof. by case: b. Qed. + +Lemma eqb_negLR b1 b2 : (~~ b1 == b2) = (b1 == ~~ b2). +Proof. by case: b1; case: b2. Qed. + +(* Equality-based predicates. *) + +Notation xpred1 := (fun a1 x => x == a1). +Notation xpred2 := (fun a1 a2 x => (x == a1) || (x == a2)). +Notation xpred3 := (fun a1 a2 a3 x => [|| x == a1, x == a2 | x == a3]). +Notation xpred4 := + (fun a1 a2 a3 a4 x => [|| x == a1, x == a2, x == a3 | x == a4]). +Notation xpredU1 := (fun a1 (p : pred _) x => (x == a1) || p x). +Notation xpredC1 := (fun a1 x => x != a1). +Notation xpredD1 := (fun (p : pred _) a1 x => (x != a1) && p x). + +Section EqPred. + +Variable T : eqType. + +Definition pred1 (a1 : T) := SimplPred (xpred1 a1). +Definition pred2 (a1 a2 : T) := SimplPred (xpred2 a1 a2). +Definition pred3 (a1 a2 a3 : T) := SimplPred (xpred3 a1 a2 a3). +Definition pred4 (a1 a2 a3 a4 : T) := SimplPred (xpred4 a1 a2 a3 a4). +Definition predU1 (a1 : T) p := SimplPred (xpredU1 a1 p). +Definition predC1 (a1 : T) := SimplPred (xpredC1 a1). +Definition predD1 p (a1 : T) := SimplPred (xpredD1 p a1). + +Lemma pred1E : pred1 =2 eq_op. Proof. move=> x y; exact: eq_sym. Qed. + +Variables (T2 : eqType) (x y : T) (z u : T2) (b : bool). + +Lemma predU1P : reflect (x = y \/ b) ((x == y) || b). +Proof. apply: (iffP orP) => [] []; by [right | move/eqP; left]. Qed. + +Lemma pred2P : reflect (x = y \/ z = u) ((x == y) || (z == u)). +Proof. by apply: (iffP orP) => [] [] /eqP; by [left | right]. Qed. + +Lemma predD1P : reflect (x <> y /\ b) ((x != y) && b). +Proof. by apply: (iffP andP)=> [] [] // /eqP. Qed. + +Lemma predU1l : x = y -> (x == y) || b. +Proof. by move->; rewrite eqxx. Qed. + +Lemma predU1r : b -> (x == y) || b. +Proof. by move->; rewrite orbT. Qed. + +Lemma eqVneq : {x = y} + {x != y}. +Proof. by case: eqP; [left | right]. Qed. + +End EqPred. + +Implicit Arguments predU1P [T x y b]. +Implicit Arguments pred2P [T T2 x y z u]. +Implicit Arguments predD1P [T x y b]. +Prenex Implicits pred1 pred2 pred3 pred4 predU1 predC1 predD1 predU1P. + +Notation "[ 'predU1' x & A ]" := (predU1 x [mem A]) + (at level 0, format "[ 'predU1' x & A ]") : fun_scope. +Notation "[ 'predD1' A & x ]" := (predD1 [mem A] x) + (at level 0, format "[ 'predD1' A & x ]") : fun_scope. + +(* Lemmas for reflected equality and functions. *) + +Section EqFun. + +Section Exo. + +Variables (aT rT : eqType) (D : pred aT) (f : aT -> rT) (g : rT -> aT). + +Lemma inj_eq : injective f -> forall x y, (f x == f y) = (x == y). +Proof. by move=> inj_f x y; apply/eqP/eqP=> [|-> //]; exact: inj_f. Qed. + +Lemma can_eq : cancel f g -> forall x y, (f x == f y) = (x == y). +Proof. move/can_inj; exact: inj_eq. Qed. + +Lemma bij_eq : bijective f -> forall x y, (f x == f y) = (x == y). +Proof. move/bij_inj; apply: inj_eq. Qed. + +Lemma can2_eq : cancel f g -> cancel g f -> forall x y, (f x == y) = (x == g y). +Proof. by move=> fK gK x y; rewrite -{1}[y]gK; exact: can_eq. Qed. + +Lemma inj_in_eq : + {in D &, injective f} -> {in D &, forall x y, (f x == f y) = (x == y)}. +Proof. by move=> inj_f x y Dx Dy; apply/eqP/eqP=> [|-> //]; exact: inj_f. Qed. + +Lemma can_in_eq : + {in D, cancel f g} -> {in D &, forall x y, (f x == f y) = (x == y)}. +Proof. by move/can_in_inj; exact: inj_in_eq. Qed. + +End Exo. + +Section Endo. + +Variable T : eqType. + +Definition frel f := [rel x y : T | f x == y]. + +Lemma inv_eq f : involutive f -> forall x y : T, (f x == y) = (x == f y). +Proof. by move=> fK; exact: can2_eq. Qed. + +Lemma eq_frel f f' : f =1 f' -> frel f =2 frel f'. +Proof. by move=> eq_f x y; rewrite /= eq_f. Qed. + +End Endo. + +Variable aT : Type. + +(* The invariant of an function f wrt a projection k is the pred of points *) +(* that have the same projection as their image. *) + +Definition invariant (rT : eqType) f (k : aT -> rT) := + [pred x | k (f x) == k x]. + +Variables (rT1 rT2 : eqType) (f : aT -> aT) (h : rT1 -> rT2) (k : aT -> rT1). + +Lemma invariant_comp : subpred (invariant f k) (invariant f (h \o k)). +Proof. by move=> x eq_kfx; rewrite /= (eqP eq_kfx). Qed. + +Lemma invariant_inj : injective h -> invariant f (h \o k) =1 invariant f k. +Proof. move=> inj_h x; exact: (inj_eq inj_h). Qed. + +End EqFun. + +Prenex Implicits frel. + +(* The coercion to rel must be explicit for derived Notations to unparse. *) +Notation coerced_frel f := (rel_of_simpl_rel (frel f)) (only parsing). + +Section FunWith. + +Variables (aT : eqType) (rT : Type). + +CoInductive fun_delta : Type := FunDelta of aT & rT. + +Definition fwith x y (f : aT -> rT) := [fun z => if z == x then y else f z]. + +Definition app_fdelta df f z := + let: FunDelta x y := df in if z == x then y else f z. + +End FunWith. + +Prenex Implicits fwith. + +Notation "x |-> y" := (FunDelta x y) + (at level 190, no associativity, + format "'[hv' x '/ ' |-> y ']'") : fun_delta_scope. + +Delimit Scope fun_delta_scope with FUN_DELTA. +Arguments Scope app_fdelta [_ type_scope fun_delta_scope _ _]. + +Notation "[ 'fun' z : T => F 'with' d1 , .. , dn ]" := + (SimplFunDelta (fun z : T => + app_fdelta d1%FUN_DELTA .. (app_fdelta dn%FUN_DELTA (fun _ => F)) ..)) + (at level 0, z ident, only parsing) : fun_scope. + +Notation "[ 'fun' z => F 'with' d1 , .. , dn ]" := + (SimplFunDelta (fun z => + app_fdelta d1%FUN_DELTA .. (app_fdelta dn%FUN_DELTA (fun _ => F)) ..)) + (at level 0, z ident, format + "'[hv' [ '[' 'fun' z => '/ ' F ']' '/' 'with' '[' d1 , '/' .. , '/' dn ']' ] ']'" + ) : fun_scope. + +Notation "[ 'eta' f 'with' d1 , .. , dn ]" := + (SimplFunDelta (fun _ => + app_fdelta d1%FUN_DELTA .. (app_fdelta dn%FUN_DELTA f) ..)) + (at level 0, format + "'[hv' [ '[' 'eta' '/ ' f ']' '/' 'with' '[' d1 , '/' .. , '/' dn ']' ] ']'" + ) : fun_scope. + +(* Various EqType constructions. *) + +Section ComparableType. + +Variable T : Type. + +Definition comparable := forall x y : T, decidable (x = y). + +Hypothesis Hcompare : comparable. + +Definition compareb x y : bool := Hcompare x y. + +Lemma compareP : Equality.axiom compareb. +Proof. by move=> x y; exact: sumboolP. Qed. + +Definition comparableClass := EqMixin compareP. + +End ComparableType. + +Definition eq_comparable (T : eqType) : comparable T := + fun x y => decP (x =P y). + +Section SubType. + +Variables (T : Type) (P : pred T). + +Structure subType : Type := SubType { + sub_sort :> Type; + val : sub_sort -> T; + Sub : forall x, P x -> sub_sort; + _ : forall K (_ : forall x Px, K (@Sub x Px)) u, K u; + _ : forall x Px, val (@Sub x Px) = x +}. + +Implicit Arguments Sub [s]. +Lemma vrefl : forall x, P x -> x = x. Proof. by []. Qed. +Definition vrefl_rect := vrefl. + +Definition clone_subType U v := + fun sT & sub_sort sT -> U => + fun c Urec cK (sT' := @SubType U v c Urec cK) & phant_id sT' sT => sT'. + +Variable sT : subType. + +CoInductive Sub_spec : sT -> Type := SubSpec x Px : Sub_spec (Sub x Px). + +Lemma SubP u : Sub_spec u. +Proof. by case: sT Sub_spec SubSpec u => T' _ C rec /= _. Qed. + +Lemma SubK x Px : @val sT (Sub x Px) = x. +Proof. by case: sT. Qed. + +Definition insub x := + if @idP (P x) is ReflectT Px then @Some sT (Sub x Px) else None. + +Definition insubd u0 x := odflt u0 (insub x). + +CoInductive insub_spec x : option sT -> Type := + | InsubSome u of P x & val u = x : insub_spec x (Some u) + | InsubNone of ~~ P x : insub_spec x None. + +Lemma insubP x : insub_spec x (insub x). +Proof. +by rewrite /insub; case: {-}_ / idP; [left; rewrite ?SubK | right; exact/negP]. +Qed. + +Lemma insubT x Px : insub x = Some (Sub x Px). +Proof. +case: insubP; last by case/negP. +case/SubP=> y Py _ def_x; rewrite -def_x SubK in Px *. +congr (Some (Sub _ _)); exact: bool_irrelevance. +Qed. + +Lemma insubF x : P x = false -> insub x = None. +Proof. by move/idP; case: insubP. Qed. + +Lemma insubN x : ~~ P x -> insub x = None. +Proof. by move/negPf/insubF. Qed. + +Lemma isSome_insub : ([eta insub] : pred T) =1 P. +Proof. by apply: fsym => x; case: insubP => // /negPf. Qed. + +Lemma insubK : ocancel insub (@val _). +Proof. by move=> x; case: insubP. Qed. + +Lemma valP (u : sT) : P (val u). +Proof. by case/SubP: u => x Px; rewrite SubK. Qed. + +Lemma valK : pcancel (@val _) insub. +Proof. case/SubP=> x Px; rewrite SubK; exact: insubT. Qed. + +Lemma val_inj : injective (@val sT). +Proof. exact: pcan_inj valK. Qed. + +Lemma valKd u0 : cancel (@val _) (insubd u0). +Proof. by move=> u; rewrite /insubd valK. Qed. + +Lemma val_insubd u0 x : val (insubd u0 x) = if P x then x else val u0. +Proof. by rewrite /insubd; case: insubP => [u -> | /negPf->]. Qed. + +Lemma insubdK u0 : {in P, cancel (insubd u0) (@val _)}. +Proof. by move=> x Px; rewrite /= val_insubd [P x]Px. Qed. + +Definition insub_eq x := + let Some_sub Px := Some (Sub x Px : sT) in + let None_sub _ := None in + (if P x as Px return P x = Px -> _ then Some_sub else None_sub) (erefl _). + +Lemma insub_eqE : insub_eq =1 insub. +Proof. +rewrite /insub_eq /insub => x; case: {2 3}_ / idP (erefl _) => // Px Px'. +by congr (Some _); apply: val_inj; rewrite !SubK. +Qed. + +End SubType. + +Implicit Arguments SubType [T P]. +Implicit Arguments Sub [T P s]. +Implicit Arguments vrefl [T P]. +Implicit Arguments vrefl_rect [T P]. +Implicit Arguments clone_subType [T P sT c Urec cK]. +Implicit Arguments insub [T P sT]. +Implicit Arguments insubT [T sT x]. +Implicit Arguments val_inj [T P sT]. +Prenex Implicits val Sub vrefl vrefl_rect insub insubd val_inj. + +Local Notation inlined_sub_rect := + (fun K K_S u => let (x, Px) as u return K u := u in K_S x Px). + +Local Notation inlined_new_rect := + (fun K K_S u => let (x) as u return K u := u in K_S x). + +Notation "[ 'subType' 'for' v ]" := (SubType _ v _ inlined_sub_rect vrefl_rect) + (at level 0, only parsing) : form_scope. + +Notation "[ 'sub' 'Type' 'for' v ]" := (SubType _ v _ _ vrefl_rect) + (at level 0, format "[ 'sub' 'Type' 'for' v ]") : form_scope. + +Notation "[ 'subType' 'for' v 'by' rec ]" := (SubType _ v _ rec vrefl) + (at level 0, format "[ 'subType' 'for' v 'by' rec ]") : form_scope. + +Notation "[ 'subType' 'of' U 'for' v ]" := (clone_subType U v id idfun) + (at level 0, format "[ 'subType' 'of' U 'for' v ]") : form_scope. + +(* +Notation "[ 'subType' 'for' v ]" := (clone_subType _ v id idfun) + (at level 0, format "[ 'subType' 'for' v ]") : form_scope. +*) +Notation "[ 'subType' 'of' U ]" := (clone_subType U _ id id) + (at level 0, format "[ 'subType' 'of' U ]") : form_scope. + +Definition NewType T U v c Urec := + let Urec' P IH := Urec P (fun x : T => IH x isT : P _) in + SubType U v (fun x _ => c x) Urec'. +Implicit Arguments NewType [T U]. + +Notation "[ 'newType' 'for' v ]" := (NewType v _ inlined_new_rect vrefl_rect) + (at level 0, only parsing) : form_scope. + +Notation "[ 'new' 'Type' 'for' v ]" := (NewType v _ _ vrefl_rect) + (at level 0, format "[ 'new' 'Type' 'for' v ]") : form_scope. + +Notation "[ 'newType' 'for' v 'by' rec ]" := (NewType v _ rec vrefl) + (at level 0, format "[ 'newType' 'for' v 'by' rec ]") : form_scope. + +Definition innew T nT x := @Sub T predT nT x (erefl true). +Implicit Arguments innew [T nT]. +Prenex Implicits innew. + +Lemma innew_val T nT : cancel val (@innew T nT). +Proof. by move=> u; apply: val_inj; exact: SubK. Qed. + +(* Prenex Implicits and renaming. *) +Notation sval := (@proj1_sig _ _). +Notation "@ 'sval'" := (@proj1_sig) (at level 10, format "@ 'sval'"). + +Section SigProj. + +Variables (T : Type) (P Q : T -> Prop). + +Lemma svalP : forall u : sig P, P (sval u). Proof. by case. Qed. + +Definition s2val (u : sig2 P Q) := let: exist2 x _ _ := u in x. + +Lemma s2valP u : P (s2val u). Proof. by case: u. Qed. + +Lemma s2valP' u : Q (s2val u). Proof. by case: u. Qed. + +End SigProj. + +Prenex Implicits svalP s2val s2valP s2valP'. + +Canonical sig_subType T (P : pred T) : subType [eta P] := + Eval hnf in [subType for @sval T [eta [eta P]]]. + +(* Shorthand for sigma types over collective predicates. *) +Notation "{ x 'in' A }" := {x | x \in A} + (at level 0, x at level 99, format "{ x 'in' A }") : type_scope. +Notation "{ x 'in' A | P }" := {x | (x \in A) && P} + (at level 0, x at level 99, format "{ x 'in' A | P }") : type_scope. + +(* Shorthand for the return type of insub. *) +Notation "{ ? x : T | P }" := (option {x : T | is_true P}) + (at level 0, x at level 99, only parsing) : type_scope. +Notation "{ ? x | P }" := {? x : _ | P} + (at level 0, x at level 99, format "{ ? x | P }") : type_scope. +Notation "{ ? x 'in' A }" := {? x | x \in A} + (at level 0, x at level 99, format "{ ? x 'in' A }") : type_scope. +Notation "{ ? x 'in' A | P }" := {? x | (x \in A) && P} + (at level 0, x at level 99, format "{ ? x 'in' A | P }") : type_scope. + +(* A variant of injection with default that infers a collective predicate *) +(* from the membership proof for the default value. *) +Definition insigd T (A : mem_pred T) x (Ax : in_mem x A) := + insubd (exist [eta A] x Ax). + +(* There should be a rel definition for the subType equality op, but this *) +(* seems to cause the simpl tactic to diverge on expressions involving == *) +(* on 4+ nested subTypes in a "strict" position (e.g., after ~~). *) +(* Definition feq f := [rel x y | f x == f y]. *) + +Section TransferEqType. + +Variables (T : Type) (eT : eqType) (f : T -> eT). + +Lemma inj_eqAxiom : injective f -> Equality.axiom (fun x y => f x == f y). +Proof. by move=> f_inj x y; apply: (iffP eqP) => [|-> //]; exact: f_inj. Qed. + +Definition InjEqMixin f_inj := EqMixin (inj_eqAxiom f_inj). + +Definition PcanEqMixin g (fK : pcancel f g) := InjEqMixin (pcan_inj fK). + +Definition CanEqMixin g (fK : cancel f g) := InjEqMixin (can_inj fK). + +End TransferEqType. + +Section SubEqType. + +Variables (T : eqType) (P : pred T) (sT : subType P). + +Notation Local ev_ax := (fun T v => @Equality.axiom T (fun x y => v x == v y)). +Lemma val_eqP : ev_ax sT val. Proof. exact: inj_eqAxiom val_inj. Qed. + +Definition sub_eqMixin := EqMixin val_eqP. +Canonical sub_eqType := Eval hnf in EqType sT sub_eqMixin. + +Definition SubEqMixin := + (let: SubType _ v _ _ _ as sT' := sT + return ev_ax sT' val -> Equality.class_of sT' in + fun vP : ev_ax _ v => EqMixin vP + ) val_eqP. + +Lemma val_eqE (u v : sT) : (val u == val v) = (u == v). +Proof. by []. Qed. + +End SubEqType. + +Implicit Arguments val_eqP [T P sT x y]. +Prenex Implicits val_eqP. + +Notation "[ 'eqMixin' 'of' T 'by' <: ]" := (SubEqMixin _ : Equality.class_of T) + (at level 0, format "[ 'eqMixin' 'of' T 'by' <: ]") : form_scope. + +Section SigEqType. + +Variables (T : eqType) (P : pred T). + +Definition sig_eqMixin := Eval hnf in [eqMixin of {x | P x} by <:]. +Canonical sig_eqType := Eval hnf in EqType {x | P x} sig_eqMixin. + +End SigEqType. + +Section ProdEqType. + +Variable T1 T2 : eqType. + +Definition pair_eq := [rel u v : T1 * T2 | (u.1 == v.1) && (u.2 == v.2)]. + +Lemma pair_eqP : Equality.axiom pair_eq. +Proof. +move=> [x1 x2] [y1 y2] /=; apply: (iffP andP) => [[]|[<- <-]] //=. +by do 2!move/eqP->. +Qed. + +Definition prod_eqMixin := EqMixin pair_eqP. +Canonical prod_eqType := Eval hnf in EqType (T1 * T2) prod_eqMixin. + +Lemma pair_eqE : pair_eq = eq_op :> rel _. Proof. by []. Qed. + +Lemma xpair_eqE (x1 y1 : T1) (x2 y2 : T2) : + ((x1, x2) == (y1, y2)) = ((x1 == y1) && (x2 == y2)). +Proof. by []. Qed. + +Lemma pair_eq1 (u v : T1 * T2) : u == v -> u.1 == v.1. +Proof. by case/andP. Qed. + +Lemma pair_eq2 (u v : T1 * T2) : u == v -> u.2 == v.2. +Proof. by case/andP. Qed. + +End ProdEqType. + +Implicit Arguments pair_eqP [T1 T2]. + +Prenex Implicits pair_eqP. + +Definition predX T1 T2 (p1 : pred T1) (p2 : pred T2) := + [pred z | p1 z.1 & p2 z.2]. + +Notation "[ 'predX' A1 & A2 ]" := (predX [mem A1] [mem A2]) + (at level 0, format "[ 'predX' A1 & A2 ]") : fun_scope. + +Section OptionEqType. + +Variable T : eqType. + +Definition opt_eq (u v : option T) : bool := + oapp (fun x => oapp (eq_op x) false v) (~~ v) u. + +Lemma opt_eqP : Equality.axiom opt_eq. +Proof. +case=> [x|] [y|] /=; by [constructor | apply: (iffP eqP) => [|[]] ->]. +Qed. + +Canonical option_eqMixin := EqMixin opt_eqP. +Canonical option_eqType := Eval hnf in EqType (option T) option_eqMixin. + +End OptionEqType. + +Definition tag := projS1. +Definition tagged I T_ : forall u, T_(tag u) := @projS2 I [eta T_]. +Definition Tagged I i T_ x := @existS I [eta T_] i x. +Implicit Arguments Tagged [I i]. +Prenex Implicits tag tagged Tagged. + +Section TaggedAs. + +Variables (I : eqType) (T_ : I -> Type). +Implicit Types u v : {i : I & T_ i}. + +Definition tagged_as u v := + if tag u =P tag v is ReflectT eq_uv then + eq_rect_r T_ (tagged v) eq_uv + else tagged u. + +Lemma tagged_asE u x : tagged_as u (Tagged T_ x) = x. +Proof. +by rewrite /tagged_as /=; case: eqP => // eq_uu; rewrite [eq_uu]eq_axiomK. +Qed. + +End TaggedAs. + +Section TagEqType. + +Variables (I : eqType) (T_ : I -> eqType). +Implicit Types u v : {i : I & T_ i}. + +Definition tag_eq u v := (tag u == tag v) && (tagged u == tagged_as u v). + +Lemma tag_eqP : Equality.axiom tag_eq. +Proof. +rewrite /tag_eq => [] [i x] [j] /=. +case: eqP => [<-|Hij] y; last by right; case. +by apply: (iffP eqP) => [->|<-]; rewrite tagged_asE. +Qed. + +Canonical tag_eqMixin := EqMixin tag_eqP. +Canonical tag_eqType := Eval hnf in EqType {i : I & T_ i} tag_eqMixin. + +Lemma tag_eqE : tag_eq = eq_op. Proof. by []. Qed. + +Lemma eq_tag u v : u == v -> tag u = tag v. +Proof. by move/eqP->. Qed. + +Lemma eq_Tagged u x :(u == Tagged _ x) = (tagged u == x). +Proof. by rewrite -tag_eqE /tag_eq eqxx tagged_asE. Qed. + +End TagEqType. + +Implicit Arguments tag_eqP [I T_ x y]. +Prenex Implicits tag_eqP. + +Section SumEqType. + +Variables T1 T2 : eqType. +Implicit Types u v : T1 + T2. + +Definition sum_eq u v := + match u, v with + | inl x, inl y | inr x, inr y => x == y + | _, _ => false + end. + +Lemma sum_eqP : Equality.axiom sum_eq. +Proof. case=> x [] y /=; by [right | apply: (iffP eqP) => [->|[->]]]. Qed. + +Canonical sum_eqMixin := EqMixin sum_eqP. +Canonical sum_eqType := Eval hnf in EqType (T1 + T2) sum_eqMixin. + +Lemma sum_eqE : sum_eq = eq_op. Proof. by []. Qed. + +End SumEqType. + +Implicit Arguments sum_eqP [T1 T2 x y]. +Prenex Implicits sum_eqP. diff --git a/mathcomp/ssreflect/plugin/trunk/ssreflect.ml4 b/mathcomp/ssreflect/plugin/trunk/ssreflect.ml4 new file mode 100644 index 0000000..931570d --- /dev/null +++ b/mathcomp/ssreflect/plugin/trunk/ssreflect.ml4 @@ -0,0 +1,6138 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +(* This line is read by the Makefile's dist target: do not remove. *) +DECLARE PLUGIN "ssreflect" +let ssrversion = "1.5";; +let ssrAstVersion = 1;; +let () = Mltop.add_known_plugin (fun () -> + if Flags.is_verbose () && not !Flags.batch_mode then begin + Printf.printf "\nSmall Scale Reflection version %s loaded.\n" ssrversion; + Printf.printf "Copyright 2005-2014 Microsoft Corporation and INRIA.\n"; + Printf.printf "Distributed under the terms of the CeCILL-B license.\n\n" + end; + (* Disable any semantics associated with bullets *) + Goptions.set_string_option_value_gen + (Some false) ["Bullet";"Behavior"] "None") + "ssreflect" +;; + +(* Defining grammar rules with "xx" in it automatically declares keywords too, + * we thus save the lexer to restore it at the end of the file *) +let frozen_lexer = Lexer.freeze () ;; + +(*i camlp4use: "pa_extend.cmo" i*) +(*i camlp4deps: "grammar/grammar.cma" i*) + +open Names +open Pp +open Pcoq +open Genarg +open Stdarg +open Constrarg +open Term +open Vars +open Context +open Topconstr +open Libnames +open Tactics +open Tacticals +open Termops +open Namegen +open Recordops +open Tacmach +open Coqlib +open Glob_term +open Util +open Evd +open Extend +open Goptions +open Tacexpr +open Tacinterp +open Pretyping +open Constr +open Tactic +open Extraargs +open Ppconstr +open Printer + +open Globnames +open Misctypes +open Decl_kinds +open Evar_kinds +open Constrexpr +open Constrexpr_ops +open Notation_term +open Notation_ops +open Locus +open Locusops + +open Ssrmatching + + +(* Tentative patch from util.ml *) + +let array_fold_right_from n f v a = + let rec fold n = + if n >= Array.length v then a else f v.(n) (fold (succ n)) + in + fold n + +let array_app_tl v l = + if Array.length v = 0 then invalid_arg "array_app_tl"; + array_fold_right_from 1 (fun e l -> e::l) v l + +let array_list_of_tl v = + if Array.length v = 0 then invalid_arg "array_list_of_tl"; + array_fold_right_from 1 (fun e l -> e::l) v [] + +(* end patch *) + +module Intset = Evar.Set + +type loc = Loc.t +let dummy_loc = Loc.ghost +let errorstrm = Errors.errorlabstrm "ssreflect" +let loc_error loc msg = Errors.user_err_loc (loc, msg, str msg) +let anomaly s = Errors.anomaly (str s) + +(** look up a name in the ssreflect internals module *) +let ssrdirpath = make_dirpath [id_of_string "ssreflect"] +let ssrqid name = make_qualid ssrdirpath (id_of_string name) +let ssrtopqid name = make_short_qualid (id_of_string name) +let locate_reference qid = + Smartlocate.global_of_extended_global (Nametab.locate_extended qid) +let mkSsrRef name = + try locate_reference (ssrqid name) with Not_found -> + try locate_reference (ssrtopqid name) with Not_found -> + Errors.error "Small scale reflection library not loaded" +let mkSsrRRef name = GRef (dummy_loc, mkSsrRef name,None), None +let mkSsrConst name env sigma = + Evd.fresh_global env sigma (mkSsrRef name) +let pf_mkSsrConst name gl = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, t = mkSsrConst name env sigma in + t, re_sig it sigma +let pf_fresh_global name gl = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma,t = Evd.fresh_global env sigma name in + t, re_sig it sigma + +(** Ssreflect load check. *) + +(* To allow ssrcoq to be fully compatible with the "plain" Coq, we only *) +(* turn on its incompatible features (the new rewrite syntax, and the *) +(* reserved identifiers) when the theory library (ssreflect.v) has *) +(* has actually been required, or is being defined. Because this check *) +(* needs to be done often (for each identifier lookup), we implement *) +(* some caching, repeating the test only when the environment changes. *) +(* We check for protect_term because it is the first constant loaded; *) +(* ssr_have would ultimately be a better choice. *) +let ssr_loaded = Summary.ref ~name:"SSR:loaded" false +let is_ssr_loaded () = + !ssr_loaded || + (if Lexer.is_keyword "SsrSyntax_is_Imported" then ssr_loaded:=true; + !ssr_loaded) + +(* 0 cost pp function. Active only if env variable SSRDEBUG is set *) +(* or if SsrDebug is Set *) +let pp_ref = ref (fun _ -> ()) +let ssr_pp s = pperrnl (str"SSR: "++Lazy.force s) +let _ = try ignore(Sys.getenv "SSRDEBUG"); pp_ref := ssr_pp with Not_found -> () +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect debugging"; + Goptions.optkey = ["SsrDebug"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !pp_ref == ssr_pp); + Goptions.optwrite = (fun b -> + Ssrmatching.debug b; + if b then pp_ref := ssr_pp else pp_ref := fun _ -> ()) } +let pp s = !pp_ref s + +(** Utils {{{ *****************************************************************) +let env_size env = List.length (Environ.named_context env) +let safeDestApp c = + match kind_of_term c with App (f, a) -> f, a | _ -> c, [| |] +let get_index = function ArgArg i -> i | _ -> + anomaly "Uninterpreted index" +(* Toplevel constr must be globalized twice ! *) +let glob_constr ist genv = function + | _, Some ce -> + let vars = Id.Map.fold (fun x _ accu -> Id.Set.add x accu) ist.lfun Id.Set.empty in + let ltacvars = { + Constrintern.empty_ltac_sign with Constrintern.ltac_vars = vars } in + Constrintern.intern_gen WithoutTypeConstraint ~ltacvars genv ce + | rc, None -> rc + +(* Term printing utilities functions for deciding bracketing. *) +let pr_paren prx x = hov 1 (str "(" ++ prx x ++ str ")") +(* String lexing utilities *) +let skip_wschars s = + let rec loop i = match s.[i] with '\n'..' ' -> loop (i + 1) | _ -> i in loop +let skip_numchars s = + let rec loop i = match s.[i] with '0'..'9' -> loop (i + 1) | _ -> i in loop +(* We also guard characters that might interfere with the ssreflect *) +(* tactic syntax. *) +let guard_term ch1 s i = match s.[i] with + | '(' -> false + | '{' | '/' | '=' -> true + | _ -> ch1 = '(' +(* The call 'guard s i' should return true if the contents of s *) +(* starting at i need bracketing to avoid ambiguities. *) +let pr_guarded guard prc c = + msg_with Format.str_formatter (prc c); + let s = Format.flush_str_formatter () ^ "$" in + if guard s (skip_wschars s 0) then pr_paren prc c else prc c +(* More sensible names for constr printers *) +let prl_constr = pr_lconstr +let pr_constr = pr_constr +let prl_glob_constr c = pr_lglob_constr_env (Global.env ()) c +let pr_glob_constr c = pr_glob_constr_env (Global.env ()) c +let prl_constr_expr = pr_lconstr_expr +let pr_constr_expr = pr_constr_expr +let prl_glob_constr_and_expr = function + | _, Some c -> prl_constr_expr c + | c, None -> prl_glob_constr c +let pr_glob_constr_and_expr = function + | _, Some c -> pr_constr_expr c + | c, None -> pr_glob_constr c +let pr_term (k, c) = pr_guarded (guard_term k) pr_glob_constr_and_expr c +let prl_term (k, c) = pr_guarded (guard_term k) prl_glob_constr_and_expr c + +(** Adding a new uninterpreted generic argument type *) +let add_genarg tag pr = + let wit = Genarg.make0 None tag in + let glob ist x = (ist, x) in + let subst _ x = x in + let interp ist gl x = (gl.Evd.sigma, x) in + let gen_pr _ _ _ = pr in + let () = Genintern.register_intern0 wit glob in + let () = Genintern.register_subst0 wit subst in + let () = Geninterp.register_interp0 wit interp in + Pptactic.declare_extra_genarg_pprule wit gen_pr gen_pr gen_pr; + wit + +(** Constructors for cast type *) +let dC t = CastConv t + +(** Constructors for constr_expr *) +let mkCProp loc = CSort (loc, GProp) +let mkCType loc = CSort (loc, GType []) +let mkCVar loc id = CRef (Ident (loc, id),None) +let isCVar = function CRef (Ident _,_) -> true | _ -> false +let destCVar = function CRef (Ident (_, id),_) -> id | _ -> + anomaly "not a CRef" +let rec mkCHoles loc n = + if n <= 0 then [] else CHole (loc, None, IntroAnonymous, None) :: mkCHoles loc (n - 1) +let mkCHole loc = CHole (loc, None, IntroAnonymous, None) +let rec isCHoles = function CHole _ :: cl -> isCHoles cl | cl -> cl = [] +let mkCExplVar loc id n = + CAppExpl (loc, (None, Ident (loc, id), None), mkCHoles loc n) +let mkCLambda loc name ty t = + CLambdaN (loc, [[loc, name], Default Explicit, ty], t) +let mkCLetIn loc name bo t = + CLetIn (loc, (loc, name), bo, t) +let mkCArrow loc ty t = + CProdN (loc, [[dummy_loc,Anonymous], Default Explicit, ty], t) +let mkCCast loc t ty = CCast (loc,t, dC ty) +(** Constructors for rawconstr *) +let mkRHole = GHole (dummy_loc, InternalHole, IntroAnonymous, None) +let rec mkRHoles n = if n > 0 then mkRHole :: mkRHoles (n - 1) else [] +let rec isRHoles = function GHole _ :: cl -> isRHoles cl | cl -> cl = [] +let mkRApp f args = if args = [] then f else GApp (dummy_loc, f, args) +let mkRVar id = GRef (dummy_loc, VarRef id,None) +let mkRltacVar id = GVar (dummy_loc, id) +let mkRCast rc rt = GCast (dummy_loc, rc, dC rt) +let mkRType = GSort (dummy_loc, GType []) +let mkRProp = GSort (dummy_loc, GProp) +let mkRArrow rt1 rt2 = GProd (dummy_loc, Anonymous, Explicit, rt1, rt2) +let mkRConstruct c = GRef (dummy_loc, ConstructRef c,None) +let mkRInd mind = GRef (dummy_loc, IndRef mind,None) +let mkRLambda n s t = GLambda (dummy_loc, n, Explicit, s, t) + +(** Constructors for constr *) +let pf_e_type_of gl t = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, ty = Typing.e_type_of env sigma t in + re_sig it sigma, ty + +let mkAppRed f c = match kind_of_term f with +| Lambda (_, _, b) -> subst1 c b +| _ -> mkApp (f, [|c|]) + +let mkProt t c gl = + let prot, gl = pf_mkSsrConst "protect_term" gl in + mkApp (prot, [|t; c|]), gl + +let mkRefl t c gl = + let refl, gl = pf_fresh_global (build_coq_eq_data()).refl gl in + mkApp (refl, [|t; c|]), gl + + +(* Application to a sequence of n rels (for building eta-expansions). *) +(* The rel indices decrease down to imin (inclusive), unless n < 0, *) +(* in which case they're incresing (from imin). *) +let mkEtaApp c n imin = + if n = 0 then c else + let nargs, mkarg = + if n < 0 then -n, (fun i -> mkRel (imin + i)) else + let imax = imin + n - 1 in n, (fun i -> mkRel (imax - i)) in + mkApp (c, Array.init nargs mkarg) +(* Same, but optimizing head beta redexes *) +let rec whdEtaApp c n = + if n = 0 then c else match kind_of_term c with + | Lambda (_, _, c') -> whdEtaApp c' (n - 1) + | _ -> mkEtaApp (lift n c) n 1 +let mkType () = Universes.new_Type (Lib.cwd ()) + +(* ssrterm conbinators *) +let combineCG t1 t2 f g = match t1, t2 with + | (x, (t1, None)), (_, (t2, None)) -> x, (g t1 t2, None) + | (x, (_, Some t1)), (_, (_, Some t2)) -> x, (mkRHole, Some (f t1 t2)) + | _, (_, (_, None)) -> anomaly "have: mixed C-G constr" + | _ -> anomaly "have: mixed G-C constr" +let loc_ofCG = function + | (_, (s, None)) -> Glob_ops.loc_of_glob_constr s + | (_, (_, Some s)) -> Constrexpr_ops.constr_loc s + +let mk_term k c = k, (mkRHole, Some c) +let mk_lterm c = mk_term ' ' c + +let map_fold_constr g f ctx acc cstr = + let array_f ctx acc x = let x, acc = f ctx acc x in acc, x in + match kind_of_term cstr with + | (Rel _ | Meta _ | Var _ | Sort _ | Const _ | Ind _ | Construct _) -> + cstr, acc + | Proj (x,c) -> + let c', acc = f ctx acc c in + (if c == c' then cstr else mkProj (x,c')), acc + | Cast (c,k, t) -> + let c', acc = f ctx acc c in + let t', acc = f ctx acc t in + (if c==c' && t==t' then cstr else mkCast (c', k, t')), acc + | Prod (na,t,c) -> + let t', acc = f ctx acc t in + let c', acc = f (g (na,None,t) ctx) acc c in + (if t==t' && c==c' then cstr else mkProd (na, t', c')), acc + | Lambda (na,t,c) -> + let t', acc = f ctx acc t in + let c', acc = f (g (na,None,t) ctx) acc c in + (if t==t' && c==c' then cstr else mkLambda (na, t', c')), acc + | LetIn (na,b,t,c) -> + let b', acc = f ctx acc b in + let t', acc = f ctx acc t in + let c', acc = f (g (na,Some b,t) ctx) acc c in + (if b==b' && t==t' && c==c' then cstr else mkLetIn (na, b', t', c')), acc + | App (c,al) -> + let c', acc = f ctx acc c in + let acc, al' = CArray.smartfoldmap (array_f ctx) acc al in + (if c==c' && Array.for_all2 (==) al al' then cstr else mkApp (c', al')), + acc + | Evar (e,al) -> + let acc, al' = CArray.smartfoldmap (array_f ctx) acc al in + (if Array.for_all2 (==) al al' then cstr else mkEvar (e, al')), acc + | Case (ci,p,c,bl) -> + let p', acc = f ctx acc p in + let c', acc = f ctx acc c in + let acc, bl' = CArray.smartfoldmap (array_f ctx) acc bl in + (if p==p' && c==c' && Array.for_all2 (==) bl bl' then cstr else + mkCase (ci, p', c', bl')), + acc + | Fix (ln,(lna,tl,bl)) -> + let acc, tl' = CArray.smartfoldmap (array_f ctx) acc tl in + let ctx' = Array.fold_left2 (fun l na t -> g (na,None,t) l) ctx lna tl in + let acc, bl' = CArray.smartfoldmap (array_f ctx') acc bl in + (if Array.for_all2 (==) tl tl' && Array.for_all2 (==) bl bl' + then cstr + else mkFix (ln,(lna,tl',bl'))), acc + | CoFix(ln,(lna,tl,bl)) -> + let acc, tl' = CArray.smartfoldmap (array_f ctx) acc tl in + let ctx' = Array.fold_left2 (fun l na t -> g (na,None,t) l) ctx lna tl in + let acc,bl' = CArray.smartfoldmap (array_f ctx') acc bl in + (if Array.for_all2 (==) tl tl' && Array.for_all2 (==) bl bl' + then cstr + else mkCoFix (ln,(lna,tl',bl'))), acc + +let pf_merge_uc_of sigma gl = + let ucst = Evd.evar_universe_context sigma in + pf_merge_uc ucst gl + +(* }}} *) + +(** Profiling {{{ *************************************************************) +type profiler = { + profile : 'a 'b. ('a -> 'b) -> 'a -> 'b; + reset : unit -> unit; + print : unit -> unit } +let profile_now = ref false +let something_profiled = ref false +let profilers = ref [] +let add_profiler f = profilers := f :: !profilers;; +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect profiling"; + Goptions.optkey = ["SsrProfiling"]; + Goptions.optread = (fun _ -> !profile_now); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> + Ssrmatching.profile b; + profile_now := b; + if b then List.iter (fun f -> f.reset ()) !profilers; + if not b then List.iter (fun f -> f.print ()) !profilers) } +let () = + let prof_total = + let init = ref 0.0 in { + profile = (fun f x -> assert false); + reset = (fun () -> init := Unix.gettimeofday ()); + print = (fun () -> if !something_profiled then + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + "total" 0 (Unix.gettimeofday() -. !init) 0.0 0.0)) } in + let prof_legenda = { + profile = (fun f x -> assert false); + reset = (fun () -> ()); + print = (fun () -> if !something_profiled then begin + prerr_endline + (Printf.sprintf "!! %39s ---------- --------- --------- ---------" + (String.make 39 '-')); + prerr_endline + (Printf.sprintf "!! %-39s %10s %9s %9s %9s" + "function" "#calls" "total" "max" "average") end) } in + add_profiler prof_legenda; + add_profiler prof_total +;; + +let mk_profiler s = + let total, calls, max = ref 0.0, ref 0, ref 0.0 in + let reset () = total := 0.0; calls := 0; max := 0.0 in + let profile f x = + if not !profile_now then f x else + let before = Unix.gettimeofday () in + try + incr calls; + let res = f x in + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + res + with exc -> + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + raise exc in + let print () = + if !calls <> 0 then begin + something_profiled := true; + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + s !calls !total !max (!total /. (float_of_int !calls))) end in + let prof = { profile = profile; reset = reset; print = print } in + add_profiler prof; + prof +;; +(* }}} *) + +let inVersion = Libobject.declare_object { + (Libobject.default_object "SSRASTVERSION") with + Libobject.load_function = (fun _ (_,v) -> + if v <> ssrAstVersion then Errors.error "Please recompile your .vo files"); + Libobject.classify_function = (fun v -> Libobject.Keep v); +} + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect version"; + Goptions.optkey = ["SsrAstVersion"]; + Goptions.optread = (fun _ -> true); + Goptions.optdepr = false; + Goptions.optwrite = (fun _ -> + Lib.add_anonymous_leaf (inVersion ssrAstVersion)) } + +let tactic_expr = Tactic.tactic_expr +let gallina_ext = Vernac_.gallina_ext +let sprintf = Printf.sprintf +let tactic_mode = G_vernac.tactic_mode + +(** 1. Utilities *) + + +let ssroldreworder = Summary.ref ~name:"SSR:oldreworder" true +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect 1.3 compatibility flag"; + Goptions.optkey = ["SsrOldRewriteGoalsOrder"]; + Goptions.optread = (fun _ -> !ssroldreworder); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssroldreworder := b) } + +let ssrhaveNOtcresolution = ref false + +let inHaveTCResolution = Libobject.declare_object { + (Libobject.default_object "SSRHAVETCRESOLUTION") with + Libobject.cache_function = (fun (_,v) -> ssrhaveNOtcresolution := v); + Libobject.load_function = (fun _ (_,v) -> ssrhaveNOtcresolution := v); + Libobject.classify_function = (fun v -> Libobject.Keep v); +} + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "have type classes"; + Goptions.optkey = ["SsrHave";"NoTCResolution"]; + Goptions.optread = (fun _ -> !ssrhaveNOtcresolution); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> + Lib.add_anonymous_leaf (inHaveTCResolution b)) } + + +(** Primitive parsing to avoid syntax conflicts with basic tactics. *) + +let accept_before_syms syms strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | _ -> raise Stream.Failure + +let accept_before_syms_or_any_id syms strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | Tok.IDENT _ -> () + | _ -> raise Stream.Failure + +let accept_before_syms_or_ids syms ids strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | Tok.IDENT id when List.mem id ids -> () + | _ -> raise Stream.Failure + +(** Pretty-printing utilities *) + +let pr_id = Ppconstr.pr_id +let pr_name = function Name id -> pr_id id | Anonymous -> str "_" +let pr_spc () = str " " +let pr_bar () = Pp.cut() ++ str "|" +let pr_list = prlist_with_sep + +let tacltop = (5,Ppextend.E) + +(** Tactic-level diagnosis *) + +let pf_pr_constr gl = pr_constr_env (pf_env gl) + +let pf_pr_glob_constr gl = pr_glob_constr_env (pf_env gl) + +(* debug *) + +let pf_msg gl = + let ppgl = pr_lconstr_env (pf_env gl) (project gl) (pf_concl gl) in + msgnl (str "goal is " ++ ppgl) + +let msgtac gl = pf_msg gl; tclIDTAC gl + +(** Tactic utilities *) + +let last_goal gls = let sigma, gll = Refiner.unpackage gls in + Refiner.repackage sigma (List.nth gll (List.length gll - 1)) + +let pf_type_id gl t = id_of_string (hdchar (pf_env gl) t) + +let not_section_id id = not (is_section_variable id) + +let is_pf_var c = isVar c && not_section_id (destVar c) + +let pf_ids_of_proof_hyps gl = + let add_hyp (id, _, _) ids = if not_section_id id then id :: ids else ids in + Context.fold_named_context add_hyp (pf_hyps gl) ~init:[] + +let pf_nf_evar gl e = Reductionops.nf_evar (project gl) e + +let pf_partial_solution gl t evl = + let sigma, g = project gl, sig_it gl in + let sigma = Goal.V82.partial_solution sigma g t in + re_sig (List.map (fun x -> (fst (destEvar x))) evl) sigma + +let pf_new_evar gl ty = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, extra = Evarutil.new_evar env sigma ty in + re_sig it sigma, extra + +(* Basic tactics *) + +let convert_concl_no_check t = convert_concl_no_check t DEFAULTcast +let convert_concl t = convert_concl t DEFAULTcast +let reduct_in_concl t = reduct_in_concl (t, DEFAULTcast) +let havetac id c = Proofview.V82.of_tactic (pose_proof (Name id) c) +let settac id c = letin_tac None (Name id) c None +let posetac id cl = Proofview.V82.of_tactic (settac id cl nowhere) +let basecuttac name c gl = + let hd, gl = pf_mkSsrConst name gl in + let t = mkApp (hd, [|c|]) in + let gl, _ = pf_e_type_of gl t in + Proofview.V82.of_tactic (apply t) gl + +(* we reduce head beta redexes *) +let betared env = + Closure.create_clos_infos + (Closure.RedFlags.mkflags [Closure.RedFlags.fBETA]) + env +;; +let introid name = tclTHEN (fun gl -> + let g, env = pf_concl gl, pf_env gl in + match kind_of_term g with + | App (hd, _) when isLambda hd -> + let g = Closure.whd_val (betared env) (Closure.inject g) in + Proofview.V82.of_tactic (convert_concl_no_check g) gl + | _ -> tclIDTAC gl) + (Proofview.V82.of_tactic (intro_mustbe_force name)) +;; + + +(** Name generation {{{ *******************************************************) + +(* Since Coq now does repeated internal checks of its external lexical *) +(* rules, we now need to carve ssreflect reserved identifiers out of *) +(* out of the user namespace. We use identifiers of the form _id_ for *) +(* this purpose, e.g., we "anonymize" an identifier id as _id_, adding *) +(* an extra leading _ if this might clash with an internal identifier. *) +(* We check for ssreflect identifiers in the ident grammar rule; *) +(* when the ssreflect Module is present this is normally an error, *) +(* but we provide a compatibility flag to reduce this to a warning. *) + +let ssr_reserved_ids = Summary.ref ~name:"SSR:idents" true + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "ssreflect identifiers"; + Goptions.optkey = ["SsrIdents"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !ssr_reserved_ids); + Goptions.optwrite = (fun b -> ssr_reserved_ids := b) + } + +let is_ssr_reserved s = + let n = String.length s in n > 2 && s.[0] = '_' && s.[n - 1] = '_' + +let internal_names = ref [] +let add_internal_name pt = internal_names := pt :: !internal_names +let is_internal_name s = List.exists (fun p -> p s) !internal_names + +let ssr_id_of_string loc s = + if is_ssr_reserved s && is_ssr_loaded () then begin + if !ssr_reserved_ids then + loc_error loc ("The identifier " ^ s ^ " is reserved.") + else if is_internal_name s then + msg_warning (str ("Conflict between " ^ s ^ " and ssreflect internal names.")) + else msg_warning (str ( + "The name " ^ s ^ " fits the _xxx_ format used for anonymous variables.\n" + ^ "Scripts with explicit references to anonymous variables are fragile.")) + end; id_of_string s + +let ssr_null_entry = Gram.Entry.of_parser "ssr_null" (fun _ -> ()) + +let (!@) = Compat.to_coqloc + +GEXTEND Gram + GLOBAL: Prim.ident; + Prim.ident: [[ s = IDENT; ssr_null_entry -> ssr_id_of_string !@loc s ]]; +END + +let mk_internal_id s = + let s' = sprintf "_%s_" s in + for i = 1 to String.length s do if s'.[i] = ' ' then s'.[i] <- '_' done; + add_internal_name ((=) s'); id_of_string s' + +let same_prefix s t n = + let rec loop i = i = n || s.[i] = t.[i] && loop (i + 1) in loop 0 + +let skip_digits s = + let n = String.length s in + let rec loop i = if i < n && is_digit s.[i] then loop (i + 1) else i in loop + +let mk_tagged_id t i = id_of_string (sprintf "%s%d_" t i) +let is_tagged t s = + let n = String.length s - 1 and m = String.length t in + m < n && s.[n] = '_' && same_prefix s t m && skip_digits s m = n + +let perm_tag = "_perm_Hyp_" +let _ = add_internal_name (is_tagged perm_tag) +let mk_perm_id = + let salt = ref 1 in + fun () -> salt := !salt mod 10000 + 1; mk_tagged_id perm_tag !salt + +let evar_tag = "_evar_" +let _ = add_internal_name (is_tagged evar_tag) +let mk_evar_name n = Name (mk_tagged_id evar_tag n) +let nb_evar_deps = function + | Name id -> + let s = string_of_id id in + if not (is_tagged evar_tag s) then 0 else + let m = String.length evar_tag in + (try int_of_string (String.sub s m (String.length s - 1 - m)) with _ -> 0) + | _ -> 0 + +let discharged_tag = "_discharged_" +let mk_discharged_id id = + id_of_string (sprintf "%s%s_" discharged_tag (string_of_id id)) +let has_discharged_tag s = + let m = String.length discharged_tag and n = String.length s - 1 in + m < n && s.[n] = '_' && same_prefix s discharged_tag m +let _ = add_internal_name has_discharged_tag +let is_discharged_id id = has_discharged_tag (string_of_id id) + +let wildcard_tag = "_the_" +let wildcard_post = "_wildcard_" +let mk_wildcard_id i = + id_of_string (sprintf "%s%s%s" wildcard_tag (String.ordinal i) wildcard_post) +let has_wildcard_tag s = + let n = String.length s in let m = String.length wildcard_tag in + let m' = String.length wildcard_post in + n < m + m' + 2 && same_prefix s wildcard_tag m && + String.sub s (n - m') m' = wildcard_post && + skip_digits s m = n - m' - 2 +let _ = add_internal_name has_wildcard_tag + +let max_suffix m (t, j0 as tj0) id = + let s = string_of_id id in let n = String.length s - 1 in + let dn = String.length t - 1 - n in let i0 = j0 - dn in + if not (i0 >= m && s.[n] = '_' && same_prefix s t m) then tj0 else + let rec loop i = + if i < i0 && s.[i] = '0' then loop (i + 1) else + if (if i < i0 then skip_digits s i = n else le_s_t i) then s, i else tj0 + and le_s_t i = + let ds = s.[i] and dt = t.[i + dn] in + if ds = dt then i = n || le_s_t (i + 1) else + dt < ds && skip_digits s i = n in + loop m + +let mk_anon_id t gl = + let m, si0, id0 = + let s = ref (sprintf "_%s_" t) in + if is_internal_name !s then s := "_" ^ !s; + let n = String.length !s - 1 in + let rec loop i j = + let d = !s.[i] in if not (is_digit d) then i + 1, j else + loop (i - 1) (if d = '0' then j else i) in + let m, j = loop (n - 1) n in m, (!s, j), id_of_string !s in + let gl_ids = pf_ids_of_hyps gl in + if not (List.mem id0 gl_ids) then id0 else + let s, i = List.fold_left (max_suffix m) si0 gl_ids in + let n = String.length s - 1 in + let rec loop i = + if s.[i] = '9' then (s.[i] <- '0'; loop (i - 1)) else + if i < m then (s.[n] <- '0'; s.[m] <- '1'; s ^ "_") else + (s.[i] <- Char.chr (Char.code s.[i] + 1); s) in + id_of_string (loop (n - 1)) + +(* We must not anonymize context names discharged by the "in" tactical. *) + +let ssr_anon_hyp = "Hyp" + +let anontac (x, _, _) gl = + let id = match x with + | Name id -> + if is_discharged_id id then id else mk_anon_id (string_of_id id) gl + | _ -> mk_anon_id ssr_anon_hyp gl in + introid id gl + +let rec constr_name c = match kind_of_term c with + | Var id -> Name id + | Cast (c', _, _) -> constr_name c' + | Const (cn,_) -> Name (id_of_label (con_label cn)) + | App (c', _) -> constr_name c' + | _ -> Anonymous + +(* }}} *) + +(** Open term to lambda-term coercion {{{ ************************************) + +(* This operation takes a goal gl and an open term (sigma, t), and *) +(* returns a term t' where all the new evars in sigma are abstracted *) +(* with the mkAbs argument, i.e., for mkAbs = mkLambda then there is *) +(* some duplicate-free array args of evars of sigma such that the *) +(* term mkApp (t', args) is convertible to t. *) +(* This makes a useful shorthand for local definitions in proofs, *) +(* i.e., pose succ := _ + 1 means pose succ := fun n : nat => n + 1, *) +(* and, in context of the the 4CT library, pose mid := maps id means *) +(* pose mid := fun d : detaSet => @maps d d (@id (datum d)) *) +(* Note that this facility does not extend to set, which tries *) +(* instead to fill holes by matching a goal subterm. *) +(* The argument to "have" et al. uses product abstraction, e.g. *) +(* have Hmid: forall s, (maps id s) = s. *) +(* stands for *) +(* have Hmid: forall (d : dataSet) (s : seq d), (maps id s) = s. *) +(* We also use this feature for rewrite rules, so that, e.g., *) +(* rewrite: (plus_assoc _ 3). *) +(* will execute as *) +(* rewrite (fun n => plus_assoc n 3) *) +(* i.e., it will rewrite some subterm .. + (3 + ..) to .. + 3 + ... *) +(* The convention is also used for the argument of the congr tactic, *) +(* e.g., congr (x + _ * 1). *) + +(* Replace new evars with lambda variables, retaining local dependencies *) +(* but stripping global ones. We use the variable names to encode the *) +(* the number of dependencies, so that the transformation is reversible. *) + +let pf_abs_evars gl (sigma, c0) = + let sigma0, ucst = project gl, Evd.evar_universe_context sigma in + let nenv = env_size (pf_env gl) in + let abs_evar n k = + let evi = Evd.find sigma k in + let dc = List.firstn n (evar_filtered_context evi) in + let abs_dc c = function + | x, Some b, t -> mkNamedLetIn x b t (mkArrow t c) + | x, None, t -> mkNamedProd x t c in + let t = Context.fold_named_context_reverse abs_dc ~init:evi.evar_concl dc in + Evarutil.nf_evar sigma t in + let rec put evlist c = match kind_of_term c with + | Evar (k, a) -> + if List.mem_assoc k evlist || Evd.mem sigma0 k then evlist else + let n = max 0 (Array.length a - nenv) in + let t = abs_evar n k in (k, (n, t)) :: put evlist t + | _ -> fold_constr put evlist c in + let evlist = put [] c0 in + if evlist = [] then 0, c0,[], ucst else + let rec lookup k i = function + | [] -> 0, 0 + | (k', (n, _)) :: evl -> if k = k' then i, n else lookup k (i + 1) evl in + let rec get i c = match kind_of_term c with + | Evar (ev, a) -> + let j, n = lookup ev i evlist in + if j = 0 then map_constr (get i) c else if n = 0 then mkRel j else + mkApp (mkRel j, Array.init n (fun k -> get i a.(n - 1 - k))) + | _ -> map_constr_with_binders ((+) 1) get i c in + let rec loop c i = function + | (_, (n, t)) :: evl -> + loop (mkLambda (mk_evar_name n, get (i - 1) t, c)) (i - 1) evl + | [] -> c in + List.length evlist, loop (get 1 c0) 1 evlist, List.map fst evlist, ucst + + + +(* As before but if (?i : T(?j)) and (?j : P : Prop), then the lambda for i + * looks like (fun evar_i : (forall pi : P. T(pi))) thanks to "loopP" and all + * occurrences of evar_i are replaced by (evar_i evar_j) thanks to "app". + * + * If P can be solved by ssrautoprop (that defaults to trivial), then + * the corresponding lambda looks like (fun evar_i : T(c)) where c is + * the solution found by ssrautoprop. + *) +let ssrautoprop_tac = ref (fun gl -> assert false) + +(* Thanks to Arnaud Spiwack for this snippet *) +let call_on_evar tac e s = + let { it = gs ; sigma = s } = + tac { it = e ; sigma = s; } in + gs, s + +let pf_abs_evars_pirrel gl (sigma, c0) = + pp(lazy(str"==PF_ABS_EVARS_PIRREL==")); + pp(lazy(str"c0= " ++ pr_constr c0)); + let sigma0 = project gl in + let c0 = Evarutil.nf_evar sigma0 (Evarutil.nf_evar sigma c0) in + let nenv = env_size (pf_env gl) in + let abs_evar n k = + let evi = Evd.find sigma k in + let dc = List.firstn n (evar_filtered_context evi) in + let abs_dc c = function + | x, Some b, t -> mkNamedLetIn x b t (mkArrow t c) + | x, None, t -> mkNamedProd x t c in + let t = Context.fold_named_context_reverse abs_dc ~init:evi.evar_concl dc in + Evarutil.nf_evar sigma0 (Evarutil.nf_evar sigma t) in + let rec put evlist c = match kind_of_term c with + | Evar (k, a) -> + if List.mem_assoc k evlist || Evd.mem sigma0 k then evlist else + let n = max 0 (Array.length a - nenv) in + let k_ty = + Retyping.get_sort_family_of + (pf_env gl) sigma (Evd.evar_concl (Evd.find sigma k)) in + let is_prop = k_ty = InProp in + let t = abs_evar n k in (k, (n, t, is_prop)) :: put evlist t + | _ -> fold_constr put evlist c in + let evlist = put [] c0 in + if evlist = [] then 0, c0 else + let pr_constr t = pr_constr (Reductionops.nf_beta (project gl) t) in + pp(lazy(str"evlist=" ++ pr_list (fun () -> str";") + (fun (k,_) -> str(Evd.string_of_existential k)) evlist)); + let evplist = + let depev = List.fold_left (fun evs (_,(_,t,_)) -> + Intset.union evs (Evarutil.undefined_evars_of_term sigma t)) Intset.empty evlist in + List.filter (fun i,(_,_,b) -> b && Intset.mem i depev) evlist in + let evlist, evplist, sigma = + if evplist = [] then evlist, [], sigma else + List.fold_left (fun (ev, evp, sigma) (i, (_,t,_) as p) -> + try + let ng, sigma = call_on_evar !ssrautoprop_tac i sigma in + if (ng <> []) then errorstrm (str "Should we tell the user?"); + List.filter (fun (j,_) -> j <> i) ev, evp, sigma + with _ -> ev, p::evp, sigma) (evlist, [], sigma) (List.rev evplist) in + let c0 = Evarutil.nf_evar sigma c0 in + let evlist = + List.map (fun (x,(y,t,z)) -> x,(y,Evarutil.nf_evar sigma t,z)) evlist in + let evplist = + List.map (fun (x,(y,t,z)) -> x,(y,Evarutil.nf_evar sigma t,z)) evplist in + pp(lazy(str"c0= " ++ pr_constr c0)); + let rec lookup k i = function + | [] -> 0, 0 + | (k', (n,_,_)) :: evl -> if k = k' then i,n else lookup k (i + 1) evl in + let rec get evlist i c = match kind_of_term c with + | Evar (ev, a) -> + let j, n = lookup ev i evlist in + if j = 0 then map_constr (get evlist i) c else if n = 0 then mkRel j else + mkApp (mkRel j, Array.init n (fun k -> get evlist i a.(n - 1 - k))) + | _ -> map_constr_with_binders ((+) 1) (get evlist) i c in + let rec app extra_args i c = match decompose_app c with + | hd, args when isRel hd && destRel hd = i -> + let j = destRel hd in + mkApp (mkRel j, Array.of_list (List.map (lift (i-1)) extra_args @ args)) + | _ -> map_constr_with_binders ((+) 1) (app extra_args) i c in + let rec loopP evlist c i = function + | (_, (n, t, _)) :: evl -> + let t = get evlist (i - 1) t in + let n = Name (id_of_string (ssr_anon_hyp ^ string_of_int n)) in + loopP evlist (mkProd (n, t, c)) (i - 1) evl + | [] -> c in + let rec loop c i = function + | (_, (n, t, _)) :: evl -> + let evs = Evarutil.undefined_evars_of_term sigma t in + let t_evplist = List.filter (fun (k,_) -> Intset.mem k evs) evplist in + let t = loopP t_evplist (get t_evplist 1 t) 1 t_evplist in + let t = get evlist (i - 1) t in + let extra_args = + List.map (fun (k,_) -> mkRel (fst (lookup k i evlist))) + (List.rev t_evplist) in + let c = if extra_args = [] then c else app extra_args 1 c in + loop (mkLambda (mk_evar_name n, t, c)) (i - 1) evl + | [] -> c in + let res = loop (get evlist 1 c0) 1 evlist in + pp(lazy(str"res= " ++ pr_constr res)); + List.length evlist, res + +(* Strip all non-essential dependencies from an abstracted term, generating *) +(* standard names for the abstracted holes. *) + +let pf_abs_cterm gl n c0 = + if n <= 0 then c0 else + let noargs = [|0|] in + let eva = Array.make n noargs in + let rec strip i c = match kind_of_term c with + | App (f, a) when isRel f -> + let j = i - destRel f in + if j >= n || eva.(j) = noargs then mkApp (f, Array.map (strip i) a) else + let dp = eva.(j) in + let nd = Array.length dp - 1 in + let mkarg k = strip i a.(if k < nd then dp.(k + 1) - j else k + dp.(0)) in + mkApp (f, Array.init (Array.length a - dp.(0)) mkarg) + | _ -> map_constr_with_binders ((+) 1) strip i c in + let rec strip_ndeps j i c = match kind_of_term c with + | Prod (x, t, c1) when i < j -> + let dl, c2 = strip_ndeps j (i + 1) c1 in + if noccurn 1 c2 then dl, lift (-1) c2 else + i :: dl, mkProd (x, strip i t, c2) + | LetIn (x, b, t, c1) when i < j -> + let _, _, c1' = destProd c1 in + let dl, c2 = strip_ndeps j (i + 1) c1' in + if noccurn 1 c2 then dl, lift (-1) c2 else + i :: dl, mkLetIn (x, strip i b, strip i t, c2) + | _ -> [], strip i c in + let rec strip_evars i c = match kind_of_term c with + | Lambda (x, t1, c1) when i < n -> + let na = nb_evar_deps x in + let dl, t2 = strip_ndeps (i + na) i t1 in + let na' = List.length dl in + eva.(i) <- Array.of_list (na - na' :: dl); + let x' = + if na' = 0 then Name (pf_type_id gl t2) else mk_evar_name na' in + mkLambda (x', t2, strip_evars (i + 1) c1) +(* if noccurn 1 c2 then lift (-1) c2 else + mkLambda (Name (pf_type_id gl t2), t2, c2) *) + | _ -> strip i c in + strip_evars 0 c0 + +(* Undo the evar abstractions. Also works for non-evar variables. *) + +let pf_unabs_evars gl ise n c0 = + if n = 0 then c0 else + let evv = Array.make n mkProp in + let nev = ref 0 in + let env0 = pf_env gl in + let nenv0 = env_size env0 in + let rec unabs i c = match kind_of_term c with + | Rel j when i - j < !nev -> evv.(i - j) + | App (f, a0) when isRel f -> + let a = Array.map (unabs i) a0 in + let j = i - destRel f in + if j >= !nev then mkApp (f, a) else + let ev, eva = destEvar evv.(j) in + let nd = Array.length eva - nenv0 in + if nd = 0 then mkApp (evv.(j), a) else + let evarg k = if k < nd then a.(nd - 1 - k) else eva.(k) in + let c' = mkEvar (ev, Array.init (nd + nenv0) evarg) in + let na = Array.length a - nd in + if na = 0 then c' else mkApp (c', Array.sub a nd na) + | _ -> map_constr_with_binders ((+) 1) unabs i c in + let push_rel = Environ.push_rel in + let rec mk_evar j env i c = match kind_of_term c with + | Prod (x, t, c1) when i < j -> + mk_evar j (push_rel (x, None, unabs i t) env) (i + 1) c1 + | LetIn (x, b, t, c1) when i < j -> + let _, _, c2 = destProd c1 in + mk_evar j (push_rel (x, Some (unabs i b), unabs i t) env) (i + 1) c2 + | _ -> Evarutil.e_new_evar env ise (unabs i c) in + let rec unabs_evars c = + if !nev = n then unabs n c else match kind_of_term c with + | Lambda (x, t, c1) when !nev < n -> + let i = !nev in + evv.(i) <- mk_evar (i + nb_evar_deps x) env0 i t; + incr nev; unabs_evars c1 + | _ -> unabs !nev c in + unabs_evars c0 + +(* }}} *) + +(** Tactical extensions. {{{ **************************************************) + +(* The TACTIC EXTEND facility can't be used for defining new user *) +(* tacticals, because: *) +(* - the concrete syntax must start with a fixed string *) +(* We use the following workaround: *) +(* - We use the (unparsable) "YouShouldNotTypeThis" token for tacticals that *) +(* don't start with a token, then redefine the grammar and *) +(* printer using GEXTEND and set_pr_ssrtac, respectively. *) + +type ssrargfmt = ArgSsr of string | ArgCoq of argument_type | ArgSep of string + +let ssrtac_name name = { + mltac_plugin = "ssreflect"; + mltac_tactic = "ssr" ^ name; +} + +let ssrtac_entry name n = { + mltac_name = ssrtac_name name; + mltac_index = n; +} + +let set_pr_ssrtac name prec afmt = + let fmt = List.map (function ArgSep s -> Some s | _ -> None) afmt in + let rec mk_akey = function + | ArgSsr s :: afmt' -> ExtraArgType ("ssr" ^ s) :: mk_akey afmt' + | ArgCoq a :: afmt' -> a :: mk_akey afmt' + | ArgSep _ :: afmt' -> mk_akey afmt' + | [] -> [] in + let tacname = ssrtac_name name in + Pptactic.declare_ml_tactic_pprule tacname [| + { Pptactic.pptac_args = mk_akey afmt; + Pptactic.pptac_prods = (prec, fmt) } + |] + +let ssrtac_atom loc name args = TacML (loc, ssrtac_entry name 0, args) +let ssrtac_expr = ssrtac_atom + + +let ssrevaltac ist gtac = + let debug = match TacStore.get ist.extra f_debug with + | None -> Tactic_debug.DebugOff | Some level -> level + in + Proofview.V82.of_tactic (interp_tac_gen ist.lfun [] debug (globTacticIn (fun _ -> gtac))) + +(* fun gl -> let lfun = [tacarg_id, val_interp ist gl gtac] in + interp_tac_gen lfun [] ist.debug tacarg_expr gl *) + +(** Generic argument-based globbing/typing utilities *) + + +let interp_wit wit ist gl x = + let globarg = in_gen (glbwit wit) x in + let sigma, arg = interp_genarg ist (pf_env gl) (project gl) (pf_concl gl) gl.Evd.it globarg in + sigma, out_gen (topwit wit) arg + +let interp_intro_pattern = interp_wit wit_intro_pattern + +let interp_constr = interp_wit wit_constr + +let interp_open_constr ist gl gc = + interp_wit wit_open_constr ist gl ((), gc) + +let interp_refine ist gl rc = + let constrvars = extract_ltac_constr_values ist (pf_env gl) in + let vars = { Pretyping.empty_lvar with + Pretyping.ltac_constrs = constrvars; ltac_genargs = ist.lfun + } in + let kind = OfType (pf_concl gl) in + let flags = { + use_typeclasses = true; + use_unif_heuristics = true; + use_hook = None; + fail_evar = false; + expand_evars = true } + in + let sigma, c = understand_ltac flags (pf_env gl) (project gl) vars kind rc in +(* pp(lazy(str"sigma@interp_refine=" ++ pr_evar_map None sigma)); *) + pp(lazy(str"c@interp_refine=" ++ pr_constr c)); + (sigma, (sigma, c)) + +let pf_match = pf_apply (fun e s c t -> understand_tcc e s ~expected_type:t c) + +(* Estimate a bound on the number of arguments of a raw constr. *) +(* This is not perfect, because the unifier may fail to *) +(* typecheck the partial application, so we use a minimum of 5. *) +(* Also, we don't handle delayed or iterated coercions to *) +(* FUNCLASS, which is probably just as well since these can *) +(* lead to infinite arities. *) + +let splay_open_constr gl (sigma, c) = + let env = pf_env gl in let t = Retyping.get_type_of env sigma c in + Reductionops.splay_prod env sigma t + +let nbargs_open_constr gl oc = + let pl, _ = splay_open_constr gl oc in List.length pl + +let interp_nbargs ist gl rc = + try + let rc6 = mkRApp rc (mkRHoles 6) in + let sigma, t = interp_open_constr ist gl (rc6, None) in + let si = sig_it gl in + let gl = re_sig si sigma in + 6 + nbargs_open_constr gl t + with _ -> 5 + +let pf_nbargs gl c = nbargs_open_constr gl (project gl, c) + +let isAppInd gl c = + try ignore (pf_reduce_to_atomic_ind gl c); true with _ -> false + +let interp_view_nbimps ist gl rc = + try + let sigma, t = interp_open_constr ist gl (rc, None) in + let si = sig_it gl in + let gl = re_sig si sigma in + let pl, c = splay_open_constr gl t in + if isAppInd gl c then List.length pl else ~-(List.length pl) + with _ -> 0 + +(* }}} *) + +(** Vernacular commands: Prenex Implicits and Search {{{ **********************) + +(* This should really be implemented as an extension to the implicit *) +(* arguments feature, but unfortuately that API is sealed. The current *) +(* workaround uses a combination of notations that works reasonably, *) +(* with the following caveats: *) +(* - The pretty-printing always elides prenex implicits, even when *) +(* they are obviously needed. *) +(* - Prenex Implicits are NEVER exported from a module, because this *) +(* would lead to faulty pretty-printing and scoping errors. *) +(* - The command "Import Prenex Implicits" can be used to reassert *) +(* Prenex Implicits for all the visible constants that had been *) +(* declared as Prenex Implicits. *) + +let declare_one_prenex_implicit locality f = + let fref = + try Smartlocate.global_with_alias f + with _ -> errorstrm (pr_reference f ++ str " is not declared") in + let rec loop = function + | a :: args' when Impargs.is_status_implicit a -> + (ExplByName (Impargs.name_of_implicit a), (true, true, true)) :: loop args' + | args' when List.exists Impargs.is_status_implicit args' -> + errorstrm (str "Expected prenex implicits for " ++ pr_reference f) + | _ -> [] in + let impls = + match Impargs.implicits_of_global fref with + | [cond,impls] -> impls + | [] -> errorstrm (str "Expected some implicits for " ++ pr_reference f) + | _ -> errorstrm (str "Multiple implicits not supported") in + match loop impls with + | [] -> + errorstrm (str "Expected some implicits for " ++ pr_reference f) + | impls -> + Impargs.declare_manual_implicits locality fref ~enriching:false [impls] + +VERNAC COMMAND EXTEND Ssrpreneximplicits CLASSIFIED AS SIDEFF + | [ "Prenex" "Implicits" ne_global_list(fl) ] + -> [ let locality = + Locality.make_section_locality (Locality.LocalityFixme.consume ()) in + List.iter (declare_one_prenex_implicit locality) fl ] +END + +(* Vernac grammar visibility patch *) + +GEXTEND Gram + GLOBAL: gallina_ext; + gallina_ext: + [ [ IDENT "Import"; IDENT "Prenex"; IDENT "Implicits" -> + Vernacexpr.VernacUnsetOption (["Printing"; "Implicit"; "Defensive"]) + ] ] + ; +END + +(** Extend Search to subsume SearchAbout, also adding hidden Type coercions. *) + +(* Main prefilter *) + +type raw_glob_search_about_item = + | RGlobSearchSubPattern of constr_expr + | RGlobSearchString of Loc.t * string * string option + +let pr_search_item = function + | RGlobSearchString (_,s,_) -> str s + | RGlobSearchSubPattern p -> pr_constr_expr p + +let wit_ssr_searchitem = add_genarg "ssr_searchitem" pr_search_item + +let interp_search_notation loc s opt_scope = + try + let interp = Notation.interp_notation_as_global_reference loc in + let ref = interp (fun _ -> true) s opt_scope in + Search.GlobSearchSubPattern (Pattern.PRef ref) + with _ -> + let diagnosis = + try + let ntns = Notation.locate_notation pr_glob_constr s opt_scope in + let ambig = "This string refers to a complex or ambiguous notation." in + str ambig ++ str "\nTry searching with one of\n" ++ ntns + with _ -> str "This string is not part of an identifier or notation." in + Errors.user_err_loc (loc, "interp_search_notation", diagnosis) + +let pr_ssr_search_item _ _ _ = pr_search_item + +(* Workaround the notation API that can only print notations *) + +let is_ident s = try Lexer.check_ident s; true with _ -> false + +let is_ident_part s = is_ident ("H" ^ s) + +let interp_search_notation loc tag okey = + let err msg = Errors.user_err_loc (loc, "interp_search_notation", msg) in + let mk_pntn s for_key = + let n = String.length s in + let s' = String.make (n + 2) ' ' in + let rec loop i i' = + if i >= n then s', i' - 2 else if s.[i] = ' ' then loop (i + 1) i' else + let j = try String.index_from s (i + 1) ' ' with _ -> n in + let m = j - i in + if s.[i] = '\'' && i < j - 2 && s.[j - 1] = '\'' then + (String.blit s (i + 1) s' i' (m - 2); loop (j + 1) (i' + m - 1)) + else if for_key && is_ident (String.sub s i m) then + (s'.[i'] <- '_'; loop (j + 1) (i' + 2)) + else (String.blit s i s' i' m; loop (j + 1) (i' + m + 1)) in + loop 0 1 in + let trim_ntn (pntn, m) = String.sub pntn 1 (max 0 m) in + let pr_ntn ntn = str "(" ++ str ntn ++ str ")" in + let pr_and_list pr = function + | [x] -> pr x + | x :: lx -> pr_list pr_comma pr lx ++ pr_comma () ++ str "and " ++ pr x + | [] -> mt () in + let pr_sc sc = str (if sc = "" then "independently" else sc) in + let pr_scs = function + | [""] -> pr_sc "" + | scs -> str "in " ++ pr_and_list pr_sc scs in + let generator, pr_tag_sc = + let ign _ = mt () in match okey with + | Some key -> + let sc = Notation.find_delimiters_scope loc key in + let pr_sc s_in = str s_in ++ spc() ++ str sc ++ pr_comma() in + Notation.pr_scope ign sc, pr_sc + | None -> Notation.pr_scopes ign, ign in + let qtag s_in = pr_tag_sc s_in ++ qstring tag ++ spc()in + let ptag, ttag = + let ptag, m = mk_pntn tag false in + if m <= 0 then err (str "empty notation fragment"); + ptag, trim_ntn (ptag, m) in + let last = ref "" and last_sc = ref "" in + let scs = ref [] and ntns = ref [] in + let push_sc sc = match !scs with + | "" :: scs' -> scs := "" :: sc :: scs' + | scs' -> scs := sc :: scs' in + let get s _ _ = match !last with + | "Scope " -> last_sc := s; last := "" + | "Lonely notation" -> last_sc := ""; last := "" + | "\"" -> + let pntn, m = mk_pntn s true in + if String.string_contains pntn ptag then begin + let ntn = trim_ntn (pntn, m) in + match !ntns with + | [] -> ntns := [ntn]; scs := [!last_sc] + | ntn' :: _ when ntn' = ntn -> push_sc !last_sc + | _ when ntn = ttag -> ntns := ntn :: !ntns; scs := [!last_sc] + | _ :: ntns' when List.mem ntn ntns' -> () + | ntn' :: ntns' -> ntns := ntn' :: ntn :: ntns' + end; + last := "" + | _ -> last := s in + pp_with (Format.make_formatter get (fun _ -> ())) generator; + let ntn = match !ntns with + | [] -> + err (hov 0 (qtag "in" ++ str "does not occur in any notation")) + | ntn :: ntns' when ntn = ttag -> + if ntns' <> [] then begin + let pr_ntns' = pr_and_list pr_ntn ntns' in + msg_warning (hov 4 (qtag "In" ++ str "also occurs in " ++ pr_ntns')) + end; ntn + | [ntn] -> + msgnl (hov 4 (qtag "In" ++ str "is part of notation " ++ pr_ntn ntn)); ntn + | ntns' -> + let e = str "occurs in" ++ spc() ++ pr_and_list pr_ntn ntns' in + err (hov 4 (str "ambiguous: " ++ qtag "in" ++ e)) in + let (nvars, body), ((_, pat), osc) = match !scs with + | [sc] -> Notation.interp_notation loc ntn (None, [sc]) + | scs' -> + try Notation.interp_notation loc ntn (None, []) with _ -> + let e = pr_ntn ntn ++ spc() ++ str "is defined " ++ pr_scs scs' in + err (hov 4 (str "ambiguous: " ++ pr_tag_sc "in" ++ e)) in + let sc = Option.default "" osc in + let _ = + let m_sc = + if osc <> None then str "In " ++ str sc ++ pr_comma() else mt() in + let ntn_pat = trim_ntn (mk_pntn pat false) in + let rbody = glob_constr_of_notation_constr loc body in + let m_body = hov 0 (Constrextern.without_symbols prl_glob_constr rbody) in + let m = m_sc ++ pr_ntn ntn_pat ++ spc () ++ str "denotes " ++ m_body in + msgnl (hov 0 m) in + if List.length !scs > 1 then + let scs' = List.remove (=) sc !scs in + let w = pr_ntn ntn ++ str " is also defined " ++ pr_scs scs' in + msg_warning (hov 4 w) + else if String.string_contains ntn " .. " then + err (pr_ntn ntn ++ str " is an n-ary notation"); + let nvars = List.filter (fun (_,(_,typ)) -> typ = NtnTypeConstr) nvars in + let rec sub () = function + | NVar x when List.mem_assoc x nvars -> GPatVar (loc, (false, x)) + | c -> + glob_constr_of_notation_constr_with_binders loc (fun _ x -> (), x) sub () c in + let _, npat = Patternops.pattern_of_glob_constr (sub () body) in + Search.GlobSearchSubPattern npat + +ARGUMENT EXTEND ssr_search_item TYPED AS ssr_searchitem + PRINTED BY pr_ssr_search_item + | [ string(s) ] -> [ RGlobSearchString (loc,s,None) ] + | [ string(s) "%" preident(key) ] -> [ RGlobSearchString (loc,s,Some key) ] + | [ constr_pattern(p) ] -> [ RGlobSearchSubPattern p ] +END + +let pr_ssr_search_arg _ _ _ = + let pr_item (b, p) = str (if b then "-" else "") ++ pr_search_item p in + pr_list spc pr_item + +ARGUMENT EXTEND ssr_search_arg TYPED AS (bool * ssr_searchitem) list + PRINTED BY pr_ssr_search_arg + | [ "-" ssr_search_item(p) ssr_search_arg(a) ] -> [ (false, p) :: a ] + | [ ssr_search_item(p) ssr_search_arg(a) ] -> [ (true, p) :: a ] + | [ ] -> [ [] ] +END + +(* Main type conclusion pattern filter *) + +let rec splay_search_pattern na = function + | Pattern.PApp (fp, args) -> splay_search_pattern (na + Array.length args) fp + | Pattern.PLetIn (_, _, bp) -> splay_search_pattern na bp + | Pattern.PRef hr -> hr, na + | _ -> Errors.error "no head constant in head search pattern" + +let coerce_search_pattern_to_sort hpat = + let env = Global.env () and sigma = Evd.empty in + let mkPApp fp n_imps args = + let args' = Array.append (Array.make n_imps (Pattern.PMeta None)) args in + Pattern.PApp (fp, args') in + let hr, na = splay_search_pattern 0 hpat in + let dc, ht = + Reductionops.splay_prod env sigma (Universes.unsafe_type_of_global hr) in + let np = List.length dc in + if np < na then Errors.error "too many arguments in head search pattern" else + let hpat' = if np = na then hpat else mkPApp hpat (np - na) [||] in + let warn () = + msg_warning (str "Listing only lemmas with conclusion matching " ++ + pr_constr_pattern hpat') in + if isSort ht then begin warn (); true, hpat' end else + let filter_head, coe_path = + try + let _, cp = + Classops.lookup_path_to_sort_from (push_rels_assum dc env) sigma ht in + warn (); + true, cp + with _ -> false, [] in + let coerce hp coe_index = + let coe = Classops.get_coercion_value coe_index in + try + let coe_ref = reference_of_constr coe in + let n_imps = Option.get (Classops.hide_coercion coe_ref) in + mkPApp (Pattern.PRef coe_ref) n_imps [|hp|] + with _ -> + errorstrm (str "need explicit coercion " ++ pr_constr coe ++ spc () + ++ str "to interpret head search pattern as type") in + filter_head, List.fold_left coerce hpat' coe_path + +let rec interp_head_pat hpat = + let filter_head, p = coerce_search_pattern_to_sort hpat in + let rec loop c = match kind_of_term c with + | Cast (c', _, _) -> loop c' + | Prod (_, _, c') -> loop c' + | LetIn (_, _, _, c') -> loop c' + | _ -> Constr_matching.is_matching (Global.env()) Evd.empty p c in + filter_head, loop + +let all_true _ = true + +let rec interp_search_about args accu = match args with +| [] -> accu +| (flag, arg) :: rem -> + fun gr env typ -> + let ans = Search.search_about_filter arg gr env typ in + (if flag then ans else not ans) && interp_search_about rem accu gr env typ + +let interp_search_arg arg = + let arg = List.map (fun (x,arg) -> x, match arg with + | RGlobSearchString (loc,s,key) -> + if is_ident_part s then Search.GlobSearchString s else + interp_search_notation loc s key + | RGlobSearchSubPattern p -> + try + let intern = Constrintern.intern_constr_pattern in + Search.GlobSearchSubPattern (snd (intern (Global.env()) p)) + with e -> let e = Errors.push e in iraise (Cerrors.process_vernac_interp_error e)) arg in + let hpat, a1 = match arg with + | (_, Search.GlobSearchSubPattern (Pattern.PMeta _)) :: a' -> all_true, a' + | (true, Search.GlobSearchSubPattern p) :: a' -> + let filter_head, p = interp_head_pat p in + if filter_head then p, a' else all_true, arg + | _ -> all_true, arg in + let is_string = + function (_, Search.GlobSearchString _) -> true | _ -> false in + let a2, a3 = List.partition is_string a1 in + interp_search_about (a2 @ a3) (fun gr env typ -> hpat typ) + +(* Module path postfilter *) + +let pr_modloc (b, m) = if b then str "-" ++ pr_reference m else pr_reference m + +let wit_ssrmodloc = add_genarg "ssrmodloc" pr_modloc + +let pr_ssr_modlocs _ _ _ ml = + if ml = [] then str "" else spc () ++ str "in " ++ pr_list spc pr_modloc ml + +ARGUMENT EXTEND ssr_modlocs TYPED AS ssrmodloc list PRINTED BY pr_ssr_modlocs + | [ ] -> [ [] ] +END + +GEXTEND Gram + GLOBAL: ssr_modlocs; + modloc: [[ "-"; m = global -> true, m | m = global -> false, m]]; + ssr_modlocs: [[ "in"; ml = LIST1 modloc -> ml ]]; +END + +let interp_modloc mr = + let interp_mod (_, mr) = + let (loc, qid) = qualid_of_reference mr in + try Nametab.full_name_module qid with Not_found -> + Errors.user_err_loc (loc, "interp_modloc", str "No Module " ++ pr_qualid qid) in + let mr_out, mr_in = List.partition fst mr in + let interp_bmod b = function + | [] -> fun _ _ _ -> true + | rmods -> Search.module_filter (List.map interp_mod rmods, b) in + let is_in = interp_bmod false mr_in and is_out = interp_bmod true mr_out in + fun gr env typ -> is_in gr env typ && is_out gr env typ + +(* The unified, extended vernacular "Search" command *) + +let ssrdisplaysearch gr env t = + let pr_res = pr_global gr ++ spc () ++ str " " ++ pr_lconstr_env env Evd.empty t in + msg (hov 2 pr_res ++ fnl ()) + +VERNAC COMMAND EXTEND SsrSearchPattern CLASSIFIED AS QUERY +| [ "Search" ssr_search_arg(a) ssr_modlocs(mr) ] -> + [ let hpat = interp_search_arg a in + let in_mod = interp_modloc mr in + let post_filter gr env typ = in_mod gr env typ && hpat gr env typ in + let display gr env typ = + if post_filter gr env typ then ssrdisplaysearch gr env typ + in + Search.generic_search None display ] +END + +(* }}} *) + +(** Alternative notations for "match" and anonymous arguments. {{{ ************) + +(* Syntax: *) +(* if <term> is <pattern> then ... else ... *) +(* if <term> is <pattern> [in ..] return ... then ... else ... *) +(* let: <pattern> := <term> in ... *) +(* let: <pattern> [in ...] := <term> return ... in ... *) +(* The scope of a top-level 'as' in the pattern extends over the *) +(* 'return' type (dependent if/let). *) +(* Note that the optional "in ..." appears next to the <pattern> *) +(* rather than the <term> in then "let:" syntax. The alternative *) +(* would lead to ambiguities in, e.g., *) +(* let: p1 := (*v---INNER LET:---v *) *) +(* let: p2 := let: p3 := e3 in k return t in k2 in k1 return t' *) +(* in b (*^--ALTERNATIVE INNER LET--------^ *) *) + +(* Caveat : There is no pretty-printing support, since this would *) +(* require a modification to the Coq kernel (adding a new match *) +(* display style -- why aren't these strings?); also, the v8.1 *) +(* pretty-printer only allows extension hooks for printing *) +(* integer or string literals. *) +(* Also note that in the v8 grammar "is" needs to be a keyword; *) +(* as this can't be done from an ML extension file, the new *) +(* syntax will only work when ssreflect.v is imported. *) + +let no_ct = None, None and no_rt = None in +let aliasvar = function + | [_, [CPatAlias (loc, _, id)]] -> Some (loc,Name id) + | _ -> None in +let mk_cnotype mp = aliasvar mp, None in +let mk_ctype mp t = aliasvar mp, Some t in +let mk_rtype t = Some t in +let mk_dthen loc (mp, ct, rt) c = (loc, mp, c), ct, rt in +let mk_let loc rt ct mp c1 = + CCases (loc, LetPatternStyle, rt, ct, [loc, mp, c1]) in +GEXTEND Gram + GLOBAL: binder_constr; + ssr_rtype: [[ "return"; t = operconstr LEVEL "100" -> mk_rtype t ]]; + ssr_mpat: [[ p = pattern -> [!@loc, [p]] ]]; + ssr_dpat: [ + [ mp = ssr_mpat; "in"; t = pattern; rt = ssr_rtype -> mp, mk_ctype mp t, rt + | mp = ssr_mpat; rt = ssr_rtype -> mp, mk_cnotype mp, rt + | mp = ssr_mpat -> mp, no_ct, no_rt + ] ]; + ssr_dthen: [[ dp = ssr_dpat; "then"; c = lconstr -> mk_dthen !@loc dp c ]]; + ssr_elsepat: [[ "else" -> [!@loc, [CPatAtom (!@loc, None)]] ]]; + ssr_else: [[ mp = ssr_elsepat; c = lconstr -> !@loc, mp, c ]]; + binder_constr: [ + [ "if"; c = operconstr LEVEL "200"; "is"; db1 = ssr_dthen; b2 = ssr_else -> + let b1, ct, rt = db1 in CCases (!@loc, MatchStyle, rt, [c, ct], [b1; b2]) + | "if"; c = operconstr LEVEL "200";"isn't";db1 = ssr_dthen; b2 = ssr_else -> + let b1, ct, rt = db1 in + let b1, b2 = + let (l1, p1, r1), (l2, p2, r2) = b1, b2 in (l1, p1, r2), (l2, p2, r1) in + CCases (!@loc, MatchStyle, rt, [c, ct], [b1; b2]) + | "let"; ":"; mp = ssr_mpat; ":="; c = lconstr; "in"; c1 = lconstr -> + mk_let (!@loc) no_rt [c, no_ct] mp c1 + | "let"; ":"; mp = ssr_mpat; ":="; c = lconstr; + rt = ssr_rtype; "in"; c1 = lconstr -> + mk_let (!@loc) rt [c, mk_cnotype mp] mp c1 + | "let"; ":"; mp = ssr_mpat; "in"; t = pattern; ":="; c = lconstr; + rt = ssr_rtype; "in"; c1 = lconstr -> + mk_let (!@loc) rt [c, mk_ctype mp t] mp c1 + ] ]; +END + +GEXTEND Gram + GLOBAL: closed_binder; + closed_binder: [ + [ ["of" | "&"]; c = operconstr LEVEL "99" -> + [LocalRawAssum ([!@loc, Anonymous], Default Explicit, c)] + ] ]; +END +(* }}} *) + +(** Tacticals (+, -, *, done, by, do, =>, first, and last). {{{ ***************) + +(** Bracketing tactical *) + +(* The tactic pretty-printer doesn't know that some extended tactics *) +(* are actually tacticals. To prevent it from improperly removing *) +(* parentheses we override the parsing rule for bracketed tactic *) +(* expressions so that the pretty-print always reflects the input. *) +(* (Removing user-specified parentheses is dubious anyway). *) + +GEXTEND Gram + GLOBAL: tactic_expr; + ssrparentacarg: [[ "("; tac = tactic_expr; ")" -> !@loc, Tacexp tac ]]; + tactic_expr: LEVEL "0" [[ arg = ssrparentacarg -> TacArg arg ]]; +END + +(** The internal "done" and "ssrautoprop" tactics. *) + +(* For additional flexibility, "done" and "ssrautoprop" are *) +(* defined in Ltac. *) +(* Although we provide a default definition in ssreflect, *) +(* we look up the definition dynamically at each call point, *) +(* to allow for user extensions. "ssrautoprop" defaults to *) +(* trivial. *) + +let donetac gl = + let tacname = + try Nametab.locate_tactic (qualid_of_ident (id_of_string "done")) + with Not_found -> try Nametab.locate_tactic (ssrqid "done") + with Not_found -> Errors.error "The ssreflect library was not loaded" in + let tacexpr = dummy_loc, Tacexpr.Reference (ArgArg (dummy_loc, tacname)) in + Proofview.V82.of_tactic (eval_tactic (Tacexpr.TacArg tacexpr)) gl + +let prof_donetac = mk_profiler "donetac";; +let donetac gl = prof_donetac.profile donetac gl;; + +let ssrautoprop gl = + try + let tacname = + try Nametab.locate_tactic (qualid_of_ident (id_of_string "ssrautoprop")) + with Not_found -> Nametab.locate_tactic (ssrqid "ssrautoprop") in + let tacexpr = dummy_loc, Tacexpr.Reference (ArgArg (dummy_loc, tacname)) in + Proofview.V82.of_tactic (eval_tactic (Tacexpr.TacArg tacexpr)) gl + with Not_found -> Proofview.V82.of_tactic (Auto.full_trivial []) gl + +let () = ssrautoprop_tac := ssrautoprop + +let tclBY tac = tclTHEN tac donetac + +(** Tactical arguments. *) + +(* We have four kinds: simple tactics, [|]-bracketed lists, hints, and swaps *) +(* The latter two are used in forward-chaining tactics (have, suffice, wlog) *) +(* and subgoal reordering tacticals (; first & ; last), respectively. *) + +(* Force use of the tactic_expr parsing entry, to rule out tick marks. *) +let pr_ssrtacarg _ _ prt = prt tacltop +ARGUMENT EXTEND ssrtacarg TYPED AS tactic PRINTED BY pr_ssrtacarg +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END +GEXTEND Gram + GLOBAL: ssrtacarg; + ssrtacarg: [[ tac = tactic_expr LEVEL "5" -> tac ]]; +END + +(* Lexically closed tactic for tacticals. *) +let pr_ssrtclarg _ _ prt tac = prt tacltop tac +ARGUMENT EXTEND ssrtclarg TYPED AS ssrtacarg + PRINTED BY pr_ssrtclarg +| [ ssrtacarg(tac) ] -> [ tac ] +END +let eval_tclarg ist tac = ssrevaltac ist tac + +let pr_ortacs prt = + let rec pr_rec = function + | [None] -> spc() ++ str "|" ++ spc() + | None :: tacs -> spc() ++ str "|" ++ pr_rec tacs + | Some tac :: tacs -> spc() ++ str "| " ++ prt tacltop tac ++ pr_rec tacs + | [] -> mt() in + function + | [None] -> spc() + | None :: tacs -> pr_rec tacs + | Some tac :: tacs -> prt tacltop tac ++ pr_rec tacs + | [] -> mt() +let pr_ssrortacs _ _ = pr_ortacs + +ARGUMENT EXTEND ssrortacs TYPED AS tactic option list PRINTED BY pr_ssrortacs +| [ ssrtacarg(tac) "|" ssrortacs(tacs) ] -> [ Some tac :: tacs ] +| [ ssrtacarg(tac) "|" ] -> [ [Some tac; None] ] +| [ ssrtacarg(tac) ] -> [ [Some tac] ] +| [ "|" ssrortacs(tacs) ] -> [ None :: tacs ] +| [ "|" ] -> [ [None; None] ] +END + +let pr_hintarg prt = function + | true, tacs -> hv 0 (str "[ " ++ pr_ortacs prt tacs ++ str " ]") + | false, [Some tac] -> prt tacltop tac + | _, _ -> mt() + +let pr_ssrhintarg _ _ = pr_hintarg + +let mk_hint tac = false, [Some tac] +let mk_orhint tacs = true, tacs +let nullhint = true, [] +let nohint = false, [] + +ARGUMENT EXTEND ssrhintarg TYPED AS bool * ssrortacs PRINTED BY pr_ssrhintarg +| [ "[" "]" ] -> [ nullhint ] +| [ "[" ssrortacs(tacs) "]" ] -> [ mk_orhint tacs ] +| [ ssrtacarg(arg) ] -> [ mk_hint arg ] +END + +ARGUMENT EXTEND ssrortacarg TYPED AS ssrhintarg PRINTED BY pr_ssrhintarg +| [ "[" ssrortacs(tacs) "]" ] -> [ mk_orhint tacs ] +END + +let hinttac ist is_by (is_or, atacs) = + let dtac = if is_by then donetac else tclIDTAC in + let mktac = function + | Some atac -> tclTHEN (ssrevaltac ist atac) dtac + | _ -> dtac in + match List.map mktac atacs with + | [] -> if is_or then dtac else tclIDTAC + | [tac] -> tac + | tacs -> tclFIRST tacs + +(** The "-"/"+"/"*" tacticals. *) + +(* These are just visual cues to flag the beginning of the script for *) +(* new subgoals, when indentation is not appropriate (typically after *) +(* tactics that generate more than two subgoals). *) + +TACTIC EXTEND ssrtclplus +| [ "YouShouldNotTypeThis" "+" ssrtclarg(arg) ] -> [ Proofview.V82.tactic (eval_tclarg ist arg) ] +END +set_pr_ssrtac "tclplus" 5 [ArgSep "+ "; ArgSsr "tclarg"] + +TACTIC EXTEND ssrtclminus +| [ "YouShouldNotTypeThis" "-" ssrtclarg(arg) ] -> [ Proofview.V82.tactic (eval_tclarg ist arg) ] +END +set_pr_ssrtac "tclminus" 5 [ArgSep "- "; ArgSsr "tclarg"] + +TACTIC EXTEND ssrtclstar +| [ "YouShouldNotTypeThis" "*" ssrtclarg(arg) ] -> [ Proofview.V82.tactic (eval_tclarg ist arg) ] +END +set_pr_ssrtac "tclstar" 5 [ArgSep "- "; ArgSsr "tclarg"] + +let gen_tclarg = in_gen (rawwit wit_ssrtclarg) + +GEXTEND Gram + GLOBAL: tactic tactic_mode; + tactic: [ + [ "+"; tac = ssrtclarg -> ssrtac_expr !@loc "tclplus" [gen_tclarg tac] + | "-"; tac = ssrtclarg -> ssrtac_expr !@loc "tclminus" [gen_tclarg tac] + | "*"; tac = ssrtclarg -> ssrtac_expr !@loc "tclstar" [gen_tclarg tac] + ] ]; + tactic_mode: [ + [ "+"; tac = G_vernac.subgoal_command -> tac None + | "-"; tac = G_vernac.subgoal_command -> tac None + | "*"; tac = G_vernac.subgoal_command -> tac None + ] ]; +END + +(** The "by" tactical. *) + +let pr_hint prt arg = + if arg = nohint then mt() else str "by " ++ pr_hintarg prt arg +let pr_ssrhint _ _ = pr_hint + +ARGUMENT EXTEND ssrhint TYPED AS ssrhintarg PRINTED BY pr_ssrhint +| [ ] -> [ nohint ] +END + +TACTIC EXTEND ssrtclby +| [ "YouShouldNotTypeThis" ssrhint(tac) ] -> [ Proofview.V82.tactic (hinttac ist true tac) ] +END +set_pr_ssrtac "tclby" 0 [ArgSsr "hint"] + +(* We can't parse "by" in ARGUMENT EXTEND because it will only be made *) +(* into a keyword in ssreflect.v; so we anticipate this in GEXTEND. *) + +GEXTEND Gram + GLOBAL: ssrhint simple_tactic; + ssrhint: [[ "by"; arg = ssrhintarg -> arg ]]; + simple_tactic: [ + [ "by"; arg = ssrhintarg -> + let garg = in_gen (rawwit wit_ssrhint) arg in + ssrtac_atom !@loc "tclby" [garg] + ] ]; +END +(* }}} *) + +(** Bound assumption argument *) + +(* The Ltac API does have a type for assumptions but it is level-dependent *) +(* and therefore impratical to use for complex arguments, so we substitute *) +(* our own to have a uniform representation. Also, we refuse to intern *) +(* idents that match global/section constants, since this would lead to *) +(* fragile Ltac scripts. *) + +type ssrhyp = SsrHyp of loc * identifier + +let hyp_id (SsrHyp (_, id)) = id +let pr_hyp (SsrHyp (_, id)) = pr_id id +let pr_ssrhyp _ _ _ = pr_hyp + +let wit_ssrhyprep = add_genarg "ssrhyprep" pr_hyp + +let hyp_err loc msg id = + Errors.user_err_loc (loc, "ssrhyp", str msg ++ pr_id id) + +let intern_hyp ist (SsrHyp (loc, id) as hyp) = + let _ = Tacintern.intern_genarg ist (in_gen (rawwit wit_var) (loc, id)) in + if not_section_id id then hyp else + hyp_err loc "Can't clear section hypothesis " id + +let interp_hyp ist gl (SsrHyp (loc, id)) = + let s, id' = interp_wit wit_var ist gl (loc, id) in + if not_section_id id' then s, SsrHyp (loc, id') else + hyp_err loc "Can't clear section hypothesis " id' + +ARGUMENT EXTEND ssrhyp TYPED AS ssrhyprep PRINTED BY pr_ssrhyp + INTERPRETED BY interp_hyp + GLOBALIZED BY intern_hyp + | [ ident(id) ] -> [ SsrHyp (loc, id) ] +END + +type ssrhyp_or_id = Hyp of ssrhyp | Id of ssrhyp + +let hoik f = function Hyp x -> f x | Id x -> f x +let hoi_id = hoik hyp_id +let pr_hoi = hoik pr_hyp +let pr_ssrhoi _ _ _ = pr_hoi + +let wit_ssrhoirep = add_genarg "ssrhoirep" pr_hoi + +let intern_ssrhoi ist = function + | Hyp h -> Hyp (intern_hyp ist h) + | Id (SsrHyp (_, id)) as hyp -> + let _ = Tacintern.intern_genarg ist (in_gen (rawwit wit_ident) id) in + hyp + +let interp_ssrhoi ist gl = function + | Hyp h -> let s, h' = interp_hyp ist gl h in s, Hyp h' + | Id (SsrHyp (loc, id)) -> + let s, id' = interp_wit wit_ident ist gl id in + s, Id (SsrHyp (loc, id')) + +ARGUMENT EXTEND ssrhoi_hyp TYPED AS ssrhoirep PRINTED BY pr_ssrhoi + INTERPRETED BY interp_ssrhoi + GLOBALIZED BY intern_ssrhoi + | [ ident(id) ] -> [ Hyp (SsrHyp(loc, id)) ] +END +ARGUMENT EXTEND ssrhoi_id TYPED AS ssrhoirep PRINTED BY pr_ssrhoi + INTERPRETED BY interp_ssrhoi + GLOBALIZED BY intern_ssrhoi + | [ ident(id) ] -> [ Id (SsrHyp(loc, id)) ] +END + +type ssrhyps = ssrhyp list + +let pr_hyps = pr_list pr_spc pr_hyp +let pr_ssrhyps _ _ _ = pr_hyps +let hyps_ids = List.map hyp_id + +let rec check_hyps_uniq ids = function + | SsrHyp (loc, id) :: _ when List.mem id ids -> + hyp_err loc "Duplicate assumption " id + | SsrHyp (_, id) :: hyps -> check_hyps_uniq (id :: ids) hyps + | [] -> () + +let check_hyp_exists hyps (SsrHyp(_, id)) = + try ignore(Context.lookup_named id hyps) + with Not_found -> errorstrm (str"No assumption is named " ++ pr_id id) + +let interp_hyps ist gl ghyps = + let hyps = List.map snd (List.map (interp_hyp ist gl) ghyps) in + check_hyps_uniq [] hyps; Tacmach.project gl, hyps + +ARGUMENT EXTEND ssrhyps TYPED AS ssrhyp list PRINTED BY pr_ssrhyps + INTERPRETED BY interp_hyps + | [ ssrhyp_list(hyps) ] -> [ check_hyps_uniq [] hyps; hyps ] +END + +(** Terms parsing. {{{ ********************************************************) + +(* Because we allow wildcards in term references, we need to stage the *) +(* interpretation of terms so that it occurs at the right time during *) +(* the execution of the tactic (e.g., so that we don't report an error *) +(* for a term that isn't actually used in the execution). *) +(* The term representation tracks whether the concrete initial term *) +(* started with an opening paren, which might avoid a conflict between *) +(* the ssrreflect term syntax and Gallina notation. *) + +(* kinds of terms *) + +type ssrtermkind = char (* print flag *) + +let input_ssrtermkind strm = match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "(" -> '(' + | Tok.KEYWORD "@" -> '@' + | _ -> ' ' + +let ssrtermkind = Gram.Entry.of_parser "ssrtermkind" input_ssrtermkind + +let id_of_Cterm t = match id_of_cpattern t with + | Some x -> x + | None -> loc_error (loc_of_cpattern t) "Only identifiers are allowed here" +let ssrhyp_of_ssrterm = function + | k, (_, Some c) as o -> + SsrHyp (constr_loc c, id_of_Cterm (cpattern_of_term o)), String.make 1 k + | _, (_, None) -> assert false + +(* terms *) +let pr_ssrterm _ _ _ = pr_term +let pf_intern_term ist gl (_, c) = glob_constr ist (pf_env gl) c +let intern_term ist sigma env (_, c) = glob_constr ist env c +let interp_term ist gl (_, c) = snd (interp_open_constr ist gl c) +let force_term ist gl (_, c) = interp_constr ist gl c +let glob_ssrterm gs = function + | k, (_, Some c) -> k, Tacintern.intern_constr gs c + | ct -> ct +let subst_ssrterm s (k, c) = k, Tacsubst.subst_glob_constr_and_expr s c +let interp_ssrterm _ gl t = Tacmach.project gl, t + +ARGUMENT EXTEND ssrterm + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_ssrterm SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "YouShouldNotTypeThis" constr(c) ] -> [ mk_lterm c ] +END + +GEXTEND Gram + GLOBAL: ssrterm; + ssrterm: [[ k = ssrtermkind; c = constr -> mk_term k c ]]; +END +(* }}} *) + +(** The "in" pseudo-tactical {{{ **********************************************) + +(* We can't make "in" into a general tactical because this would create a *) +(* crippling conflict with the ltac let .. in construct. Hence, we add *) +(* explicitly an "in" suffix to all the extended tactics for which it is *) +(* relevant (including move, case, elim) and to the extended do tactical *) +(* below, which yields a general-purpose "in" of the form do [...] in ... *) + +(* This tactical needs to come before the intro tactics because the latter *) +(* must take precautions in order not to interfere with the discharged *) +(* assumptions. This is especially difficult for discharged "let"s, which *) +(* the default simpl and unfold tactics would erase blindly. *) + +(** Clear switch *) + +type ssrclear = ssrhyps + +let pr_clear_ne clr = str "{" ++ pr_hyps clr ++ str "}" +let pr_clear sep clr = if clr = [] then mt () else sep () ++ pr_clear_ne clr + +let pr_ssrclear _ _ _ = pr_clear mt + +ARGUMENT EXTEND ssrclear_ne TYPED AS ssrhyps PRINTED BY pr_ssrclear +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ check_hyps_uniq [] clr; clr ] +END + +ARGUMENT EXTEND ssrclear TYPED AS ssrclear_ne PRINTED BY pr_ssrclear +| [ ssrclear_ne(clr) ] -> [ clr ] +| [ ] -> [ [] ] +END + +let cleartac clr = check_hyps_uniq [] clr; clear (hyps_ids clr) + +(* type ssrwgen = ssrclear * ssrhyp * string *) + +let pr_wgen = function + | (clr, Some((id,k),None)) -> spc() ++ pr_clear mt clr ++ str k ++ pr_hoi id + | (clr, Some((id,k),Some p)) -> + spc() ++ pr_clear mt clr ++ str"(" ++ str k ++ pr_hoi id ++ str ":=" ++ + pr_cpattern p ++ str ")" + | (clr, None) -> spc () ++ pr_clear mt clr +let pr_ssrwgen _ _ _ = pr_wgen + +(* no globwith for char *) +ARGUMENT EXTEND ssrwgen + TYPED AS ssrclear * ((ssrhoi_hyp * string) * cpattern option) option + PRINTED BY pr_ssrwgen +| [ ssrclear_ne(clr) ] -> [ clr, None ] +| [ ssrhoi_hyp(hyp) ] -> [ [], Some((hyp, " "), None) ] +| [ "@" ssrhoi_hyp(hyp) ] -> [ [], Some((hyp, "@"), None) ] +| [ "(" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id," "),Some p) ] +| [ "(" ssrhoi_id(id) ")" ] -> [ [], Some ((id,"("), None) ] +| [ "(@" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id,"@"),Some p) ] +| [ "(" "@" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id,"@"),Some p) ] +END + +type ssrclseq = InGoal | InHyps + | InHypsGoal | InHypsSeqGoal | InSeqGoal | InHypsSeq | InAll | InAllHyps + +let pr_clseq = function + | InGoal | InHyps -> mt () + | InSeqGoal -> str "|- *" + | InHypsSeqGoal -> str " |- *" + | InHypsGoal -> str " *" + | InAll -> str "*" + | InHypsSeq -> str " |-" + | InAllHyps -> str "* |-" + +let wit_ssrclseq = add_genarg "ssrclseq" pr_clseq +let pr_clausehyps = pr_list pr_spc pr_wgen +let pr_ssrclausehyps _ _ _ = pr_clausehyps + +ARGUMENT EXTEND ssrclausehyps +TYPED AS ssrwgen list PRINTED BY pr_ssrclausehyps +| [ ssrwgen(hyp) "," ssrclausehyps(hyps) ] -> [ hyp :: hyps ] +| [ ssrwgen(hyp) ssrclausehyps(hyps) ] -> [ hyp :: hyps ] +| [ ssrwgen(hyp) ] -> [ [hyp] ] +END + +(* type ssrclauses = ssrahyps * ssrclseq *) + +let pr_clauses (hyps, clseq) = + if clseq = InGoal then mt () + else str "in " ++ pr_clausehyps hyps ++ pr_clseq clseq +let pr_ssrclauses _ _ _ = pr_clauses + +ARGUMENT EXTEND ssrclauses TYPED AS ssrwgen list * ssrclseq + PRINTED BY pr_ssrclauses + | [ "in" ssrclausehyps(hyps) "|-" "*" ] -> [ hyps, InHypsSeqGoal ] + | [ "in" ssrclausehyps(hyps) "|-" ] -> [ hyps, InHypsSeq ] + | [ "in" ssrclausehyps(hyps) "*" ] -> [ hyps, InHypsGoal ] + | [ "in" ssrclausehyps(hyps) ] -> [ hyps, InHyps ] + | [ "in" "|-" "*" ] -> [ [], InSeqGoal ] + | [ "in" "*" ] -> [ [], InAll ] + | [ "in" "*" "|-" ] -> [ [], InAllHyps ] + | [ ] -> [ [], InGoal ] +END + +let nohide = mkRel 0 +let hidden_goal_tag = "the_hidden_goal" + +(* Reduction that preserves the Prod/Let spine of the "in" tactical. *) + +let inc_safe n = if n = 0 then n else n + 1 +let rec safe_depth c = match kind_of_term c with +| LetIn (Name x, _, _, c') when is_discharged_id x -> safe_depth c' + 1 +| LetIn (_, _, _, c') | Prod (_, _, c') -> inc_safe (safe_depth c') +| _ -> 0 + +let red_safe r e s c0 = + let rec red_to e c n = match kind_of_term c with + | Prod (x, t, c') when n > 0 -> + let t' = r e s t in let e' = Environ.push_rel (x, None, t') e in + mkProd (x, t', red_to e' c' (n - 1)) + | LetIn (x, b, t, c') when n > 0 -> + let t' = r e s t in let e' = Environ.push_rel (x, None, t') e in + mkLetIn (x, r e s b, t', red_to e' c' (n - 1)) + | _ -> r e s c in + red_to e c0 (safe_depth c0) + +let check_wgen_uniq gens = + let clears = List.flatten (List.map fst gens) in + check_hyps_uniq [] clears; + let ids = CList.map_filter + (function (_,Some ((id,_),_)) -> Some (hoi_id id) | _ -> None) gens in + let rec check ids = function + | id :: _ when List.mem id ids -> + errorstrm (str"Duplicate generalization " ++ pr_id id) + | id :: hyps -> check (id :: ids) hyps + | [] -> () in + check [] ids + +let pf_clauseids gl gens clseq = + let keep_clears = List.map (fun x, _ -> x, None) in + if gens <> [] then (check_wgen_uniq gens; gens) else + if clseq <> InAll && clseq <> InAllHyps then keep_clears gens else + Errors.error "assumptions should be named explicitly" + +let hidden_clseq = function InHyps | InHypsSeq | InAllHyps -> true | _ -> false + +let hidetacs clseq idhide cl0 = + if not (hidden_clseq clseq) then [] else + [posetac idhide cl0; + Proofview.V82.of_tactic (convert_concl_no_check (mkVar idhide))] + +let discharge_hyp (id', (id, mode)) gl = + let cl' = subst_var id (pf_concl gl) in + match pf_get_hyp gl id, mode with + | (_, None, t), _ | (_, Some _, t), "(" -> + apply_type (mkProd (Name id', t, cl')) [mkVar id] gl + | (_, Some v, t), _ -> + Proofview.V82.of_tactic (convert_concl (mkLetIn (Name id', v, t, cl'))) gl + +let endclausestac id_map clseq gl_id cl0 gl = + let not_hyp' id = not (List.mem_assoc id id_map) in + let orig_id id = try List.assoc id id_map with _ -> id in + let dc, c = Term.decompose_prod_assum (pf_concl gl) in + let hide_goal = hidden_clseq clseq in + let c_hidden = hide_goal && c = mkVar gl_id in + let rec fits forced = function + | (id, _) :: ids, (Name id', _, _) :: dc' when id' = id -> + fits true (ids, dc') + | ids, dc' -> + forced && ids = [] && (not hide_goal || dc' = [] && c_hidden) in + let rec unmark c = match kind_of_term c with + | Var id when hidden_clseq clseq && id = gl_id -> cl0 + | Prod (Name id, t, c') when List.mem_assoc id id_map -> + mkProd (Name (orig_id id), unmark t, unmark c') + | LetIn (Name id, v, t, c') when List.mem_assoc id id_map -> + mkLetIn (Name (orig_id id), unmark v, unmark t, unmark c') + | _ -> map_constr unmark c in + let utac hyp = + Proofview.V82.of_tactic + (convert_hyp_no_check (map_named_declaration unmark hyp)) in + let utacs = List.map utac (pf_hyps gl) in + let ugtac gl' = + Proofview.V82.of_tactic + (convert_concl_no_check (unmark (pf_concl gl'))) gl' in + let ctacs = if hide_goal then [clear [gl_id]] else [] in + let mktac itacs = tclTHENLIST (itacs @ utacs @ ugtac :: ctacs) in + let itac (_, id) = Proofview.V82.of_tactic (introduction id) in + if fits false (id_map, List.rev dc) then mktac (List.map itac id_map) gl else + let all_ids = ids_of_rel_context dc @ pf_ids_of_hyps gl in + if List.for_all not_hyp' all_ids && not c_hidden then mktac [] gl else + Errors.error "tampering with discharged assumptions of \"in\" tactical" + +let is_id_constr c = match kind_of_term c with + | Lambda(_,_,c) when isRel c -> 1 = destRel c + | _ -> false + +let red_product_skip_id env sigma c = match kind_of_term c with + | App(hd,args) when Array.length args = 1 && is_id_constr hd -> args.(0) + | _ -> try Tacred.red_product env sigma c with _ -> c + +let abs_wgen keep_let ist f gen (gl,args,c) = + let sigma, env = project gl, pf_env gl in + let evar_closed t p = + if occur_existential t then + Errors.user_err_loc (loc_of_cpattern p,"ssreflect", + pr_constr_pat t ++ + str" contains holes and matches no subterm of the goal") in + match gen with + | _, Some ((x, mode), None) when mode = "@" || (mode = " " && keep_let) -> + let x = hoi_id x in + let _, bo, ty = pf_get_hyp gl x in + gl, + (if bo <> None then args else mkVar x :: args), + mkProd_or_LetIn (Name (f x),bo,ty) (subst_var x c) + | _, Some ((x, _), None) -> + let x = hoi_id x in + gl, mkVar x :: args, mkProd (Name (f x), pf_get_hyp_typ gl x, subst_var x c) + | _, Some ((x, "@"), Some p) -> + let x = hoi_id x in + let cp = interp_cpattern ist gl p None in + let (t, ucst), c = + try fill_occ_pattern ~raise_NoMatch:true env sigma c cp None 1 + with NoMatch -> redex_of_pattern env cp, c in + evar_closed t p; + let ut = red_product_skip_id env sigma t in + pf_merge_uc ucst gl, args, mkLetIn(Name (f x), ut, pf_type_of gl t, c) + | _, Some ((x, _), Some p) -> + let x = hoi_id x in + let cp = interp_cpattern ist gl p None in + let (t, ucst), c = + try fill_occ_pattern ~raise_NoMatch:true env sigma c cp None 1 + with NoMatch -> redex_of_pattern env cp, c in + evar_closed t p; + pf_merge_uc ucst gl, t :: args, mkProd(Name (f x), pf_type_of gl t, c) + | _ -> gl, args, c + +let clr_of_wgen gen clrs = match gen with + | clr, Some ((x, _), None) -> + let x = hoi_id x in + cleartac clr :: cleartac [SsrHyp(Loc.ghost,x)] :: clrs + | clr, _ -> cleartac clr :: clrs + +let tclCLAUSES ist tac (gens, clseq) gl = + if clseq = InGoal || clseq = InSeqGoal then tac gl else + let clr_gens = pf_clauseids gl gens clseq in + let clear = tclTHENLIST (List.rev(List.fold_right clr_of_wgen clr_gens [])) in + let gl_id = mk_anon_id hidden_goal_tag gl in + let cl0 = pf_concl gl in + let dtac gl = + let c = pf_concl gl in + let gl, args, c = + List.fold_right (abs_wgen true ist mk_discharged_id) gens (gl,[], c) in + apply_type c args gl in + let endtac = + let id_map = CList.map_filter (function + | _, Some ((x,_),_) -> let id = hoi_id x in Some (mk_discharged_id id, id) + | _, None -> None) gens in + endclausestac id_map clseq gl_id cl0 in + tclTHENLIST (hidetacs clseq gl_id cl0 @ [dtac; clear; tac; endtac]) gl +(* }}} *) + +(** Simpl switch *) + +type ssrsimpl = Simpl | Cut | SimplCut | Nop + +let pr_simpl = function + | Simpl -> str "/=" + | Cut -> str "//" + | SimplCut -> str "//=" + | Nop -> mt () + +let pr_ssrsimpl _ _ _ = pr_simpl + +let wit_ssrsimplrep = add_genarg "ssrsimplrep" pr_simpl + +ARGUMENT EXTEND ssrsimpl_ne TYPED AS ssrsimplrep PRINTED BY pr_ssrsimpl +| [ "/=" ] -> [ Simpl ] +| [ "//" ] -> [ Cut ] +| [ "//=" ] -> [ SimplCut ] +END + +ARGUMENT EXTEND ssrsimpl TYPED AS ssrsimplrep PRINTED BY pr_ssrsimpl +| [ ssrsimpl_ne(sim) ] -> [ sim ] +| [ ] -> [ Nop ] +END + +(* We must avoid zeta-converting any "let"s created by the "in" tactical. *) + +let safe_simpltac gl = + let cl' = red_safe Tacred.simpl (pf_env gl) (project gl) (pf_concl gl) in + Proofview.V82.of_tactic (convert_concl_no_check cl') gl + +let simpltac = function + | Simpl -> safe_simpltac + | Cut -> tclTRY donetac + | SimplCut -> tclTHEN safe_simpltac (tclTRY donetac) + | Nop -> tclIDTAC + +(** Rewriting direction *) + +let pr_dir = function L2R -> str "->" | R2L -> str "<-" +let pr_rwdir = function L2R -> mt() | R2L -> str "-" + +let rewritetac dir c = + (* Due to the new optional arg ?tac, application shouldn't be too partial *) + Proofview.V82.of_tactic begin + Equality.general_rewrite (dir = L2R) AllOccurrences true false c + end + +let wit_ssrdir = add_genarg "ssrdir" pr_dir + +let dir_org = function L2R -> 1 | R2L -> 2 + +(** Indexes *) + +(* Since SSR indexes are always positive numbers, we use the 0 value *) +(* to encode an omitted index. We reuse the in or_var type, but we *) +(* supply our own interpretation function, which checks for non *) +(* positive values, and allows the use of constr numerals, so that *) +(* e.g., "let n := eval compute in (1 + 3) in (do n!clear)" works. *) + +type ssrindex = int or_var + +let pr_index = function + | ArgVar (_, id) -> pr_id id + | ArgArg n when n > 0 -> int n + | _ -> mt () +let pr_ssrindex _ _ _ = pr_index + +let noindex = ArgArg 0 +let allocc = Some(false,[]) + +let check_index loc i = + if i > 0 then i else loc_error loc "Index not positive" +let mk_index loc = function ArgArg i -> ArgArg (check_index loc i) | iv -> iv + +let interp_index ist gl idx = + Tacmach.project gl, + match idx with + | ArgArg _ -> idx + | ArgVar (loc, id) -> + let i = + try + let v = Id.Map.find id ist.lfun in + begin match Value.to_int v with + | Some i -> i + | None -> + begin match Value.to_constr v with + | Some c -> + let rc = Detyping.detype false [] (pf_env gl) (project gl) c in + begin match Notation.uninterp_prim_token rc with + | _, Numeral bigi -> int_of_string (Bigint.to_string bigi) + | _ -> raise Not_found + end + | None -> raise Not_found + end end + with _ -> loc_error loc "Index not a number" in + ArgArg (check_index loc i) + +ARGUMENT EXTEND ssrindex TYPED AS int_or_var PRINTED BY pr_ssrindex + INTERPRETED BY interp_index +| [ int_or_var(i) ] -> [ mk_index loc i ] +END + +(** Occurrence switch *) + +(* The standard syntax of complemented occurrence lists involves a single *) +(* initial "-", e.g., {-1 3 5}. An initial *) +(* "+" may be used to indicate positive occurrences (the default). The *) +(* "+" is optional, except if the list of occurrences starts with a *) +(* variable or is empty (to avoid confusion with a clear switch). The *) +(* empty positive switch "{+}" selects no occurrences, while the empty *) +(* negative switch "{-}" selects all occurrences explicitly; this is the *) +(* default, but "{-}" prevents the implicit clear, and can be used to *) +(* force dependent elimination -- see ndefectelimtac below. *) + +type ssrocc = occ + +let pr_occ = function + | Some (true, occ) -> str "{-" ++ pr_list pr_spc int occ ++ str "}" + | Some (false, occ) -> str "{+" ++ pr_list pr_spc int occ ++ str "}" + | None -> str "{}" + +let pr_ssrocc _ _ _ = pr_occ + +ARGUMENT EXTEND ssrocc TYPED AS (bool * int list) option PRINTED BY pr_ssrocc +| [ natural(n) natural_list(occ) ] -> [ + Some (false, List.map (check_index loc) (n::occ)) ] +| [ "-" natural_list(occ) ] -> [ Some (true, occ) ] +| [ "+" natural_list(occ) ] -> [ Some (false, occ) ] +END + +let pf_mkprod gl c ?(name=constr_name c) cl = + let t = pf_type_of gl c in + if name <> Anonymous || noccurn 1 cl then mkProd (name, t, cl) else + mkProd (Name (pf_type_id gl t), t, cl) + +let pf_abs_prod name gl c cl = pf_mkprod gl c ~name (subst_term c cl) + +(** Discharge occ switch (combined occurrence / clear switch *) + +type ssrdocc = ssrclear option * ssrocc option + +let mkocc occ = None, occ +let noclr = mkocc None +let mkclr clr = Some clr, None +let nodocc = mkclr [] + +let pr_docc = function + | None, occ -> pr_occ occ + | Some clr, _ -> pr_clear mt clr + +let pr_ssrdocc _ _ _ = pr_docc + +ARGUMENT EXTEND ssrdocc TYPED AS ssrclear option * ssrocc PRINTED BY pr_ssrdocc +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ mkclr clr ] +| [ "{" ssrocc(occ) "}" ] -> [ mkocc occ ] +END + +(** View hint database and View application. {{{ ******************************) + +(* There are three databases of lemmas used to mediate the application *) +(* of reflection lemmas: one for forward chaining, one for backward *) +(* chaining, and one for secondary backward chaining. *) + +(* View hints *) + +let rec isCxHoles = function (CHole _, None) :: ch -> isCxHoles ch | _ -> false + +let pr_raw_ssrhintref prc _ _ = function + | CAppExpl (_, (None, r,x), args) when isCHoles args -> + prc (CRef (r,x)) ++ str "|" ++ int (List.length args) + | CApp (_, (_, CRef _), _) as c -> prc c + | CApp (_, (_, c), args) when isCxHoles args -> + prc c ++ str "|" ++ int (List.length args) + | c -> prc c + +let pr_rawhintref = function + | GApp (_, f, args) when isRHoles args -> + pr_glob_constr f ++ str "|" ++ int (List.length args) + | c -> pr_glob_constr c + +let pr_glob_ssrhintref _ _ _ (c, _) = pr_rawhintref c + +let pr_ssrhintref prc _ _ = prc + +let mkhintref loc c n = match c with + | CRef (r,x) -> CAppExpl (loc, (None, r, x), mkCHoles loc n) + | _ -> mkAppC (c, mkCHoles loc n) + +ARGUMENT EXTEND ssrhintref + PRINTED BY pr_ssrhintref + RAW_TYPED AS constr RAW_PRINTED BY pr_raw_ssrhintref + GLOB_TYPED AS constr GLOB_PRINTED BY pr_glob_ssrhintref + | [ constr(c) ] -> [ c ] + | [ constr(c) "|" natural(n) ] -> [ mkhintref loc c n ] +END + +(* View purpose *) + +let pr_viewpos = function + | 0 -> str " for move/" + | 1 -> str " for apply/" + | 2 -> str " for apply//" + | _ -> mt () + +let pr_ssrviewpos _ _ _ = pr_viewpos + +ARGUMENT EXTEND ssrviewpos TYPED AS int PRINTED BY pr_ssrviewpos + | [ "for" "move" "/" ] -> [ 0 ] + | [ "for" "apply" "/" ] -> [ 1 ] + | [ "for" "apply" "/" "/" ] -> [ 2 ] + | [ "for" "apply" "//" ] -> [ 2 ] + | [ ] -> [ 3 ] +END + +let pr_ssrviewposspc _ _ _ i = pr_viewpos i ++ spc () + +ARGUMENT EXTEND ssrviewposspc TYPED AS ssrviewpos PRINTED BY pr_ssrviewposspc + | [ ssrviewpos(i) ] -> [ i ] +END + +(* The table and its display command *) + +let viewtab : glob_constr list array = Array.make 3 [] + +let _ = + let init () = Array.fill viewtab 0 3 [] in + let freeze _ = Array.copy viewtab in + let unfreeze vt = Array.blit vt 0 viewtab 0 3 in + Summary.declare_summary "ssrview" + { Summary.freeze_function = freeze; + Summary.unfreeze_function = unfreeze; + Summary.init_function = init } + +let mapviewpos f n k = if n < 3 then f n else for i = 0 to k - 1 do f i done + +let print_view_hints i = + let pp_viewname = str "Hint View" ++ pr_viewpos i ++ str " " in + let pp_hints = pr_list spc pr_rawhintref viewtab.(i) in + ppnl (pp_viewname ++ hov 0 pp_hints ++ Pp.cut ()) + +VERNAC COMMAND EXTEND PrintView CLASSIFIED AS QUERY +| [ "Print" "Hint" "View" ssrviewpos(i) ] -> [ mapviewpos print_view_hints i 3 ] +END + +(* Populating the table *) + +let cache_viewhint (_, (i, lvh)) = + let mem_raw h = List.exists (Notation_ops.eq_glob_constr h) in + let add_hint h hdb = if mem_raw h hdb then hdb else h :: hdb in + viewtab.(i) <- List.fold_right add_hint lvh viewtab.(i) + +let subst_viewhint ( subst, (i, lvh as ilvh)) = + let lvh' = List.smartmap (Detyping.subst_glob_constr subst) lvh in + if lvh' == lvh then ilvh else i, lvh' + +let classify_viewhint x = Libobject.Substitute x + +let in_viewhint = + Libobject.declare_object {(Libobject.default_object "VIEW_HINTS") with + Libobject.open_function = (fun i o -> if i = 1 then cache_viewhint o); + Libobject.cache_function = cache_viewhint; + Libobject.subst_function = subst_viewhint; + Libobject.classify_function = classify_viewhint } + +let glob_view_hints lvh = + List.map (Constrintern.intern_constr (Global.env ())) lvh + +let add_view_hints lvh i = Lib.add_anonymous_leaf (in_viewhint (i, lvh)) + +VERNAC COMMAND EXTEND HintView CLASSIFIED AS QUERY + | [ "Hint" "View" ssrviewposspc(n) ne_ssrhintref_list(lvh) ] -> + [ mapviewpos (add_view_hints (glob_view_hints lvh)) n 2 ] +END + +(** Views *) + +(* Views for the "move" and "case" commands are actually open *) +(* terms, but this is handled by interp_view, which is called *) +(* by interp_casearg. We use lists, to support the *) +(* "double-view" feature of the apply command. *) + +(* type ssrview = ssrterm list *) + +let pr_view = pr_list mt (fun c -> str "/" ++ pr_term c) + +let pr_ssrview _ _ _ = pr_view + +ARGUMENT EXTEND ssrview TYPED AS ssrterm list + PRINTED BY pr_ssrview +| [ "/" constr(c) ] -> [ [mk_term ' ' c] ] +| [ "/" constr(c) ssrview(w) ] -> [ (mk_term ' ' c) :: w ] +END + +(* There are two ways of "applying" a view to term: *) +(* 1- using a view hint if the view is an instance of some *) +(* (reflection) inductive predicate. *) +(* 2- applying the view if it coerces to a function, adding *) +(* implicit arguments. *) +(* They require guessing the view hints and the number of *) +(* implicits, respectively, which we do by brute force. *) + +let view_error s gv = + errorstrm (str ("Cannot " ^ s ^ " view ") ++ pr_term gv) + +let interp_view ist si env sigma gv rid = + match intern_term ist sigma env gv with + | GApp (loc, GHole _, rargs) -> + let rv = GApp (loc, rid, rargs) in + snd (interp_open_constr ist (re_sig si sigma) (rv, None)) + | rv -> + let interp rc rargs = + interp_open_constr ist (re_sig si sigma) (mkRApp rc rargs, None) in + let rec simple_view rargs n = + if n < 0 then view_error "use" gv else + try interp rv rargs with _ -> simple_view (mkRHole :: rargs) (n - 1) in + let view_nbimps = interp_view_nbimps ist (re_sig si sigma) rv in + let view_args = [mkRApp rv (mkRHoles view_nbimps); rid] in + let rec view_with = function + | [] -> simple_view [rid] (interp_nbargs ist (re_sig si sigma) rv) + | hint :: hints -> try interp hint view_args with _ -> view_with hints in + snd (view_with (if view_nbimps < 0 then [] else viewtab.(0))) + +let top_id = mk_internal_id "top assumption" + +let with_view ist si env gl0 c name cl prune = + let c2r ist x = { ist with lfun = + Id.Map.add top_id (Value.of_constr x) ist.lfun } in + let rec loop (sigma, c') = function + | f :: view -> + let rid, ist = match kind_of_term c' with + | Var id -> mkRVar id, ist + | _ -> mkRltacVar top_id, c2r ist c' in + loop (interp_view ist si env sigma f rid) view + | [] -> + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + let c' = Reductionops.nf_evar sigma c' in + let n, c', _, ucst = pf_abs_evars gl0 (sigma, c') in + let c' = if not prune then c' else pf_abs_cterm gl0 n c' in + let gl0 = pf_merge_uc ucst gl0 in + pf_abs_prod name gl0 c' (prod_applist cl [c]), c', pf_merge_uc_of sigma gl0 + in loop + +let pf_with_view ist gl (prune, view) cl c = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + with_view ist si env gl c (constr_name c) cl prune (sigma, c) view +(* }}} *) + +(** Extended intro patterns {{{ ***********************************************) + +type ssrtermrep = char * glob_constr_and_expr +type ssripat = + | IpatSimpl of ssrclear * ssrsimpl + | IpatId of identifier + | IpatWild + | IpatCase of ssripats list + | IpatRw of ssrocc * ssrdir + | IpatAll + | IpatAnon + | IpatView of ssrtermrep list + | IpatNoop + | IpatNewHidden of identifier list +and ssripats = ssripat list + +let remove_loc = snd + +let rec ipat_of_intro_pattern = function + | IntroNaming (IntroIdentifier id) -> IpatId id + | IntroAction IntroWildcard -> IpatWild + | IntroAction (IntroOrAndPattern iorpat) -> + IpatCase + (List.map (List.map ipat_of_intro_pattern) + (List.map (List.map remove_loc) iorpat)) + | IntroNaming IntroAnonymous -> IpatAnon + | IntroAction (IntroRewrite b) -> IpatRw (allocc, if b then L2R else R2L) + | IntroNaming (IntroFresh id) -> IpatAnon + | IntroAction (IntroApplyOn _) -> (* to do *) Errors.error "TO DO" + | IntroAction (IntroInjection ips) -> + IpatCase [List.map ipat_of_intro_pattern (List.map remove_loc ips)] + | IntroForthcoming _ -> (* Unable to determine which kind of ipat interp_introid could return [HH] *) + assert false + +let rec pr_ipat = function + | IpatId id -> pr_id id + | IpatSimpl (clr, sim) -> pr_clear mt clr ++ pr_simpl sim + | IpatCase iorpat -> hov 1 (str "[" ++ pr_iorpat iorpat ++ str "]") + | IpatRw (occ, dir) -> pr_occ occ ++ pr_dir dir + | IpatAll -> str "*" + | IpatWild -> str "_" + | IpatAnon -> str "?" + | IpatView v -> pr_view v + | IpatNoop -> str "-" + | IpatNewHidden l -> str "[:" ++ pr_list spc pr_id l ++ str "]" +and pr_iorpat iorpat = pr_list pr_bar pr_ipats iorpat +and pr_ipats ipats = pr_list spc pr_ipat ipats + +let wit_ssripatrep = add_genarg "ssripatrep" pr_ipat + +let pr_ssripat _ _ _ = pr_ipat +let pr_ssripats _ _ _ = pr_ipats +let pr_ssriorpat _ _ _ = pr_iorpat + +let intern_ipat ist ipat = + let rec check_pat = function + | IpatSimpl (clr, _) -> ignore (List.map (intern_hyp ist) clr) + | IpatCase iorpat -> List.iter (List.iter check_pat) iorpat + | _ -> () in + check_pat ipat; ipat + +let intern_ipats ist = List.map (intern_ipat ist) + +let interp_introid ist gl id = + try IntroNaming (IntroIdentifier (hyp_id (snd (interp_hyp ist gl (SsrHyp (dummy_loc, id)))))) + with _ -> snd(snd (interp_intro_pattern ist gl (dummy_loc,IntroNaming (IntroIdentifier id)))) + +let rec add_intro_pattern_hyps (loc, ipat) hyps = match ipat with + | IntroNaming (IntroIdentifier id) -> + if not_section_id id then SsrHyp (loc, id) :: hyps else + hyp_err loc "Can't delete section hypothesis " id + | IntroAction IntroWildcard -> hyps + | IntroAction (IntroOrAndPattern iorpat) -> + List.fold_right (List.fold_right add_intro_pattern_hyps) iorpat hyps + | IntroNaming IntroAnonymous -> [] + | IntroNaming (IntroFresh _) -> [] + | IntroAction (IntroRewrite _) -> hyps + | IntroAction (IntroInjection ips) -> List.fold_right add_intro_pattern_hyps ips hyps + | IntroAction (IntroApplyOn (c,pat)) -> add_intro_pattern_hyps pat hyps + | IntroForthcoming _ -> + (* As in ipat_of_intro_pattern, was unable to determine which kind + of ipat interp_introid could return [HH] *) assert false + +let rec interp_ipat ist gl = + let ltacvar id = Id.Map.mem id ist.lfun in + let rec interp = function + | IpatId id when ltacvar id -> + ipat_of_intro_pattern (interp_introid ist gl id) + | IpatSimpl (clr, sim) -> + let add_hyps (SsrHyp (loc, id) as hyp) hyps = + if not (ltacvar id) then hyp :: hyps else + add_intro_pattern_hyps (loc, (interp_introid ist gl id)) hyps in + let clr' = List.fold_right add_hyps clr [] in + check_hyps_uniq [] clr'; IpatSimpl (clr', sim) + | IpatCase iorpat -> IpatCase (List.map (List.map interp) iorpat) + | IpatNewHidden l -> + IpatNewHidden + (List.map (function + | IntroNaming (IntroIdentifier id) -> id + | _ -> assert false) + (List.map (interp_introid ist gl) l)) + | ipat -> ipat in + interp + +let interp_ipats ist gl l = project gl, List.map (interp_ipat ist gl) l + +let pushIpatRw = function + | pats :: orpat -> (IpatRw (allocc, L2R) :: pats) :: orpat + | [] -> [] + +let pushIpatNoop = function + | pats :: orpat -> (IpatNoop :: pats) :: orpat + | [] -> [] + +ARGUMENT EXTEND ssripat TYPED AS ssripatrep list PRINTED BY pr_ssripats + INTERPRETED BY interp_ipats + GLOBALIZED BY intern_ipats + | [ "_" ] -> [ [IpatWild] ] + | [ "*" ] -> [ [IpatAll] ] + | [ ident(id) ] -> [ [IpatId id] ] + | [ "?" ] -> [ [IpatAnon] ] + | [ ssrsimpl_ne(sim) ] -> [ [IpatSimpl ([], sim)] ] + | [ ssrdocc(occ) "->" ] -> [ match occ with + | None, occ -> [IpatRw (occ, L2R)] + | Some clr, _ -> [IpatSimpl (clr, Nop); IpatRw (allocc, L2R)]] + | [ ssrdocc(occ) "<-" ] -> [ match occ with + | None, occ -> [IpatRw (occ, R2L)] + | Some clr, _ -> [IpatSimpl (clr, Nop); IpatRw (allocc, R2L)]] + | [ ssrdocc(occ) ] -> [ match occ with + | Some cl, _ -> check_hyps_uniq [] cl; [IpatSimpl (cl, Nop)] + | _ -> loc_error loc "Only identifiers are allowed here"] + | [ "->" ] -> [ [IpatRw (allocc, L2R)] ] + | [ "<-" ] -> [ [IpatRw (allocc, R2L)] ] + | [ "-" ] -> [ [IpatNoop] ] + | [ "-/" "=" ] -> [ [IpatNoop;IpatSimpl([],Simpl)] ] + | [ "-/=" ] -> [ [IpatNoop;IpatSimpl([],Simpl)] ] + | [ "-/" "/" ] -> [ [IpatNoop;IpatSimpl([],Cut)] ] + | [ "-//" ] -> [ [IpatNoop;IpatSimpl([],Cut)] ] + | [ "-/" "/=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ "-//" "=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ "-//=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ ssrview(v) ] -> [ [IpatView v] ] + | [ "[" ":" ident_list(idl) "]" ] -> [ [IpatNewHidden idl] ] +END + +ARGUMENT EXTEND ssripats TYPED AS ssripat PRINTED BY pr_ssripats + | [ ssripat(i) ssripats(tl) ] -> [ i @ tl ] + | [ ] -> [ [] ] +END + +ARGUMENT EXTEND ssriorpat TYPED AS ssripat list PRINTED BY pr_ssriorpat +| [ ssripats(pats) "|" ssriorpat(orpat) ] -> [ pats :: orpat ] +| [ ssripats(pats) "|-" ">" ssriorpat(orpat) ] -> [ pats :: pushIpatRw orpat ] +| [ ssripats(pats) "|-" ssriorpat(orpat) ] -> [ pats :: pushIpatNoop orpat ] +| [ ssripats(pats) "|->" ssriorpat(orpat) ] -> [ pats :: pushIpatRw orpat ] +| [ ssripats(pats) "||" ssriorpat(orpat) ] -> [ pats :: [] :: orpat ] +| [ ssripats(pats) "|||" ssriorpat(orpat) ] -> [ pats :: [] :: [] :: orpat ] +| [ ssripats(pats) "||||" ssriorpat(orpat) ] -> [ [pats; []; []; []] @ orpat ] +| [ ssripats(pats) ] -> [ [pats] ] +END + +let reject_ssrhid strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "[" -> + (match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD ":" -> raise Stream.Failure + | _ -> ()) + | _ -> () + +let test_nohidden = Gram.Entry.of_parser "test_ssrhid" reject_ssrhid + +ARGUMENT EXTEND ssrcpat TYPED AS ssripatrep PRINTED BY pr_ssripat + | [ "YouShouldNotTypeThis" ssriorpat(x) ] -> [ IpatCase x ] +END + +GEXTEND Gram + GLOBAL: ssrcpat; + ssrcpat: [[ test_nohidden; "["; iorpat = ssriorpat; "]" -> IpatCase iorpat ]]; +END + +GEXTEND Gram + GLOBAL: ssripat; + ssripat: [[ pat = ssrcpat -> [pat] ]]; +END + +ARGUMENT EXTEND ssripats_ne TYPED AS ssripat PRINTED BY pr_ssripats + | [ ssripat(i) ssripats(tl) ] -> [ i @ tl ] +END + +(* subsets of patterns *) +let check_ssrhpats loc w_binders ipats = + let err_loc s = Errors.user_err_loc (loc, "ssreflect", s) in + let clr, ipats = + let rec aux clr = function + | IpatSimpl (cl, Nop) :: tl -> aux (clr @ cl) tl + | IpatSimpl (cl, sim) :: tl -> clr @ cl, IpatSimpl ([], sim) :: tl + | tl -> clr, tl + in aux [] ipats in + let simpl, ipats = + match List.rev ipats with + | IpatSimpl ([],_) as s :: tl -> [s], List.rev tl + | _ -> [], ipats in + if simpl <> [] && not w_binders then + err_loc (str "No s-item allowed here: " ++ pr_ipats simpl); + let ipat, binders = + let rec loop ipat = function + | [] -> ipat, [] + | ( IpatId _| IpatAnon| IpatCase _| IpatRw _ as i) :: tl -> + if w_binders then + if simpl <> [] && tl <> [] then + err_loc(str"binders XOR s-item allowed here: "++pr_ipats(tl@simpl)) + else if not (List.for_all (function IpatId _ -> true | _ -> false) tl) + then err_loc (str "Only binders allowed here: " ++ pr_ipats tl) + else ipat @ [i], tl + else + if tl = [] then ipat @ [i], [] + else err_loc (str "No binder or s-item allowed here: " ++ pr_ipats tl) + | hd :: tl -> loop (ipat @ [hd]) tl + in loop [] ipats in + ((clr, ipat), binders), simpl + +let single loc = + function [x] -> x | _ -> loc_error loc "Only one intro pattern is allowed" + +let pr_hpats (((clr, ipat), binders), simpl) = + pr_clear mt clr ++ pr_ipats ipat ++ pr_ipats binders ++ pr_ipats simpl +let pr_ssrhpats _ _ _ = pr_hpats +let pr_ssrhpats_wtransp _ _ _ (_, x) = pr_hpats x + +ARGUMENT EXTEND ssrhpats TYPED AS ((ssrclear * ssripat) * ssripat) * ssripat +PRINTED BY pr_ssrhpats + | [ ssripats(i) ] -> [ check_ssrhpats loc true i ] +END + +ARGUMENT EXTEND ssrhpats_wtransp + TYPED AS bool * (((ssrclear * ssripat) * ssripat) * ssripat) + PRINTED BY pr_ssrhpats_wtransp + | [ ssripats(i) ] -> [ false,check_ssrhpats loc true i ] + | [ ssripats(i) "@" ssripats(j) ] -> [ true,check_ssrhpats loc true (i @ j) ] +END + +ARGUMENT EXTEND ssrhpats_nobs +TYPED AS ((ssrclear * ssripat) * ssripat) * ssripat PRINTED BY pr_ssrhpats + | [ ssripats(i) ] -> [ check_ssrhpats loc false i ] +END + +ARGUMENT EXTEND ssrrpat TYPED AS ssripatrep PRINTED BY pr_ssripat + | [ "->" ] -> [ IpatRw (allocc, L2R) ] + | [ "<-" ] -> [ IpatRw (allocc, R2L) ] +END + +type ssrintros = ssripats + +let pr_intros sep intrs = + if intrs = [] then mt() else sep () ++ str "=> " ++ pr_ipats intrs +let pr_ssrintros _ _ _ = pr_intros mt + +ARGUMENT EXTEND ssrintros_ne TYPED AS ssripat + PRINTED BY pr_ssrintros + | [ "=>" ssripats_ne(pats) ] -> [ pats ] +END + +ARGUMENT EXTEND ssrintros TYPED AS ssrintros_ne PRINTED BY pr_ssrintros + | [ ssrintros_ne(intrs) ] -> [ intrs ] + | [ ] -> [ [] ] +END + +let injecteq_id = mk_internal_id "injection equation" + +let pf_nb_prod gl = nb_prod (pf_concl gl) + +let rev_id = mk_internal_id "rev concl" + +let revtoptac n0 gl = + let n = pf_nb_prod gl - n0 in + let dc, cl = decompose_prod_n n (pf_concl gl) in + let dc' = dc @ [Name rev_id, compose_prod (List.rev dc) cl] in + let f = compose_lam dc' (mkEtaApp (mkRel (n + 1)) (-n) 1) in + refine (mkApp (f, [|Evarutil.mk_new_meta ()|])) gl + +let equality_inj l b id c gl = + let msg = ref "" in + try Proofview.V82.of_tactic (Equality.inj l b None c) gl + with + | Compat.Exc_located(_,Errors.UserError (_,s)) + | Errors.UserError (_,s) + when msg := Pp.string_of_ppcmds s; + !msg = "Not a projectable equality but a discriminable one." || + !msg = "Nothing to inject." -> + msg_warning (str !msg); + discharge_hyp (id, (id, "")) gl + +let injectidl2rtac id c gl = + tclTHEN (equality_inj None true id c) (revtoptac (pf_nb_prod gl)) gl + +let injectl2rtac c = match kind_of_term c with +| Var id -> injectidl2rtac id (mkVar id, NoBindings) +| _ -> + let id = injecteq_id in + tclTHENLIST [havetac id c; injectidl2rtac id (mkVar id, NoBindings); clear [id]] + +let is_injection_case c gl = + let (mind,_), _ = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in + eq_gr (IndRef mind) (build_coq_eq ()) + +let perform_injection c gl = + let mind, t = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in + let dc, eqt = decompose_prod t in + if dc = [] then injectl2rtac c gl else + if not (closed0 eqt) then + Errors.error "can't decompose a quantified equality" else + let cl = pf_concl gl in let n = List.length dc in + let c_eq = mkEtaApp c n 2 in + let cl1 = mkLambda (Anonymous, mkArrow eqt cl, mkApp (mkRel 1, [|c_eq|])) in + let id = injecteq_id in + let id_with_ebind = (mkVar id, NoBindings) in + let injtac = tclTHEN (introid id) (injectidl2rtac id id_with_ebind) in + tclTHENLAST (Proofview.V82.of_tactic (apply (compose_lam dc cl1))) injtac gl + +let simplest_newcase_ref = ref (fun t gl -> assert false) +let simplest_newcase x gl = !simplest_newcase_ref x gl + +let ssrscasetac c gl = + if is_injection_case c gl then perform_injection c gl + else simplest_newcase c gl + +let intro_all gl = + let dc, _ = Term.decompose_prod_assum (pf_concl gl) in + tclTHENLIST (List.map anontac (List.rev dc)) gl + +let rec intro_anon gl = + try anontac (List.hd (fst (Term.decompose_prod_n_assum 1 (pf_concl gl)))) gl + with err0 -> try tclTHEN red_in_concl intro_anon gl with _ -> raise err0 + (* with _ -> Errors.error "No product even after reduction" *) + +let with_top tac = + tclTHENLIST [introid top_id; tac (mkVar top_id); clear [top_id]] + +let rec mapLR f = function [] -> [] | x :: s -> let y = f x in y :: mapLR f s + +let wild_ids = ref [] + +let new_wild_id () = + let i = 1 + List.length !wild_ids in + let id = mk_wildcard_id i in + wild_ids := id :: !wild_ids; + id + +let clear_wilds wilds gl = + clear (List.filter (fun id -> List.mem id wilds) (pf_ids_of_hyps gl)) gl + +let clear_with_wilds wilds clr0 gl = + let extend_clr clr (id, _, _ as nd) = + if List.mem id clr || not (List.mem id wilds) then clr else + let vars = global_vars_set_of_decl (pf_env gl) nd in + let occurs id' = Idset.mem id' vars in + if List.exists occurs clr then id :: clr else clr in + clear (Context.fold_named_context_reverse extend_clr ~init:clr0 (pf_hyps gl)) gl + +let tclTHENS_nonstrict tac tacl taclname gl = + let tacres = tac gl in + let n_gls = List.length (sig_it tacres) in + let n_tac = List.length tacl in + if n_gls = n_tac then tclTHENS (fun _ -> tacres) tacl gl else + if n_gls = 0 then tacres else + let pr_only n1 n2 = if n1 < n2 then str "only " else mt () in + let pr_nb n1 n2 name = + pr_only n1 n2 ++ int n1 ++ str (" " ^ String.plural n1 name) in + errorstrm (pr_nb n_tac n_gls taclname ++ spc () + ++ str "for " ++ pr_nb n_gls n_tac "subgoal") + +(* Forward reference to extended rewrite *) +let ipat_rewritetac = ref (fun _ -> rewritetac) + +let rec is_name_in_ipats name = function + | IpatSimpl(clr,_) :: tl -> + List.exists (function SsrHyp(_,id) -> id = name) clr + || is_name_in_ipats name tl + | IpatId id :: tl -> id = name || is_name_in_ipats name tl + | IpatCase l :: tl -> is_name_in_ipats name (List.flatten l @ tl) + | _ :: tl -> is_name_in_ipats name tl + | [] -> false + +let move_top_with_view = ref (fun _ -> assert false) + +let rec nat_of_n n = + if n = 0 then mkConstruct path_of_O + else mkApp (mkConstruct path_of_S, [|nat_of_n (n-1)|]) + +let ssr_abstract_id = Summary.ref "~name:SSR:abstractid" 0 + +let mk_abstract_id () = incr ssr_abstract_id; nat_of_n !ssr_abstract_id + +let ssrmkabs id gl = + let env, concl = pf_env gl, pf_concl gl in + let step sigma = + let sigma, abstract_proof, abstract_ty = + let sigma, (ty, _) = + Evarutil.new_type_evar env sigma Evd.univ_flexible_alg in + let sigma, ablock = mkSsrConst "abstract_lock" env sigma in + let sigma, lock = Evarutil.new_evar env sigma ablock in + let sigma, abstract = mkSsrConst "abstract" env sigma in + let abstract_ty = mkApp(abstract, [|ty;mk_abstract_id ();lock|]) in + let sigma, m = Evarutil.new_evar env sigma abstract_ty in + sigma, m, abstract_ty in + let sigma, kont = + let rd = Name id, None, abstract_ty in + Evarutil.new_evar (Environ.push_rel rd env) sigma concl in + pp(lazy(pr_constr concl)); + let term = mkApp (mkLambda(Name id,abstract_ty,kont) ,[|abstract_proof|]) in + let sigma, _ = Typing.e_type_of env sigma term in + sigma, term in + Proofview.V82.of_tactic + (Proofview.tclTHEN + (Tactics.New.refine step) + (Proofview.tclFOCUS 1 3 Proofview.shelve)) gl + +let ssrmkabstac ids = + List.fold_right (fun id tac -> tclTHENFIRST (ssrmkabs id) tac) ids tclIDTAC + +(* introstac: for "move" and "clear", tclEQINTROS: for "case" and "elim" *) +(* This block hides the spaghetti-code needed to implement the only two *) +(* tactics that should be used to process intro patters. *) +(* The difficulty is that we don't want to always rename, but we can *) +(* compute needeed renamings only at runtime, so we theread a tree like *) +(* imperativestructure so that outer renamigs are inherited by inner *) +(* ipts and that the cler performed at the end of ipatstac clears hyps *) +(* eventually renamed at runtime. *) +(* TODO: hide wild_ids in this block too *) +let introstac, tclEQINTROS = + let rec map_acc_k f k = function + | [] -> (* tricky: we save wilds now, we get to_cler (aka k) later *) + let clear_ww = clear_with_wilds !wild_ids in + [fun gl -> clear_ww (hyps_ids (List.flatten (List.map (!) k))) gl] + | x :: xs -> let k, x = f k xs x in x :: map_acc_k f k xs in + let rename force to_clr rest clr gl = + let hyps = pf_hyps gl in + pp(lazy(str"rename " ++ pr_clear spc clr)); + let () = if not force then List.iter (check_hyp_exists hyps) clr in + if List.exists (fun x -> force || is_name_in_ipats (hyp_id x) rest) clr then + let ren_clr, ren = + List.split (List.map (fun x -> let x = hyp_id x in + let x' = mk_anon_id (string_of_id x) gl in + SsrHyp (dummy_loc, x'), (x, x')) clr) in + let () = to_clr := ren_clr in + Proofview.V82.of_tactic (rename_hyp ren) gl + else + let () = to_clr := clr in + tclIDTAC gl in + let rec ipattac ?ist k rest = function + | IpatWild -> k, introid (new_wild_id ()) + | IpatCase iorpat -> k, tclIORPAT ?ist k (with_top ssrscasetac) iorpat + | IpatRw (occ, dir) -> k, with_top (!ipat_rewritetac occ dir) + | IpatId id -> k, introid id + | IpatNewHidden idl -> k, ssrmkabstac idl + | IpatSimpl (clr, sim) -> + let to_clr = ref [] in + to_clr :: k, tclTHEN (rename false to_clr rest clr) (simpltac sim) + | IpatAll -> k, intro_all + | IpatAnon -> k, intro_anon + | IpatNoop -> k, tclIDTAC + | IpatView v -> match ist with + | None -> anomaly "ipattac with no ist but view" + | Some ist -> match rest with + | (IpatCase _ | IpatRw _)::_ -> + let to_clr = ref [] in let top_id = ref top_id in + to_clr :: k, + tclTHEN + (!move_top_with_view false top_id (false,v) ist) + (fun gl -> + rename true to_clr rest [SsrHyp (dummy_loc, !top_id)]gl) + | _ -> k, !move_top_with_view true (ref top_id) (true,v) ist + and tclIORPAT ?ist k tac = function + | [[]] -> tac + | orp -> + tclTHENS_nonstrict tac (mapLR (ipatstac ?ist k) orp) "intro pattern" + and ipatstac ?ist k ipats = + tclTHENLIST (map_acc_k (ipattac ?ist) k ipats) in + let introstac ?ist ipats = + wild_ids := []; + let tac = ipatstac ?ist [] ipats in + tclTHENLIST [tac; clear_wilds !wild_ids] in + let tclEQINTROS ?ist tac eqtac ipats = + wild_ids := []; + let rec split_itacs to_clr tac' = function + | (IpatSimpl _ as spat) :: ipats' -> + let to_clr, tac = ipattac ?ist to_clr ipats' spat in + split_itacs to_clr (tclTHEN tac' tac) ipats' + | IpatCase iorpat :: ipats' -> + to_clr, tclIORPAT ?ist to_clr tac' iorpat, ipats' + | ipats' -> to_clr, tac', ipats' in + let to_clr, tac1, ipats' = split_itacs [] tac ipats in + let tac2 = ipatstac ?ist to_clr ipats' in + tclTHENLIST [tac1; eqtac; tac2; clear_wilds !wild_ids] in + introstac, tclEQINTROS +;; + +let rec eqmoveipats eqpat = function + | (IpatSimpl _ as ipat) :: ipats -> ipat :: eqmoveipats eqpat ipats + | (IpatAll :: _ | []) as ipats -> IpatAnon :: eqpat :: ipats + | ipat :: ipats -> ipat :: eqpat :: ipats + +(* General case *) +let tclINTROS ist tac ipats = + tclEQINTROS ~ist (tac ist) tclIDTAC ipats + +(** The "=>" tactical *) + +let ssrintros_sep = + let atom_sep = function + | TacSplit (_, [NoBindings]) -> mt + (* | TacExtend (_, "ssrapply", []) -> mt *) + | _ -> spc in + function + | TacId [] -> mt + | TacArg (_, Tacexp _) -> mt + | TacArg (_, Reference _) -> mt + | TacAtom (_, atom) -> atom_sep atom + | _ -> spc + +let pr_ssrintrosarg _ _ prt (tac, ipats) = + prt tacltop tac ++ pr_intros (ssrintros_sep tac) ipats + +ARGUMENT EXTEND ssrintrosarg TYPED AS tactic * ssrintros + PRINTED BY pr_ssrintrosarg +| [ "YouShouldNotTypeThis" ssrtacarg(arg) ssrintros_ne(ipats) ] -> [ arg, ipats ] +END + +TACTIC EXTEND ssrtclintros +| [ "YouShouldNotTypeThis" ssrintrosarg(arg) ] -> + [ let tac, intros = arg in + Proofview.V82.tactic (tclINTROS ist (fun ist -> ssrevaltac ist tac) intros) ] +END +set_pr_ssrtac "tclintros" 0 [ArgSsr "introsarg"] + +let tclintros_expr loc tac ipats = + let args = [in_gen (rawwit wit_ssrintrosarg) (tac, ipats)] in + ssrtac_expr loc "tclintros" args + +GEXTEND Gram + GLOBAL: tactic_expr; + tactic_expr: LEVEL "1" [ RIGHTA + [ tac = tactic_expr; intros = ssrintros_ne -> tclintros_expr !@loc tac intros + ] ]; +END +(* }}} *) + +(** Multipliers {{{ ***********************************************************) + +(* modality *) + +type ssrmmod = May | Must | Once + +let pr_mmod = function May -> str "?" | Must -> str "!" | Once -> mt () + +let wit_ssrmmod = add_genarg "ssrmmod" pr_mmod +let ssrmmod = Pcoq.create_generic_entry "ssrmmod" (Genarg.rawwit wit_ssrmmod) +GEXTEND Gram + GLOBAL: ssrmmod; + ssrmmod: [[ "!" -> Must | LEFTQMARK -> May | "?" -> May]]; +END + +(* tactical *) + +let tclID tac = tac + +let tclDOTRY n tac = + if n <= 0 then tclIDTAC else + let rec loop i gl = + if i = n then tclTRY tac gl else + tclTRY (tclTHEN tac (loop (i + 1))) gl in + loop 1 + +let tclDO n tac = + let prefix i = str"At iteration " ++ int i ++ str": " in + let tac_err_at i gl = + try tac gl + with + | Errors.UserError (l, s) as e -> + let _, info = Errors.push e in + let e' = Errors.UserError (l, prefix i ++ s) in + Util.iraise (e', info) + | Compat.Exc_located(loc, Errors.UserError (l, s)) -> + raise (Compat.Exc_located(loc, Errors.UserError (l, prefix i ++ s))) in + let rec loop i gl = + if i = n then tac_err_at i gl else + (tclTHEN (tac_err_at i) (loop (i + 1))) gl in + loop 1 + +let tclMULT = function + | 0, May -> tclREPEAT + | 1, May -> tclTRY + | n, May -> tclDOTRY n + | 0, Must -> tclAT_LEAST_ONCE + | n, Must when n > 1 -> tclDO n + | _ -> tclID + +(** The "do" tactical. ********************************************************) + +(* +type ssrdoarg = ((ssrindex * ssrmmod) * ssrhint) * ssrclauses +*) + +let pr_ssrdoarg prc _ prt (((n, m), tac), clauses) = + pr_index n ++ pr_mmod m ++ pr_hintarg prt tac ++ pr_clauses clauses + +ARGUMENT EXTEND ssrdoarg + TYPED AS ((ssrindex * ssrmmod) * ssrhintarg) * ssrclauses + PRINTED BY pr_ssrdoarg +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let ssrdotac ist (((n, m), tac), clauses) = + let mul = get_index n, m in + tclCLAUSES ist (tclMULT mul (hinttac ist false tac)) clauses + +TACTIC EXTEND ssrtcldo +| [ "YouShouldNotTypeThis" "do" ssrdoarg(arg) ] -> [ Proofview.V82.tactic (ssrdotac ist arg) ] +END +set_pr_ssrtac "tcldo" 3 [ArgSep "do "; ArgSsr "doarg"] + +let ssrdotac_expr loc n m tac clauses = + let arg = ((n, m), tac), clauses in + ssrtac_expr loc "tcldo" [in_gen (rawwit wit_ssrdoarg) arg] + +GEXTEND Gram + GLOBAL: tactic_expr; + ssrdotac: [ + [ tac = tactic_expr LEVEL "3" -> mk_hint tac + | tacs = ssrortacarg -> tacs + ] ]; + tactic_expr: LEVEL "3" [ RIGHTA + [ IDENT "do"; m = ssrmmod; tac = ssrdotac; clauses = ssrclauses -> + ssrdotac_expr !@loc noindex m tac clauses + | IDENT "do"; tac = ssrortacarg; clauses = ssrclauses -> + ssrdotac_expr !@loc noindex Once tac clauses + | IDENT "do"; n = int_or_var; m = ssrmmod; + tac = ssrdotac; clauses = ssrclauses -> + ssrdotac_expr !@loc (mk_index !@loc n) m tac clauses + ] ]; +END +(* }}} *) + +(** The "first" and "last" tacticals. {{{ *************************************) + +(* type ssrseqarg = ssrindex * (ssrtacarg * ssrtac option) *) + +let pr_seqtacarg prt = function + | (is_first, []), _ -> str (if is_first then "first" else "last") + | tac, Some dtac -> + hv 0 (pr_hintarg prt tac ++ spc() ++ str "|| " ++ prt tacltop dtac) + | tac, _ -> pr_hintarg prt tac + +let pr_ssrseqarg _ _ prt = function + | ArgArg 0, tac -> pr_seqtacarg prt tac + | i, tac -> pr_index i ++ str " " ++ pr_seqtacarg prt tac + +(* We must parse the index separately to resolve the conflict with *) +(* an unindexed tactic. *) +ARGUMENT EXTEND ssrseqarg TYPED AS ssrindex * (ssrhintarg * tactic option) + PRINTED BY pr_ssrseqarg +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let sq_brace_tacnames = + ["first"; "solve"; "do"; "rewrite"; "have"; "suffices"; "wlog"] + (* "by" is a keyword *) +let accept_ssrseqvar strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT id when not (List.mem id sq_brace_tacnames) -> + accept_before_syms_or_ids ["["] ["first";"last"] strm + | _ -> raise Stream.Failure + +let test_ssrseqvar = Gram.Entry.of_parser "test_ssrseqvar" accept_ssrseqvar + +let swaptacarg (loc, b) = (b, []), Some (TacId []) + +let check_seqtacarg dir arg = match snd arg, dir with + | ((true, []), Some (TacAtom (loc, _))), L2R -> + loc_error loc "expected \"last\"" + | ((false, []), Some (TacAtom (loc, _))), R2L -> + loc_error loc "expected \"first\"" + | _, _ -> arg + +let ssrorelse = Gram.Entry.create "ssrorelse" +GEXTEND Gram + GLOBAL: ssrorelse ssrseqarg; + ssrseqidx: [ + [ test_ssrseqvar; id = Prim.ident -> ArgVar (!@loc, id) + | n = Prim.natural -> ArgArg (check_index !@loc n) + ] ]; + ssrswap: [[ IDENT "first" -> !@loc, true | IDENT "last" -> !@loc, false ]]; + ssrorelse: [[ "||"; tac = tactic_expr LEVEL "2" -> tac ]]; + ssrseqarg: [ + [ arg = ssrswap -> noindex, swaptacarg arg + | i = ssrseqidx; tac = ssrortacarg; def = OPT ssrorelse -> i, (tac, def) + | i = ssrseqidx; arg = ssrswap -> i, swaptacarg arg + | tac = tactic_expr LEVEL "3" -> noindex, (mk_hint tac, None) + ] ]; +END + +let tclPERM perm tac gls = + let subgls = tac gls in + let sigma, subgll = Refiner.unpackage subgls in + let subgll' = perm subgll in + Refiner.repackage sigma subgll' +(* +let tclPERM perm tac gls = + let mkpft n g r = + {Proof_type.open_subgoals = n; Proof_type.goal = g; Proof_type.ref = r} in + let mkleaf g = mkpft 0 g None in + let mkprpft n g pr a = mkpft n g (Some (Proof_type.Prim pr, a)) in + let mkrpft n g c = mkprpft n g (Proof_type.Refine c) in + let mkipft n g = + let mki pft (id, _, _ as d) = + let g' = {g with evar_concl = mkNamedProd_or_LetIn d g.evar_concl} in + mkprpft n g' (Proof_type.Intro id) [pft] in + List.fold_left mki in + let gl = Refiner.sig_it gls in + let mkhyp subgl = + let rec chop_section = function + | (x, _, _ as d) :: e when not_section_id x -> d :: chop_section e + | _ -> [] in + let lhyps = Environ.named_context_of_val subgl.evar_hyps in + mk_perm_id (), subgl, chop_section lhyps in + let mkpfvar (hyp, subgl, lhyps) = + let mkarg args (lhyp, body, _) = + if body = None then mkVar lhyp :: args else args in + mkrpft 0 subgl (applist (mkVar hyp, List.fold_left mkarg [] lhyps)) [] in + let mkpfleaf (_, subgl, lhyps) = mkipft 1 gl (mkleaf subgl) lhyps in + let mkmeta _ = Evarutil.mk_new_meta () in + let mkhypdecl (hyp, subgl, lhyps) = + hyp, None, it_mkNamedProd_or_LetIn subgl.evar_concl lhyps in + let subgls, v as res0 = tac gls in + let sigma, subgll = Refiner.unpackage subgls in + let n = List.length subgll in if n = 0 then res0 else + let hyps = List.map mkhyp subgll in + let hyp_decls = List.map mkhypdecl (List.rev (perm hyps)) in + let c = applist (mkmeta (), List.map mkmeta subgll) in + let pft0 = mkipft 0 gl (v (List.map mkpfvar hyps)) hyp_decls in + let pft1 = mkrpft n gl c (pft0 :: List.map mkpfleaf (perm hyps)) in + let subgll', v' = Refiner.frontier pft1 in + Refiner.repackage sigma subgll', v' +*) + +let tclREV tac gl = tclPERM List.rev tac gl + +let rot_hyps dir i hyps = + let n = List.length hyps in + if i = 0 then List.rev hyps else + if i > n then Errors.error "Not enough subgoals" else + let rec rot i l_hyps = function + | hyp :: hyps' when i > 0 -> rot (i - 1) (hyp :: l_hyps) hyps' + | hyps' -> hyps' @ (List.rev l_hyps) in + rot (match dir with L2R -> i | R2L -> n - i) [] hyps + +let tclSEQAT ist atac1 dir (ivar, ((_, atacs2), atac3)) = + let i = get_index ivar in + let evtac = ssrevaltac ist in + let tac1 = evtac atac1 in + if atacs2 = [] && atac3 <> None then tclPERM (rot_hyps dir i) tac1 else + let evotac = function Some atac -> evtac atac | _ -> tclIDTAC in + let tac3 = evotac atac3 in + let rec mk_pad n = if n > 0 then tac3 :: mk_pad (n - 1) else [] in + match dir, mk_pad (i - 1), List.map evotac atacs2 with + | L2R, [], [tac2] when atac3 = None -> tclTHENFIRST tac1 tac2 + | L2R, [], [tac2] when atac3 = None -> tclTHENLAST tac1 tac2 + | L2R, pad, tacs2 -> tclTHENSFIRSTn tac1 (Array.of_list (pad @ tacs2)) tac3 + | R2L, pad, tacs2 -> tclTHENSLASTn tac1 tac3 (Array.of_list (tacs2 @ pad)) + +(* We can't actually parse the direction separately because this *) +(* would introduce conflicts with the basic ltac syntax. *) +let pr_ssrseqdir _ _ _ = function + | L2R -> str ";" ++ spc () ++ str "first " + | R2L -> str ";" ++ spc () ++ str "last " + +ARGUMENT EXTEND ssrseqdir TYPED AS ssrdir PRINTED BY pr_ssrseqdir +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +TACTIC EXTEND ssrtclseq +| [ "YouShouldNotTypeThis" ssrtclarg(tac) ssrseqdir(dir) ssrseqarg(arg) ] -> + [ Proofview.V82.tactic (tclSEQAT ist tac dir arg) ] +END +set_pr_ssrtac "tclseq" 5 [ArgSsr "tclarg"; ArgSsr "seqdir"; ArgSsr "seqarg"] + +let tclseq_expr loc tac dir arg = + let arg1 = in_gen (rawwit wit_ssrtclarg) tac in + let arg2 = in_gen (rawwit wit_ssrseqdir) dir in + let arg3 = in_gen (rawwit wit_ssrseqarg) (check_seqtacarg dir arg) in + ssrtac_expr loc "tclseq" [arg1; arg2; arg3] + +GEXTEND Gram + GLOBAL: tactic_expr; + ssr_first: [ + [ tac = ssr_first; ipats = ssrintros_ne -> tclintros_expr !@loc tac ipats + | "["; tacl = LIST0 tactic_expr SEP "|"; "]" -> TacFirst tacl + ] ]; + ssr_first_else: [ + [ tac1 = ssr_first; tac2 = ssrorelse -> TacOrelse (tac1, tac2) + | tac = ssr_first -> tac ]]; + tactic_expr: LEVEL "4" [ LEFTA + [ tac1 = tactic_expr; ";"; IDENT "first"; tac2 = ssr_first_else -> + TacThen (tac1, tac2) + | tac = tactic_expr; ";"; IDENT "first"; arg = ssrseqarg -> + tclseq_expr !@loc tac L2R arg + | tac = tactic_expr; ";"; IDENT "last"; arg = ssrseqarg -> + tclseq_expr !@loc tac R2L arg + ] ]; +END +(* }}} *) + +(** 5. Bookkeeping tactics (clear, move, case, elim) *) + +(* post-interpretation of terms *) +let all_ok _ _ = true + +let pf_abs_ssrterm ?(resolve_typeclasses=false) ist gl t = + let sigma, ct as t = interp_term ist gl t in + let sigma, _ as t = + let env = pf_env gl in + if not resolve_typeclasses then t + else + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + sigma, Evarutil.nf_evar sigma ct in + let n, c, abstracted_away, ucst = pf_abs_evars gl t in + List.fold_left Evd.remove sigma abstracted_away, pf_abs_cterm gl n c, ucst + +let pf_interp_ty ?(resolve_typeclasses=false) ist gl ty = + let n_binders = ref 0 in + let ty = match ty with + | a, (t, None) -> + let rec force_type = function + | GProd (l, x, k, s, t) -> incr n_binders; GProd (l, x, k, s, force_type t) + | GLetIn (l, x, v, t) -> incr n_binders; GLetIn (l, x, v, force_type t) + | ty -> mkRCast ty mkRType in + a, (force_type t, None) + | _, (_, Some ty) -> + let rec force_type = function + | CProdN (l, abs, t) -> + n_binders := !n_binders + List.length (List.flatten (List.map pi1 abs)); + CProdN (l, abs, force_type t) + | CLetIn (l, n, v, t) -> incr n_binders; CLetIn (l, n, v, force_type t) + | ty -> mkCCast dummy_loc ty (mkCType dummy_loc) in + mk_term ' ' (force_type ty) in + let strip_cast (sigma, t) = + let rec aux t = match kind_of_type t with + | CastType (t, ty) when !n_binders = 0 && isSort ty -> t + | ProdType(n,s,t) -> decr n_binders; mkProd (n, s, aux t) + | LetInType(n,v,ty,t) -> decr n_binders; mkLetIn (n, v, ty, aux t) + | _ -> anomaly "pf_interp_ty: ssr Type cast deleted by typecheck" in + sigma, aux t in + let sigma, cty as ty = strip_cast (interp_term ist gl ty) in + let ty = + let env = pf_env gl in + if not resolve_typeclasses then ty + else + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + sigma, Evarutil.nf_evar sigma cty in + let n, c, _, ucst = pf_abs_evars gl ty in + let lam_c = pf_abs_cterm gl n c in + let ctx, c = decompose_lam_n n lam_c in + n, compose_prod ctx c, lam_c, ucst +;; + +let whd_app f args = Reductionops.whd_betaiota Evd.empty (mkApp (f, args)) + +let pr_cargs a = + str "[" ++ pr_list pr_spc pr_constr (Array.to_list a) ++ str "]" + +let pp_term gl t = + let t = Reductionops.nf_evar (project gl) t in pr_constr t +let pp_concat hd ?(sep=str", ") = function [] -> hd | x :: xs -> + hd ++ List.fold_left (fun acc x -> acc ++ sep ++ x) x xs + +let fake_pmatcher_end () = + mkProp, L2R, (Evd.empty, Evd.empty_evar_universe_context, mkProp) + +(* TASSI: given (c : ty), generates (c ??? : ty[???/...]) with m evars *) +exception NotEnoughProducts +let prof_saturate_whd = mk_profiler "saturate.whd";; +let saturate ?(beta=false) ?(bi_types=false) env sigma c ?(ty=Retyping.get_type_of env sigma c) m += + let rec loop ty args sigma n = + if n = 0 then + let args = List.rev args in + (if beta then Reductionops.whd_beta sigma else fun x -> x) + (mkApp (c, Array.of_list (List.map snd args))), ty, args, sigma + else match kind_of_type ty with + | ProdType (_, src, tgt) -> + let sigma, x = + Evarutil.new_evar env (create_evar_defs sigma) + (if bi_types then Reductionops.nf_betaiota sigma src else src) in + loop (subst1 x tgt) ((m - n,x) :: args) sigma (n-1) + | CastType (t, _) -> loop t args sigma n + | LetInType (_, v, _, t) -> loop (subst1 v t) args sigma n + | SortType _ -> assert false + | AtomicType _ -> + let ty = + prof_saturate_whd.profile + (Reductionops.whd_betadeltaiota env sigma) ty in + match kind_of_type ty with + | ProdType _ -> loop ty args sigma n + | _ -> anomaly "saturate did not find enough products" + in + loop ty [] sigma m + +let pf_saturate ?beta ?bi_types gl c ?ty m = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + let t, ty, args, sigma = saturate ?beta ?bi_types env sigma c ?ty m in + t, ty, args, re_sig si sigma + +(** Rewrite redex switch *) + +(** Generalization (discharge) item *) + +(* An item is a switch + term pair. *) + +(* type ssrgen = ssrdocc * ssrterm *) + +let pr_gen (docc, dt) = pr_docc docc ++ pr_cpattern dt + +let pr_ssrgen _ _ _ = pr_gen + +ARGUMENT EXTEND ssrgen TYPED AS ssrdocc * cpattern PRINTED BY pr_ssrgen +| [ ssrdocc(docc) cpattern(dt) ] -> [ docc, dt ] +| [ cpattern(dt) ] -> [ nodocc, dt ] +END + +let has_occ ((_, occ), _) = occ <> None +let hyp_of_var v = SsrHyp (dummy_loc, destVar v) + +let interp_clr = function +| Some clr, (k, c) + when (k = ' ' || k = '@') && is_pf_var c -> hyp_of_var c :: clr +| Some clr, _ -> clr +| None, _ -> [] + +(* XXX the k of the redex should percolate out *) +let pf_interp_gen_aux ist gl to_ind ((oclr, occ), t) = + let pat = interp_cpattern ist gl t None in (* UGLY API *) + let cl, env, sigma = pf_concl gl, pf_env gl, project gl in + let (c, ucst), cl = + try fill_occ_pattern ~raise_NoMatch:true env sigma cl pat occ 1 + with NoMatch -> redex_of_pattern env pat, cl in + let clr = interp_clr (oclr, (tag_of_cpattern t, c)) in + if not(occur_existential c) then + if tag_of_cpattern t = '@' then + if not (isVar c) then + errorstrm (str "@ can be used with variables only") + else match pf_get_hyp gl (destVar c) with + | _, None, _ -> errorstrm (str "@ can be used with let-ins only") + | name, Some bo, ty -> true, pat, mkLetIn (Name name,bo,ty,cl),c,clr,ucst + else false, pat, pf_mkprod gl c cl, c, clr,ucst + else if to_ind && occ = None then + let nv, p, _, ucst' = pf_abs_evars gl (fst pat, c) in + let ucst = Evd.union_evar_universe_context ucst ucst' in + if nv = 0 then anomaly "occur_existential but no evars" else + false, pat, mkProd (constr_name c, pf_type_of gl p, pf_concl gl), p, clr,ucst + else loc_error (loc_of_cpattern t) "generalized term didn't match" + +let genclrtac cl cs clr = + let tclmyORELSE tac1 tac2 gl = + try tac1 gl + with e when Errors.noncritical e -> tac2 e gl in + (* apply_type may give a type error, but the useful message is + * the one of clear. You type "move: x" and you get + * "x is used in hyp H" instead of + * "The term H has type T x but is expected to have type T x0". *) + tclTHEN + (tclmyORELSE + (apply_type cl cs) + (fun type_err gl -> + tclTHEN + (tclTHEN (Proofview.V82.of_tactic (elim_type (build_coq_False ()))) (cleartac clr)) + (fun gl -> raise type_err) + gl)) + (cleartac clr) + +let gentac ist gen gl = +(* pp(lazy(str"sigma@gentac=" ++ pr_evar_map None (project gl))); *) + let conv, _, cl, c, clr, ucst = pf_interp_gen_aux ist gl false gen in + pp(lazy(str"c@gentac=" ++ pr_constr c)); + let gl = pf_merge_uc ucst gl in + if conv + then tclTHEN (Proofview.V82.of_tactic (convert_concl cl)) (cleartac clr) gl + else genclrtac cl [c] clr gl + +let pf_interp_gen ist gl to_ind gen = + let _, _, a, b, c, ucst = pf_interp_gen_aux ist gl to_ind gen in + a, b ,c, pf_merge_uc ucst gl + +(** Generalization (discharge) sequence *) + +(* A discharge sequence is represented as a list of up to two *) +(* lists of d-items, plus an ident list set (the possibly empty *) +(* final clear switch). The main list is empty iff the command *) +(* is defective, and has length two if there is a sequence of *) +(* dependent terms (and in that case it is the first of the two *) +(* lists). Thus, the first of the two lists is never empty. *) + +(* type ssrgens = ssrgen list *) +(* type ssrdgens = ssrgens list * ssrclear *) + +let gens_sep = function [], [] -> mt | _ -> spc + +let pr_dgens pr_gen (gensl, clr) = + let prgens s gens = str s ++ pr_list spc pr_gen gens in + let prdeps deps = prgens ": " deps ++ spc () ++ str "/" in + match gensl with + | [deps; []] -> prdeps deps ++ pr_clear pr_spc clr + | [deps; gens] -> prdeps deps ++ prgens " " gens ++ pr_clear spc clr + | [gens] -> prgens ": " gens ++ pr_clear spc clr + | _ -> pr_clear pr_spc clr + +let pr_ssrdgens _ _ _ = pr_dgens pr_gen + +let cons_gen gen = function + | gens :: gensl, clr -> (gen :: gens) :: gensl, clr + | _ -> anomaly "missing gen list" + +let cons_dep (gensl, clr) = + if List.length gensl = 1 then ([] :: gensl, clr) else + Errors.error "multiple dependents switches '/'" + +ARGUMENT EXTEND ssrdgens_tl TYPED AS ssrgen list list * ssrclear + PRINTED BY pr_ssrdgens +| [ "{" ne_ssrhyp_list(clr) "}" cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (mkclr clr, dt) dgens ] +| [ "{" ne_ssrhyp_list(clr) "}" ] -> + [ [[]], clr ] +| [ "{" ssrocc(occ) "}" cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (mkocc occ, dt) dgens ] +| [ "/" ssrdgens_tl(dgens) ] -> + [ cons_dep dgens ] +| [ cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (nodocc, dt) dgens ] +| [ ] -> + [ [[]], [] ] +END + +ARGUMENT EXTEND ssrdgens TYPED AS ssrdgens_tl PRINTED BY pr_ssrdgens +| [ ":" ssrgen(gen) ssrdgens_tl(dgens) ] -> [ cons_gen gen dgens ] +END + +let genstac (gens, clr) ist = + tclTHENLIST (cleartac clr :: List.rev_map (gentac ist) gens) + +(* Common code to handle generalization lists along with the defective case *) + +let with_defective maintac deps clr ist gl = + let top_id = + match kind_of_type (pf_concl gl) with + | ProdType (Name id, _, _) + when has_discharged_tag (string_of_id id) -> id + | _ -> top_id in + let top_gen = mkclr clr, cpattern_of_id top_id in + tclTHEN (introid top_id) (maintac deps top_gen ist) gl + +let with_dgens (gensl, clr) maintac ist = match gensl with + | [deps; []] -> with_defective maintac deps clr ist + | [deps; gen :: gens] -> + tclTHEN (genstac (gens, clr) ist) (maintac deps gen ist) + | [gen :: gens] -> tclTHEN (genstac (gens, clr) ist) (maintac [] gen ist) + | _ -> with_defective maintac [] clr ist + +let first_goal gls = + let gl = gls.Evd.it and sig_0 = gls.Evd.sigma in + if List.is_empty gl then Errors.error "first_goal"; + { Evd.it = List.hd gl; Evd.sigma = sig_0; } + +let with_deps deps0 maintac cl0 cs0 clr0 ist gl0 = + let rec loop gl cl cs clr args clrs = function + | [] -> + let n = List.length args in + maintac (if n > 0 then applist (to_lambda n cl, args) else cl) clrs ist gl0 + | dep :: deps -> + let gl' = first_goal (genclrtac cl cs clr gl) in + let cl', c', clr',gl' = pf_interp_gen ist gl' false dep in + loop gl' cl' [c'] clr' (c' :: args) (clr' :: clrs) deps in + loop gl0 cl0 cs0 clr0 cs0 [clr0] (List.rev deps0) + +(** Equations *) + +(* argument *) + +type ssreqid = ssripat option + +let pr_eqid = function Some pat -> str " " ++ pr_ipat pat | None -> mt () +let pr_ssreqid _ _ _ = pr_eqid + +(* We must use primitive parsing here to avoid conflicts with the *) +(* basic move, case, and elim tactics. *) +ARGUMENT EXTEND ssreqid TYPED AS ssripatrep option PRINTED BY pr_ssreqid +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let accept_ssreqid strm = + match Compat.get_tok (Util.stream_nth 0 strm) with + | Tok.IDENT _ -> accept_before_syms [":"] strm + | Tok.KEYWORD ":" -> () + | Tok.KEYWORD pat when List.mem pat ["_"; "?"; "->"; "<-"] -> + accept_before_syms [":"] strm + | _ -> raise Stream.Failure + +let test_ssreqid = Gram.Entry.of_parser "test_ssreqid" accept_ssreqid + +GEXTEND Gram + GLOBAL: ssreqid; + ssreqpat: [ + [ id = Prim.ident -> IpatId id + | "_" -> IpatWild + | "?" -> IpatAnon + | occ = ssrdocc; "->" -> (match occ with + | None, occ -> IpatRw (occ, L2R) + | _ -> loc_error !@loc "Only occurrences are allowed here") + | occ = ssrdocc; "<-" -> (match occ with + | None, occ -> IpatRw (occ, R2L) + | _ -> loc_error !@loc "Only occurrences are allowed here") + | "->" -> IpatRw (allocc, L2R) + | "<-" -> IpatRw (allocc, R2L) + ]]; + ssreqid: [ + [ test_ssreqid; pat = ssreqpat -> Some pat + | test_ssreqid -> None + ]]; +END + +(* creation *) + +let mkEq dir cl c t n gl = + let eqargs = [|t; c; c|] in eqargs.(dir_org dir) <- mkRel n; + let eq, gl = pf_fresh_global (build_coq_eq()) gl in + let refl, gl = mkRefl t c gl in + mkArrow (mkApp (eq, eqargs)) (lift 1 cl), refl, gl + +let pushmoveeqtac cl c gl = + let x, t, cl1 = destProd cl in + let cl2, eqc, gl = mkEq R2L cl1 c t 1 gl in + apply_type (mkProd (x, t, cl2)) [c; eqc] gl + +let pushcaseeqtac cl gl = + let cl1, args = destApplication cl in + let n = Array.length args in + let dc, cl2 = decompose_lam_n n cl1 in + let _, t = List.nth dc (n - 1) in + let cl3, eqc, gl = mkEq R2L cl2 args.(0) t n gl in + let prot, gl = mkProt (pf_type_of gl cl) cl3 gl in + let cl4 = mkApp (compose_lam dc prot, args) in + let gl, _ = pf_e_type_of gl cl4 in + tclTHEN (apply_type cl4 [eqc]) + (Proofview.V82.of_tactic (convert_concl cl4)) gl + +let pushelimeqtac gl = + let _, args = destApplication (pf_concl gl) in + let x, t, _ = destLambda args.(1) in + let cl1 = mkApp (args.(1), Array.sub args 2 (Array.length args - 2)) in + let cl2, eqc, gl = mkEq L2R cl1 args.(2) t 1 gl in + tclTHEN (apply_type (mkProd (x, t, cl2)) [args.(2); eqc]) + (Proofview.V82.of_tactic intro) gl + +(** Bookkeeping (discharge-intro) argument *) + +(* Since all bookkeeping ssr commands have the same discharge-intro *) +(* argument format we use a single grammar entry point to parse them. *) +(* the entry point parses only non-empty arguments to avoid conflicts *) +(* with the basic Coq tactics. *) + +(* type ssrarg = ssrview * (ssreqid * (ssrdgens * ssripats)) *) + +let pr_ssrarg _ _ _ (view, (eqid, (dgens, ipats))) = + let pri = pr_intros (gens_sep dgens) in + pr_view view ++ pr_eqid eqid ++ pr_dgens pr_gen dgens ++ pri ipats + +ARGUMENT EXTEND ssrarg TYPED AS ssrview * (ssreqid * (ssrdgens * ssrintros)) + PRINTED BY pr_ssrarg +| [ ssrview(view) ssreqid(eqid) ssrdgens(dgens) ssrintros(ipats) ] -> + [ view, (eqid, (dgens, ipats)) ] +| [ ssrview(view) ssrclear(clr) ssrintros(ipats) ] -> + [ view, (None, (([], clr), ipats)) ] +| [ ssreqid(eqid) ssrdgens(dgens) ssrintros(ipats) ] -> + [ [], (eqid, (dgens, ipats)) ] +| [ ssrclear_ne(clr) ssrintros(ipats) ] -> + [ [], (None, (([], clr), ipats)) ] +| [ ssrintros_ne(ipats) ] -> + [ [], (None, (([], []), ipats)) ] +END + +(** The "clear" tactic *) + +(* We just add a numeric version that clears the n top assumptions. *) + +let poptac ist n = introstac ~ist (List.init n (fun _ -> IpatWild)) + +TACTIC EXTEND ssrclear + | [ "clear" natural(n) ] -> [ Proofview.V82.tactic (poptac ist n) ] +END + +(** The "move" tactic *) + +let rec improper_intros = function + | IpatSimpl _ :: ipats -> improper_intros ipats + | (IpatId _ | IpatAnon | IpatCase _ | IpatAll) :: _ -> false + | _ -> true + +let check_movearg = function + | view, (eqid, _) when view <> [] && eqid <> None -> + Errors.error "incompatible view and equation in move tactic" + | view, (_, (([gen :: _], _), _)) when view <> [] && has_occ gen -> + Errors.error "incompatible view and occurrence switch in move tactic" + | _, (_, ((dgens, _), _)) when List.length dgens > 1 -> + Errors.error "dependents switch `/' in move tactic" + | _, (eqid, (_, ipats)) when eqid <> None && improper_intros ipats -> + Errors.error "no proper intro pattern for equation in move tactic" + | arg -> arg + +ARGUMENT EXTEND ssrmovearg TYPED AS ssrarg PRINTED BY pr_ssrarg +| [ ssrarg(arg) ] -> [ check_movearg arg ] +END + +let viewmovetac_aux clear name_ref (_, vl as v) _ gen ist gl = + let cl, c, clr, gl = pf_interp_gen ist gl false gen in + let cl, c, gl = if vl = [] then cl, c, gl else pf_with_view ist gl v cl c in + let clr = if clear then clr else [] in + name_ref := (match id_of_cpattern (snd gen) with Some id -> id | _ -> top_id); + genclrtac cl [c] clr gl + +let () = move_top_with_view := + (fun c r v -> with_defective (viewmovetac_aux c r v) [] []) + +let viewmovetac v deps gen ist gl = + viewmovetac_aux true (ref top_id) v deps gen ist gl + +let eqmovetac _ gen ist gl = + let cl, c, _, gl = pf_interp_gen ist gl false gen in pushmoveeqtac cl c gl + +let movehnftac gl = match kind_of_term (pf_concl gl) with + | Prod _ | LetIn _ -> tclIDTAC gl + | _ -> hnf_in_concl gl + +let ssrmovetac ist = function + | _::_ as view, (_, (dgens, ipats)) -> + let dgentac = with_dgens dgens (viewmovetac (true, view)) ist in + tclTHEN dgentac (introstac ~ist ipats) + | _, (Some pat, (dgens, ipats)) -> + let dgentac = with_dgens dgens eqmovetac ist in + tclTHEN dgentac (introstac ~ist (eqmoveipats pat ipats)) + | _, (_, (([gens], clr), ipats)) -> + let gentac = genstac (gens, clr) ist in + tclTHEN gentac (introstac ~ist ipats) + | _, (_, ((_, clr), ipats)) -> + tclTHENLIST [movehnftac; cleartac clr; introstac ~ist ipats] + +TACTIC EXTEND ssrmove +| [ "move" ssrmovearg(arg) ssrrpat(pat) ] -> + [ Proofview.V82.tactic (tclTHEN (ssrmovetac ist arg) (introstac ~ist [pat])) ] +| [ "move" ssrmovearg(arg) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrmovetac ist arg) clauses) ] +| [ "move" ssrrpat(pat) ] -> [ Proofview.V82.tactic (introstac ~ist [pat]) ] +| [ "move" ] -> [ Proofview.V82.tactic (movehnftac) ] +END + +(* TASSI: given the type of an elimination principle, it finds the higher order + * argument (index), it computes it's arity and the arity of the eliminator and + * checks if the eliminator is recursive or not *) +let analyze_eliminator elimty env sigma = + let rec loop ctx t = match kind_of_type t with + | AtomicType (hd, args) when isRel hd -> + ctx, destRel hd, not (noccurn 1 t), Array.length args + | CastType (t, _) -> loop ctx t + | ProdType (x, ty, t) -> loop ((x,None,ty) :: ctx) t + | LetInType (x,b,ty,t) -> loop ((x,Some b,ty) :: ctx) (subst1 b t) + | _ -> + let env' = Environ.push_rel_context ctx env in + let t' = Reductionops.whd_betadeltaiota env' sigma t in + if not (Term.eq_constr t t') then loop ctx t' else + errorstrm (str"The eliminator has the wrong shape."++spc()++ + str"A (applied) bound variable was expected as the conclusion of "++ + str"the eliminator's"++Pp.cut()++str"type:"++spc()++pr_constr elimty) in + let ctx, pred_id, elim_is_dep, n_pred_args = loop [] elimty in + let n_elim_args = rel_context_nhyps ctx in + let is_rec_elim = + let count_occurn n term = + let count = ref 0 in + let rec occur_rec n c = match kind_of_term c with + | Rel m -> if m = n then incr count + | _ -> iter_constr_with_binders succ occur_rec n c + in + occur_rec n term; !count in + let occurr2 n t = count_occurn n t > 1 in + not (List.for_all_i + (fun i (_,rd) -> pred_id <= i || not (occurr2 (pred_id - i) rd)) + 1 (assums_of_rel_context ctx)) + in + n_elim_args - pred_id, n_elim_args, is_rec_elim, elim_is_dep, n_pred_args + +(* TASSI: This version of unprotects inlines the unfold tactic definition, + * since we don't want to wipe out let-ins, and it seems there is no flag + * to change that behaviour in the standard unfold code *) +let unprotecttac gl = + let c, gl = pf_mkSsrConst "protect_term" gl in + let prot, _ = destConst c in + onClause (fun idopt -> + let hyploc = Option.map (fun id -> id, InHyp) idopt in + reduct_option + (Reductionops.clos_norm_flags + (Closure.RedFlags.mkflags + [Closure.RedFlags.fBETA; + Closure.RedFlags.fCONST prot; + Closure.RedFlags.fIOTA]), DEFAULTcast) hyploc) + allHypsAndConcl gl + +let dependent_apply_error = + try Errors.error "Could not fill dependent hole in \"apply\"" with err -> err + +(* TASSI: Sometimes Coq's apply fails. According to my experience it may be + * related to goals that are products and with beta redexes. In that case it + * guesses the wrong number of implicit arguments for your lemma. What follows + * is just like apply, but with a user-provided number n of implicits. + * + * Refine.refine function that handles type classes and evars but fails to + * handle "dependently typed higher order evars". + * + * Refiner.refiner that does not handle metas with a non ground type but works + * with dependently typed higher order metas. *) +let applyn ~with_evars ?beta ?(with_shelve=false) n t gl = + if with_evars then + let refine gl = + let t, ty, args, gl = pf_saturate ?beta ~bi_types:true gl t n in +(* pp(lazy(str"sigma@saturate=" ++ pr_evar_map None (project gl))); *) + let gl = pf_unify_HO gl ty (pf_concl gl) in + let gs = CList.map_filter (fun (_, e) -> + if isEvar (pf_nf_evar gl e) then Some e else None) + args in + pf_partial_solution gl t gs + in + Proofview.(V82.of_tactic + (tclTHEN (V82.tactic refine) + (if with_shelve then shelve_unifiable else tclUNIT ()))) gl + else + let t, gl = if n = 0 then t, gl else + let sigma, si = project gl, sig_it gl in + let rec loop sigma bo args = function (* saturate with metas *) + | 0 -> mkApp (t, Array.of_list (List.rev args)), re_sig si sigma + | n -> match kind_of_term bo with + | Lambda (_, ty, bo) -> + if not (closed0 ty) then raise dependent_apply_error; + let m = Evarutil.new_meta () in + loop (meta_declare m ty sigma) bo ((mkMeta m)::args) (n-1) + | _ -> assert false + in loop sigma t [] n in + pp(lazy(str"Refiner.refiner " ++ pr_constr t)); + Refiner.refiner (Proof_type.Refine t) gl + +let refine_with ?(first_goes_last=false) ?beta ?(with_evars=true) oc gl = + let rec mkRels = function 1 -> [] | n -> mkRel n :: mkRels (n-1) in + let uct = Evd.evar_universe_context (fst oc) in + let n, oc = pf_abs_evars_pirrel gl oc in + let gl = pf_merge_uc uct gl in + let oc = if not first_goes_last || n <= 1 then oc else + let l, c = decompose_lam oc in + if not (List.for_all_i (fun i (_,t) -> closedn ~-i t) (1-n) l) then oc else + compose_lam (let xs,y = List.chop (n-1) l in y @ xs) + (mkApp (compose_lam l c, Array.of_list (mkRel 1 :: mkRels n))) + in + pp(lazy(str"after: " ++ pr_constr oc)); + try applyn ~with_evars ~with_shelve:true ?beta n oc gl with _ -> raise dependent_apply_error + +let pf_fresh_inductive_instance ind gl = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, indu = Evd.fresh_inductive_instance env sigma ind in + indu, re_sig it sigma + +(** The "case" and "elim" tactic *) + +(* A case without explicit dependent terms but with both a view and an *) +(* occurrence switch and/or an equation is treated as dependent, with the *) +(* viewed term as the dependent term (the occurrence switch would be *) +(* meaningless otherwise). When both a view and explicit dependents are *) +(* present, it is forbidden to put a (meaningless) occurrence switch on *) +(* the viewed term. *) + +(* This is both elim and case (defaulting to the former). If ~elim is omitted + * the standard eliminator is chosen. The code is made of 4 parts: + * 1. find the eliminator if not given as ~elim and analyze it + * 2. build the patterns to be matched against the conclusion, looking at + * (occ, c), deps and the pattern inferred from the type of the eliminator + * 3. build the new predicate matching the patterns, and the tactic to + * generalize the equality in case eqid is not None + * 4. build the tactic handle intructions and clears as required in ipats and + * by eqid *) +let ssrelim ?(is_case=false) ?ist deps what ?elim eqid ipats gl = + (* some sanity checks *) + let oc, orig_clr, occ, c_gen, gl = match what with + | `EConstr(_,_,t) when isEvar t -> + anomaly "elim called on a constr evar" + | `EGen _ when ist = None -> + anomaly "no ist and non simple elimination" + | `EGen (_, g) when elim = None && is_wildcard g -> + errorstrm(str"Indeterminate pattern and no eliminator") + | `EGen ((Some clr,occ), g) when is_wildcard g -> + None, clr, occ, None, gl + | `EGen ((None, occ), g) when is_wildcard g -> None,[],occ,None,gl + | `EGen ((_, occ), p as gen) -> + let _, c, clr,gl = pf_interp_gen (Option.get ist) gl true gen in + Some c, clr, occ, Some p,gl + | `EConstr (clr, occ, c) -> Some c, clr, occ, None,gl in + let orig_gl, concl, env = gl, pf_concl gl, pf_env gl in + pp(lazy(str(if is_case then "==CASE==" else "==ELIM=="))); + (* Utils of local interest only *) + let iD s ?t gl = let t = match t with None -> pf_concl gl | Some x -> x in + pp(lazy(str s ++ pr_constr t)); tclIDTAC gl in + let eq, gl = pf_fresh_global (build_coq_eq ()) gl in + let protectC, gl = pf_mkSsrConst "protect_term" gl in + let fire_subst gl t = Reductionops.nf_evar (project gl) t in + let fire_sigma sigma t = Reductionops.nf_evar sigma t in + let is_undef_pat = function + | sigma, T t -> + (match kind_of_term (fire_sigma sigma t) with Evar _ -> true | _ -> false) + | _ -> false in + let match_pat env p occ h cl = + let sigma0 = project orig_gl in + pp(lazy(str"matching: " ++ pp_pattern p)); + let (c,ucst), cl = + fill_occ_pattern ~raise_NoMatch:true env sigma0 cl p occ h in + pp(lazy(str" got: " ++ pr_constr c)); + c, cl, ucst in + let mkTpat gl t = (* takes a term, refreshes it and makes a T pattern *) + let n, t, _, ucst = pf_abs_evars orig_gl (project gl, fire_subst gl t) in + let t, _, _, sigma = saturate ~beta:true env (project gl) t n in + Evd.merge_universe_context sigma ucst, T t in + let unif_redex gl (sigma, r as p) t = (* t is a hint for the redex of p *) + let n, t, _, ucst = pf_abs_evars orig_gl (project gl, fire_subst gl t) in + let t, _, _, sigma = saturate ~beta:true env sigma t n in + let sigma = Evd.merge_universe_context sigma ucst in + match r with + | X_In_T (e, p) -> sigma, E_As_X_In_T (t, e, p) + | _ -> + try unify_HO env sigma t (fst (redex_of_pattern env p)), r + with e when Errors.noncritical e -> p in + (* finds the eliminator applies it to evars and c saturated as needed *) + (* obtaining "elim ??? (c ???)". pred is the higher order evar *) + let cty, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl = + match elim with + | Some elim -> + let gl, elimty = pf_e_type_of gl elim in + let pred_id, n_elim_args, is_rec, elim_is_dep, n_pred_args = + analyze_eliminator elimty env (project gl) in + let elim, elimty, elim_args, gl = + pf_saturate ~beta:is_case gl elim ~ty:elimty n_elim_args in + let pred = List.assoc pred_id elim_args in + let elimty = Reductionops.whd_betadeltaiota env (project gl) elimty in + None, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl + | None -> + let c = Option.get oc in let c_ty = pf_type_of gl c in + let ((kn, i) as ind, _ as indu), unfolded_c_ty = + pf_reduce_to_quantified_ind gl c_ty in + let sort = elimination_sort_of_goal gl in + let gl, elim = + if not is_case then + let t, gl = pf_fresh_global (Indrec.lookup_eliminator ind sort) gl in + gl, t + else + pf_eapply (fun env sigma -> + Indrec.build_case_analysis_scheme env sigma indu true) gl sort in + let elimty = pf_type_of gl elim in + let pred_id,n_elim_args,is_rec,elim_is_dep,n_pred_args = + analyze_eliminator elimty env (project gl) in + let rctx = fst (decompose_prod_assum unfolded_c_ty) in + let n_c_args = rel_context_length rctx in + let c, c_ty, t_args, gl = pf_saturate gl c ~ty:c_ty n_c_args in + let elim, elimty, elim_args, gl = + pf_saturate ~beta:is_case gl elim ~ty:elimty n_elim_args in + let pred = List.assoc pred_id elim_args in + let pc = match n_c_args, c_gen with + | 0, Some p -> interp_cpattern (Option.get ist) orig_gl p None + | _ -> mkTpat gl c in + let cty = Some (c, c_ty, pc) in + let elimty = Reductionops.whd_betadeltaiota env (project gl) elimty in + cty, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl + in + pp(lazy(str"elim= "++ pr_constr_pat elim)); + pp(lazy(str"elimty= "++ pr_constr elimty)); + let inf_deps_r = match kind_of_type elimty with + | AtomicType (_, args) -> List.rev (Array.to_list args) + | _ -> assert false in + let saturate_until gl c c_ty f = + let rec loop n = try + let c, c_ty, _, gl = pf_saturate gl c ~ty:c_ty n in + let gl' = f c c_ty gl in + Some (c, c_ty, gl, gl') + with NotEnoughProducts -> None | _ -> loop (n+1) in loop 0 in + let elim_is_dep, gl = match cty with + | None -> true, gl + | Some (c, c_ty, _) -> + let res = + if elim_is_dep then None else + let arg = List.assoc (n_elim_args - 1) elim_args in + let arg_ty = pf_type_of gl arg in + match saturate_until gl c c_ty (fun c c_ty gl -> + pf_unify_HO (pf_unify_HO gl c_ty arg_ty) arg c) with + | Some (c, _, _, gl) -> Some (false, gl) + | None -> None in + match res with + | Some x -> x + | None -> + let inf_arg = List.hd inf_deps_r in + let inf_arg_ty = pf_type_of gl inf_arg in + match saturate_until gl c c_ty (fun _ c_ty gl -> + pf_unify_HO gl c_ty inf_arg_ty) with + | Some (c, _, _,gl) -> true, gl + | None -> + errorstrm (str"Unable to apply the eliminator to the term"++ + spc()++pr_constr c++spc()++str"or to unify it's type with"++ + pr_constr inf_arg_ty) in + pp(lazy(str"elim_is_dep= " ++ bool elim_is_dep)); + let predty = pf_type_of gl pred in + (* Patterns for the inductive types indexes to be bound in pred are computed + * looking at the ones provided by the user and the inferred ones looking at + * the type of the elimination principle *) + let pp_pat (_,p,_,_) = pp_pattern p in + let pp_inf_pat gl (_,_,t,_) = pr_constr_pat (fire_subst gl t) in + let patterns, clr, gl = + let rec loop patterns clr i = function + | [],[] -> patterns, clr, gl + | ((oclr, occ), t):: deps, inf_t :: inf_deps -> + let ist = match ist with Some x -> x | None -> assert false in + let p = interp_cpattern ist orig_gl t None in + let clr_t = + interp_clr (oclr,(tag_of_cpattern t,fst (redex_of_pattern env p)))in + (* if we are the index for the equation we do not clear *) + let clr_t = if deps = [] && eqid <> None then [] else clr_t in + let p = if is_undef_pat p then mkTpat gl inf_t else p in + loop (patterns @ [i, p, inf_t, occ]) + (clr_t @ clr) (i+1) (deps, inf_deps) + | [], c :: inf_deps -> + pp(lazy(str"adding inferred pattern " ++ pr_constr_pat c)); + loop (patterns @ [i, mkTpat gl c, c, allocc]) + clr (i+1) ([], inf_deps) + | _::_, [] -> errorstrm (str "Too many dependent abstractions") in + let deps, head_p, inf_deps_r = match what, elim_is_dep, cty with + | `EConstr _, _, None -> anomaly "Simple welim with no term" + | _, false, _ -> deps, [], inf_deps_r + | `EGen gen, true, None -> deps @ [gen], [], inf_deps_r + | _, true, Some (c, _, pc) -> + let occ = if occ = None then allocc else occ in + let inf_p, inf_deps_r = List.hd inf_deps_r, List.tl inf_deps_r in + deps, [1, pc, inf_p, occ], inf_deps_r in + let patterns, clr, gl = + loop [] orig_clr (List.length head_p+1) (List.rev deps, inf_deps_r) in + head_p @ patterns, Util.List.uniquize clr, gl + in + pp(lazy(pp_concat (str"patterns=") (List.map pp_pat patterns))); + pp(lazy(pp_concat (str"inf. patterns=") (List.map (pp_inf_pat gl) patterns))); + (* Predicate generation, and (if necessary) tactic to generalize the + * equation asked by the user *) + let elim_pred, gen_eq_tac, clr, gl = + let error gl t inf_t = errorstrm (str"The given pattern matches the term"++ + spc()++pp_term gl t++spc()++str"while the inferred pattern"++ + spc()++pr_constr_pat (fire_subst gl inf_t)++spc()++ str"doesn't") in + let match_or_postpone (cl, gl, post) (h, p, inf_t, occ) = + let p = unif_redex gl p inf_t in + if is_undef_pat p then + let () = pp(lazy(str"postponing " ++ pp_pattern p)) in + cl, gl, post @ [h, p, inf_t, occ] + else try + let c, cl, ucst = match_pat env p occ h cl in + let gl = pf_merge_uc ucst gl in + let gl = try pf_unify_HO gl inf_t c with _ -> error gl c inf_t in + cl, gl, post + with + | NoMatch | NoProgress -> + let e, ucst = redex_of_pattern env p in + let gl = pf_merge_uc ucst gl in + let n, e, _, _ucst = pf_abs_evars gl (fst p, e) in + let e, _, _, gl = pf_saturate ~beta:true gl e n in + let gl = try pf_unify_HO gl inf_t e with _ -> error gl e inf_t in + cl, gl, post + in + let rec match_all concl gl patterns = + let concl, gl, postponed = + List.fold_left match_or_postpone (concl, gl, []) patterns in + if postponed = [] then concl, gl + else if List.length postponed = List.length patterns then + errorstrm (str "Some patterns are undefined even after all"++spc()++ + str"the defined ones matched") + else match_all concl gl postponed in + let concl, gl = match_all concl gl patterns in + let pred_rctx, _ = decompose_prod_assum (fire_subst gl predty) in + let concl, gen_eq_tac, clr, gl = match eqid with + | Some (IpatId _) when not is_rec -> + let k = List.length deps in + let c = fire_subst gl (List.assoc (n_elim_args - k - 1) elim_args) in + let t = pf_type_of gl c in + let gen_eq_tac, gl = + let refl = mkApp (eq, [|t; c; c|]) in + let new_concl = mkArrow refl (lift 1 (pf_concl orig_gl)) in + let new_concl = fire_subst gl new_concl in + let erefl, gl = mkRefl t c gl in + let erefl = fire_subst gl erefl in + apply_type new_concl [erefl], gl in + let rel = k + if elim_is_dep then 1 else 0 in + let src, gl = mkProt mkProp (mkApp (eq,[|t; c; mkRel rel|])) gl in + let concl = mkArrow src (lift 1 concl) in + let clr = if deps <> [] then clr else [] in + concl, gen_eq_tac, clr, gl + | _ -> concl, tclIDTAC, clr, gl in + let mk_lam t r = mkLambda_or_LetIn r t in + let concl = List.fold_left mk_lam concl pred_rctx in + let gl, concl = + if eqid <> None && is_rec then + let concl, gl = mkProt (pf_type_of gl concl) concl gl in + let gl, _ = pf_e_type_of gl concl in + gl, concl + else gl, concl in + concl, gen_eq_tac, clr, gl in + let gl, pty = pf_e_type_of gl elim_pred in + pp(lazy(str"elim_pred=" ++ pp_term gl elim_pred)); + pp(lazy(str"elim_pred_ty=" ++ pp_term gl pty)); + let gl = pf_unify_HO gl pred elim_pred in + let elim = fire_subst gl elim in + let gl, _ = pf_e_type_of gl elim in + (* check that the patterns do not contain non instantiated dependent metas *) + let () = + let evars_of_term = Evarutil.undefined_evars_of_term (project gl) in + let patterns = List.map (fun (_,_,t,_) -> fire_subst gl t) patterns in + let patterns_ev = List.map evars_of_term patterns in + let ev = List.fold_left Intset.union Intset.empty patterns_ev in + let ty_ev = Intset.fold (fun i e -> + let ex = i in + let i_ty = Evd.evar_concl (Evd.find (project gl) ex) in + Intset.union e (evars_of_term i_ty)) + ev Intset.empty in + let inter = Intset.inter ev ty_ev in + if not (Intset.is_empty inter) then begin + let i = Intset.choose inter in + let pat = List.find (fun t -> Intset.mem i (evars_of_term t)) patterns in + errorstrm(str"Pattern"++spc()++pr_constr_pat pat++spc()++ + str"was not completely instantiated and one of its variables"++spc()++ + str"occurs in the type of another non instantieted pattern variable"); + end + in + (* the elim tactic, with the eliminator and the predicated we computed *) + let elim = project gl, elim in + let elim_tac gl = + tclTHENLIST [refine_with ~with_evars:false elim; cleartac clr] gl in + (* handling of following intro patterns and equation introduction if any *) + let elim_intro_tac gl = + let intro_eq = + match eqid with + | Some (IpatId ipat) when not is_rec -> + let rec intro_eq gl = match kind_of_type (pf_concl gl) with + | ProdType (_, src, tgt) -> + (match kind_of_type src with + | AtomicType (hd, _) when Term.eq_constr hd protectC -> + tclTHENLIST [unprotecttac; introid ipat] gl + | _ -> tclTHENLIST [ iD "IA"; introstac [IpatAnon]; intro_eq] gl) + |_ -> errorstrm (str "Too many names in intro pattern") in + intro_eq + | Some (IpatId ipat) -> + let name gl = mk_anon_id "K" gl in + let intro_lhs gl = + let elim_name = match orig_clr, what with + | [SsrHyp(_, x)], _ -> x + | _, `EConstr(_,_,t) when isVar t -> destVar t + | _ -> name gl in + if is_name_in_ipats elim_name ipats then introid (name gl) gl + else introid elim_name gl + in + let rec gen_eq_tac gl = + let concl = pf_concl gl in + let ctx, last = decompose_prod_assum concl in + let args = match kind_of_type last with + | AtomicType (hd, args) -> assert(Term.eq_constr hd protectC); args + | _ -> assert false in + let case = args.(Array.length args-1) in + if not(closed0 case) then tclTHEN (introstac [IpatAnon]) gen_eq_tac gl + else + let case_ty = pf_type_of gl case in + let refl = mkApp (eq, [|lift 1 case_ty; mkRel 1; lift 1 case|]) in + let new_concl = fire_subst gl + (mkProd (Name (name gl), case_ty, mkArrow refl (lift 2 concl))) in + let erefl, gl = mkRefl case_ty case gl in + let erefl = fire_subst gl erefl in + apply_type new_concl [case;erefl] gl in + tclTHENLIST [gen_eq_tac; intro_lhs; introid ipat] + | _ -> tclIDTAC in + let unprot = if eqid <> None && is_rec then unprotecttac else tclIDTAC in + tclEQINTROS ?ist elim_tac (tclTHENLIST [intro_eq; unprot]) ipats gl + in + tclTHENLIST [gen_eq_tac; elim_intro_tac] orig_gl +;; + +let simplest_newelim x= ssrelim ~is_case:false [] (`EConstr ([],None,x)) None [] +let simplest_newcase x= ssrelim ~is_case:true [] (`EConstr ([],None,x)) None [] +let _ = simplest_newcase_ref := simplest_newcase + +let check_casearg = function +| view, (_, (([_; gen :: _], _), _)) when view <> [] && has_occ gen -> + Errors.error "incompatible view and occurrence switch in dependent case tactic" +| arg -> arg + +ARGUMENT EXTEND ssrcasearg TYPED AS ssrarg PRINTED BY pr_ssrarg +| [ ssrarg(arg) ] -> [ check_casearg arg ] +END + +let ssrcasetac ist (view, (eqid, (dgens, ipats))) = + let ndefectcasetac view eqid ipats deps ((_, occ), _ as gen) ist gl = + let simple = (eqid = None && deps = [] && occ = None) in + let cl, c, clr, gl = pf_interp_gen ist gl true gen in + let _, vc, gl = + if view = [] then c, c, gl else pf_with_view ist gl (false, view) cl c in + if simple && is_injection_case vc gl then + tclTHENLIST [perform_injection vc; cleartac clr; introstac ~ist ipats] gl + else + (* macro for "case/v E: x" ---> "case E: x / (v x)" *) + let deps, clr, occ = + if view <> [] && eqid <> None && deps = [] then [gen], [], None + else deps, clr, occ in + ssrelim ~is_case:true ~ist deps (`EConstr (clr,occ, vc)) eqid ipats gl + in + with_dgens dgens (ndefectcasetac view eqid ipats) ist + +TACTIC EXTEND ssrcase +| [ "case" ssrcasearg(arg) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrcasetac ist arg) clauses) ] +| [ "case" ] -> [ Proofview.V82.tactic (with_top ssrscasetac) ] +END + +(** The "elim" tactic *) + +(* Elim views are elimination lemmas, so the eliminated term is not addded *) +(* to the dependent terms as for "case", unless it actually occurs in the *) +(* goal, the "all occurrences" {+} switch is used, or the equation switch *) +(* is used and there are no dependents. *) + +let ssrelimtac ist (view, (eqid, (dgens, ipats))) = + let ndefectelimtac view eqid ipats deps gen ist gl = + let elim = match view with [v] -> Some (snd(force_term ist gl v)) | _ -> None in + ssrelim ~ist deps (`EGen gen) ?elim eqid ipats gl + in + with_dgens dgens (ndefectelimtac view eqid ipats) ist + +TACTIC EXTEND ssrelim +| [ "elim" ssrarg(arg) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrelimtac ist arg) clauses) ] +| [ "elim" ] -> [ Proofview.V82.tactic (with_top simplest_newelim) ] +END + +(** 6. Backward chaining tactics: apply, exact, congr. *) + +(** The "apply" tactic *) + +let pr_agen (docc, dt) = pr_docc docc ++ pr_term dt +let pr_ssragen _ _ _ = pr_agen +let pr_ssragens _ _ _ = pr_dgens pr_agen + +ARGUMENT EXTEND ssragen TYPED AS ssrdocc * ssrterm PRINTED BY pr_ssragen +| [ "{" ne_ssrhyp_list(clr) "}" ssrterm(dt) ] -> [ mkclr clr, dt ] +| [ ssrterm(dt) ] -> [ nodocc, dt ] +END + +ARGUMENT EXTEND ssragens TYPED AS ssragen list list * ssrclear +PRINTED BY pr_ssragens +| [ "{" ne_ssrhyp_list(clr) "}" ssrterm(dt) ssragens(agens) ] -> + [ cons_gen (mkclr clr, dt) agens ] +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ [[]], clr] +| [ ssrterm(dt) ssragens(agens) ] -> + [ cons_gen (nodocc, dt) agens ] +| [ ] -> [ [[]], [] ] +END + +let mk_applyarg views agens intros = views, (None, (agens, intros)) + +let pr_ssraarg _ _ _ (view, (eqid, (dgens, ipats))) = + let pri = pr_intros (gens_sep dgens) in + pr_view view ++ pr_eqid eqid ++ pr_dgens pr_agen dgens ++ pri ipats + +ARGUMENT EXTEND ssrapplyarg +TYPED AS ssrview * (ssreqid * (ssragens * ssrintros)) +PRINTED BY pr_ssraarg +| [ ":" ssragen(gen) ssragens(dgens) ssrintros(intros) ] -> + [ mk_applyarg [] (cons_gen gen dgens) intros ] +| [ ssrclear_ne(clr) ssrintros(intros) ] -> + [ mk_applyarg [] ([], clr) intros ] +| [ ssrintros_ne(intros) ] -> + [ mk_applyarg [] ([], []) intros ] +| [ ssrview(view) ":" ssragen(gen) ssragens(dgens) ssrintros(intros) ] -> + [ mk_applyarg view (cons_gen gen dgens) intros ] +| [ ssrview(view) ssrclear(clr) ssrintros(intros) ] -> + [ mk_applyarg view ([], clr) intros ] +END + +let interp_agen ist gl ((goclr, _), (k, gc)) (clr, rcs) = +(* pp(lazy(str"sigma@interp_agen=" ++ pr_evar_map None (project gl))); *) + let rc = glob_constr ist (pf_env gl) gc in + let rcs' = rc :: rcs in + match goclr with + | None -> clr, rcs' + | Some ghyps -> + let clr' = snd (interp_hyps ist gl ghyps) @ clr in + if k <> ' ' then clr', rcs' else + match rc with + | GVar (loc, id) when not_section_id id -> SsrHyp (loc, id) :: clr', rcs' + | GRef (loc, VarRef id, _) when not_section_id id -> + SsrHyp (loc, id) :: clr', rcs' + | _ -> clr', rcs' + +let interp_agens ist gl gagens = + match List.fold_right (interp_agen ist gl) gagens ([], []) with + | clr, rlemma :: args -> + let n = interp_nbargs ist gl rlemma - List.length args in + let rec loop i = + if i > n then + errorstrm (str "Cannot apply lemma " ++ pf_pr_glob_constr gl rlemma) + else + try interp_refine ist gl (mkRApp rlemma (mkRHoles i @ args)) + with _ -> loop (i + 1) in + clr, loop 0 + | _ -> assert false + +let apply_rconstr ?ist t gl = +(* pp(lazy(str"sigma@apply_rconstr=" ++ pr_evar_map None (project gl))); *) + let n = match ist, t with + | None, (GVar (_, id) | GRef (_, VarRef id,_)) -> pf_nbargs gl (mkVar id) + | Some ist, _ -> interp_nbargs ist gl t + | _ -> anomaly "apply_rconstr without ist and not RVar" in + let mkRlemma i = mkRApp t (mkRHoles i) in + let cl = pf_concl gl in + let rec loop i = + if i > n then + errorstrm (str"Cannot apply lemma "++pf_pr_glob_constr gl t) + else try pf_match gl (mkRlemma i) (OfType cl) with _ -> loop (i + 1) in + refine_with (loop 0) gl + +let mkRAppView ist gl rv gv = + let nb_view_imps = interp_view_nbimps ist gl rv in + mkRApp rv (mkRHoles (abs nb_view_imps)) + +let prof_apply_interp_with = mk_profiler "ssrapplytac.interp_with";; + +let refine_interp_apply_view i ist gl gv = + let pair i = List.map (fun x -> i, x) in + let rv = pf_intern_term ist gl gv in + let v = mkRAppView ist gl rv gv in + let interp_with (i, hint) = + interp_refine ist gl (mkRApp hint (v :: mkRHoles i)) in + let interp_with x = prof_apply_interp_with.profile interp_with x in + let rec loop = function + | [] -> (try apply_rconstr ~ist rv gl with _ -> view_error "apply" gv) + | h :: hs -> (try refine_with (snd (interp_with h)) gl with _ -> loop hs) in + loop (pair i viewtab.(i) @ if i = 2 then pair 1 viewtab.(1) else []) + +let apply_top_tac gl = + tclTHENLIST [introid top_id; apply_rconstr (mkRVar top_id); clear [top_id]] gl + +let inner_ssrapplytac gviews ggenl gclr ist gl = + let _, clr = interp_hyps ist gl gclr in + let vtac gv i gl' = refine_interp_apply_view i ist gl' gv in + let ggenl, tclGENTAC = + if gviews <> [] && ggenl <> [] then + let ggenl= List.map (fun (x,g) -> x, cpattern_of_term g) (List.hd ggenl) in + [], tclTHEN (genstac (ggenl,[]) ist) + else ggenl, tclTHEN tclIDTAC in + tclGENTAC (fun gl -> + match gviews, ggenl with + | v :: tl, [] -> + let dbl = if List.length tl = 1 then 2 else 1 in + tclTHEN + (List.fold_left (fun acc v -> tclTHENLAST acc (vtac v dbl)) (vtac v 1) tl) + (cleartac clr) gl + | [], [agens] -> + let clr', (sigma, lemma) = interp_agens ist gl agens in + let gl = pf_merge_uc_of sigma gl in + tclTHENLIST [cleartac clr; refine_with ~beta:true lemma; cleartac clr'] gl + | _, _ -> tclTHEN apply_top_tac (cleartac clr) gl) gl + +let ssrapplytac ist (views, (_, ((gens, clr), intros))) = + tclINTROS ist (inner_ssrapplytac views gens clr) intros + +TACTIC EXTEND ssrapply +| [ "apply" ssrapplyarg(arg) ] -> [ Proofview.V82.tactic (ssrapplytac ist arg) ] +| [ "apply" ] -> [ Proofview.V82.tactic apply_top_tac ] +END + +(** The "exact" tactic *) + +let mk_exactarg views dgens = mk_applyarg views dgens [] + +ARGUMENT EXTEND ssrexactarg TYPED AS ssrapplyarg PRINTED BY pr_ssraarg +| [ ":" ssragen(gen) ssragens(dgens) ] -> + [ mk_exactarg [] (cons_gen gen dgens) ] +| [ ssrview(view) ssrclear(clr) ] -> + [ mk_exactarg view ([], clr) ] +| [ ssrclear_ne(clr) ] -> + [ mk_exactarg [] ([], clr) ] +END + +let vmexacttac pf gl = exact_no_check (mkCast (pf, VMcast, pf_concl gl)) gl + +TACTIC EXTEND ssrexact +| [ "exact" ssrexactarg(arg) ] -> [ Proofview.V82.tactic (tclBY (ssrapplytac ist arg)) ] +| [ "exact" ] -> [ Proofview.V82.tactic (tclORELSE donetac (tclBY apply_top_tac)) ] +| [ "exact" "<:" lconstr(pf) ] -> [ Proofview.V82.tactic (vmexacttac pf) ] +END + +(** The "congr" tactic *) + +(* type ssrcongrarg = open_constr * (int * constr) *) + +let pr_ssrcongrarg _ _ _ ((n, f), dgens) = + (if n <= 0 then mt () else str " " ++ int n) ++ + str " " ++ pr_term f ++ pr_dgens pr_gen dgens + +ARGUMENT EXTEND ssrcongrarg TYPED AS (int * ssrterm) * ssrdgens + PRINTED BY pr_ssrcongrarg +| [ natural(n) constr(c) ssrdgens(dgens) ] -> [ (n, mk_term ' ' c), dgens ] +| [ natural(n) constr(c) ] -> [ (n, mk_term ' ' c),([[]],[]) ] +| [ constr(c) ssrdgens(dgens) ] -> [ (0, mk_term ' ' c), dgens ] +| [ constr(c) ] -> [ (0, mk_term ' ' c), ([[]],[]) ] +END + +let rec mkRnat n = + if n <= 0 then GRef (dummy_loc, glob_O, None) else + mkRApp (GRef (dummy_loc, glob_S, None)) [mkRnat (n - 1)] + +let interp_congrarg_at ist gl n rf ty m = + pp(lazy(str"===interp_congrarg_at===")); + let congrn, _ = mkSsrRRef "nary_congruence" in + let args1 = mkRnat n :: mkRHoles n @ [ty] in + let args2 = mkRHoles (3 * n) in + let rec loop i = + if i + n > m then None else + try + let rt = mkRApp congrn (args1 @ mkRApp rf (mkRHoles i) :: args2) in + pp(lazy(str"rt=" ++ pr_glob_constr rt)); + Some (interp_refine ist gl rt) + with _ -> loop (i + 1) in + loop 0 + +let pattern_id = mk_internal_id "pattern value" + +let congrtac ((n, t), ty) ist gl = + pp(lazy(str"===congr===")); + pp(lazy(str"concl=" ++ pr_constr (pf_concl gl))); + let sigma, _ as it = interp_term ist gl t in + let gl = pf_merge_uc_of sigma gl in + let _, f, _, _ucst = pf_abs_evars gl it in + let ist' = {ist with lfun = + Id.Map.add pattern_id (Value.of_constr f) Id.Map.empty } in + let rf = mkRltacVar pattern_id in + let m = pf_nbargs gl f in + let _, cf = if n > 0 then + match interp_congrarg_at ist' gl n rf ty m with + | Some cf -> cf + | None -> errorstrm (str "No " ++ int n ++ str "-congruence with " + ++ pr_term t) + else let rec loop i = + if i > m then errorstrm (str "No congruence with " ++ pr_term t) + else match interp_congrarg_at ist' gl i rf ty m with + | Some cf -> cf + | None -> loop (i + 1) in + loop 1 in + tclTHEN (refine_with cf) (tclTRY (Proofview.V82.of_tactic reflexivity)) gl + +let newssrcongrtac arg ist gl = + pp(lazy(str"===newcongr===")); + pp(lazy(str"concl=" ++ pr_constr (pf_concl gl))); + (* utils *) + let fs gl t = Reductionops.nf_evar (project gl) t in + let tclMATCH_GOAL (c, gl_c) proj t_ok t_fail gl = + match try Some (pf_unify_HO gl_c (pf_concl gl) c) with _ -> None with + | Some gl_c -> + tclTHEN (Proofview.V82.of_tactic (convert_concl (fs gl_c c))) + (t_ok (proj gl_c)) gl + | None -> t_fail () gl in + let mk_evar gl ty = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + let sigma, x = Evarutil.new_evar env (create_evar_defs sigma) ty in + x, re_sig si sigma in + let arr, gl = pf_mkSsrConst "ssr_congr_arrow" gl in + let ssr_congr lr = mkApp (arr, lr) in + (* here thw two cases: simple equality or arrow *) + let equality, _, eq_args, gl' = + let eq, gl = pf_fresh_global (build_coq_eq ()) gl in + pf_saturate gl eq 3 in + tclMATCH_GOAL (equality, gl') (fun gl' -> fs gl' (List.assoc 0 eq_args)) + (fun ty -> congrtac (arg, Detyping.detype false [] (pf_env gl) (project gl) ty) ist) + (fun () -> + let lhs, gl' = mk_evar gl mkProp in let rhs, gl' = mk_evar gl' mkProp in + let arrow = mkArrow lhs (lift 1 rhs) in + tclMATCH_GOAL (arrow, gl') (fun gl' -> [|fs gl' lhs;fs gl' rhs|]) + (fun lr -> tclTHEN (Proofview.V82.of_tactic (apply (ssr_congr lr))) (congrtac (arg, mkRType) ist)) + (fun _ _ -> errorstrm (str"Conclusion is not an equality nor an arrow"))) + gl +;; + +TACTIC EXTEND ssrcongr +| [ "congr" ssrcongrarg(arg) ] -> +[ let arg, dgens = arg in + Proofview.V82.tactic begin + match dgens with + | [gens], clr -> tclTHEN (genstac (gens,clr) ist) (newssrcongrtac arg ist) + | _ -> errorstrm (str"Dependent family abstractions not allowed in congr") + end] +END + +(** 7. Rewriting tactics (rewrite, unlock) *) + +(** Coq rewrite compatibility flag *) + +let ssr_strict_match = ref false + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "strict redex matching"; + Goptions.optkey = ["Match"; "Strict"]; + Goptions.optread = (fun () -> !ssr_strict_match); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssr_strict_match := b) } + +(** Rewrite multiplier *) + +type ssrmult = int * ssrmmod + +let notimes = 0 +let nomult = 1, Once + +let pr_mult (n, m) = + if n > 0 && m <> Once then int n ++ pr_mmod m else pr_mmod m + +let pr_ssrmult _ _ _ = pr_mult + +ARGUMENT EXTEND ssrmult_ne TYPED AS int * ssrmmod PRINTED BY pr_ssrmult + | [ natural(n) ssrmmod(m) ] -> [ check_index loc n, m ] + | [ ssrmmod(m) ] -> [ notimes, m ] +END + +ARGUMENT EXTEND ssrmult TYPED AS ssrmult_ne PRINTED BY pr_ssrmult + | [ ssrmult_ne(m) ] -> [ m ] + | [ ] -> [ nomult ] +END + +(** Rewrite clear/occ switches *) + +let pr_rwocc = function + | None, None -> mt () + | None, occ -> pr_occ occ + | Some clr, _ -> pr_clear_ne clr + +let pr_ssrrwocc _ _ _ = pr_rwocc + +ARGUMENT EXTEND ssrrwocc TYPED AS ssrdocc PRINTED BY pr_ssrrwocc +| [ "{" ssrhyp_list(clr) "}" ] -> [ mkclr clr ] +| [ "{" ssrocc(occ) "}" ] -> [ mkocc occ ] +| [ ] -> [ noclr ] +END + +(** Rewrite rules *) + +type ssrwkind = RWred of ssrsimpl | RWdef | RWeq +(* type ssrrule = ssrwkind * ssrterm *) + +let pr_rwkind = function + | RWred s -> pr_simpl s + | RWdef -> str "/" + | RWeq -> mt () + +let wit_ssrrwkind = add_genarg "ssrrwkind" pr_rwkind + +let pr_rule = function + | RWred s, _ -> pr_simpl s + | RWdef, r-> str "/" ++ pr_term r + | RWeq, r -> pr_term r + +let pr_ssrrule _ _ _ = pr_rule + +let noruleterm loc = mk_term ' ' (mkCProp loc) + +ARGUMENT EXTEND ssrrule_ne TYPED AS ssrrwkind * ssrterm PRINTED BY pr_ssrrule + | [ ssrsimpl_ne(s) ] -> [ RWred s, noruleterm loc ] + | [ "/" ssrterm(t) ] -> [ RWdef, t ] + | [ ssrterm(t) ] -> [ RWeq, t ] +END + +ARGUMENT EXTEND ssrrule TYPED AS ssrrule_ne PRINTED BY pr_ssrrule + | [ ssrrule_ne(r) ] -> [ r ] + | [ ] -> [ RWred Nop, noruleterm loc ] +END + +(** Rewrite arguments *) + +(* type ssrrwarg = (ssrdir * ssrmult) * ((ssrdocc * ssrpattern) * ssrrule) *) + +let pr_option f = function None -> mt() | Some x -> f x +let pr_pattern_squarep= pr_option (fun r -> str "[" ++ pr_rpattern r ++ str "]") +let pr_ssrpattern_squarep _ _ _ = pr_pattern_squarep +let pr_rwarg ((d, m), ((docc, rx), r)) = + pr_rwdir d ++ pr_mult m ++ pr_rwocc docc ++ pr_pattern_squarep rx ++ pr_rule r + +let pr_ssrrwarg _ _ _ = pr_rwarg + +let mk_rwarg (d, (n, _ as m)) ((clr, occ as docc), rx) (rt, _ as r) = + if rt <> RWeq then begin + if rt = RWred Nop && not (m = nomult && occ = None && rx = None) + && (clr = None || clr = Some []) then + anomaly "Improper rewrite clear switch"; + if d = R2L && rt <> RWdef then + Errors.error "Right-to-left switch on simplification"; + if n <> 1 && rt = RWred Cut then + Errors.error "Bad or useless multiplier"; + if occ <> None && rx = None && rt <> RWdef then + Errors.error "Missing redex for simplification occurrence" + end; (d, m), ((docc, rx), r) + +let norwmult = L2R, nomult +let norwocc = noclr, None + +(* +let pattern_ident = Prim.pattern_ident in +GEXTEND Gram +GLOBAL: pattern_ident; +pattern_ident: +[[c = pattern_ident -> (CRef (Ident (loc,c)), NoBindings)]]; +END +*) + +ARGUMENT EXTEND ssrpattern_squarep +TYPED AS rpattern option PRINTED BY pr_ssrpattern_squarep + | [ "[" rpattern(rdx) "]" ] -> [ Some rdx ] + | [ ] -> [ None ] +END + +ARGUMENT EXTEND ssrpattern_ne_squarep +TYPED AS rpattern option PRINTED BY pr_ssrpattern_squarep + | [ "[" rpattern(rdx) "]" ] -> [ Some rdx ] +END + + +ARGUMENT EXTEND ssrrwarg + TYPED AS (ssrdir * ssrmult) * ((ssrdocc * rpattern option) * ssrrule) + PRINTED BY pr_ssrrwarg + | [ "-" ssrmult(m) ssrrwocc(docc) ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg (R2L, m) (docc, rx) r ] + | [ "-/" ssrterm(t) ] -> (* just in case '-/' should become a token *) + [ mk_rwarg (R2L, nomult) norwocc (RWdef, t) ] + | [ ssrmult_ne(m) ssrrwocc(docc) ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg (L2R, m) (docc, rx) r ] + | [ "{" ne_ssrhyp_list(clr) "}" ssrpattern_ne_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (mkclr clr, rx) r ] + | [ "{" ne_ssrhyp_list(clr) "}" ssrrule(r) ] -> + [ mk_rwarg norwmult (mkclr clr, None) r ] + | [ "{" ssrocc(occ) "}" ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (mkocc occ, rx) r ] + | [ "{" "}" ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (nodocc, rx) r ] + | [ ssrpattern_ne_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (noclr, rx) r ] + | [ ssrrule_ne(r) ] -> + [ mk_rwarg norwmult norwocc r ] +END + +let simplintac occ rdx sim gl = + let simptac gl = + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let simp env c _ = red_safe Tacred.simpl env sigma0 c in + Proofview.V82.of_tactic + (convert_concl_no_check (eval_pattern env0 sigma0 concl0 rdx occ simp)) + gl in + match sim with + | Simpl -> simptac gl + | SimplCut -> tclTHEN simptac (tclTRY donetac) gl + | _ -> simpltac sim gl + +let rec get_evalref c = match kind_of_term c with + | Var id -> EvalVarRef id + | Const (k,_) -> EvalConstRef k + | App (c', _) -> get_evalref c' + | Cast (c', _, _) -> get_evalref c' + | _ -> errorstrm (str "The term " ++ pr_constr c ++ str " is not unfoldable") + +(* Strip a pattern generated by a prenex implicit to its constant. *) +let strip_unfold_term ((sigma, t) as p) kt = match kind_of_term t with + | App (f, a) when kt = ' ' && Array.for_all isEvar a && isConst f -> + (sigma, f), true + | Const _ | Var _ -> p, true + | _ -> p, false + +let unfoldintac occ rdx t (kt,_) gl = + let fs sigma x = Reductionops.nf_evar sigma x in + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let (sigma, t), const = strip_unfold_term t kt in + let body env t c = + Tacred.unfoldn [OnlyOccurrences [1], get_evalref t] env sigma0 c in + let easy = occ = None && rdx = None in + let red_flags = if easy then Closure.betaiotazeta else Closure.betaiota in + let beta env = Reductionops.clos_norm_flags red_flags env sigma0 in + let unfold, conclude = match rdx with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let ise = create_evar_defs sigma in + let ise, u = mk_tpattern env0 sigma0 (ise,t) all_ok L2R t in + let find_T, end_T = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[u]) in + (fun env c h -> + try find_T env c h (fun env c _ -> body env t c) + with NoMatch when easy -> c + | NoMatch | NoProgress -> errorstrm (str"No occurrence of " + ++ pr_constr_pat t ++ spc() ++ str "in " ++ pr_constr c)), + (fun () -> try end_T () with + | NoMatch when easy -> fake_pmatcher_end () + | NoMatch -> anomaly "unfoldintac") + | _ -> + (fun env (c as orig_c) h -> + if const then + let rec aux c = + match kind_of_term c with + | Const _ when Term.eq_constr c t -> body env t t + | App (f,a) when Term.eq_constr f t -> mkApp (body env f f,a) + | _ -> let c = Reductionops.whd_betaiotazeta sigma0 c in + match kind_of_term c with + | Const _ when Term.eq_constr c t -> body env t t + | App (f,a) when Term.eq_constr f t -> mkApp (body env f f,a) + | Const f -> aux (body env c c) + | App (f, a) -> aux (mkApp (body env f f, a)) + | _ -> errorstrm (str "The term "++pr_constr orig_c++ + str" contains no " ++ pr_constr t ++ str" even after unfolding") + in aux c + else + try body env t (fs (unify_HO env sigma c t) t) + with _ -> errorstrm (str "The term " ++ + pr_constr c ++spc()++ str "does not unify with " ++ pr_constr_pat t)), + fake_pmatcher_end in + let concl = + try beta env0 (eval_pattern env0 sigma0 concl0 rdx occ unfold) + with Option.IsNone -> errorstrm (str"Failed to unfold " ++ pr_constr_pat t) in + let _ = conclude () in + Proofview.V82.of_tactic (convert_concl concl) gl +;; + +let foldtac occ rdx ft gl = + let fs sigma x = Reductionops.nf_evar sigma x in + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let sigma, t = ft in + let fold, conclude = match rdx with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let ise = create_evar_defs sigma in + let ut = red_product_skip_id env0 sigma t in + let ise, ut = mk_tpattern env0 sigma0 (ise,t) all_ok L2R ut in + let find_T, end_T = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[ut]) in + (fun env c h -> try find_T env c h (fun env t _ -> t) with NoMatch -> c), + (fun () -> try end_T () with NoMatch -> fake_pmatcher_end ()) + | _ -> + (fun env c h -> try let sigma = unify_HO env sigma c t in fs sigma t + with _ -> errorstrm (str "fold pattern " ++ pr_constr_pat t ++ spc () + ++ str "does not match redex " ++ pr_constr_pat c)), + fake_pmatcher_end in + let concl = eval_pattern env0 sigma0 concl0 rdx occ fold in + let _ = conclude () in + Proofview.V82.of_tactic (convert_concl concl) gl +;; + +let converse_dir = function L2R -> R2L | R2L -> L2R + +let rw_progress rhs lhs ise = not (Term.eq_constr lhs (Evarutil.nf_evar ise rhs)) + +(* Coq has a more general form of "equation" (any type with a single *) +(* constructor with no arguments with_rect_r elimination lemmas). *) +(* However there is no clear way of determining the LHS and RHS of *) +(* such a generic Leibnitz equation -- short of inspecting the type *) +(* of the elimination lemmas. *) + +let rec strip_prod_assum c = match kind_of_term c with + | Prod (_, _, c') -> strip_prod_assum c' + | LetIn (_, v, _, c') -> strip_prod_assum (subst1 v c) + | Cast (c', _, _) -> strip_prod_assum c' + | _ -> c + +let rule_id = mk_internal_id "rewrite rule" + +exception PRtype_error +exception PRindetermined_rhs of constr + +let pirrel_rewrite pred rdx rdx_ty new_rdx dir (sigma, c) c_ty gl = +(* pp(lazy(str"sigma@pirrel_rewrite=" ++ pr_evar_map None sigma)); *) + let env = pf_env gl in + let beta = Reductionops.clos_norm_flags Closure.beta env sigma in + let sigma, p = + let sigma = create_evar_defs sigma in + Evarutil.new_evar env sigma (beta (subst1 new_rdx pred)) in + let pred = mkNamedLambda pattern_id rdx_ty pred in + let elim, gl = + let ((kn, i) as ind, _), unfolded_c_ty = pf_reduce_to_quantified_ind gl c_ty in + let sort = elimination_sort_of_goal gl in + let elim, gl = pf_fresh_global (Indrec.lookup_eliminator ind sort) gl in + if dir = R2L then elim, gl else (* taken from Coq's rewrite *) + let elim, _ = destConst elim in + let mp,dp,l = repr_con (constant_of_kn (canonical_con elim)) in + let l' = label_of_id (Nameops.add_suffix (id_of_label l) "_r") in + let c1' = Global.constant_of_delta_kn (canonical_con (make_con mp dp l')) in + mkConst c1', gl in + let proof = mkApp (elim, [| rdx_ty; new_rdx; pred; p; rdx; c |]) in + (* We check the proof is well typed *) + let sigma, proof_ty = + try Typing.e_type_of env sigma proof with _ -> raise PRtype_error in + pp(lazy(str"pirrel_rewrite proof term of type: " ++ pr_constr proof_ty)); + try refine_with + ~first_goes_last:(not !ssroldreworder) ~with_evars:false (sigma, proof) gl + with _ -> + (* we generate a msg like: "Unable to find an instance for the variable" *) + let c = Reductionops.nf_evar sigma c in + let hd_ty, miss = match kind_of_term c with + | App (hd, args) -> + let hd_ty = Retyping.get_type_of env sigma hd in + let names = let rec aux t = function 0 -> [] | n -> + let t = Reductionops.whd_betadeltaiota env sigma t in + match kind_of_type t with + | ProdType (name, _, t) -> name :: aux t (n-1) + | _ -> assert false in aux hd_ty (Array.length args) in + hd_ty, Util.List.map_filter (fun (t, name) -> + let evs = Intset.elements (Evarutil.undefined_evars_of_term sigma t) in + let open_evs = List.filter (fun k -> + InProp <> Retyping.get_sort_family_of + env sigma (Evd.evar_concl (Evd.find sigma k))) + evs in + if open_evs <> [] then Some name else None) + (List.combine (Array.to_list args) names) + | _ -> anomaly "rewrite rule not an application" in + errorstrm (Himsg.explain_refiner_error (Logic.UnresolvedBindings miss)++ + (Pp.fnl()++str"Rule's type:" ++ spc() ++ pr_constr hd_ty)) +;; + +let is_const_ref c r = isConst c && eq_gr (ConstRef (fst(destConst c))) r +let is_construct_ref c r = + isConstruct c && eq_gr (ConstructRef (fst(destConstruct c))) r +let is_ind_ref c r = isInd c && eq_gr (IndRef (fst(destInd c))) r + +let rwcltac cl rdx dir sr gl = + let n, r_n,_, ucst = pf_abs_evars gl sr in + let r_n' = pf_abs_cterm gl n r_n in + let r' = subst_var pattern_id r_n' in + let gl = pf_merge_uc ucst gl in + let rdxt = Retyping.get_type_of (pf_env gl) (fst sr) rdx in +(* pp(lazy(str"sigma@rwcltac=" ++ pr_evar_map None (fst sr))); *) + pp(lazy(str"r@rwcltac=" ++ pr_constr (snd sr))); + let cvtac, rwtac, gl = + if closed0 r' then + let env, sigma, c, c_eq = pf_env gl, fst sr, snd sr, build_coq_eq () in + let sigma, c_ty = Typing.e_type_of env sigma c in + pp(lazy(str"c_ty@rwcltac=" ++ pr_constr c_ty)); + match kind_of_type (Reductionops.whd_betadeltaiota env sigma c_ty) with + | AtomicType(e, a) when is_ind_ref e c_eq -> + let new_rdx = if dir = L2R then a.(2) else a.(1) in + pirrel_rewrite cl rdx rdxt new_rdx dir (sigma,c) c_ty, tclIDTAC, gl + | _ -> + let cl' = mkApp (mkNamedLambda pattern_id rdxt cl, [|rdx|]) in + let sigma, _ = Typing.e_type_of env sigma cl' in + let gl = pf_merge_uc_of sigma gl in + Proofview.V82.of_tactic (convert_concl cl'), rewritetac dir r', gl + else + let dc, r2 = decompose_lam_n n r' in + let r3, _, r3t = + try destCast r2 with _ -> + errorstrm (str "no cast from " ++ pr_constr_pat (snd sr) + ++ str " to " ++ pr_constr r2) in + let cl' = mkNamedProd rule_id (compose_prod dc r3t) (lift 1 cl) in + let cl'' = mkNamedProd pattern_id rdxt cl' in + let itacs = [introid pattern_id; introid rule_id] in + let cltac = clear [pattern_id; rule_id] in + let rwtacs = [rewritetac dir (mkVar rule_id); cltac] in + apply_type cl'' [rdx; compose_lam dc r3], tclTHENLIST (itacs @ rwtacs), gl + in + let cvtac' _ = + try cvtac gl with + | PRtype_error -> + if occur_existential (pf_concl gl) + then errorstrm (str "Rewriting impacts evars") + else errorstrm (str "Dependent type error in rewrite of " + ++ pf_pr_constr gl (project gl) (mkNamedLambda pattern_id rdxt cl)) + | Errors.UserError _ as e -> raise e + | e -> anomaly ("cvtac's exception: " ^ Printexc.to_string e); + in + tclTHEN cvtac' rwtac gl + +let prof_rwcltac = mk_profiler "rwrxtac.rwcltac";; +let rwcltac cl rdx dir sr gl = + prof_rwcltac.profile (rwcltac cl rdx dir sr) gl +;; + + +let lz_coq_prod = + let prod = lazy (build_prod ()) in fun () -> Lazy.force prod + +let lz_setoid_relation = + let sdir = ["Classes"; "RelationClasses"] in + let last_srel = ref (Environ.empty_env, None) in + fun env -> match !last_srel with + | env', srel when env' == env -> srel + | _ -> + let srel = + try Some (coq_constant "Class_setoid" sdir "RewriteRelation") + with _ -> None in + last_srel := (env, srel); srel + +let ssr_is_setoid env = + match lz_setoid_relation env with + | None -> fun _ _ _ -> false + | Some srel -> + fun sigma r args -> + Rewrite.is_applied_rewrite_relation env + sigma [] (mkApp (r, args)) <> None + +let prof_rwxrtac_find_rule = mk_profiler "rwrxtac.find_rule";; + +let closed0_check cl p gl = + if closed0 cl then + errorstrm (str"No occurrence of redex "++pf_pr_constr gl (project gl) p) + +let rwrxtac occ rdx_pat dir rule gl = + let env = pf_env gl in + let coq_prod = lz_coq_prod () in + let is_setoid = ssr_is_setoid env in + let r_sigma, rules = + let rec loop d sigma r t0 rs red = + let t = + if red = 1 then Tacred.hnf_constr env sigma t0 + else Reductionops.whd_betaiotazeta sigma t0 in + pp(lazy(str"rewrule="++pr_constr t)); + match kind_of_term t with + | Prod (_, xt, at) -> + let ise, x = Evarutil.new_evar env (create_evar_defs sigma) xt in + loop d ise (mkApp (r, [|x|])) (subst1 x at) rs 0 + | App (pr, a) when is_ind_ref pr coq_prod.Coqlib.typ -> + let sr sigma = match kind_of_term (Tacred.hnf_constr env sigma r) with + | App (c, ra) when is_construct_ref c coq_prod.Coqlib.intro -> + fun i -> ra.(i + 1), sigma + | _ -> let ra = Array.append a [|r|] in + function 1 -> + let sigma, pi1 = Evd.fresh_global env sigma coq_prod.Coqlib.proj1 in + mkApp (pi1, ra), sigma + | _ -> + let sigma, pi2 = Evd.fresh_global env sigma coq_prod.Coqlib.proj2 in + mkApp (pi2, ra), sigma in + if Term.eq_constr a.(0) (build_coq_True ()) then + let s, sigma = sr sigma 2 in + loop (converse_dir d) sigma s a.(1) rs 0 + else + let s, sigma = sr sigma 2 in + let sigma, rs2 = loop d sigma s a.(1) rs 0 in + let s, sigma = sr sigma 1 in + loop d sigma s a.(0) rs2 0 + | App (r_eq, a) when Hipattern.match_with_equality_type t != None -> + let indu = destInd r_eq and rhs = Array.last a in + let np = Inductiveops.inductive_nparamdecls (fst indu) in + let ind_ct = Inductiveops.type_of_constructors env indu in + let lhs0 = last_arg (strip_prod_assum ind_ct.(0)) in + let rdesc = match kind_of_term lhs0 with + | Rel i -> + let lhs = a.(np - i) in + let lhs, rhs = if d = L2R then lhs, rhs else rhs, lhs in +(* msgnl (str "RW: " ++ pr_rwdir d ++ str " " ++ pr_constr_pat r ++ str " : " + ++ pr_constr_pat lhs ++ str " ~> " ++ pr_constr_pat rhs); *) + d, r, lhs, rhs +(* + let l_i, r_i = if d = L2R then i, 1 - ndep else 1 - ndep, i in + let lhs = a.(np - l_i) and rhs = a.(np - r_i) in + let a' = Array.copy a in let _ = a'.(np - l_i) <- mkVar pattern_id in + let r' = mkCast (r, DEFAULTcast, mkApp (r_eq, a')) in + (d, r', lhs, rhs) +*) + | _ -> + let lhs = substl (array_list_of_tl (Array.sub a 0 np)) lhs0 in + let lhs, rhs = if d = R2L then lhs, rhs else rhs, lhs in + let d' = if Array.length a = 1 then d else converse_dir d in + d', r, lhs, rhs in + sigma, rdesc :: rs + | App (s_eq, a) when is_setoid sigma s_eq a -> + let np = Array.length a and i = 3 - dir_org d in + let lhs = a.(np - i) and rhs = a.(np + i - 3) in + let a' = Array.copy a in let _ = a'.(np - i) <- mkVar pattern_id in + let r' = mkCast (r, DEFAULTcast, mkApp (s_eq, a')) in + sigma, (d, r', lhs, rhs) :: rs + | _ -> + if red = 0 then loop d sigma r t rs 1 + else errorstrm (str "not a rewritable relation: " ++ pr_constr_pat t + ++ spc() ++ str "in rule " ++ pr_constr_pat (snd rule)) + in + let sigma, r = rule in + let t = Retyping.get_type_of env sigma r in + loop dir sigma r t [] 0 in + let find_rule rdx = + let rec rwtac = function + | [] -> + errorstrm (str "pattern " ++ pr_constr_pat rdx ++ + str " does not match " ++ pr_dir_side dir ++ + str " of " ++ pr_constr_pat (snd rule)) + | (d, r, lhs, rhs) :: rs -> + try + let ise = unify_HO env (create_evar_defs r_sigma) lhs rdx in + if not (rw_progress rhs rdx ise) then raise NoMatch else + d, (ise, Evd.evar_universe_context ise, Reductionops.nf_evar ise r) + with _ -> rwtac rs in + rwtac rules in + let find_rule rdx = prof_rwxrtac_find_rule.profile find_rule rdx in + let sigma0, env0, concl0 = project gl, pf_env gl, pf_concl gl in + let find_R, conclude = match rdx_pat with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let upats_origin = dir, snd rule in + let rpat env sigma0 (sigma, pats) (d, r, lhs, rhs) = + let sigma, pat = + mk_tpattern env sigma0 (sigma,r) (rw_progress rhs) d lhs in + sigma, pats @ [pat] in + let rpats = List.fold_left (rpat env0 sigma0) (r_sigma,[]) rules in + let find_R, end_R = mk_tpattern_matcher sigma0 occ ~upats_origin rpats in + find_R ~k:(fun _ _ h -> mkRel h), + fun cl -> let rdx,d,r = end_R () in closed0_check cl rdx gl; (d,r),rdx + | Some(_, (T e | X_In_T (_,e) | E_As_X_In_T (e,_,_) | E_In_X_In_T (e,_,_))) -> + let r = ref None in + (fun env c h -> do_once r (fun () -> find_rule c, c); mkRel h), + (fun concl -> closed0_check concl e gl; assert_done r) in + let concl = eval_pattern env0 sigma0 concl0 rdx_pat occ find_R in + let (d, r), rdx = conclude concl in + let r = Evd.merge_universe_context (pi1 r) (pi2 r), pi3 r in + rwcltac concl rdx d r gl +;; + +let prof_rwxrtac = mk_profiler "rwrxtac";; +let rwrxtac occ rdx_pat dir rule gl = + prof_rwxrtac.profile (rwrxtac occ rdx_pat dir rule) gl +;; + + +(* Resolve forward reference *) +let _ = + ipat_rewritetac := fun occ dir c gl -> rwrxtac occ None dir (project gl, c) gl + +let rwargtac ist ((dir, mult), (((oclr, occ), grx), (kind, gt))) gl = + let fail = ref false in + let interp_rpattern ist gl gc = + try interp_rpattern ist gl gc + with _ when snd mult = May -> fail := true; project gl, T mkProp in + let interp gc gl = + try interp_term ist gl gc + with _ when snd mult = May -> fail := true; (project gl, mkProp) in + let rwtac gl = + let rx = Option.map (interp_rpattern ist gl) grx in + let t = interp gt gl in + (match kind with + | RWred sim -> simplintac occ rx sim + | RWdef -> if dir = R2L then foldtac occ rx t else unfoldintac occ rx t gt + | RWeq -> rwrxtac occ rx dir t) gl in + let ctac = cleartac (interp_clr (oclr, (fst gt, snd (interp gt gl)))) in + if !fail then ctac gl else tclTHEN (tclMULT mult rwtac) ctac gl + +(** Rewrite argument sequence *) + +(* type ssrrwargs = ssrrwarg list *) + +let pr_ssrrwargs _ _ _ rwargs = pr_list spc pr_rwarg rwargs + +ARGUMENT EXTEND ssrrwargs TYPED AS ssrrwarg list PRINTED BY pr_ssrrwargs + | [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let ssr_rw_syntax = Summary.ref ~name:"SSR:rewrite" true + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "ssreflect rewrite"; + Goptions.optkey = ["SsrRewrite"]; + Goptions.optread = (fun _ -> !ssr_rw_syntax); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssr_rw_syntax := b) } + +let test_ssr_rw_syntax = + let test strm = + if not !ssr_rw_syntax then raise Stream.Failure else + if is_ssr_loaded () then () else + match Compat.get_tok (Util.stream_nth 0 strm) with + | Tok.KEYWORD key when List.mem key.[0] ['{'; '['; '/'] -> () + | _ -> raise Stream.Failure in + Gram.Entry.of_parser "test_ssr_rw_syntax" test + +GEXTEND Gram + GLOBAL: ssrrwargs; + ssrrwargs: [[ test_ssr_rw_syntax; a = LIST1 ssrrwarg -> a ]]; +END + +(** The "rewrite" tactic *) + +let ssrrewritetac ist rwargs = + tclTHENLIST (List.map (rwargtac ist) rwargs) + +TACTIC EXTEND ssrrewrite + | [ "rewrite" ssrrwargs(args) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrrewritetac ist args) clauses) ] +END + +(** The "unlock" tactic *) + +let pr_unlockarg (occ, t) = pr_occ occ ++ pr_term t +let pr_ssrunlockarg _ _ _ = pr_unlockarg + +ARGUMENT EXTEND ssrunlockarg TYPED AS ssrocc * ssrterm + PRINTED BY pr_ssrunlockarg + | [ "{" ssrocc(occ) "}" ssrterm(t) ] -> [ occ, t ] + | [ ssrterm(t) ] -> [ None, t ] +END + +let pr_ssrunlockargs _ _ _ args = pr_list spc pr_unlockarg args + +ARGUMENT EXTEND ssrunlockargs TYPED AS ssrunlockarg list + PRINTED BY pr_ssrunlockargs + | [ ssrunlockarg_list(args) ] -> [ args ] +END + +let unfoldtac occ ko t kt gl = + let cl, c = pf_fill_occ_term gl occ (fst (strip_unfold_term t kt)) in + let cl' = subst1 (pf_unfoldn [OnlyOccurrences [1], get_evalref c] gl c) cl in + let f = if ko = None then Closure.betaiotazeta else Closure.betaiota in + Proofview.V82.of_tactic + (convert_concl (pf_reduce (Reductionops.clos_norm_flags f) gl cl')) gl + +let unlocktac ist args gl = + let utac (occ, gt) gl = + unfoldtac occ occ (interp_term ist gl gt) (fst gt) gl in + let locked, gl = pf_mkSsrConst "locked" gl in + let key, gl = pf_mkSsrConst "master_key" gl in + let ktacs = [ + (fun gl -> unfoldtac None None (project gl,locked) '(' gl); + simplest_newcase key ] in + tclTHENLIST (List.map utac args @ ktacs) gl + +TACTIC EXTEND ssrunlock + | [ "unlock" ssrunlockargs(args) ssrclauses(clauses) ] -> +[ Proofview.V82.tactic (tclCLAUSES ist (unlocktac ist args) clauses) ] +END + +(** 8. Forward chaining tactics (pose, set, have, suffice, wlog) *) + +(** Defined identifier *) + +type ssrfwdid = identifier + +let pr_ssrfwdid _ _ _ id = pr_spc () ++ pr_id id + +(* We use a primitive parser for the head identifier of forward *) +(* tactis to avoid syntactic conflicts with basic Coq tactics. *) +ARGUMENT EXTEND ssrfwdid TYPED AS ident PRINTED BY pr_ssrfwdid + | [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let accept_ssrfwdid strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT id -> accept_before_syms_or_any_id [":"; ":="; "("] strm + | _ -> raise Stream.Failure + + +let test_ssrfwdid = Gram.Entry.of_parser "test_ssrfwdid" accept_ssrfwdid + +GEXTEND Gram + GLOBAL: ssrfwdid; + ssrfwdid: [[ test_ssrfwdid; id = Prim.ident -> id ]]; + END + + + +(** Definition value formatting *) + +(* We use an intermediate structure to correctly render the binder list *) +(* abbreviations. We use a list of hints to extract the binders and *) +(* base term from a term, for the two first levels of representation of *) +(* of constr terms. *) + +type 'term ssrbind = + | Bvar of name + | Bdecl of name list * 'term + | Bdef of name * 'term option * 'term + | Bstruct of name + | Bcast of 'term + +let pr_binder prl = function + | Bvar x -> + pr_name x + | Bdecl (xs, t) -> + str "(" ++ pr_list pr_spc pr_name xs ++ str " : " ++ prl t ++ str ")" + | Bdef (x, None, v) -> + str "(" ++ pr_name x ++ str " := " ++ prl v ++ str ")" + | Bdef (x, Some t, v) -> + str "(" ++ pr_name x ++ str " : " ++ prl t ++ + str " := " ++ prl v ++ str ")" + | Bstruct x -> + str "{struct " ++ pr_name x ++ str "}" + | Bcast t -> + str ": " ++ prl t + +type 'term ssrbindval = 'term ssrbind list * 'term + +type ssrbindfmt = + | BFvar + | BFdecl of int (* #xs *) + | BFcast (* final cast *) + | BFdef of bool (* has cast? *) + | BFrec of bool * bool (* has struct? * has cast? *) + +let rec mkBstruct i = function + | Bvar x :: b -> + if i = 0 then [Bstruct x] else mkBstruct (i - 1) b + | Bdecl (xs, _) :: b -> + let i' = i - List.length xs in + if i' < 0 then [Bstruct (List.nth xs i)] else mkBstruct i' b + | _ :: b -> mkBstruct i b + | [] -> [] + +let rec format_local_binders h0 bl0 = match h0, bl0 with + | BFvar :: h, LocalRawAssum ([_, x], _, _) :: bl -> + Bvar x :: format_local_binders h bl + | BFdecl _ :: h, LocalRawAssum (lxs, _, t) :: bl -> + Bdecl (List.map snd lxs, t) :: format_local_binders h bl + | BFdef false :: h, LocalRawDef ((_, x), v) :: bl -> + Bdef (x, None, v) :: format_local_binders h bl + | BFdef true :: h, + LocalRawDef ((_, x), CCast (_, v, CastConv t)) :: bl -> + Bdef (x, Some t, v) :: format_local_binders h bl + | _ -> [] + +let rec format_constr_expr h0 c0 = match h0, c0 with + | BFvar :: h, CLambdaN (_, [[_, x], _, _], c) -> + let bs, c' = format_constr_expr h c in + Bvar x :: bs, c' + | BFdecl _:: h, CLambdaN (_, [lxs, _, t], c) -> + let bs, c' = format_constr_expr h c in + Bdecl (List.map snd lxs, t) :: bs, c' + | BFdef false :: h, CLetIn(_, (_, x), v, c) -> + let bs, c' = format_constr_expr h c in + Bdef (x, None, v) :: bs, c' + | BFdef true :: h, CLetIn(_, (_, x), CCast (_, v, CastConv t), c) -> + let bs, c' = format_constr_expr h c in + Bdef (x, Some t, v) :: bs, c' + | [BFcast], CCast (_, c, CastConv t) -> + [Bcast t], c + | BFrec (has_str, has_cast) :: h, + CFix (_, _, [_, (Some locn, CStructRec), bl, t, c]) -> + let bs = format_local_binders h bl in + let bstr = if has_str then [Bstruct (Name (snd locn))] else [] in + bs @ bstr @ (if has_cast then [Bcast t] else []), c + | BFrec (_, has_cast) :: h, CCoFix (_, _, [_, bl, t, c]) -> + format_local_binders h bl @ (if has_cast then [Bcast t] else []), c + | _, c -> + [], c + +let rec format_glob_decl h0 d0 = match h0, d0 with + | BFvar :: h, (x, _, None, _) :: d -> + Bvar x :: format_glob_decl h d + | BFdecl 1 :: h, (x, _, None, t) :: d -> + Bdecl ([x], t) :: format_glob_decl h d + | BFdecl n :: h, (x, _, None, t) :: d when n > 1 -> + begin match format_glob_decl (BFdecl (n - 1) :: h) d with + | Bdecl (xs, _) :: bs -> Bdecl (x :: xs, t) :: bs + | bs -> Bdecl ([x], t) :: bs + end + | BFdef false :: h, (x, _, Some v, _) :: d -> + Bdef (x, None, v) :: format_glob_decl h d + | BFdef true:: h, (x, _, Some (GCast (_, v, CastConv t)), _) :: d -> + Bdef (x, Some t, v) :: format_glob_decl h d + | _, (x, _, None, t) :: d -> + Bdecl ([x], t) :: format_glob_decl [] d + | _, (x, _, Some v, _) :: d -> + Bdef (x, None, v) :: format_glob_decl [] d + | _, [] -> [] + +let rec format_glob_constr h0 c0 = match h0, c0 with + | BFvar :: h, GLambda (_, x, _, _, c) -> + let bs, c' = format_glob_constr h c in + Bvar x :: bs, c' + | BFdecl 1 :: h, GLambda (_, x, _, t, c) -> + let bs, c' = format_glob_constr h c in + Bdecl ([x], t) :: bs, c' + | BFdecl n :: h, GLambda (_, x, _, t, c) when n > 1 -> + begin match format_glob_constr (BFdecl (n - 1) :: h) c with + | Bdecl (xs, _) :: bs, c' -> Bdecl (x :: xs, t) :: bs, c' + | _ -> [Bdecl ([x], t)], c + end + | BFdef false :: h, GLetIn(_, x, v, c) -> + let bs, c' = format_glob_constr h c in + Bdef (x, None, v) :: bs, c' + | BFdef true :: h, GLetIn(_, x, GCast (_, v, CastConv t), c) -> + let bs, c' = format_glob_constr h c in + Bdef (x, Some t, v) :: bs, c' + | [BFcast], GCast (_, c, CastConv t) -> + [Bcast t], c + | BFrec (has_str, has_cast) :: h, GRec (_, f, _, bl, t, c) + when Array.length c = 1 -> + let bs = format_glob_decl h bl.(0) in + let bstr = match has_str, f with + | true, GFix ([|Some i, GStructRec|], _) -> mkBstruct i bs + | _ -> [] in + bs @ bstr @ (if has_cast then [Bcast t.(0)] else []), c.(0) + | _, c -> + [], c + +(** Forward chaining argument *) + +(* There are three kinds of forward definitions: *) +(* - Hint: type only, cast to Type, may have proof hint. *) +(* - Have: type option + value, no space before type *) +(* - Pose: binders + value, space before binders. *) + +type ssrfwdkind = FwdHint of string * bool | FwdHave | FwdPose + +type ssrfwdfmt = ssrfwdkind * ssrbindfmt list + +let pr_fwdkind = function + | FwdHint (s,_) -> str (s ^ " ") | _ -> str " :=" ++ spc () +let pr_fwdfmt (fk, _ : ssrfwdfmt) = pr_fwdkind fk + +let wit_ssrfwdfmt = add_genarg "ssrfwdfmt" pr_fwdfmt + +(* type ssrfwd = ssrfwdfmt * ssrterm *) + +let mkFwdVal fk c = ((fk, []), mk_term ' ' c) +let mkssrFwdVal fk c = ((fk, []), (c,None)) + +let mkFwdCast fk loc t c = ((fk, [BFcast]), mk_term ' ' (CCast (loc, c, dC t))) +let mkssrFwdCast fk loc t c = ((fk, [BFcast]), (c, Some t)) + +let mkFwdHint s t = + let loc = constr_loc t in + mkFwdCast (FwdHint (s,false)) loc t (mkCHole loc) +let mkFwdHintNoTC s t = + let loc = constr_loc t in + mkFwdCast (FwdHint (s,true)) loc t (mkCHole loc) + +let pr_gen_fwd prval prc prlc fk (bs, c) = + let prc s = str s ++ spc () ++ prval prc prlc c in + match fk, bs with + | FwdHint (s,_), [Bcast t] -> str s ++ spc () ++ prlc t + | FwdHint (s,_), _ -> prc (s ^ "(* typeof *)") + | FwdHave, [Bcast t] -> str ":" ++ spc () ++ prlc t ++ prc " :=" + | _, [] -> prc " :=" + | _, _ -> spc () ++ pr_list spc (pr_binder prlc) bs ++ prc " :=" + +let pr_fwd_guarded prval prval' = function +| (fk, h), (_, (_, Some c)) -> + pr_gen_fwd prval pr_constr_expr prl_constr_expr fk (format_constr_expr h c) +| (fk, h), (_, (c, None)) -> + pr_gen_fwd prval' pr_glob_constr prl_glob_constr fk (format_glob_constr h c) + +let pr_unguarded prc prlc = prlc + +let pr_fwd = pr_fwd_guarded pr_unguarded pr_unguarded +let pr_ssrfwd _ _ _ = pr_fwd + +ARGUMENT EXTEND ssrfwd TYPED AS (ssrfwdfmt * ssrterm) PRINTED BY pr_ssrfwd + | [ ":=" lconstr(c) ] -> [ mkFwdVal FwdPose c ] + | [ ":" lconstr (t) ":=" lconstr(c) ] -> [ mkFwdCast FwdPose loc t c ] +END + +(** Independent parsing for binders *) + +(* The pose, pose fix, and pose cofix tactics use these internally to *) +(* parse argument fragments. *) + +let pr_ssrbvar prc _ _ v = prc v + +ARGUMENT EXTEND ssrbvar TYPED AS constr PRINTED BY pr_ssrbvar +| [ ident(id) ] -> [ mkCVar loc id ] +| [ "_" ] -> [ mkCHole loc ] +END + +let bvar_lname = function + | CRef (Ident (loc, id), _) -> loc, Name id + | c -> constr_loc c, Anonymous + +let pr_ssrbinder prc _ _ (_, c) = prc c + +ARGUMENT EXTEND ssrbinder TYPED AS ssrfwdfmt * constr PRINTED BY pr_ssrbinder + | [ ssrbvar(bv) ] -> + [ let xloc, _ as x = bvar_lname bv in + (FwdPose, [BFvar]), + CLambdaN (loc,[[x],Default Explicit,mkCHole xloc],mkCHole loc) ] + | [ "(" ssrbvar(bv) ")" ] -> + [ let xloc, _ as x = bvar_lname bv in + (FwdPose, [BFvar]), + CLambdaN (loc,[[x],Default Explicit,mkCHole xloc],mkCHole loc) ] + | [ "(" ssrbvar(bv) ":" lconstr(t) ")" ] -> + [ let x = bvar_lname bv in + (FwdPose, [BFdecl 1]), + CLambdaN (loc, [[x], Default Explicit, t], mkCHole loc) ] + | [ "(" ssrbvar(bv) ne_ssrbvar_list(bvs) ":" lconstr(t) ")" ] -> + [ let xs = List.map bvar_lname (bv :: bvs) in + let n = List.length xs in + (FwdPose, [BFdecl n]), + CLambdaN (loc, [xs, Default Explicit, t], mkCHole loc) ] + | [ "(" ssrbvar(id) ":" lconstr(t) ":=" lconstr(v) ")" ] -> + [ let loc' = Loc.join_loc (constr_loc t) (constr_loc v) in + let v' = CCast (loc', v, dC t) in + (FwdPose,[BFdef true]), CLetIn (loc,bvar_lname id, v',mkCHole loc) ] + | [ "(" ssrbvar(id) ":=" lconstr(v) ")" ] -> + [ (FwdPose,[BFdef false]), CLetIn (loc,bvar_lname id, v,mkCHole loc) ] +END + +GEXTEND Gram + GLOBAL: ssrbinder; + ssrbinder: [ + [ ["of" | "&"]; c = operconstr LEVEL "99" -> + let loc = !@loc in + (FwdPose, [BFvar]), + CLambdaN (loc,[[loc,Anonymous],Default Explicit,c],mkCHole loc) ] + ]; +END + +let rec binders_fmts = function + | ((_, h), _) :: bs -> h @ binders_fmts bs + | _ -> [] + +let push_binders c2 bs = + let loc2 = constr_loc c2 in let mkloc loc1 = Loc.join_loc loc1 loc2 in + let rec loop ty c = function + | (_, CLambdaN (loc1, b, _)) :: bs when ty -> + CProdN (mkloc loc1, b, loop ty c bs) + | (_, CLambdaN (loc1, b, _)) :: bs -> + CLambdaN (mkloc loc1, b, loop ty c bs) + | (_, CLetIn (loc1, x, v, _)) :: bs -> + CLetIn (mkloc loc1, x, v, loop ty c bs) + | [] -> c + | _ -> anomaly "binder not a lambda nor a let in" in + match c2 with + | CCast (x, ct, CastConv cty) -> + (CCast (x, loop false ct bs, CastConv (loop true cty bs))) + | ct -> loop false ct bs + +let rec fix_binders = function + | (_, CLambdaN (_, [xs, _, t], _)) :: bs -> + LocalRawAssum (xs, Default Explicit, t) :: fix_binders bs + | (_, CLetIn (_, x, v, _)) :: bs -> + LocalRawDef (x, v) :: fix_binders bs + | _ -> [] + +let pr_ssrstruct _ _ _ = function + | Some id -> str "{struct " ++ pr_id id ++ str "}" + | None -> mt () + +ARGUMENT EXTEND ssrstruct TYPED AS ident option PRINTED BY pr_ssrstruct +| [ "{" "struct" ident(id) "}" ] -> [ Some id ] +| [ ] -> [ None ] +END + +(** The "pose" tactic *) + +(* The plain pose form. *) + +let bind_fwd bs = function + | (fk, h), (ck, (rc, Some c)) -> + (fk,binders_fmts bs @ h), (ck,(rc,Some (push_binders c bs))) + | fwd -> fwd + +ARGUMENT EXTEND ssrposefwd TYPED AS ssrfwd PRINTED BY pr_ssrfwd + | [ ssrbinder_list(bs) ssrfwd(fwd) ] -> [ bind_fwd bs fwd ] +END + +(* The pose fix form. *) + +let pr_ssrfixfwd _ _ _ (id, fwd) = str " fix " ++ pr_id id ++ pr_fwd fwd + +let bvar_locid = function + | CRef (Ident (loc, id), _) -> loc, id + | _ -> Errors.error "Missing identifier after \"(co)fix\"" + + +ARGUMENT EXTEND ssrfixfwd TYPED AS ident * ssrfwd PRINTED BY pr_ssrfixfwd + | [ "fix" ssrbvar(bv) ssrbinder_list(bs) ssrstruct(sid) ssrfwd(fwd) ] -> + [ let (_, id) as lid = bvar_locid bv in + let (fk, h), (ck, (rc, oc)) = fwd in + let c = Option.get oc in + let has_cast, t', c' = match format_constr_expr h c with + | [Bcast t'], c' -> true, t', c' + | _ -> false, mkCHole (constr_loc c), c in + let lb = fix_binders bs in + let has_struct, i = + let rec loop = function + (l', Name id') :: _ when Option.equal Id.equal sid (Some id') -> true, (l', id') + | [l', Name id'] when sid = None -> false, (l', id') + | _ :: bn -> loop bn + | [] -> Errors.error "Bad structural argument" in + loop (names_of_local_assums lb) in + let h' = BFrec (has_struct, has_cast) :: binders_fmts bs in + let fix = CFix (loc, lid, [lid, (Some i, CStructRec), lb, t', c']) in + id, ((fk, h'), (ck, (rc, Some fix))) ] +END + + +(* The pose cofix form. *) + +let pr_ssrcofixfwd _ _ _ (id, fwd) = str " cofix " ++ pr_id id ++ pr_fwd fwd + +ARGUMENT EXTEND ssrcofixfwd TYPED AS ssrfixfwd PRINTED BY pr_ssrcofixfwd + | [ "cofix" ssrbvar(bv) ssrbinder_list(bs) ssrfwd(fwd) ] -> + [ let _, id as lid = bvar_locid bv in + let (fk, h), (ck, (rc, oc)) = fwd in + let c = Option.get oc in + let has_cast, t', c' = match format_constr_expr h c with + | [Bcast t'], c' -> true, t', c' + | _ -> false, mkCHole (constr_loc c), c in + let h' = BFrec (false, has_cast) :: binders_fmts bs in + let cofix = CCoFix (loc, lid, [lid, fix_binders bs, t', c']) in + id, ((fk, h'), (ck, (rc, Some cofix))) + ] +END + +let ssrposetac ist (id, (_, t)) gl = + let sigma, t, ucst = pf_abs_ssrterm ist gl t in + posetac id t (pf_merge_uc ucst gl) + + +let prof_ssrposetac = mk_profiler "ssrposetac";; +let ssrposetac arg gl = prof_ssrposetac.profile (ssrposetac arg) gl;; + +TACTIC EXTEND ssrpose +| [ "pose" ssrfixfwd(ffwd) ] -> [ Proofview.V82.tactic (ssrposetac ist ffwd) ] +| [ "pose" ssrcofixfwd(ffwd) ] -> [ Proofview.V82.tactic (ssrposetac ist ffwd) ] +| [ "pose" ssrfwdid(id) ssrposefwd(fwd) ] -> [ Proofview.V82.tactic (ssrposetac ist (id, fwd)) ] +END + +(** The "set" tactic *) + +(* type ssrsetfwd = ssrfwd * ssrdocc *) + +let guard_setrhs s i = s.[i] = '{' + +let pr_setrhs occ prc prlc c = + if occ = nodocc then pr_guarded guard_setrhs prlc c else pr_docc occ ++ prc c + +let pr_fwd_guarded prval prval' = function +| (fk, h), (_, (_, Some c)) -> + pr_gen_fwd prval pr_constr_expr prl_constr_expr fk (format_constr_expr h c) +| (fk, h), (_, (c, None)) -> + pr_gen_fwd prval' pr_glob_constr prl_glob_constr fk (format_glob_constr h c) + +(* This does not print the type, it should be fixed... *) +let pr_ssrsetfwd _ _ _ (((fk,_),(t,_)), docc) = + pr_gen_fwd (fun _ _ -> pr_cpattern) + (fun _ -> mt()) (fun _ -> mt()) fk ([Bcast ()],t) + +ARGUMENT EXTEND ssrsetfwd +TYPED AS (ssrfwdfmt * (lcpattern * ssrterm option)) * ssrdocc +PRINTED BY pr_ssrsetfwd +| [ ":" lconstr(t) ":=" "{" ssrocc(occ) "}" cpattern(c) ] -> + [ mkssrFwdCast FwdPose loc (mk_lterm t) c, mkocc occ ] +| [ ":" lconstr(t) ":=" lcpattern(c) ] -> + [ mkssrFwdCast FwdPose loc (mk_lterm t) c, nodocc ] +| [ ":=" "{" ssrocc(occ) "}" cpattern(c) ] -> + [ mkssrFwdVal FwdPose c, mkocc occ ] +| [ ":=" lcpattern(c) ] -> [ mkssrFwdVal FwdPose c, nodocc ] +END + +let ssrsettac ist id ((_, (pat, pty)), (_, occ)) gl = + let pat = interp_cpattern ist gl pat (Option.map snd pty) in + let cl, sigma, env = pf_concl gl, project gl, pf_env gl in + let (c, ucst), cl = + try fill_occ_pattern ~raise_NoMatch:true env sigma cl pat occ 1 + with NoMatch -> redex_of_pattern ~resolve_typeclasses:true env pat, cl in + if occur_existential c then errorstrm(str"The pattern"++spc()++ + pr_constr_pat c++spc()++str"did not match and has holes."++spc()++ + str"Did you mean pose?") else + let c, cty = match kind_of_term c with + | Cast(t, DEFAULTcast, ty) -> t, ty + | _ -> c, pf_type_of gl c in + let cl' = mkLetIn (Name id, c, cty, cl) in + let gl = pf_merge_uc ucst gl in + tclTHEN (Proofview.V82.of_tactic (convert_concl cl')) (introid id) gl + +TACTIC EXTEND ssrset +| [ "set" ssrfwdid(id) ssrsetfwd(fwd) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrsettac ist id fwd) clauses) ] +END + +(** The "have" tactic *) + +(* type ssrhavefwd = ssrfwd * ssrhint *) + +let pr_ssrhavefwd _ _ prt (fwd, hint) = pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrhavefwd TYPED AS ssrfwd * ssrhint PRINTED BY pr_ssrhavefwd +| [ ":" lconstr(t) ssrhint(hint) ] -> [ mkFwdHint ":" t, hint ] +| [ ":" lconstr(t) ":=" lconstr(c) ] -> [ mkFwdCast FwdHave loc t c, nohint ] +| [ ":" lconstr(t) ":=" ] -> [ mkFwdHintNoTC ":" t, nohint ] +| [ ":=" lconstr(c) ] -> [ mkFwdVal FwdHave c, nohint ] +END + +let intro_id_to_binder = List.map (function + | IpatId id -> + let xloc, _ as x = bvar_lname (mkCVar dummy_loc id) in + (FwdPose, [BFvar]), + CLambdaN (dummy_loc, [[x], Default Explicit, mkCHole xloc], + mkCHole dummy_loc) + | _ -> anomaly "non-id accepted as binder") + +let binder_to_intro_id = List.map (function + | (FwdPose, [BFvar]), CLambdaN (_,[ids,_,_],_) + | (FwdPose, [BFdecl _]), CLambdaN (_,[ids,_,_],_) -> + List.map (function (_, Name id) -> IpatId id | _ -> IpatAnon) ids + | (FwdPose, [BFdef _]), CLetIn (_,(_,Name id),_,_) -> [IpatId id] + | (FwdPose, [BFdef _]), CLetIn (_,(_,Anonymous),_,_) -> [IpatAnon] + | _ -> anomaly "ssrbinder is not a binder") + +let pr_ssrhavefwdwbinders _ _ prt (tr,((hpats, (fwd, hint)))) = + pr_hpats hpats ++ pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrhavefwdwbinders + TYPED AS bool * (ssrhpats * (ssrfwd * ssrhint)) + PRINTED BY pr_ssrhavefwdwbinders +| [ ssrhpats_wtransp(trpats) ssrbinder_list(bs) ssrhavefwd(fwd) ] -> + [ let tr, pats = trpats in + let ((clr, pats), binders), simpl = pats in + let allbs = intro_id_to_binder binders @ bs in + let allbinders = binders @ List.flatten (binder_to_intro_id bs) in + let hint = bind_fwd allbs (fst fwd), snd fwd in + tr, ((((clr, pats), allbinders), simpl), hint) ] +END + +(* Tactic. *) + +let is_Evar_or_CastedMeta x = + isEvar_or_Meta x || + (isCast x && isEvar_or_Meta (pi1 (destCast x))) + +let occur_existential_or_casted_meta c = + let rec occrec c = match kind_of_term c with + | Evar _ -> raise Not_found + | Cast (m,_,_) when isMeta m -> raise Not_found + | _ -> iter_constr occrec c + in try occrec c; false with Not_found -> true + +let examine_abstract id gl = + let tid = pf_type_of gl id in + let abstract, gl = pf_mkSsrConst "abstract" gl in + if not (isApp tid) || not (Term.eq_constr (fst(destApp tid)) abstract) then + errorstrm(strbrk"not an abstract constant: "++pr_constr id); + let _, args_id = destApp tid in + if Array.length args_id <> 3 then + errorstrm(strbrk"not a proper abstract constant: "++pr_constr id); + if not (is_Evar_or_CastedMeta args_id.(2)) then + errorstrm(strbrk"abstract constant "++pr_constr id++str" already used"); + tid, args_id + +let pf_find_abstract_proof check_lock gl abstract_n = + let fire gl t = Reductionops.nf_evar (project gl) t in + let abstract, gl = pf_mkSsrConst "abstract" gl in + let l = Evd.fold_undefined (fun e ei l -> + match kind_of_term ei.Evd.evar_concl with + | App(hd, [|ty; n; lock|]) + when (not check_lock || + (occur_existential_or_casted_meta (fire gl ty) && + is_Evar_or_CastedMeta (fire gl lock))) && + Term.eq_constr hd abstract && Term.eq_constr n abstract_n -> e::l + | _ -> l) (project gl) [] in + match l with + | [e] -> e + | _ -> errorstrm(strbrk"abstract constant "++pr_constr abstract_n++ + strbrk" not found in the evar map exactly once. "++ + strbrk"Did you tamper with it?") + +let unfold cl = + let module R = Reductionops in let module F = Closure.RedFlags in + reduct_in_concl (R.clos_norm_flags (F.mkflags + (List.map (fun c -> F.fCONST (fst (destConst c))) cl @ [F.fBETA; F.fIOTA]))) + +let havegentac ist t gl = + let sigma, c, ucst = pf_abs_ssrterm ist gl t in + let gl = pf_merge_uc ucst gl in + apply_type (mkArrow (pf_type_of gl c) (pf_concl gl)) [c] gl + +let havetac ist + (transp,((((clr, pats), binders), simpl), (((fk, _), t), hint))) + suff namefst gl += + let concl = pf_concl gl in + let skols, pats = + List.partition (function IpatNewHidden _ -> true | _ -> false) pats in + let itac_mkabs = introstac ~ist skols in + let itac_c = introstac ~ist (IpatSimpl(clr,Nop) :: pats) in + let itac, id, clr = introstac ~ist pats, tclIDTAC, cleartac clr in + let binderstac n = + let rec aux = function 0 -> [] | n -> IpatAnon :: aux (n-1) in + tclTHEN (if binders <> [] then introstac ~ist (aux n) else tclIDTAC) + (introstac ~ist binders) in + let simpltac = introstac ~ist simpl in + let fixtc = + not !ssrhaveNOtcresolution && + match fk with FwdHint(_,true) -> false | _ -> true in + let hint = hinttac ist true hint in + let cuttac t gl = + if transp then + let have_let, gl = pf_mkSsrConst "ssr_have_let" gl in + let step = mkApp (have_let, [|concl;t|]) in + let gl, _ = pf_e_type_of gl step in + applyn ~with_evars:true ~with_shelve:false 2 step gl + else basecuttac "ssr_have" t gl in + (* Introduce now abstract constants, so that everything sees them *) + let abstract_key, gl = pf_mkSsrConst "abstract_key" gl in + let unlock_abs (idty,args_id) gl = + let gl, _ = pf_e_type_of gl idty in + pf_unify_HO gl args_id.(2) abstract_key in + tclTHENFIRST itac_mkabs (fun gl -> + let mkt t = mk_term ' ' t in + let mkl t = (' ', (t, None)) in + let interp gl rtc t = pf_abs_ssrterm ~resolve_typeclasses:rtc ist gl t in + let interp_ty gl rtc t = + let a,b,_,u = pf_interp_ty ~resolve_typeclasses:rtc ist gl t in a,b,u in + let ct, cty, hole, loc = match t with + | _, (_, Some (CCast (loc, ct, CastConv cty))) -> + mkt ct, mkt cty, mkt (mkCHole dummy_loc), loc + | _, (_, Some ct) -> + mkt ct, mkt (mkCHole dummy_loc), mkt (mkCHole dummy_loc), dummy_loc + | _, (GCast (loc, ct, CastConv cty), None) -> + mkl ct, mkl cty, mkl mkRHole, loc + | _, (t, None) -> mkl t, mkl mkRHole, mkl mkRHole, dummy_loc in + let gl, cut, sol, itac1, itac2 = + match fk, namefst, suff with + | FwdHave, true, true -> + errorstrm (str"Suff have does not accept a proof term") + | FwdHave, false, true -> + let cty = combineCG cty hole (mkCArrow loc) mkRArrow in + let _, t, uc = interp gl false (combineCG ct cty (mkCCast loc) mkRCast) in + let gl = pf_merge_uc uc gl in + let ty = pf_type_of gl t in + let ctx, _ = decompose_prod_n 1 ty in + let assert_is_conv gl = + try Proofview.V82.of_tactic (convert_concl (compose_prod ctx concl)) gl + with _ -> errorstrm (str "Given proof term is not of type " ++ + pr_constr (mkArrow (mkVar (id_of_string "_")) concl)) in + gl, ty, tclTHEN assert_is_conv (Proofview.V82.of_tactic (apply t)), id, itac_c + | FwdHave, false, false -> + let skols = List.flatten (List.map (function + | IpatNewHidden ids -> ids + | _ -> assert false) skols) in + let skols_args = + List.map (fun id -> examine_abstract (mkVar id) gl) skols in + let gl = List.fold_right unlock_abs skols_args gl in + let sigma, t, uc = + interp gl false (combineCG ct cty (mkCCast loc) mkRCast) in + let gl = re_sig (sig_it gl) (Evd.merge_universe_context sigma uc) in + let gs = + List.map (fun (_,a) -> + pf_find_abstract_proof false gl a.(1)) skols_args in + let tacopen_skols gl = + let stuff, g = Refiner.unpackage gl in + Refiner.repackage stuff (gs @ [g]) in + let gl, ty = pf_e_type_of gl t in + gl, ty, Proofview.V82.of_tactic (apply t), id, + tclTHEN (tclTHEN itac_c simpltac) + (tclTHEN tacopen_skols (fun gl -> + let abstract, gl = pf_mkSsrConst "abstract" gl in + unfold [abstract; abstract_key] gl)) + | _,true,true -> + let _, ty, uc = interp_ty gl fixtc cty in let gl = pf_merge_uc uc gl in + gl, mkArrow ty concl, hint, itac, clr + | _,false,true -> + let _, ty, uc = interp_ty gl fixtc cty in let gl = pf_merge_uc uc gl in + gl, mkArrow ty concl, hint, id, itac_c + | _, false, false -> + let n, cty, uc = interp_ty gl fixtc cty in let gl = pf_merge_uc uc gl in + gl, cty, tclTHEN (binderstac n) hint, id, tclTHEN itac_c simpltac + | _, true, false -> assert false in + tclTHENS (cuttac cut) [ tclTHEN sol itac1; itac2 ] gl) + gl +;; + +(* to extend the abstract value one needs: + Utility lemma to partially instantiate an abstract constant type. + Lemma use_abstract T n l (x : abstract T n l) : T. + Proof. by case: l x. Qed. +*) +let ssrabstract ist gens (*last*) gl = + let main _ (_,cid) ist gl = +(* + let proj1, proj2, prod = + let pdata = build_prod () in + pdata.Coqlib.proj1, pdata.Coqlib.proj2, pdata.Coqlib.typ in +*) + let concl, env = pf_concl gl, pf_env gl in + let fire gl t = Reductionops.nf_evar (project gl) t in + let abstract, gl = pf_mkSsrConst "abstract" gl in + let abstract_key, gl = pf_mkSsrConst "abstract_key" gl in + let id = mkVar (Option.get (id_of_cpattern cid)) in + let idty, args_id = examine_abstract id gl in + let abstract_n = args_id.(1) in + let abstract_proof = pf_find_abstract_proof true gl abstract_n in + let gl, proof = + let pf_unify_HO gl a b = + try pf_unify_HO gl a b + with _ -> errorstrm(strbrk"The abstract variable "++pr_constr id++ + strbrk" cannot abstract this goal. Did you generalize it?") in + let rec find_hole p t = + match kind_of_term t with + | Evar _ (*when last*) -> pf_unify_HO gl concl t, p + | Meta _ (*when last*) -> pf_unify_HO gl concl t, p + | Cast(m,_,_) when isEvar_or_Meta m (*when last*) -> pf_unify_HO gl concl t, p +(* + | Evar _ -> + let sigma, it = project gl, sig_it gl in + let sigma, ty = Evarutil.new_type_evar sigma env in + let gl = re_sig it sigma in + let p = mkApp (proj2,[|ty;concl;p|]) in + let concl = mkApp(prod,[|ty; concl|]) in + pf_unify_HO gl concl t, p + | App(hd, [|left; right|]) when Term.eq_constr hd prod -> + find_hole (mkApp (proj1,[|left;right;p|])) left +*) + | _ -> errorstrm(strbrk"abstract constant "++pr_constr abstract_n++ + strbrk" has an unexpected shape. Did you tamper with it?") + in + find_hole + ((*if last then*) id + (*else mkApp(mkSsrConst "use_abstract",Array.append args_id [|id|])*)) + (fire gl args_id.(0)) in + let gl = (*if last then*) pf_unify_HO gl abstract_key args_id.(2) (*else gl*) in + let gl, _ = pf_e_type_of gl idty in + let proof = fire gl proof in +(* if last then *) + let tacopen gl = + let stuff, g = Refiner.unpackage gl in + Refiner.repackage stuff [ g; abstract_proof ] in + tclTHENS tacopen [tclSOLVE [Proofview.V82.of_tactic (apply proof)];unfold[abstract;abstract_key]] gl +(* else apply proof gl *) + in + let introback ist (gens, _) = + introstac ~ist + (List.map (fun (_,cp) -> match id_of_cpattern cp with + | None -> IpatAnon + | Some id -> IpatId id) + (List.tl (List.hd gens))) in + tclTHEN (with_dgens gens main ist) (introback ist gens) gl + +(* The standard TACTIC EXTEND does not work for abstract *) +GEXTEND Gram + GLOBAL: tactic_expr; + tactic_expr: LEVEL "3" + [ RIGHTA [ IDENT "abstract"; gens = ssrdgens -> + ssrtac_expr !@loc "abstract" + [Genarg.in_gen (Genarg.rawwit wit_ssrdgens) gens] ]]; +END +TACTIC EXTEND ssrabstract +| [ "abstract" ssrdgens(gens) ] -> [ + if List.length (fst gens) <> 1 then + errorstrm (str"dependents switches '/' not allowed here"); + Proofview.V82.tactic (ssrabstract ist gens) ] +END + +let prof_havetac = mk_profiler "havetac";; +let havetac arg a b gl = prof_havetac.profile (havetac arg a b) gl;; + +TACTIC EXTEND ssrhave +| [ "have" ssrhavefwdwbinders(fwd) ] -> + [ Proofview.V82.tactic (havetac ist fwd false false) ] +END + +TACTIC EXTEND ssrhavesuff +| [ "have" "suff" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ Proofview.V82.tactic (havetac ist (false,(pats,fwd)) true false) ] +END + +TACTIC EXTEND ssrhavesuffices +| [ "have" "suffices" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ Proofview.V82.tactic (havetac ist (false,(pats,fwd)) true false) ] +END + +TACTIC EXTEND ssrsuffhave +| [ "suff" "have" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ Proofview.V82.tactic (havetac ist (false,(pats,fwd)) true true) ] +END + +TACTIC EXTEND ssrsufficeshave +| [ "suffices" "have" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ Proofview.V82.tactic (havetac ist (false,(pats,fwd)) true true) ] +END + +(** The "suffice" tactic *) + +let pr_ssrsufffwdwbinders _ _ prt (hpats, (fwd, hint)) = + pr_hpats hpats ++ pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrsufffwd + TYPED AS ssrhpats * (ssrfwd * ssrhint) PRINTED BY pr_ssrsufffwdwbinders +| [ ssrhpats(pats) ssrbinder_list(bs) ":" lconstr(t) ssrhint(hint) ] -> + [ let ((clr, pats), binders), simpl = pats in + let allbs = intro_id_to_binder binders @ bs in + let allbinders = binders @ List.flatten (binder_to_intro_id bs) in + let fwd = mkFwdHint ":" t in + (((clr, pats), allbinders), simpl), (bind_fwd allbs fwd, hint) ] +END + +let sufftac ist ((((clr, pats),binders),simpl), ((_, c), hint)) = + let htac = tclTHEN (introstac ~ist pats) (hinttac ist true hint) in + let c = match c with + | (a, (b, Some (CCast (_, _, CastConv cty)))) -> a, (b, Some cty) + | (a, (GCast (_, _, CastConv cty), None)) -> a, (cty, None) + | _ -> anomaly "suff: ssr cast hole deleted by typecheck" in + let ctac gl = + let _,ty,_,uc = pf_interp_ty ist gl c in let gl = pf_merge_uc uc gl in + basecuttac "ssr_suff" ty gl in + tclTHENS ctac [htac; tclTHEN (cleartac clr) (introstac ~ist (binders@simpl))] + +TACTIC EXTEND ssrsuff +| [ "suff" ssrsufffwd(fwd) ] -> [ Proofview.V82.tactic (sufftac ist fwd) ] +END + +TACTIC EXTEND ssrsuffices +| [ "suffices" ssrsufffwd(fwd) ] -> [ Proofview.V82.tactic (sufftac ist fwd) ] +END + +(** The "wlog" (Without Loss Of Generality) tactic *) + +(* type ssrwlogfwd = ssrwgen list * ssrfwd *) + +let pr_ssrwlogfwd _ _ _ (gens, t) = + str ":" ++ pr_list mt pr_wgen gens ++ spc() ++ pr_fwd t + +ARGUMENT EXTEND ssrwlogfwd TYPED AS ssrwgen list * ssrfwd + PRINTED BY pr_ssrwlogfwd +| [ ":" ssrwgen_list(gens) "/" lconstr(t) ] -> [ gens, mkFwdHint "/" t] +END + +let destProd_or_LetIn c = + match kind_of_term c with + | Prod (n,ty,c) -> (n,None,ty), c + | LetIn (n,bo,ty,c) -> (n,Some bo,ty), c + | _ -> raise DestKO + +let wlogtac ist (((clr0, pats),_),_) (gens, ((_, ct))) hint suff ghave gl = + let mkabs gen = abs_wgen false ist (fun x -> x) gen in + let mkclr gen clrs = clr_of_wgen gen clrs in + let mkpats = function + | _, Some ((x, _), _) -> fun pats -> IpatId (hoi_id x) :: pats + | _ -> fun x -> x in + let ct = match ct with + | (a, (b, Some (CCast (_, _, CastConv cty)))) -> a, (b, Some cty) + | (a, (GCast (_, _, CastConv cty), None)) -> a, (cty, None) + | _ -> anomaly "wlog: ssr cast hole deleted by typecheck" in + let cut_implies_goal = not (suff || ghave <> `NoGen) in + let c, args, ct, gl = + let gens = List.filter (function _, Some _ -> true | _ -> false) gens in + let concl = pf_concl gl in + let c = mkProp in + let c = if cut_implies_goal then mkArrow c concl else c in + let gl, args, c = List.fold_right mkabs gens (gl,[],c) in + let env, _ = + List.fold_left (fun (env, c) _ -> + let rd, c = destProd_or_LetIn c in + Environ.push_rel rd env, c) (pf_env gl, c) gens in + let sigma, ev = Evarutil.new_evar env (project gl) Term.mkProp in + let k, _ = Term.destEvar ev in + let fake_gl = {Evd.it = k; Evd.sigma = sigma} in + let _, ct, _, uc = pf_interp_ty ist fake_gl ct in + let rec var2rel c g s = match kind_of_term c, g with + | Prod(Anonymous,_,c), [] -> mkProd(Anonymous, Vars.subst_vars s ct, c) + | Sort _, [] -> Vars.subst_vars s ct + | LetIn(Name id as n,b,ty,c), _::g -> mkLetIn (n,b,ty,var2rel c g (id::s)) + | Prod(Name id as n,ty,c), _::g -> mkProd (n,ty,var2rel c g (id::s)) + | _ -> Errors.anomaly(str"SSR: wlog: var2rel: " ++ pr_constr c) in + let c = var2rel c gens [] in + let rec pired c = function + | [] -> c + | t::ts as args -> match kind_of_term c with + | Prod(_,_,c) -> pired (subst1 t c) ts + | LetIn(id,b,ty,c) -> mkLetIn (id,b,ty,pired c args) + | _ -> Errors.anomaly(str"SSR: wlog: pired: " ++ pr_constr c) in + c, args, pired c args, pf_merge_uc uc gl in + let tacipat pats = introstac ~ist pats in + let tacigens = + tclTHEN + (tclTHENLIST(List.rev(List.fold_right mkclr gens [cleartac clr0]))) + (introstac ~ist (List.fold_right mkpats gens [])) in + let hinttac = hinttac ist true hint in + let cut_kind, fst_goal_tac, snd_goal_tac = + match suff, ghave with + | true, `NoGen -> "ssr_wlog", tclTHEN hinttac (tacipat pats), tacigens + | false, `NoGen -> "ssr_wlog", hinttac, tclTHEN tacigens (tacipat pats) + | true, `Gen _ -> assert false + | false, `Gen id -> + if gens = [] then errorstrm(str"gen have requires some generalizations"); + let clear0 = cleartac clr0 in + let id, name_general_hyp, cleanup, pats = match id, pats with + | None, (IpatId id as ip)::pats -> Some id, tacipat [ip], clear0, pats + | None, _ -> None, tclIDTAC, clear0, pats + | Some (Some id),_ -> Some id, introid id, clear0, pats + | Some _,_ -> + let id = mk_anon_id "tmp" gl in + Some id, introid id, tclTHEN clear0 (clear [id]), pats in + let tac_specialize = match id with + | None -> tclIDTAC + | Some id -> + if pats = [] then tclIDTAC else + let args = Array.of_list args in + pp(lazy(str"specialized="++pr_constr (mkApp (mkVar id,args)))); + pp(lazy(str"specialized_ty="++pr_constr ct)); + tclTHENS (basecuttac "ssr_have" ct) + [Proofview.V82.of_tactic (apply (mkApp (mkVar id,args))); tclIDTAC] in + "ssr_have", + (if hint = nohint then tacigens else hinttac), + tclTHENLIST [name_general_hyp; tac_specialize; tacipat pats; cleanup] + in + tclTHENS (basecuttac cut_kind c) [fst_goal_tac; snd_goal_tac] gl + +TACTIC EXTEND ssrwlog +| [ "wlog" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint false `NoGen) ] +END + +TACTIC EXTEND ssrwlogs +| [ "wlog" "suff" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint true `NoGen) ] +END + +TACTIC EXTEND ssrwlogss +| [ "wlog" "suffices" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ]-> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint true `NoGen) ] +END + +TACTIC EXTEND ssrwithoutloss +| [ "without" "loss" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint false `NoGen) ] +END + +TACTIC EXTEND ssrwithoutlosss +| [ "without" "loss" "suff" + ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint true `NoGen) ] +END + +TACTIC EXTEND ssrwithoutlossss +| [ "without" "loss" "suffices" + ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ]-> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint true `NoGen) ] +END + +(* Generally have *) +let pr_idcomma _ _ _ = function + | None -> mt() + | Some None -> str"_, " + | Some (Some id) -> pr_id id ++ str", " + +ARGUMENT EXTEND ssr_idcomma TYPED AS ident option option PRINTED BY pr_idcomma + | [ ] -> [ None ] +END + +let accept_idcomma strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT _ | Tok.KEYWORD "_" -> accept_before_syms [","] strm + | _ -> raise Stream.Failure + +let test_idcomma = Gram.Entry.of_parser "test_idcomma" accept_idcomma + +GEXTEND Gram + GLOBAL: ssr_idcomma; + ssr_idcomma: [ [ test_idcomma; + ip = [ id = IDENT -> Some (id_of_string id) | "_" -> None ]; "," -> + Some ip + ] ]; +END + +let augment_preclr clr1 (((clr0, x),y),z) = (((clr1 @ clr0, x),y),z) + +TACTIC EXTEND ssrgenhave +| [ "gen" "have" ssrclear(clr) + ssr_idcomma(id) ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ let pats = augment_preclr clr pats in + Proofview.V82.tactic (wlogtac ist pats fwd hint false (`Gen id)) ] +END + +TACTIC EXTEND ssrgenhave2 +| [ "generally" "have" ssrclear(clr) + ssr_idcomma(id) ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ let pats = augment_preclr clr pats in + Proofview.V82.tactic (wlogtac ist pats fwd hint false (`Gen id)) ] +END + +(** Canonical Structure alias *) + +let def_body : Vernacexpr.definition_expr Gram.Entry.e = Obj.magic + (Grammar.Entry.find (Obj.magic gallina_ext) "vernac:def_body") in + +GEXTEND Gram + GLOBAL: gallina_ext; + + gallina_ext: + (* Canonical structure *) + [[ IDENT "Canonical"; qid = Constr.global -> + Vernacexpr.VernacCanonical (AN qid) + | IDENT "Canonical"; ntn = Prim.by_notation -> + Vernacexpr.VernacCanonical (ByNotation ntn) + | IDENT "Canonical"; qid = Constr.global; + d = def_body -> + let s = coerce_reference_to_id qid in + Vernacexpr.VernacDefinition + ((Some Decl_kinds.Global,Decl_kinds.CanonicalStructure), + (dummy_loc,s),(d )) + ]]; +END + +(** 9. Keyword compatibility fixes. *) + +(* Coq v8.1 notation uses "by" and "of" quasi-keywords, i.e., reserved *) +(* identifiers used as keywords. This is incompatible with ssreflect.v *) +(* which makes "by" and "of" true keywords, because of technicalities *) +(* in the internal lexer-parser API of Coq. We patch this here by *) +(* adding new parsing rules that recognize the new keywords. *) +(* To make matters worse, the Coq grammar for tactics fails to *) +(* export the non-terminals we need to patch. Fortunately, the CamlP5 *) +(* API provides a backdoor access (with loads of Obj.magic trickery). *) + +(* Coq v8.3 defines "by" as a keyword, some hacks are not needed any *) +(* longer and thus comment out. Such comments are marked with v8.3 *) + +let tac_ent = List.fold_left Grammar.Entry.find (Obj.magic simple_tactic) in +let hypident_ent = + tac_ent ["clause_dft_all"; "in_clause"; "hypident_occ"; "hypident"] in +let id_or_meta : Obj.t Gram.Entry.e = Obj.magic + (Grammar.Entry.find hypident_ent "id_or_meta") in +let hypident : (Obj.t * hyp_location_flag) Gram.Entry.e = + Obj.magic hypident_ent in +GEXTEND Gram + GLOBAL: hypident; +hypident: [ + [ "("; IDENT "type"; "of"; id = id_or_meta; ")" -> id, InHypTypeOnly + | "("; IDENT "value"; "of"; id = id_or_meta; ")" -> id, InHypValueOnly + ] ]; +END + +GEXTEND Gram + GLOBAL: hloc; +hloc: [ + [ "in"; "("; "Type"; "of"; id = ident; ")" -> + HypLocation ((dummy_loc,id), InHypTypeOnly) + | "in"; "("; IDENT "Value"; "of"; id = ident; ")" -> + HypLocation ((dummy_loc,id), InHypValueOnly) + ] ]; +END + +let constr_eval + : (Constrexpr.constr_expr,Obj.t,Obj.t) Genredexpr.may_eval Gram.Entry.e + = Obj.magic (Grammar.Entry.find (Obj.magic constr_may_eval) "constr_eval") + +GEXTEND Gram + GLOBAL: constr_eval; + constr_eval: [ + [ IDENT "type"; "of"; c = Constr.constr -> Genredexpr.ConstrTypeOf c ] + ]; +END + +(* We wipe out all the keywords generated by the grammar rules we defined. *) +(* The user is supposed to Require Import ssreflect or Require ssreflect *) +(* and Import ssreflect.SsrSyntax to obtain these keywords and as a *) +(* consequence the extended ssreflect grammar. *) +let () = Lexer.unfreeze frozen_lexer ;; + +(* vim: set filetype=ocaml foldmethod=marker: *) diff --git a/mathcomp/ssreflect/plugin/trunk/ssreflect.mllib b/mathcomp/ssreflect/plugin/trunk/ssreflect.mllib new file mode 100644 index 0000000..006b70f --- /dev/null +++ b/mathcomp/ssreflect/plugin/trunk/ssreflect.mllib @@ -0,0 +1,2 @@ +Ssrmatching +Ssreflect diff --git a/mathcomp/ssreflect/plugin/trunk/ssrmatching.ml4 b/mathcomp/ssreflect/plugin/trunk/ssrmatching.ml4 new file mode 100644 index 0000000..29580a6 --- /dev/null +++ b/mathcomp/ssreflect/plugin/trunk/ssrmatching.ml4 @@ -0,0 +1,1238 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +(* Defining grammar rules with "xx" in it automatically declares keywords too, + * we thus save the lexer to restore it at the end of the file *) +let frozen_lexer = Lexer.freeze () ;; + +(*i camlp4use: "pa_extend.cmo" i*) +(*i camlp4deps: "grammar/grammar.cma" i*) + +open Names +open Pp +open Pcoq +open Genarg +open Constrarg +open Term +open Vars +open Topconstr +open Libnames +open Tactics +open Tacticals +open Termops +open Namegen +open Recordops +open Tacmach +open Coqlib +open Glob_term +open Util +open Evd +open Extend +open Goptions +open Tacexpr +open Tacinterp +open Pretyping +open Constr +open Tactic +open Extraargs +open Ppconstr +open Printer + +open Globnames +open Misctypes +open Decl_kinds +open Evar_kinds +open Constrexpr +open Constrexpr_ops +open Notation_term +open Notation_ops +open Locus +open Locusops + +DECLARE PLUGIN "ssreflect" + +type loc = Loc.t +let dummy_loc = Loc.ghost +let errorstrm = Errors.errorlabstrm "ssreflect" +let loc_error loc msg = Errors.user_err_loc (loc, msg, str msg) + +(* 0 cost pp function. Active only if env variable SSRDEBUG is set *) +(* or if SsrDebug is Set *) +let pp_ref = ref (fun _ -> ()) +let ssr_pp s = pperrnl (str"SSR: "++Lazy.force s) +let _ = try ignore(Sys.getenv "SSRDEBUG"); pp_ref := ssr_pp with Not_found -> () +let debug b = + if b then pp_ref := ssr_pp else pp_ref := fun _ -> () +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssrmatching debugging"; + Goptions.optkey = ["SsrMatchingDebug"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !pp_ref == ssr_pp); + Goptions.optwrite = debug } +let pp s = !pp_ref s + +(** Utils {{{ *****************************************************************) +let env_size env = List.length (Environ.named_context env) +let safeDestApp c = + match kind_of_term c with App (f, a) -> f, a | _ -> c, [| |] +let get_index = function ArgArg i -> i | _ -> + Errors.anomaly (str"Uninterpreted index") +(* Toplevel constr must be globalized twice ! *) +let glob_constr ist genv = function + | _, Some ce -> + let vars = Id.Map.fold (fun x _ accu -> Id.Set.add x accu) ist.lfun Id.Set.empty in + let ltacvars = { Constrintern.empty_ltac_sign with Constrintern.ltac_vars = vars } in + Constrintern.intern_gen WithoutTypeConstraint ~ltacvars:ltacvars genv ce + | rc, None -> rc + +(* Term printing utilities functions for deciding bracketing. *) +let pr_paren prx x = hov 1 (str "(" ++ prx x ++ str ")") +(* String lexing utilities *) +let skip_wschars s = + let rec loop i = match s.[i] with '\n'..' ' -> loop (i + 1) | _ -> i in loop +(* We also guard characters that might interfere with the ssreflect *) +(* tactic syntax. *) +let guard_term ch1 s i = match s.[i] with + | '(' -> false + | '{' | '/' | '=' -> true + | _ -> ch1 = '(' +(* The call 'guard s i' should return true if the contents of s *) +(* starting at i need bracketing to avoid ambiguities. *) +let pr_guarded guard prc c = + msg_with Format.str_formatter (prc c); + let s = Format.flush_str_formatter () ^ "$" in + if guard s (skip_wschars s 0) then pr_paren prc c else prc c +(* More sensible names for constr printers *) +let pr_constr = pr_constr +let prl_glob_constr c = pr_lglob_constr_env (Global.env ()) c +let pr_glob_constr c = pr_glob_constr_env (Global.env ()) c +let prl_constr_expr = pr_lconstr_expr +let pr_constr_expr = pr_constr_expr +let prl_glob_constr_and_expr = function + | _, Some c -> prl_constr_expr c + | c, None -> prl_glob_constr c +let pr_glob_constr_and_expr = function + | _, Some c -> pr_constr_expr c + | c, None -> pr_glob_constr c +let pr_term (k, c) = pr_guarded (guard_term k) pr_glob_constr_and_expr c +let prl_term (k, c) = pr_guarded (guard_term k) prl_glob_constr_and_expr c + +(** Adding a new uninterpreted generic argument type *) +let add_genarg tag pr = + let wit = Genarg.make0 None tag in + let glob ist x = (ist, x) in + let subst _ x = x in + let interp ist gl x = (gl.Evd.sigma, x) in + let gen_pr _ _ _ = pr in + let () = Genintern.register_intern0 wit glob in + let () = Genintern.register_subst0 wit subst in + let () = Geninterp.register_interp0 wit interp in + Pptactic.declare_extra_genarg_pprule wit gen_pr gen_pr gen_pr; + wit + +(** Constructors for cast type *) +let dC t = CastConv t +(** Constructors for constr_expr *) +let isCVar = function CRef (Ident _, _) -> true | _ -> false +let destCVar = function CRef (Ident (_, id), _) -> id | _ -> + Errors.anomaly (str"not a CRef") +let mkCHole loc = CHole (loc, None, IntroAnonymous, None) +let mkCLambda loc name ty t = + CLambdaN (loc, [[loc, name], Default Explicit, ty], t) +let mkCLetIn loc name bo t = + CLetIn (loc, (loc, name), bo, t) +let mkCCast loc t ty = CCast (loc,t, dC ty) +(** Constructors for rawconstr *) +let mkRHole = GHole (dummy_loc, InternalHole, IntroAnonymous, None) +let mkRApp f args = if args = [] then f else GApp (dummy_loc, f, args) +let mkRCast rc rt = GCast (dummy_loc, rc, dC rt) +let mkRLambda n s t = GLambda (dummy_loc, n, Explicit, s, t) + +(* ssrterm conbinators *) +let combineCG t1 t2 f g = match t1, t2 with + | (x, (t1, None)), (_, (t2, None)) -> x, (g t1 t2, None) + | (x, (_, Some t1)), (_, (_, Some t2)) -> x, (mkRHole, Some (f t1 t2)) + | _, (_, (_, None)) -> Errors.anomaly (str"have: mixed C-G constr") + | _ -> Errors.anomaly (str"have: mixed G-C constr") +let loc_ofCG = function + | (_, (s, None)) -> Glob_ops.loc_of_glob_constr s + | (_, (_, Some s)) -> Constrexpr_ops.constr_loc s + +let mk_term k c = k, (mkRHole, Some c) +let mk_lterm = mk_term ' ' + +(* }}} *) + +(** Profiling {{{ *************************************************************) +type profiler = { + profile : 'a 'b. ('a -> 'b) -> 'a -> 'b; + reset : unit -> unit; + print : unit -> unit } +let profile_now = ref false +let something_profiled = ref false +let profilers = ref [] +let add_profiler f = profilers := f :: !profilers;; +let profile b = + profile_now := b; + if b then List.iter (fun f -> f.reset ()) !profilers; + if not b then List.iter (fun f -> f.print ()) !profilers +;; +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssrmatching profiling"; + Goptions.optkey = ["SsrMatchingProfiling"]; + Goptions.optread = (fun _ -> !profile_now); + Goptions.optdepr = false; + Goptions.optwrite = profile } +let () = + let prof_total = + let init = ref 0.0 in { + profile = (fun f x -> assert false); + reset = (fun () -> init := Unix.gettimeofday ()); + print = (fun () -> if !something_profiled then + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + "total" 0 (Unix.gettimeofday() -. !init) 0.0 0.0)) } in + let prof_legenda = { + profile = (fun f x -> assert false); + reset = (fun () -> ()); + print = (fun () -> if !something_profiled then begin + prerr_endline + (Printf.sprintf "!! %39s ---------- --------- --------- ---------" + (String.make 39 '-')); + prerr_endline + (Printf.sprintf "!! %-39s %10s %9s %9s %9s" + "function" "#calls" "total" "max" "average") end) } in + add_profiler prof_legenda; + add_profiler prof_total +;; + +let mk_profiler s = + let total, calls, max = ref 0.0, ref 0, ref 0.0 in + let reset () = total := 0.0; calls := 0; max := 0.0 in + let profile f x = + if not !profile_now then f x else + let before = Unix.gettimeofday () in + try + incr calls; + let res = f x in + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + res + with exc -> + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + raise exc in + let print () = + if !calls <> 0 then begin + something_profiled := true; + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + s !calls !total !max (!total /. (float_of_int !calls))) end in + let prof = { profile = profile; reset = reset; print = print } in + add_profiler prof; + prof +;; +(* }}} *) + +exception NoProgress + +(** Unification procedures. *) + +(* To enforce the rigidity of the rooted match we always split *) +(* top applications, so the unification procedures operate on *) +(* arrays of patterns and terms. *) +(* We perform three kinds of unification: *) +(* EQ: exact conversion check *) +(* FO: first-order unification of evars, without conversion *) +(* HO: higher-order unification with conversion *) +(* The subterm unification strategy is to find the first FO *) +(* match, if possible, and the first HO match otherwise, then *) +(* compute all the occurrences that are EQ matches for the *) +(* relevant subterm. *) +(* Additional twists: *) +(* - If FO/HO fails then we attempt to fill evars using *) +(* typeclasses before raising an outright error. We also *) +(* fill typeclasses even after a successful match, since *) +(* beta-reduction and canonical instances may leave *) +(* undefined evars. *) +(* - We do postchecks to rule out matches that are not *) +(* closed or that assign to a global evar; these can be *) +(* disabled for rewrite or dependent family matches. *) +(* - We do a full FO scan before turning to HO, as the FO *) +(* comparison can be much faster than the HO one. *) + +let unif_EQ env sigma p c = + let evars = existential_opt_value sigma in + try let _ = Reduction.conv env p ~evars c in true with _ -> false + +let unif_EQ_args env sigma pa a = + let n = Array.length pa in + let rec loop i = (i = n) || unif_EQ env sigma pa.(i) a.(i) && loop (i + 1) in + loop 0 + +let prof_unif_eq_args = mk_profiler "unif_EQ_args";; +let unif_EQ_args env sigma pa a = + prof_unif_eq_args.profile (unif_EQ_args env sigma pa) a +;; + +let unif_HO env ise p c = Evarconv.the_conv_x env p c ise + +let unif_HOtype env ise p c = Evarconv.the_conv_x_leq env p c ise + +let unif_HO_args env ise0 pa i ca = + let n = Array.length pa in + let rec loop ise j = + if j = n then ise else loop (unif_HO env ise pa.(j) ca.(i + j)) (j + 1) in + loop ise0 0 + +(* FO unification should boil down to calling w_unify with no_delta, but *) +(* alas things are not so simple: w_unify does partial type-checking, *) +(* which breaks down when the no-delta flag is on (as the Coq type system *) +(* requires full convertibility. The workaround here is to convert all *) +(* evars into metas, since 8.2 does not TC metas. This means some lossage *) +(* for HO evars, though hopefully Miller patterns can pick up some of *) +(* those cases, and HO matching will mop up the rest. *) +let flags_FO = + let flags = + { (Unification.default_no_delta_unify_flags ()).Unification.core_unify_flags + with + Unification.modulo_conv_on_closed_terms = None; + Unification.modulo_eta = true; + Unification.modulo_betaiota = true; + Unification.modulo_delta_types = full_transparent_state} + in + { Unification.core_unify_flags = flags; + Unification.merge_unify_flags = flags; + Unification.subterm_unify_flags = flags; + Unification.allow_K_in_toplevel_higher_order_unification = false; + Unification.resolve_evars = + (Unification.default_no_delta_unify_flags ()).Unification.resolve_evars + } +let unif_FO env ise p c = + Unification.w_unify env ise Reduction.CONV ~flags:flags_FO p c + +(* Perform evar substitution in main term and prune substitution. *) +let nf_open_term sigma0 ise c = + let s = ise and s' = ref sigma0 in + let rec nf c' = match kind_of_term c' with + | Evar ex -> + begin try nf (existential_value s ex) with _ -> + let k, a = ex in let a' = Array.map nf a in + if not (Evd.mem !s' k) then + s' := Evd.add !s' k (Evarutil.nf_evar_info s (Evd.find s k)); + mkEvar (k, a') + end + | _ -> map_constr nf c' in + let copy_def k evi () = + if evar_body evi != Evd.Evar_empty then () else + match Evd.evar_body (Evd.find s k) with + | Evar_defined c' -> s' := Evd.define k (nf c') !s' + | _ -> () in + let c' = nf c in let _ = Evd.fold copy_def sigma0 () in + !s', Evd.evar_universe_context s, c' + +let unif_end env sigma0 ise0 pt ok = + let ise = Evarconv.consider_remaining_unif_problems env ise0 in + let s, uc, t = nf_open_term sigma0 ise pt in + let ise1 = create_evar_defs s in + let ise2 = Typeclasses.resolve_typeclasses ~fail:true env ise1 in + if not (ok ise) then raise NoProgress else + if ise2 == ise1 then (s, uc, t) + else + let s, uc', t = nf_open_term sigma0 ise2 t in + s, Evd.union_evar_universe_context uc uc', t + +let pf_unif_HO gl sigma pt p c = + let env = pf_env gl in + let ise = unif_HO env (create_evar_defs sigma) p c in + unif_end env (project gl) ise pt (fun _ -> true) + +let unify_HO env sigma0 t1 t2 = + let sigma = unif_HO env sigma0 t1 t2 in + let sigma, uc, _ = unif_end env sigma0 sigma t2 (fun _ -> true) in + Evd.merge_universe_context sigma uc + +let pf_unify_HO gl t1 t2 = + let env, sigma0, si = pf_env gl, project gl, sig_it gl in + let sigma = unify_HO env sigma0 t1 t2 in + re_sig si sigma + +(* This is what the definition of iter_constr should be... *) +let iter_constr_LR f c = match kind_of_term c with + | Evar (k, a) -> Array.iter f a + | Cast (cc, _, t) -> f cc; f t + | Prod (_, t, b) | Lambda (_, t, b) -> f t; f b + | LetIn (_, v, t, b) -> f v; f t; f b + | App (cf, a) -> f cf; Array.iter f a + | Case (_, p, v, b) -> f v; f p; Array.iter f b + | Fix (_, (_, t, b)) | CoFix (_, (_, t, b)) -> + for i = 0 to Array.length t - 1 do f t.(i); f b.(i) done + | _ -> () + +(* The comparison used to determine which subterms matches is KEYED *) +(* CONVERSION. This looks for convertible terms that either have the same *) +(* same head constant as pat if pat is an application (after beta-iota), *) +(* or start with the same constr constructor (esp. for LetIn); this is *) +(* disregarded if the head term is let x := ... in x, and casts are always *) +(* ignored and removed). *) +(* Record projections get special treatment: in addition to the projection *) +(* constant itself, ssreflect also recognizes head constants of canonical *) +(* projections. *) + +exception NoMatch +type ssrdir = L2R | R2L +let pr_dir_side = function L2R -> str "LHS" | R2L -> str "RHS" +let inv_dir = function L2R -> R2L | R2L -> L2R + + +type pattern_class = + | KpatFixed + | KpatConst + | KpatEvar of existential_key + | KpatLet + | KpatLam + | KpatRigid + | KpatFlex + | KpatProj of constant + +type tpattern = { + up_k : pattern_class; + up_FO : constr; + up_f : constr; + up_a : constr array; + up_t : constr; (* equation proof term or matched term *) + up_dir : ssrdir; (* direction of the rule *) + up_ok : constr -> evar_map -> bool; (* progess test for rewrite *) + } + +let all_ok _ _ = true + +let proj_nparams c = + try 1 + Recordops.find_projection_nparams (ConstRef c) with _ -> 0 + +let isFixed c = match kind_of_term c with + | Var _ | Ind _ | Construct _ | Const _ -> true + | _ -> false + +let isRigid c = match kind_of_term c with + | Prod _ | Sort _ | Lambda _ | Case _ | Fix _ | CoFix _ -> true + | _ -> false + +exception UndefPat + +let hole_var = mkVar (id_of_string "_") +let pr_constr_pat c0 = + let rec wipe_evar c = + if isEvar c then hole_var else map_constr wipe_evar c in + pr_constr (wipe_evar c0) + +(* Turn (new) evars into metas *) +let evars_for_FO ~hack env sigma0 (ise0:evar_map) c0 = + let ise = ref ise0 in + let sigma = ref ise0 in + let nenv = env_size env + if hack then 1 else 0 in + let rec put c = match kind_of_term c with + | Evar (k, a as ex) -> + begin try put (existential_value !sigma ex) + with NotInstantiatedEvar -> + if Evd.mem sigma0 k then map_constr put c else + let evi = Evd.find !sigma k in + let dc = List.firstn (max 0 (Array.length a - nenv)) (evar_filtered_context evi) in + let abs_dc (d, c) = function + | x, Some b, t -> d, mkNamedLetIn x (put b) (put t) c + | x, None, t -> mkVar x :: d, mkNamedProd x (put t) c in + let a, t = + Context.fold_named_context_reverse abs_dc ~init:([], (put evi.evar_concl)) dc in + let m = Evarutil.new_meta () in + ise := meta_declare m t !ise; + sigma := Evd.define k (applist (mkMeta m, a)) !sigma; + put (existential_value !sigma ex) + end + | _ -> map_constr put c in + let c1 = put c0 in !ise, c1 + +(* Compile a match pattern from a term; t is the term to fill. *) +(* p_origin can be passed to obtain a better error message *) +let mk_tpattern ?p_origin ?(hack=false) env sigma0 (ise, t) ok dir p = + let k, f, a = + let f, a = Reductionops.whd_betaiota_stack ise p in + match kind_of_term f with + | Const (p,_) -> + let np = proj_nparams p in + if np = 0 || np > List.length a then KpatConst, f, a else + let a1, a2 = List.chop np a in KpatProj p, applist(f, a1), a2 + | Var _ | Ind _ | Construct _ -> KpatFixed, f, a + | Evar (k, _) -> + if Evd.mem sigma0 k then KpatEvar k, f, a else + if a <> [] then KpatFlex, f, a else + (match p_origin with None -> Errors.error "indeterminate pattern" + | Some (dir, rule) -> + errorstrm (str "indeterminate " ++ pr_dir_side dir + ++ str " in " ++ pr_constr_pat rule)) + | LetIn (_, v, _, b) -> + if b <> mkRel 1 then KpatLet, f, a else KpatFlex, v, a + | Lambda _ -> KpatLam, f, a + | _ -> KpatRigid, f, a in + let aa = Array.of_list a in + let ise', p' = evars_for_FO ~hack env sigma0 ise (mkApp (f, aa)) in + ise', + { up_k = k; up_FO = p'; up_f = f; + up_a = aa; up_ok = ok; up_dir = dir; up_t = t} + +(* Specialize a pattern after a successful match: assign a precise head *) +(* kind and arity for Proj and Flex patterns. *) +let ungen_upat lhs (sigma, uc, t) u = + let f, a = safeDestApp lhs in + let k = match kind_of_term f with + | Var _ | Ind _ | Construct _ -> KpatFixed + | Const _ -> KpatConst + | Evar (k, _) -> if is_defined sigma k then raise NoMatch else KpatEvar k + | LetIn _ -> KpatLet + | Lambda _ -> KpatLam + | _ -> KpatRigid in + sigma, uc, {u with up_k = k; up_FO = lhs; up_f = f; up_a = a; up_t = t} + +let nb_cs_proj_args pc f u = + let na k = + List.length (snd (lookup_canonical_conversion (ConstRef pc, k))).o_TCOMPS in + try match kind_of_term f with + | Prod _ -> na Prod_cs + | Sort s -> na (Sort_cs (family_of_sort s)) + | Const (c',_) when Constant.equal c' pc -> Array.length (snd (destApp u.up_f)) + | Var _ | Ind _ | Construct _ | Const _ -> na (Const_cs (global_of_constr f)) + | _ -> -1 + with Not_found -> -1 + +let isEvar_k k f = + match kind_of_term f with Evar (k', _) -> k = k' | _ -> false + +let nb_args c = + match kind_of_term c with App (_, a) -> Array.length a | _ -> 0 + +let mkSubArg i a = if i = Array.length a then a else Array.sub a 0 i +let mkSubApp f i a = if i = 0 then f else mkApp (f, mkSubArg i a) + +let splay_app ise = + let rec loop c a = match kind_of_term c with + | App (f, a') -> loop f (Array.append a' a) + | Cast (c', _, _) -> loop c' a + | Evar ex -> + (try loop (existential_value ise ex) a with _ -> c, a) + | _ -> c, a in + fun c -> match kind_of_term c with + | App (f, a) -> loop f a + | Cast _ | Evar _ -> loop c [| |] + | _ -> c, [| |] + +let filter_upat i0 f n u fpats = + let na = Array.length u.up_a in + if n < na then fpats else + let np = match u.up_k with + | KpatConst when Term.eq_constr u.up_f f -> na + | KpatFixed when Term.eq_constr u.up_f f -> na + | KpatEvar k when isEvar_k k f -> na + | KpatLet when isLetIn f -> na + | KpatLam when isLambda f -> na + | KpatRigid when isRigid f -> na + | KpatFlex -> na + | KpatProj pc -> + let np = na + nb_cs_proj_args pc f u in if n < np then -1 else np + | _ -> -1 in + if np < na then fpats else + let () = if !i0 < np then i0 := n in (u, np) :: fpats + +let filter_upat_FO i0 f n u fpats = + let np = nb_args u.up_FO in + if n < np then fpats else + let ok = match u.up_k with + | KpatConst -> Term.eq_constr u.up_f f + | KpatFixed -> Term.eq_constr u.up_f f + | KpatEvar k -> isEvar_k k f + | KpatLet -> isLetIn f + | KpatLam -> isLambda f + | KpatRigid -> isRigid f + | KpatProj pc -> Term.eq_constr f (mkConst pc) + | KpatFlex -> i0 := n; true in + if ok then begin if !i0 < np then i0 := np; (u, np) :: fpats end else fpats + +exception FoundUnif of (evar_map * evar_universe_context * tpattern) +(* Note: we don't update env as we descend into the term, as the primitive *) +(* unification procedure always rejects subterms with bound variables. *) + +(* We are forced to duplicate code between the FO/HO matching because we *) +(* have to work around several kludges in unify.ml: *) +(* - w_unify drops into second-order unification when the pattern is an *) +(* application whose head is a meta. *) +(* - w_unify tries to unify types without subsumption when the pattern *) +(* head is an evar or meta (e.g., it fails on ?1 = nat when ?1 : Type). *) +(* - w_unify expands let-in (zeta conversion) eagerly, whereas we want to *) +(* match a head let rigidly. *) +let match_upats_FO upats env sigma0 ise = + let rec loop c = + let f, a = splay_app ise c in let i0 = ref (-1) in + let fpats = + List.fold_right (filter_upat_FO i0 f (Array.length a)) upats [] in + while !i0 >= 0 do + let i = !i0 in i0 := -1; + let c' = mkSubApp f i a in + let one_match (u, np) = + let skip = + if i <= np then i < np else + if u.up_k == KpatFlex then begin i0 := i - 1; false end else + begin if !i0 < np then i0 := np; true end in + if skip || not (closed0 c') then () else try + let _ = match u.up_k with + | KpatFlex -> + let kludge v = mkLambda (Anonymous, mkProp, v) in + unif_FO env ise (kludge u.up_FO) (kludge c') + | KpatLet -> + let kludge vla = + let vl, a = safeDestApp vla in + let x, v, t, b = destLetIn vl in + mkApp (mkLambda (x, t, b), Array.cons v a) in + unif_FO env ise (kludge u.up_FO) (kludge c') + | _ -> unif_FO env ise u.up_FO c' in + let ise' = (* Unify again using HO to assign evars *) + let p = mkApp (u.up_f, u.up_a) in + try unif_HO env ise p c' with _ -> raise NoMatch in + let lhs = mkSubApp f i a in + let pt' = unif_end env sigma0 ise' u.up_t (u.up_ok lhs) in + raise (FoundUnif (ungen_upat lhs pt' u)) + with FoundUnif _ as sigma_u -> raise sigma_u + | Not_found -> Errors.anomaly (str"incomplete ise in match_upats_FO") + | _ -> () in + List.iter one_match fpats + done; + iter_constr_LR loop f; Array.iter loop a in + fun c -> try loop c with Invalid_argument _ -> Errors.anomaly (str"IN FO") + +let prof_FO = mk_profiler "match_upats_FO";; +let match_upats_FO upats env sigma0 ise c = + prof_FO.profile (match_upats_FO upats env sigma0) ise c +;; + + +let match_upats_HO upats env sigma0 ise c = + let it_did_match = ref false in + let rec aux upats env sigma0 ise c = + let f, a = splay_app ise c in let i0 = ref (-1) in + let fpats = List.fold_right (filter_upat i0 f (Array.length a)) upats [] in + while !i0 >= 0 do + let i = !i0 in i0 := -1; + let one_match (u, np) = + let skip = + if i <= np then i < np else + if u.up_k == KpatFlex then begin i0 := i - 1; false end else + begin if !i0 < np then i0 := np; true end in + if skip then () else try + let ise' = match u.up_k with + | KpatFixed | KpatConst -> ise + | KpatEvar _ -> + let _, pka = destEvar u.up_f and _, ka = destEvar f in + unif_HO_args env ise pka 0 ka + | KpatLet -> + let x, v, t, b = destLetIn f in + let _, pv, _, pb = destLetIn u.up_f in + let ise' = unif_HO env ise pv v in + unif_HO (Environ.push_rel (x, None, t) env) ise' pb b + | KpatFlex | KpatProj _ -> + unif_HO env ise u.up_f (mkSubApp f (i - Array.length u.up_a) a) + | _ -> unif_HO env ise u.up_f f in + let ise'' = unif_HO_args env ise' u.up_a (i - Array.length u.up_a) a in + let lhs = mkSubApp f i a in + let pt' = unif_end env sigma0 ise'' u.up_t (u.up_ok lhs) in + raise (FoundUnif (ungen_upat lhs pt' u)) + with FoundUnif _ as sigma_u -> raise sigma_u + | NoProgress -> it_did_match := true + | _ -> () in + List.iter one_match fpats + done; + iter_constr_LR (aux upats env sigma0 ise) f; + Array.iter (aux upats env sigma0 ise) a + in + aux upats env sigma0 ise c; + if !it_did_match then raise NoProgress + +let prof_HO = mk_profiler "match_upats_HO";; +let match_upats_HO upats env sigma0 ise c = + prof_HO.profile (match_upats_HO upats env sigma0) ise c +;; + + +let fixed_upat = function +| {up_k = KpatFlex | KpatEvar _ | KpatProj _} -> false +| {up_t = t} -> not (occur_existential t) + +let do_once r f = match !r with Some _ -> () | None -> r := Some (f ()) + +let assert_done r = + match !r with Some x -> x | None -> Errors.anomaly (str"do_once never called") + +type subst = Environ.env -> Term.constr -> int -> Term.constr +type find_P = + Environ.env -> Term.constr -> int -> + k:subst -> + Term.constr +type conclude = unit -> + Term.constr * ssrdir * (Evd.evar_map * Evd.evar_universe_context * Term.constr) + +(* upats_origin makes a better error message only *) +let mk_tpattern_matcher + ?(raise_NoMatch=false) ?upats_origin sigma0 occ (ise, upats) += + let nocc = ref 0 and skip_occ = ref false in + let use_occ, occ_list = match occ with + | Some (true, ol) -> ol = [], ol + | Some (false, ol) -> ol <> [], ol + | None -> false, [] in + let max_occ = List.fold_right max occ_list 0 in + let subst_occ = + let occ_set = Array.make max_occ (not use_occ) in + let _ = List.iter (fun i -> occ_set.(i - 1) <- use_occ) occ_list in + let _ = if max_occ = 0 then skip_occ := use_occ in + fun () -> incr nocc; + if !nocc = max_occ then skip_occ := use_occ; + if !nocc <= max_occ then occ_set.(!nocc - 1) else not use_occ in + let upat_that_matched = ref None in + let match_EQ env sigma u = + match u.up_k with + | KpatLet -> + let x, pv, t, pb = destLetIn u.up_f in + let env' = Environ.push_rel (x, None, t) env in + let match_let f = match kind_of_term f with + | LetIn (_, v, _, b) -> unif_EQ env sigma pv v && unif_EQ env' sigma pb b + | _ -> false in match_let + | KpatFixed -> Term.eq_constr u.up_f + | KpatConst -> Term.eq_constr u.up_f + | KpatLam -> fun c -> + (match kind_of_term c with + | Lambda _ -> unif_EQ env sigma u.up_f c + | _ -> false) + | _ -> unif_EQ env sigma u.up_f in +let p2t p = mkApp(p.up_f,p.up_a) in +let source () = match upats_origin, upats with + | None, [p] -> + (if fixed_upat p then str"term " else str"partial term ") ++ + pr_constr_pat (p2t p) ++ spc() + | Some (dir,rule), [p] -> str"The " ++ pr_dir_side dir ++ str" of " ++ + pr_constr_pat rule ++ fnl() ++ ws 4 ++ pr_constr_pat (p2t p) ++ fnl() + | Some (dir,rule), _ -> str"The " ++ pr_dir_side dir ++ str" of " ++ + pr_constr_pat rule ++ spc() + | _, [] | None, _::_::_ -> + Errors.anomaly (str"mk_tpattern_matcher with no upats_origin") in +((fun env c h ~k -> + do_once upat_that_matched (fun () -> + try + match_upats_FO upats env sigma0 ise c; + match_upats_HO upats env sigma0 ise c; + raise NoMatch + with FoundUnif sigma_u -> sigma_u + | NoMatch when (not raise_NoMatch) -> + errorstrm (source () ++ str "does not match any subterm of the goal") + | NoProgress when (not raise_NoMatch) -> + let dir = match upats_origin with Some (d,_) -> d | _ -> + Errors.anomaly (str"mk_tpattern_matcher with no upats_origin") in + errorstrm (str"all matches of "++source()++ + str"are equal to the " ++ pr_dir_side (inv_dir dir)) + | NoProgress -> raise NoMatch); + let sigma, _, ({up_f = pf; up_a = pa} as u) = assert_done upat_that_matched in + pp(lazy(str"sigma@tmatch=" ++ pr_evar_map None sigma)); + if !skip_occ then (ignore(k env u.up_t 0); c) else + let match_EQ = match_EQ env sigma u in + let pn = Array.length pa in + let rec subst_loop (env,h as acc) c' = + if !skip_occ then c' else + let f, a = splay_app sigma c' in + if Array.length a >= pn && match_EQ f && unif_EQ_args env sigma pa a then + let a1, a2 = Array.chop (Array.length pa) a in + let fa1 = mkApp (f, a1) in + let f' = if subst_occ () then k env u.up_t h else fa1 in + mkApp (f', Array.map_left (subst_loop acc) a2) + else + (* TASSI: clear letin values to avoid unfolding *) + let inc_h (n,_,ty) (env,h') = Environ.push_rel (n,None,ty) env, h' + 1 in + let f' = map_constr_with_binders_left_to_right inc_h subst_loop acc f in + mkApp (f', Array.map_left (subst_loop acc) a) in + subst_loop (env,h) c) : find_P), +((fun () -> + let sigma, uc, ({up_f = pf; up_a = pa} as u) = + match !upat_that_matched with + | Some x -> x | None when raise_NoMatch -> raise NoMatch + | None -> Errors.anomaly (str"companion function never called") in + let p' = mkApp (pf, pa) in + if max_occ <= !nocc then p', u.up_dir, (sigma, uc, u.up_t) + else errorstrm (str"Only " ++ int !nocc ++ str" < " ++ int max_occ ++ + str(String.plural !nocc " occurence") ++ match upats_origin with + | None -> str" of" ++ spc() ++ pr_constr_pat p' + | Some (dir,rule) -> str" of the " ++ pr_dir_side dir ++ fnl() ++ + ws 4 ++ pr_constr_pat p' ++ fnl () ++ + str"of " ++ pr_constr_pat rule)) : conclude) + +type ('ident, 'term) ssrpattern = + | T of 'term + | In_T of 'term + | X_In_T of 'ident * 'term + | In_X_In_T of 'ident * 'term + | E_In_X_In_T of 'term * 'ident * 'term + | E_As_X_In_T of 'term * 'ident * 'term + +let pr_pattern = function + | T t -> prl_term t + | In_T t -> str "in " ++ prl_term t + | X_In_T (x,t) -> prl_term x ++ str " in " ++ prl_term t + | In_X_In_T (x,t) -> str "in " ++ prl_term x ++ str " in " ++ prl_term t + | E_In_X_In_T (e,x,t) -> + prl_term e ++ str " in " ++ prl_term x ++ str " in " ++ prl_term t + | E_As_X_In_T (e,x,t) -> + prl_term e ++ str " as " ++ prl_term x ++ str " in " ++ prl_term t + +let pr_pattern_w_ids = function + | T t -> prl_term t + | In_T t -> str "in " ++ prl_term t + | X_In_T (x,t) -> pr_id x ++ str " in " ++ prl_term t + | In_X_In_T (x,t) -> str "in " ++ pr_id x ++ str " in " ++ prl_term t + | E_In_X_In_T (e,x,t) -> + prl_term e ++ str " in " ++ pr_id x ++ str " in " ++ prl_term t + | E_As_X_In_T (e,x,t) -> + prl_term e ++ str " as " ++ pr_id x ++ str " in " ++ prl_term t + +let pr_pattern_aux pr_constr = function + | T t -> pr_constr t + | In_T t -> str "in " ++ pr_constr t + | X_In_T (x,t) -> pr_constr x ++ str " in " ++ pr_constr t + | In_X_In_T (x,t) -> str "in " ++ pr_constr x ++ str " in " ++ pr_constr t + | E_In_X_In_T (e,x,t) -> + pr_constr e ++ str " in " ++ pr_constr x ++ str " in " ++ pr_constr t + | E_As_X_In_T (e,x,t) -> + pr_constr e ++ str " as " ++ pr_constr x ++ str " in " ++ pr_constr t +let pp_pattern (sigma, p) = + pr_pattern_aux (fun t -> pr_constr (pi3 (nf_open_term sigma sigma t))) p +let pr_cpattern = pr_term +let pr_rpattern _ _ _ = pr_pattern + +let pr_option f = function None -> mt() | Some x -> f x +let pr_ssrpattern _ _ _ = pr_option pr_pattern +let pr_pattern_squarep = pr_option (fun r -> str "[" ++ pr_pattern r ++ str "]") +let pr_ssrpattern_squarep _ _ _ = pr_pattern_squarep +let pr_pattern_roundp = pr_option (fun r -> str "(" ++ pr_pattern r ++ str ")") +let pr_ssrpattern_roundp _ _ _ = pr_pattern_roundp + +let wit_rpatternty = add_genarg "rpatternty" pr_pattern + +ARGUMENT EXTEND rpattern TYPED AS rpatternty PRINTED BY pr_rpattern + | [ lconstr(c) ] -> [ T (mk_lterm c) ] + | [ "in" lconstr(c) ] -> [ In_T (mk_lterm c) ] + | [ lconstr(x) "in" lconstr(c) ] -> + [ X_In_T (mk_lterm x, mk_lterm c) ] + | [ "in" lconstr(x) "in" lconstr(c) ] -> + [ In_X_In_T (mk_lterm x, mk_lterm c) ] + | [ lconstr(e) "in" lconstr(x) "in" lconstr(c) ] -> + [ E_In_X_In_T (mk_lterm e, mk_lterm x, mk_lterm c) ] + | [ lconstr(e) "as" lconstr(x) "in" lconstr(c) ] -> + [ E_As_X_In_T (mk_lterm e, mk_lterm x, mk_lterm c) ] +END + +type cpattern = char * glob_constr_and_expr +let tag_of_cpattern = fst +let loc_of_cpattern = loc_ofCG +let cpattern_of_term t = t +type occ = (bool * int list) option + +type rpattern = (cpattern, cpattern) ssrpattern +let pr_rpattern = pr_pattern + +type pattern = Evd.evar_map * (Term.constr, Term.constr) ssrpattern + + +let id_of_cpattern = function + | _,(_,Some (CRef (Ident (_, x), _))) -> Some x + | _,(_,Some (CAppExpl (_, (_, Ident (_, x), _), []))) -> Some x + | _,(GRef (_, VarRef x, _) ,None) -> Some x + | _ -> None +let id_of_Cterm t = match id_of_cpattern t with + | Some x -> x + | None -> loc_error (loc_of_cpattern t) "Only identifiers are allowed here" + +let interp_wit wit ist gl x = + let globarg = in_gen (glbwit wit) x in + let sigma, arg = interp_genarg ist (pf_env gl) (project gl) (pf_concl gl) gl.Evd.it globarg in + sigma, out_gen (topwit wit) arg +let interp_constr = interp_wit wit_constr +let interp_open_constr ist gl gc = + interp_wit wit_open_constr ist gl ((), gc) +let pf_intern_term ist gl (_, c) = glob_constr ist (pf_env gl) c +let interp_term ist gl (_, c) = snd (interp_open_constr ist gl c) +let glob_ssrterm gs = function + | k, (_, Some c) -> k, Tacintern.intern_constr gs c + | ct -> ct +let subst_ssrterm s (k, c) = k, Tacsubst.subst_glob_constr_and_expr s c +let pr_ssrterm _ _ _ = pr_term +let input_ssrtermkind strm = match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "(" -> '(' + | Tok.KEYWORD "@" -> '@' + | _ -> ' ' +let ssrtermkind = Gram.Entry.of_parser "ssrtermkind" input_ssrtermkind + +(* This piece of code asserts the following notations are reserved *) +(* Reserved Notation "( a 'in' b )" (at level 0). *) +(* Reserved Notation "( a 'as' b )" (at level 0). *) +(* Reserved Notation "( a 'in' b 'in' c )" (at level 0). *) +(* Reserved Notation "( a 'as' b 'in' c )" (at level 0). *) +let glob_cpattern gs p = + pp(lazy(str"globbing pattern: " ++ pr_term p)); + let glob x = snd (glob_ssrterm gs (mk_lterm x)) in + let encode k s l = + let name = Name (id_of_string ("_ssrpat_" ^ s)) in + k, (mkRCast mkRHole (mkRLambda name mkRHole (mkRApp mkRHole l)), None) in + let bind_in t1 t2 = + let d = dummy_loc in let n = Name (destCVar t1) in + fst (glob (mkCCast d (mkCHole d) (mkCLambda d n (mkCHole d) t2))) in + let check_var t2 = if not (isCVar t2) then + loc_error (constr_loc t2) "Only identifiers are allowed here" in + match p with + | _, (_, None) as x -> x + | k, (v, Some t) as orig -> + if k = 'x' then glob_ssrterm gs ('(', (v, Some t)) else + match t with + | CNotation(_, "( _ in _ )", ([t1; t2], [], [])) -> + (try match glob t1, glob t2 with + | (r1, None), (r2, None) -> encode k "In" [r1;r2] + | (r1, Some _), (r2, Some _) when isCVar t1 -> + encode k "In" [r1; r2; bind_in t1 t2] + | (r1, Some _), (r2, Some _) -> encode k "In" [r1; r2] + | _ -> Errors.anomaly (str"where are we?") + with _ when isCVar t1 -> encode k "In" [bind_in t1 t2]) + | CNotation(_, "( _ in _ in _ )", ([t1; t2; t3], [], [])) -> + check_var t2; encode k "In" [fst (glob t1); bind_in t2 t3] + | CNotation(_, "( _ as _ )", ([t1; t2], [], [])) -> + encode k "As" [fst (glob t1); fst (glob t2)] + | CNotation(_, "( _ as _ in _ )", ([t1; t2; t3], [], [])) -> + check_var t2; encode k "As" [fst (glob t1); bind_in t2 t3] + | _ -> glob_ssrterm gs orig +;; + +let interp_ssrterm _ gl t = Tacmach.project gl, t + +ARGUMENT EXTEND cpattern + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_cpattern SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "Qed" constr(c) ] -> [ mk_lterm c ] +END + +let (!@) = Compat.to_coqloc + +GEXTEND Gram + GLOBAL: cpattern; + cpattern: [[ k = ssrtermkind; c = constr -> + let pattern = mk_term k c in + if loc_ofCG pattern <> !@loc && k = '(' then mk_term 'x' c else pattern ]]; +END + +ARGUMENT EXTEND lcpattern + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_cpattern SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "Qed" lconstr(c) ] -> [ mk_lterm c ] +END + +GEXTEND Gram + GLOBAL: lcpattern; + lcpattern: [[ k = ssrtermkind; c = lconstr -> + let pattern = mk_term k c in + if loc_ofCG pattern <> !@loc && k = '(' then mk_term 'x' c else pattern ]]; +END + +let interp_pattern ist gl red redty = + pp(lazy(str"interpreting: " ++ pr_pattern red)); + let xInT x y = X_In_T(x,y) and inXInT x y = In_X_In_T(x,y) in + let inT x = In_T x and eInXInT e x t = E_In_X_In_T(e,x,t) in + let eAsXInT e x t = E_As_X_In_T(e,x,t) in + let mkG ?(k=' ') x = k,(x,None) in + let decode t f g = + try match (pf_intern_term ist gl t) with + | GCast(_,GHole _,CastConv(GLambda(_,Name x,_,_,c))) -> f x (' ',(c,None)) + | it -> g t with _ -> g t in + let decodeG t f g = decode (mkG t) f g in + let bad_enc id _ = Errors.anomaly (str"bad encoding for pattern "++str id) in + let cleanup_XinE h x rp sigma = + let h_k = match kind_of_term h with Evar (k,_) -> k | _ -> assert false in + let to_clean, update = (* handle rename if x is already used *) + let ctx = pf_hyps gl in + let len = Context.named_context_length ctx in + let name = ref None in + try ignore(Context.lookup_named x ctx); (name, fun k -> + if !name = None then + let nctx = Evd.evar_context (Evd.find sigma k) in + let nlen = Context.named_context_length nctx in + if nlen > len then begin + name := Some (pi1 (List.nth nctx (nlen - len - 1))) + end) + with Not_found -> ref (Some x), fun _ -> () in + let sigma0 = project gl in + let new_evars = + let rec aux acc t = match kind_of_term t with + | Evar (k,_) -> + if k = h_k || List.mem k acc || Evd.mem sigma0 k then acc else + (update k; k::acc) + | _ -> fold_constr aux acc t in + aux [] (Evarutil.nf_evar sigma rp) in + let sigma = + List.fold_left (fun sigma e -> + if Evd.is_defined sigma e then sigma else (* clear may be recursiv *) + let name = Option.get !to_clean in + pp(lazy(pr_id name)); + try snd(Logic.prim_refiner (Proof_type.Thin [name]) sigma e) + with Evarutil.ClearDependencyError _ -> sigma) + sigma new_evars in + sigma in + let red = match red with + | T(k,(GCast (_,GHole _,(CastConv(GLambda (_,Name id,_,_,t)))),None)) + when let id = string_of_id id in let len = String.length id in + (len > 8 && String.sub id 0 8 = "_ssrpat_") -> + let id = string_of_id id in let len = String.length id in + (match String.sub id 8 (len - 8), t with + | "In", GApp(_, _, [t]) -> decodeG t xInT (fun x -> T x) + | "In", GApp(_, _, [e; t]) -> decodeG t (eInXInT (mkG e)) (bad_enc id) + | "In", GApp(_, _, [e; t; e_in_t]) -> + decodeG t (eInXInT (mkG e)) + (fun _ -> decodeG e_in_t xInT (fun _ -> assert false)) + | "As", GApp(_, _, [e; t]) -> decodeG t (eAsXInT (mkG e)) (bad_enc id) + | _ -> bad_enc id ()) + | T t -> decode t xInT (fun x -> T x) + | In_T t -> decode t inXInT inT + | X_In_T (e,t) -> decode t (eInXInT e) (fun x -> xInT (id_of_Cterm e) x) + | In_X_In_T (e,t) -> inXInT (id_of_Cterm e) t + | E_In_X_In_T (e,x,rp) -> eInXInT e (id_of_Cterm x) rp + | E_As_X_In_T (e,x,rp) -> eAsXInT e (id_of_Cterm x) rp in + pp(lazy(str"decoded as: " ++ pr_pattern_w_ids red)); + let red = match redty with None -> red | Some ty -> let ty = ' ', ty in + match red with + | T t -> T (combineCG t ty (mkCCast (loc_ofCG t)) mkRCast) + | X_In_T (x,t) -> + let ty = pf_intern_term ist gl ty in + E_As_X_In_T (mkG (mkRCast mkRHole ty), x, t) + | E_In_X_In_T (e,x,t) -> + let ty = mkG (pf_intern_term ist gl ty) in + E_In_X_In_T (combineCG e ty (mkCCast (loc_ofCG t)) mkRCast, x, t) + | E_As_X_In_T (e,x,t) -> + let ty = mkG (pf_intern_term ist gl ty) in + E_As_X_In_T (combineCG e ty (mkCCast (loc_ofCG t)) mkRCast, x, t) + | red -> red in + pp(lazy(str"typed as: " ++ pr_pattern_w_ids red)); + let mkXLetIn loc x (a,(g,c)) = match c with + | Some b -> a,(g,Some (mkCLetIn loc x (mkCHole loc) b)) + | None -> a,(GLetIn (loc,x,(GHole (loc, BinderType x, IntroAnonymous, None)), g), None) in + match red with + | T t -> let sigma, t = interp_term ist gl t in sigma, T t + | In_T t -> let sigma, t = interp_term ist gl t in sigma, In_T t + | X_In_T (x, rp) | In_X_In_T (x, rp) -> + let mk x p = match red with X_In_T _ -> X_In_T(x,p) | _ -> In_X_In_T(x,p) in + let rp = mkXLetIn dummy_loc (Name x) rp in + let sigma, rp = interp_term ist gl rp in + let _, h, _, rp = destLetIn rp in + let sigma = cleanup_XinE h x rp sigma in + let rp = subst1 h (Evarutil.nf_evar sigma rp) in + sigma, mk h rp + | E_In_X_In_T(e, x, rp) | E_As_X_In_T (e, x, rp) -> + let mk e x p = + match red with E_In_X_In_T _ ->E_In_X_In_T(e,x,p)|_->E_As_X_In_T(e,x,p) in + let rp = mkXLetIn dummy_loc (Name x) rp in + let sigma, rp = interp_term ist gl rp in + let _, h, _, rp = destLetIn rp in + let sigma = cleanup_XinE h x rp sigma in + let rp = subst1 h (Evarutil.nf_evar sigma rp) in + let sigma, e = interp_term ist (re_sig (sig_it gl) sigma) e in + sigma, mk e h rp +;; +let interp_cpattern ist gl red redty = interp_pattern ist gl (T red) redty;; +let interp_rpattern ist gl red = interp_pattern ist gl red None;; + +(* The full occurrence set *) +let noindex = Some(false,[]) + +(* calls do_subst on every sub-term identified by (pattern,occ) *) +let eval_pattern ?raise_NoMatch env0 sigma0 concl0 pattern occ do_subst = + let fs sigma x = Reductionops.nf_evar sigma x in + let pop_evar sigma e p = + let { Evd.evar_body = e_body } as e_def = Evd.find sigma e in + let e_body = match e_body with Evar_defined c -> c + | _ -> errorstrm (str "Matching the pattern " ++ pr_constr p ++ + str " did not instantiate ?" ++ int (Evar.repr e) ++ spc () ++ + str "Does the variable bound by the \"in\" construct occur "++ + str "in the pattern?") in + let sigma = + Evd.add (Evd.remove sigma e) e {e_def with Evd.evar_body = Evar_empty} in + sigma, e_body in + let ex_value hole = + match kind_of_term hole with Evar (e,_) -> e | _ -> assert false in + let mk_upat_for ?hack env sigma0 (sigma, t) ?(p=t) ok = + let sigma,pat= mk_tpattern ?hack env sigma0 (sigma,p) ok L2R (fs sigma t) in + sigma, [pat] in + match pattern with + | None -> do_subst env0 concl0 1 + | Some (sigma, (T rp | In_T rp)) -> + let rp = fs sigma rp in + let ise = create_evar_defs sigma in + let occ = match pattern with Some (_, T _) -> occ | _ -> noindex in + let rp = mk_upat_for env0 sigma0 (ise, rp) all_ok in + let find_T, end_T = mk_tpattern_matcher ?raise_NoMatch sigma0 occ rp in + let concl = find_T env0 concl0 1 do_subst in + let _ = end_T () in + concl + | Some (sigma, (X_In_T (hole, p) | In_X_In_T (hole, p))) -> + let p = fs sigma p in + let occ = match pattern with Some (_, X_In_T _) -> occ | _ -> noindex in + let ex = ex_value hole in + let rp = mk_upat_for ~hack:true env0 sigma0 (sigma, p) all_ok in + let find_T, end_T = mk_tpattern_matcher sigma0 noindex rp in + (* we start from sigma, so hole is considered a rigid head *) + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher ?raise_NoMatch sigma occ holep in + let concl = find_T env0 concl0 1 (fun env c h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h + (fun env _ -> do_subst env e_body))) in + let _ = end_X () in let _ = end_T () in + concl + | Some (sigma, E_In_X_In_T (e, hole, p)) -> + let p, e = fs sigma p, fs sigma e in + let ex = ex_value hole in + let rp = mk_upat_for ~hack:true env0 sigma0 (sigma, p) all_ok in + let find_T, end_T = mk_tpattern_matcher sigma0 noindex rp in + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher sigma noindex holep in + let re = mk_upat_for env0 sigma0 (sigma, e) all_ok in + let find_E, end_E = mk_tpattern_matcher ?raise_NoMatch sigma0 occ re in + let concl = find_T env0 concl0 1 (fun env c h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h (fun env c h -> + find_E env e_body h do_subst))) in + let _ = end_E () in let _ = end_X () in let _ = end_T () in + concl + | Some (sigma, E_As_X_In_T (e, hole, p)) -> + let p, e = fs sigma p, fs sigma e in + let ex = ex_value hole in + let rp = + let e_sigma = unify_HO env0 sigma hole e in + e_sigma, fs e_sigma p in + let rp = mk_upat_for ~hack:true env0 sigma0 rp all_ok in + let find_TE, end_TE = mk_tpattern_matcher sigma0 noindex rp in + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher sigma occ holep in + let concl = find_TE env0 concl0 1 (fun env c h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h (fun env c h -> + let e_sigma = unify_HO env sigma e_body e in + let e_body = fs e_sigma e in + do_subst env e_body h))) in + let _ = end_X () in let _ = end_TE () in + concl +;; + +let redex_of_pattern ?(resolve_typeclasses=false) env (sigma, p) = + let e = match p with + | In_T _ | In_X_In_T _ -> Errors.anomaly (str"pattern without redex") + | T e | X_In_T (e, _) | E_As_X_In_T (e, _, _) | E_In_X_In_T (e, _, _) -> e in + let sigma = + if not resolve_typeclasses then sigma + else Typeclasses.resolve_typeclasses ~fail:false env sigma in + Reductionops.nf_evar sigma e, Evd.evar_universe_context sigma + +let fill_occ_pattern ?raise_NoMatch env sigma cl pat occ h = + let find_R, conclude = let r = ref None in + (fun env c h' -> do_once r (fun () -> c, Evd.empty_evar_universe_context); + mkRel (h'+h-1)), + (fun _ -> if !r = None then redex_of_pattern env pat else assert_done r) in + let cl = eval_pattern ?raise_NoMatch env sigma cl (Some pat) occ find_R in + let e = conclude cl in + e, cl +;; + +(* clenup interface for external use *) +let mk_tpattern ?p_origin env sigma0 sigma_t f dir c = + mk_tpattern ?p_origin env sigma0 sigma_t f dir c +;; + +let pf_fill_occ env concl occ sigma0 p (sigma, t) ok h = + let ise = create_evar_defs sigma in + let ise, u = mk_tpattern env sigma0 (ise,t) ok L2R p in + let find_U, end_U = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[u]) in + let concl = find_U env concl h (fun _ _ -> mkRel) in + let rdx, _, (sigma, uc, p) = end_U () in + sigma, uc, p, concl, rdx + +let fill_occ_term env cl occ sigma0 (sigma, t) = + try + let sigma',uc,t',cl,_= pf_fill_occ env cl occ sigma0 t (sigma, t) all_ok 1 in + if sigma' != sigma0 then Errors.error "matching impacts evars" + else cl, (Evd.merge_universe_context sigma' uc, t') + with NoMatch -> try + let sigma', uc, t' = + unif_end env sigma0 (create_evar_defs sigma) t (fun _ -> true) in + if sigma' != sigma0 then raise NoMatch + else cl, (Evd.merge_universe_context sigma' uc, t') + with _ -> + errorstrm (str "partial term " ++ pr_constr_pat t + ++ str " does not match any subterm of the goal") + +let pf_fill_occ_term gl occ t = + let sigma0 = project gl and env = pf_env gl and concl = pf_concl gl in + let cl,(_,t) = fill_occ_term env concl occ sigma0 t in + cl, t + +let cpattern_of_id id = ' ', (GRef (dummy_loc, VarRef id, None), None) + +let is_wildcard = function + | _,(_,Some (CHole _)|GHole _,None) -> true + | _ -> false + +(* "ssrpattern" *) +let pr_ssrpatternarg _ _ _ cpat = pr_rpattern cpat + +ARGUMENT EXTEND ssrpatternarg + TYPED AS rpattern + PRINTED BY pr_ssrpatternarg +| [ "[" rpattern(pat) "]" ] -> [ pat ] +END + +let pf_merge_uc uc gl = + re_sig (sig_it gl) (Evd.merge_universe_context (project gl) uc) + +let ssrpatterntac ist arg gl = + let pat = interp_rpattern ist gl arg in + let sigma0 = project gl in + let concl0 = pf_concl gl in + let (t, uc), concl_x = + fill_occ_pattern (Global.env()) sigma0 concl0 pat noindex 1 in + let tty = pf_type_of gl t in + let concl = mkLetIn (Name (id_of_string "toto"), t, tty, concl_x) in + Proofview.V82.of_tactic (convert_concl concl DEFAULTcast) gl + +TACTIC EXTEND ssrat +| [ "ssrpattern" ssrpatternarg(arg) ] -> [ Proofview.V82.tactic (ssrpatterntac ist arg) ] +END + + +(* We wipe out all the keywords generated by the grammar rules we defined. *) +(* The user is supposed to Require Import ssreflect or Require ssreflect *) +(* and Import ssreflect.SsrSyntax to obtain these keywords and as a *) +(* consequence the extended ssreflect grammar. *) +let () = Lexer.unfreeze frozen_lexer ;; + +(* vim: set filetype=ocaml foldmethod=marker: *) diff --git a/mathcomp/ssreflect/plugin/trunk/ssrmatching.mli b/mathcomp/ssreflect/plugin/trunk/ssrmatching.mli new file mode 100644 index 0000000..b355dc1 --- /dev/null +++ b/mathcomp/ssreflect/plugin/trunk/ssrmatching.mli @@ -0,0 +1,238 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +open Genarg +open Tacexpr +open Environ +open Evd +open Proof_type +open Term + +(** ******** Small Scale Reflection pattern matching facilities ************* *) + +(** Pattern parsing *) + +(** The type of context patterns, the patterns of the [set] tactic and + [:] tactical. These are patterns that identify a precise subterm. *) +type cpattern +val pr_cpattern : cpattern -> Pp.std_ppcmds + +(** CS cpattern: (f _), (X in t), (t in X in t), (t as X in t) *) +val cpattern : cpattern Pcoq.Gram.entry +val wit_cpattern : cpattern uniform_genarg_type + +(** OS cpattern: f _, (X in t), (t in X in t), (t as X in t) *) +val lcpattern : cpattern Pcoq.Gram.entry +val wit_lcpattern : cpattern uniform_genarg_type + +(** The type of rewrite patterns, the patterns of the [rewrite] tactic. + These patterns also include patterns that identify all the subterms + of a context (i.e. "in" prefix) *) +type rpattern +val pr_rpattern : rpattern -> Pp.std_ppcmds + +(** OS rpattern: f _, in t, X in t, in X in t, t in X in t, t as X in t *) +val rpattern : rpattern Pcoq.Gram.entry +val wit_rpattern : rpattern uniform_genarg_type + +(** Pattern interpretation and matching *) + +exception NoMatch +exception NoProgress + +(** AST for [rpattern] (and consequently [cpattern]) *) +type ('ident, 'term) ssrpattern = + | T of 'term + | In_T of 'term + | X_In_T of 'ident * 'term + | In_X_In_T of 'ident * 'term + | E_In_X_In_T of 'term * 'ident * 'term + | E_As_X_In_T of 'term * 'ident * 'term + +type pattern = evar_map * (constr, constr) ssrpattern +val pp_pattern : pattern -> Pp.std_ppcmds + +(** Extracts the redex and applies to it the substitution part of the pattern. + @raise Anomaly if called on [In_T] or [In_X_In_T] *) +val redex_of_pattern : + ?resolve_typeclasses:bool -> env -> pattern -> + constr Evd.in_evar_universe_context + +(** [interp_rpattern ise gl rpat] "internalizes" and "interprets" [rpat] + in the current [Ltac] interpretation signature [ise] and tactic input [gl]*) +val interp_rpattern : + Tacinterp.interp_sign -> goal Tacmach.sigma -> + rpattern -> + pattern + +(** [interp_cpattern ise gl cpat ty] "internalizes" and "interprets" [cpat] + in the current [Ltac] interpretation signature [ise] and tactic input [gl]. + [ty] is an optional type for the redex of [cpat] *) +val interp_cpattern : + Tacinterp.interp_sign -> goal Tacmach.sigma -> + cpattern -> glob_constr_and_expr option -> + pattern + +(** The set of occurrences to be matched. The boolean is set to true + * to signal the complement of this set (i.e. {-1 3}) *) +type occ = (bool * int list) option + +(** Substitution function. The [int] argument is the number of binders + traversed so far *) +type subst = env -> constr -> int -> constr + +(** [eval_pattern b env sigma t pat occ subst] maps [t] calling [subst] on every + [occ] occurrence of [pat]. The [int] argument is the number of + binders traversed. If [pat] is [None] then then subst is called on [t]. + [t] must live in [env] and [sigma], [pat] must have been interpreted in + (an extension of) [sigma]. + @raise NoMatch if [pat] has no occurrence and [b] is [true] (default [false]) + @return [t] where all [occ] occurrences of [pat] have been mapped using + [subst] *) +val eval_pattern : + ?raise_NoMatch:bool -> + env -> evar_map -> constr -> + pattern option -> occ -> subst -> + constr + +(** [fill_occ_pattern b env sigma t pat occ h] is a simplified version of + [eval_pattern]. + It replaces all [occ] occurrences of [pat] in [t] with Rel [h]. + [t] must live in [env] and [sigma], [pat] must have been interpreted in + (an extension of) [sigma]. + @raise NoMatch if [pat] has no occurrence and [b] is [true] (default [false]) + @return the instance of the redex of [pat] that was matched and [t] + transformed as described above. *) +val fill_occ_pattern : + ?raise_NoMatch:bool -> + env -> evar_map -> constr -> + pattern -> occ -> int -> + constr Evd.in_evar_universe_context * constr + +(** *************************** Low level APIs ****************************** *) + +(* The primitive matching facility. It matches of a term with holes, like + the T pattern above, and calls a continuation on its occurrences. *) + +type ssrdir = L2R | R2L +val pr_dir_side : ssrdir -> Pp.std_ppcmds + +(** a pattern for a term with wildcards *) +type tpattern + +(** [mk_tpattern env sigma0 sigma_t ok p_origin dir p] compiles a term [t] + living in [env] [sigma] (an extension of [sigma0]) intro a [tpattern]. + The [tpattern] can hold a (proof) term [p] and a diction [dir]. The [ok] + callback is used to filter occurrences. + @return the compiled [tpattern] and its [evar_map] + @raise UserEerror is the pattern is a wildcard *) +val mk_tpattern : + ?p_origin:ssrdir * constr -> + env -> evar_map -> + evar_map * constr -> + (constr -> evar_map -> bool) -> + ssrdir -> constr -> + evar_map * tpattern + +(** [findP env t i k] is a stateful function that finds the next occurrence + of a tpattern and calls the callback [k] to map the subterm matched. + The [int] argument passed to [k] is the number of binders traversed so far + plus the initial value [i]. + @return [t] where the subterms identified by the selected occurrences of + the patter have been mapped using [k] + @raise NoMatch if the raise_NoMatch flag given to [mk_tpattern_matcher] is + [true] and if the pattern did not match + @raise UserEerror if the raise_NoMatch flag given to [mk_tpattern_matcher] is + [false] and if the pattern did not match *) +type find_P = + env -> constr -> int -> k:subst -> constr + +(** [conclude ()] asserts that all mentioned ocurrences have been visited. + @return the instance of the pattern, the evarmap after the pattern + instantiation, the proof term and the ssrdit stored in the tpattern + @raise UserEerror if too many occurrences were specified *) +type conclude = + unit -> constr * ssrdir * (evar_map * Evd.evar_universe_context * constr) + +(** [mk_tpattern_matcher b o sigma0 occ sigma_tplist] creates a pair + a function [find_P] and [conclude] with the behaviour explained above. + The flag [b] (default [false]) changes the error reporting behaviour + of [find_P] if none of the [tpattern] matches. The argument [o] can + be passed to tune the [UserError] eventually raised (useful if the + pattern is coming from the LHS/RHS of an equation) *) +val mk_tpattern_matcher : + ?raise_NoMatch:bool -> + ?upats_origin:ssrdir * constr -> + evar_map -> occ -> evar_map * tpattern list -> + find_P * conclude + +(** Example of [mk_tpattern_matcher] to implement + [rewrite \{occ\}\[in t\]rules]. + It first matches "in t" (called [pat]), then in all matched subterms + it matches the LHS of the rules using [find_R]. + [concl0] is the initial goal, [concl] will be the goal where some terms + are replaced by a De Bruijn index. The [rw_progress] extra check + selects only occurrences that are not rewritten to themselves (e.g. + an occurrence "x + x" rewritten with the commutativity law of addition + is skipped) {[ + let find_R, conclude = match pat with + | Some (_, In_T _) -> + let aux (sigma, pats) (d, r, lhs, rhs) = + let sigma, pat = + mk_tpattern env0 sigma0 (sigma, r) (rw_progress rhs) d lhs in + sigma, pats @ [pat] in + let rpats = List.fold_left aux (r_sigma, []) rules in + let find_R, end_R = mk_tpattern_matcher sigma0 occ rpats in + find_R ~k:(fun _ _ h -> mkRel h), + fun cl -> let rdx, d, r = end_R () in (d,r),rdx + | _ -> ... in + let concl = eval_pattern env0 sigma0 concl0 pat occ find_R in + let (d, r), rdx = conclude concl in ]} *) + +(* convenience shortcut: [pf_fill_occ_term gl occ (sigma,t)] returns + * the conclusion of [gl] where [occ] occurrences of [t] have been replaced + * by [Rel 1] and the instance of [t] *) +val pf_fill_occ_term : + goal Tacmach.sigma -> occ -> evar_map * constr -> constr * constr + +(* It may be handy to inject a simple term into the first form of cpattern *) +val cpattern_of_term : char * glob_constr_and_expr -> cpattern + +(** Helpers to make stateful closures. Example: a [find_P] function may be + called many times, but the pattern instantiation phase is performed only the + first time. The corresponding [conclude] has to return the instantiated + pattern redex. Since it is up to [find_P] to raise [NoMatch] if the pattern + has no instance, [conclude] considers it an anomaly if the pattern did + not match *) + +(** [do_once r f] calls [f] and updates the ref only once *) +val do_once : 'a option ref -> (unit -> 'a) -> unit +(** [assert_done r] return the content of r. @raise Anomaly is r is [None] *) +val assert_done : 'a option ref -> 'a + +(** Very low level APIs. + these are calls to evarconv's [the_conv_x] followed by + [consider_remaining_unif_problems] and [resolve_typeclasses]. + In case of failure they raise [NoMatch] *) + +val unify_HO : env -> evar_map -> constr -> constr -> evar_map +val pf_unify_HO : goal Tacmach.sigma -> constr -> constr -> goal Tacmach.sigma + +(** Some more low level functions needed to implement the full SSR language + on top of the former APIs *) +val tag_of_cpattern : cpattern -> char +val loc_of_cpattern : cpattern -> Loc.t +val id_of_cpattern : cpattern -> Names.variable option +val is_wildcard : cpattern -> bool +val cpattern_of_id : Names.variable -> cpattern +val cpattern_of_id : Names.variable -> cpattern +val pr_constr_pat : constr -> Pp.std_ppcmds +val pf_merge_uc : Evd.evar_universe_context -> goal Evd.sigma -> goal Evd.sigma + +(* One can also "Set SsrMatchingDebug" from a .v *) +val debug : bool -> unit + +(* One should delimit a snippet with "Set SsrMatchingProfiling" and + * "Unset SsrMatchingProfiling" to get timings *) +val profile : bool -> unit + +(* eof *) diff --git a/mathcomp/ssreflect/plugin/v8.4/ssreflect.ml4 b/mathcomp/ssreflect/plugin/v8.4/ssreflect.ml4 new file mode 100644 index 0000000..c7a104b --- /dev/null +++ b/mathcomp/ssreflect/plugin/v8.4/ssreflect.ml4 @@ -0,0 +1,6030 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +(* This line is read by the Makefile's dist target: do not remove. *) +let ssrversion = "1.5";; +let ssrAstVersion = 1;; +let () = Mltop.add_known_plugin (fun () -> + if Flags.is_verbose () && not !Flags.batch_mode then begin + Printf.printf "\nSmall Scale Reflection version %s loaded.\n" ssrversion; + Printf.printf "Copyright 2005-2012 Microsoft Corporation and INRIA.\n"; + Printf.printf "Distributed under the terms of the CeCILL-B license.\n\n" + end; + (* Disable any semantics associated with bullets *) + Goptions.set_string_option_value_gen + (Some false) ["Bullet";"Behavior"] "None") + "ssreflect" +;; +(* Defining grammar rules with "xx" in it automatically declares keywords too *) +let frozen_lexer = Lexer.freeze () ;; + +(*i camlp4use: "pa_extend.cmo" i*) +(*i camlp4deps: "parsing/grammar.cma" i*) + +open Names +open Pp +open Pcoq +open Genarg +open Term +open Topconstr +open Libnames +open Tactics +open Tacticals +open Termops +open Namegen +open Recordops +open Tacmach +open Coqlib +open Glob_term +open Util +open Evd +open Extend +open Goptions +open Tacexpr +open Tacinterp +open Pretyping.Default +open Constr +open Tactic +open Extraargs +open Ppconstr +open Printer + +open Ssrmatching + +(** look up a name in the ssreflect internals module *) +let ssrdirpath = make_dirpath [id_of_string "ssreflect"] +let ssrqid name = make_qualid ssrdirpath (id_of_string name) +let ssrtopqid name = make_short_qualid (id_of_string name) +let locate_reference qid = + Smartlocate.global_of_extended_global (Nametab.locate_extended qid) +let mkSsrRef name = + try locate_reference (ssrqid name) with Not_found -> + try locate_reference (ssrtopqid name) with Not_found -> + error "Small scale reflection library not loaded" +let mkSsrRRef name = GRef (dummy_loc, mkSsrRef name) +let mkSsrConst name = constr_of_reference (mkSsrRef name) +let loc_error loc msg = user_err_loc (loc, msg, str msg) +let errorstrm = errorlabstrm "ssreflect" + +(** Ssreflect load check. *) + +(* To allow ssrcoq to be fully compatible with the "plain" Coq, we only *) +(* turn on its incompatible features (the new rewrite syntax, and the *) +(* reserved identifiers) when the theory library (ssreflect.v) has *) +(* has actually been required, or is being defined. Because this check *) +(* needs to be done often (for each identifier lookup), we implement *) +(* some caching, repeating the test only when the environment changes. *) +(* We check for protect_term because it is the first constant loaded; *) +(* ssr_have would ultimately be a better choice. *) +let ssr_loaded = + let cache = ref (Global.safe_env (), false) in + fun () -> + Lexer.is_keyword "is" && + let new_lbl = Global.safe_env () in + match !cache with + | lbl, loaded when lbl == new_lbl -> loaded + | _ -> + let loaded = + (try ignore (mkSsrRef "protect_term"); true with _ -> false) in + cache := new_lbl, loaded; loaded + +(* 0 cost pp function. Active only if env variable SSRDEBUG is set *) +(* or if SsrDebug is Set *) +let pp_ref = ref (fun _ -> ()) +let ssr_pp s = pperrnl (str"SSR: "++Lazy.force s) +let _ = try ignore(Sys.getenv "SSRDEBUG"); pp_ref := ssr_pp with Not_found -> () +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect debugging"; + Goptions.optkey = ["SsrDebug"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !pp_ref == ssr_pp); + Goptions.optwrite = (fun b -> + Ssrmatching.debug b; + if b then pp_ref := ssr_pp else pp_ref := fun _ -> ()) } +let pp s = !pp_ref s + +(** Utils {{{ *****************************************************************) +let env_size env = List.length (Environ.named_context env) +let safeDestApp c = + match kind_of_term c with App (f, a) -> f, a | _ -> c, [| |] +let get_index = function ArgArg i -> i | _ -> + anomaly "Uninterpreted index" +(* Toplevel constr must be globalized twice ! *) +let glob_constr ist gsigma genv = function + | _, Some ce -> + let ltacvars = List.map fst ist.lfun, [] in + Constrintern.intern_gen false ~ltacvars:ltacvars gsigma genv ce + | rc, None -> rc + +(* Term printing utilities functions for deciding bracketing. *) +let pr_paren prx x = hov 1 (str "(" ++ prx x ++ str ")") +(* String lexing utilities *) +let skip_wschars s = + let rec loop i = match s.[i] with '\n'..' ' -> loop (i + 1) | _ -> i in loop +let skip_numchars s = + let rec loop i = match s.[i] with '0'..'9' -> loop (i + 1) | _ -> i in loop +(* We also guard characters that might interfere with the ssreflect *) +(* tactic syntax. *) +let guard_term ch1 s i = match s.[i] with + | '(' -> false + | '{' | '/' | '=' -> true + | _ -> ch1 = '(' +(* The call 'guard s i' should return true if the contents of s *) +(* starting at i need bracketing to avoid ambiguities. *) +let pr_guarded guard prc c = + msg_with Format.str_formatter (prc c); + let s = Format.flush_str_formatter () ^ "$" in + if guard s (skip_wschars s 0) then pr_paren prc c else prc c +(* More sensible names for constr printers *) +let prl_constr = pr_lconstr +let pr_constr = pr_constr +let prl_glob_constr c = pr_lglob_constr_env (Global.env ()) c +let pr_glob_constr c = pr_glob_constr_env (Global.env ()) c +let prl_constr_expr = pr_lconstr_expr +let pr_constr_expr = pr_constr_expr +let prl_glob_constr_and_expr = function + | _, Some c -> prl_constr_expr c + | c, None -> prl_glob_constr c +let pr_glob_constr_and_expr = function + | _, Some c -> pr_constr_expr c + | c, None -> pr_glob_constr c +let pr_term (k, c) = pr_guarded (guard_term k) pr_glob_constr_and_expr c +let prl_term (k, c) = pr_guarded (guard_term k) prl_glob_constr_and_expr c + +(** Adding a new uninterpreted generic argument type *) +let add_genarg tag pr = + let wit, globwit, rawwit as wits = create_arg None tag in + let glob _ rarg = in_gen globwit (out_gen rawwit rarg) in + let interp _ gl garg = Tacmach.project gl,in_gen wit (out_gen globwit garg) in + let subst _ garg = garg in + add_interp_genarg tag (glob, interp, subst); + let gen_pr _ _ _ = pr in + Pptactic.declare_extra_genarg_pprule + (rawwit, gen_pr) (globwit, gen_pr) (wit, gen_pr); + wits + +(** Constructors for cast type *) +let dC t = CastConv (DEFAULTcast,t) + +(** Constructors for constr_expr *) +let mkCProp loc = CSort (loc, GProp Null) +let mkCType loc = CSort (loc, GType None) +let mkCVar loc id = CRef (Ident (loc, id)) +let isCVar = function CRef (Ident _) -> true | _ -> false +let destCVar = function CRef (Ident (_, id)) -> id | _ -> + anomaly "not a CRef" +let rec mkCHoles loc n = + if n <= 0 then [] else CHole (loc, None) :: mkCHoles loc (n - 1) +let mkCHole loc = CHole (loc, None) +let rec isCHoles = function CHole _ :: cl -> isCHoles cl | cl -> cl = [] +let mkCExplVar loc id n = + CAppExpl (loc, (None, Ident (loc, id)), mkCHoles loc n) +let mkCLambda loc name ty t = + CLambdaN (loc, [[loc, name], Default Explicit, ty], t) +let mkCLetIn loc name bo t = + CLetIn (loc, (loc, name), bo, t) +let mkCArrow loc ty t = + CProdN (loc, [[dummy_loc,Anonymous], Default Explicit, ty], t) +let mkCCast loc t ty = CCast (loc,t, dC ty) +(** Constructors for rawconstr *) +let mkRHole = GHole (dummy_loc, InternalHole) +let rec mkRHoles n = if n > 0 then mkRHole :: mkRHoles (n - 1) else [] +let rec isRHoles = function GHole _ :: cl -> isRHoles cl | cl -> cl = [] +let mkRApp f args = if args = [] then f else GApp (dummy_loc, f, args) +let mkRVar id = GRef (dummy_loc, VarRef id) +let mkRltacVar id = GVar (dummy_loc, id) +let mkRCast rc rt = GCast (dummy_loc, rc, dC rt) +let mkRType = GSort (dummy_loc, GType None) +let mkRProp = GSort (dummy_loc, GProp Null) +let mkRArrow rt1 rt2 = GProd (dummy_loc, Anonymous, Explicit, rt1, rt2) +let mkRConstruct c = GRef (dummy_loc, ConstructRef c) +let mkRInd mind = GRef (dummy_loc, IndRef mind) +let mkRLambda n s t = GLambda (dummy_loc, n, Explicit, s, t) + +(** Constructors for constr *) + +let mkAppRed f c = match kind_of_term f with +| Lambda (_, _, b) -> subst1 c b +| _ -> mkApp (f, [|c|]) +let mkProt t c = mkApp (mkSsrConst "protect_term", [|t; c|]) +let mkRefl t c = mkApp ((build_coq_eq_data()).refl, [|t; c|]) +(* Application to a sequence of n rels (for building eta-expansions). *) +(* The rel indices decrease down to imin (inclusive), unless n < 0, *) +(* in which case they're incresing (from imin). *) +let mkEtaApp c n imin = + if n = 0 then c else + let nargs, mkarg = + if n < 0 then -n, (fun i -> mkRel (imin + i)) else + let imax = imin + n - 1 in n, (fun i -> mkRel (imax - i)) in + mkApp (c, Array.init nargs mkarg) +(* Same, but optimizing head beta redexes *) +let rec whdEtaApp c n = + if n = 0 then c else match kind_of_term c with + | Lambda (_, _, c') -> whdEtaApp c' (n - 1) + | _ -> mkEtaApp (lift n c) n 1 +let mkType () = mkType (Univ.fresh_local_univ ()) + +(* ssrterm conbinators *) +let combineCG t1 t2 f g = match t1, t2 with + | (x, (t1, None)), (_, (t2, None)) -> x, (g t1 t2, None) + | (x, (_, Some t1)), (_, (_, Some t2)) -> x, (mkRHole, Some (f t1 t2)) + | _, (_, (_, None)) -> anomaly "have: mixed C-G constr" + | _ -> anomaly "have: mixed G-C constr" +let loc_ofCG = function + | (_, (s, None)) -> loc_of_glob_constr s + | (_, (_, Some s)) -> constr_loc s + +let mk_term k c = k, (mkRHole, Some c) +let mk_lterm = mk_term ' ' + +(* Backport from coq trunk *) +let array_smartfoldmap f accu (ar : 'a array) = + let len = Array.length ar in + let i = ref 0 in + let break = ref true in + let r = ref accu in + (** This variable is never accessed unset *) + let temp = ref None in + while !break && (!i < len) do + let v = Array.unsafe_get ar !i in + let (accu, v') = f !r v in + r := accu; + if v == v' then incr i + else begin + break := false; + temp := Some v'; + end + done; + if !i < len then begin + let ans : 'a array = Array.make len (Array.unsafe_get ar 0) in + (** TODO: use unsafe_blit in 4.01 *) + Array.blit ar 0 ans 0 !i; + let v = match !temp with None -> assert false | Some x -> x in + Array.unsafe_set ans !i v; + incr i; + while !i < len do + let v = Array.unsafe_get ar !i in + let (accu, v') = f !r v in + r := accu; + Array.unsafe_set ans !i v'; + incr i + done; + !r, ans + end else !r, ar + +let array_for_all2 f v1 v2 = + let rec allrec = function + | -1 -> true + | n -> + let ans = f (Array.unsafe_get v1 n) (Array.unsafe_get v2 n) in + ans && (allrec (n-1)) + in + let lv1 = Array.length v1 in + lv1 = Array.length v2 && allrec (pred lv1) + +let fold_left2 f a v1 v2 = + let lv1 = Array.length v1 in + let rec fold a n = + if n >= lv1 then a + else fold (f a (Array.unsafe_get v1 n) (Array.unsafe_get v2 n)) (succ n) + in + if Array.length v2 <> lv1 then invalid_arg "Array.fold_left2"; + fold a 0 +(* /Backport *) + +let map_fold_constr g f ctx acc cstr = + let array_f ctx acc x = let x, acc = f ctx acc x in acc, x in + match kind_of_term cstr with + | (Rel _ | Meta _ | Var _ | Sort _ | Const _ | Ind _ | Construct _) -> + cstr, acc + | Cast (c,k, t) -> + let c', acc = f ctx acc c in + let t', acc = f ctx acc t in + (if c==c' && t==t' then cstr else mkCast (c', k, t')), acc + | Prod (na,t,c) -> + let t', acc = f ctx acc t in + let c', acc = f (g (na,None,t) ctx) acc c in + (if t==t' && c==c' then cstr else mkProd (na, t', c')), acc + | Lambda (na,t,c) -> + let t', acc = f ctx acc t in + let c', acc = f (g (na,None,t) ctx) acc c in + (if t==t' && c==c' then cstr else mkLambda (na, t', c')), acc + | LetIn (na,b,t,c) -> + let b', acc = f ctx acc b in + let t', acc = f ctx acc t in + let c', acc = f (g (na,Some b,t) ctx) acc c in + (if b==b' && t==t' && c==c' then cstr else mkLetIn (na, b', t', c')), acc + | App (c,al) -> + let c', acc = f ctx acc c in + let acc, al' = array_smartfoldmap (array_f ctx) acc al in + (if c==c' && array_for_all2 (==) al al' then cstr else mkApp (c', al')), + acc + | Evar (e,al) -> + let acc, al' = array_smartfoldmap (array_f ctx) acc al in + (if array_for_all2 (==) al al' then cstr else mkEvar (e, al')), acc + | Case (ci,p,c,bl) -> + let p', acc = f ctx acc p in + let c', acc = f ctx acc c in + let acc, bl' = array_smartfoldmap (array_f ctx) acc bl in + (if p==p' && c==c' && array_for_all2 (==) bl bl' then cstr else + mkCase (ci, p', c', bl')), + acc + | Fix (ln,(lna,tl,bl)) -> + let acc, tl' = array_smartfoldmap (array_f ctx) acc tl in + let ctx' = array_fold_left2 (fun l na t -> g (na,None,t) l) ctx lna tl in + let acc, bl' = array_smartfoldmap (array_f ctx') acc bl in + (if array_for_all2 (==) tl tl' && array_for_all2 (==) bl bl' + then cstr + else mkFix (ln,(lna,tl',bl'))), acc + | CoFix(ln,(lna,tl,bl)) -> + let acc, tl' = array_smartfoldmap (array_f ctx) acc tl in + let ctx' = array_fold_left2 (fun l na t -> g (na,None,t) l) ctx lna tl in + let acc,bl' = array_smartfoldmap (array_f ctx') acc bl in + (if array_for_all2 (==) tl tl' && array_for_all2 (==) bl bl' + then cstr + else mkCoFix (ln,(lna,tl',bl'))), acc + + +(* }}} *) + +(** Profiling {{{ *************************************************************) +type profiler = { + profile : 'a 'b. ('a -> 'b) -> 'a -> 'b; + reset : unit -> unit; + print : unit -> unit } +let profile_now = ref false +let something_profiled = ref false +let profilers = ref [] +let add_profiler f = profilers := f :: !profilers;; +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect profiling"; + Goptions.optkey = ["SsrProfiling"]; + Goptions.optread = (fun _ -> !profile_now); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> + Ssrmatching.profile b; + profile_now := b; + if b then List.iter (fun f -> f.reset ()) !profilers; + if not b then List.iter (fun f -> f.print ()) !profilers) } +let () = + let prof_total = + let init = ref 0.0 in { + profile = (fun f x -> assert false); + reset = (fun () -> init := Unix.gettimeofday ()); + print = (fun () -> if !something_profiled then + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + "total" 0 (Unix.gettimeofday() -. !init) 0.0 0.0)) } in + let prof_legenda = { + profile = (fun f x -> assert false); + reset = (fun () -> ()); + print = (fun () -> if !something_profiled then begin + prerr_endline + (Printf.sprintf "!! %39s ---------- --------- --------- ---------" + (String.make 39 '-')); + prerr_endline + (Printf.sprintf "!! %-39s %10s %9s %9s %9s" + "function" "#calls" "total" "max" "average") end) } in + add_profiler prof_legenda; + add_profiler prof_total +;; + +let mk_profiler s = + let total, calls, max = ref 0.0, ref 0, ref 0.0 in + let reset () = total := 0.0; calls := 0; max := 0.0 in + let profile f x = + if not !profile_now then f x else + let before = Unix.gettimeofday () in + try + incr calls; + let res = f x in + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + res + with exc -> + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + raise exc in + let print () = + if !calls <> 0 then begin + something_profiled := true; + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + s !calls !total !max (!total /. (float_of_int !calls))) end in + let prof = { profile = profile; reset = reset; print = print } in + add_profiler prof; + prof +;; +(* }}} *) + +let inVersion = Libobject.declare_object { + (Libobject.default_object "SSRASTVERSION") with + Libobject.load_function = (fun _ (_,v) -> + if v <> ssrAstVersion then error "Please recompile your .vo files"); + Libobject.classify_function = (fun v -> Libobject.Keep v); +} + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect version"; + Goptions.optkey = ["SsrAstVersion"]; + Goptions.optread = (fun _ -> true); + Goptions.optdepr = false; + Goptions.optwrite = (fun _ -> + Lib.add_anonymous_leaf (inVersion ssrAstVersion)) } + +let tactic_expr = Tactic.tactic_expr +let gallina_ext = Vernac_.gallina_ext +let sprintf = Printf.sprintf +let tactic_mode = G_vernac.tactic_mode + +(** 1. Utilities *) + + +let ssroldreworder = ref true +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect 1.3 compatibility flag"; + Goptions.optkey = ["SsrOldRewriteGoalsOrder"]; + Goptions.optread = (fun _ -> !ssroldreworder); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssroldreworder := b) } + +let ssrhaveNOtcresolution = ref false + +let inHaveTCResolution = Libobject.declare_object { + (Libobject.default_object "SSRHAVETCRESOLUTION") with + Libobject.cache_function = (fun (_,v) -> ssrhaveNOtcresolution := v); + Libobject.load_function = (fun _ (_,v) -> ssrhaveNOtcresolution := v); + Libobject.classify_function = (fun v -> Libobject.Keep v); +} + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "have type classes"; + Goptions.optkey = ["SsrHave";"NoTCResolution"]; + Goptions.optread = (fun _ -> !ssrhaveNOtcresolution); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> + Lib.add_anonymous_leaf (inHaveTCResolution b)) } + + +(** Primitive parsing to avoid syntax conflicts with basic tactics. *) + +let accept_before_syms syms strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | _ -> raise Stream.Failure + +let accept_before_syms_or_any_id syms strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | Tok.IDENT _ -> () + | _ -> raise Stream.Failure + +let accept_before_syms_or_ids syms ids strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | Tok.IDENT id when List.mem id ids -> () + | _ -> raise Stream.Failure + +(** Pretty-printing utilities *) + +let pr_id = Ppconstr.pr_id +let pr_name = function Name id -> pr_id id | Anonymous -> str "_" +let pr_spc () = str " " +let pr_bar () = Pp.cut() ++ str "|" +let pr_list = prlist_with_sep + +let tacltop = (5,Ppextend.E) + +(** Tactic-level diagnosis *) + +let pf_pr_constr gl = pr_constr_env (pf_env gl) + +let pf_pr_glob_constr gl = pr_glob_constr_env (pf_env gl) + +(* debug *) + +let pf_msg gl = + let ppgl = pr_lconstr_env (pf_env gl) (pf_concl gl) in + msgnl (str "goal is " ++ ppgl) + +let msgtac gl = pf_msg gl; tclIDTAC gl + +(** Tactic utilities *) + +let last_goal gls = let sigma, gll = Refiner.unpackage gls in + Refiner.repackage sigma (List.nth gll (List.length gll - 1)) + +let pf_type_id gl t = id_of_string (hdchar (pf_env gl) t) + +let not_section_id id = not (is_section_variable id) + +let is_pf_var c = isVar c && not_section_id (destVar c) + +let pf_ids_of_proof_hyps gl = + let add_hyp (id, _, _) ids = if not_section_id id then id :: ids else ids in + Sign.fold_named_context add_hyp (pf_hyps gl) ~init:[] + +let pf_nf_evar gl e = Reductionops.nf_evar (project gl) e + +let pf_partial_solution ?(shelve=false) gl t evl = + let sigma, g = project gl, sig_it gl in + let evars_of_term c = + let rec evrec acc c = + match kind_of_term c with + | Evar (n, l) -> Evd.ExistentialSet.add n (Array.fold_left evrec acc l) + | _ -> fold_constr evrec acc c + in + evrec Evd.ExistentialSet.empty c in + let evars_of_named_context nc = + List.fold_right (fun (_, b, t) s -> + Option.fold_left (fun s t -> + Evd.ExistentialSet.union s (evars_of_term t)) + (Evd.ExistentialSet.union s (evars_of_term t)) b) + nc Evd.ExistentialSet.empty in + let evars_of_filtered_evar_info evi = + Evd.ExistentialSet.union (evars_of_term evi.evar_concl) + (Evd.ExistentialSet.union + (match evi.evar_body with + | Evar_empty -> Evd.ExistentialSet.empty + | Evar_defined b -> evars_of_term b) + (evars_of_named_context (evar_filtered_context evi))) in + let depends_on src tgt = + let evi = Evd.find sigma tgt in + Evd.ExistentialSet.mem src + (evars_of_filtered_evar_info (Evarutil.nf_evar_info sigma evi)) in + let occurs_in_another_statement ev = + List.exists (fun x -> + let ev', _ = destEvar x in + not (ev = ev') && depends_on ev ev') evl in + let sigma = Goal.V82.partial_solution sigma g t in + re_sig (Util.list_map_filter (fun x -> + let ev, _ = destEvar x in + if shelve && occurs_in_another_statement ev then None + else Some (Goal.build ev)) evl) sigma + +let pf_new_evar gl ty = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, extra = Evarutil.new_evar sigma env ty in + re_sig it sigma, extra + +(* Basic tactics *) + +let convert_concl_no_check t = convert_concl_no_check t DEFAULTcast +let convert_concl t = convert_concl t DEFAULTcast +let reduct_in_concl t = reduct_in_concl (t, DEFAULTcast) +let havetac id = pose_proof (Name id) +let settac id c = letin_tac None (Name id) c None +let posetac id cl = settac id cl nowhere +let basecuttac name c = apply (mkApp (mkSsrConst name, [|c|])) + +(* we reduce head beta redexes *) +let betared env = + Closure.create_clos_infos + (Closure.RedFlags.mkflags [Closure.RedFlags.fBETA]) + env +;; +let introid name = tclTHEN (fun gl -> + let g, env = pf_concl gl, pf_env gl in + match kind_of_term g with + | App (hd, _) when isLambda hd -> + let g = Closure.whd_val (betared env) (Closure.inject g) in + convert_concl_no_check g gl + | _ -> tclIDTAC gl) + (intro_mustbe_force name) +;; + +(** Name generation {{{ *******************************************************) + +(* Since Coq now does repeated internal checks of its external lexical *) +(* rules, we now need to carve ssreflect reserved identifiers out of *) +(* out of the user namespace. We use identifiers of the form _id_ for *) +(* this purpose, e.g., we "anonymize" an identifier id as _id_, adding *) +(* an extra leading _ if this might clash with an internal identifier. *) +(* We check for ssreflect identifiers in the ident grammar rule; *) +(* when the ssreflect Module is present this is normally an error, *) +(* but we provide a compatibility flag to reduce this to a warning. *) + +let ssr_reserved_ids = ref true + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "ssreflect identifiers"; + Goptions.optkey = ["SsrIdents"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !ssr_reserved_ids); + Goptions.optwrite = (fun b -> ssr_reserved_ids := b) + } + +let is_ssr_reserved s = + let n = String.length s in n > 2 && s.[0] = '_' && s.[n - 1] = '_' + +let internal_names = ref [] +let add_internal_name pt = internal_names := pt :: !internal_names +let is_internal_name s = List.exists (fun p -> p s) !internal_names + +let ssr_id_of_string loc s = + if is_ssr_reserved s && ssr_loaded () then begin + if !ssr_reserved_ids then + loc_error loc ("The identifier " ^ s ^ " is reserved.") + else if is_internal_name s then + warning ("Conflict between " ^ s ^ " and ssreflect internal names.") + else warning ( + "The name " ^ s ^ " fits the _xxx_ format used for anonymous variables.\n" + ^ "Scripts with explicit references to anonymous variables are fragile.") + end; id_of_string s + +let ssr_null_entry = Gram.Entry.of_parser "ssr_null" (fun _ -> ()) + +GEXTEND Gram + GLOBAL: Prim.ident; + Prim.ident: [[ s = IDENT; ssr_null_entry -> ssr_id_of_string loc s ]]; +END + +let mk_internal_id s = + let s' = sprintf "_%s_" s in + for i = 1 to String.length s do if s'.[i] = ' ' then s'.[i] <- '_' done; + add_internal_name ((=) s'); id_of_string s' + +let same_prefix s t n = + let rec loop i = i = n || s.[i] = t.[i] && loop (i + 1) in loop 0 + +let skip_digits s = + let n = String.length s in + let rec loop i = if i < n && is_digit s.[i] then loop (i + 1) else i in loop + +let mk_tagged_id t i = id_of_string (sprintf "%s%d_" t i) +let is_tagged t s = + let n = String.length s - 1 and m = String.length t in + m < n && s.[n] = '_' && same_prefix s t m && skip_digits s m = n + +let perm_tag = "_perm_Hyp_" +let _ = add_internal_name (is_tagged perm_tag) +let mk_perm_id = + let salt = ref 1 in + fun () -> salt := !salt mod 10000 + 1; mk_tagged_id perm_tag !salt + +let evar_tag = "_evar_" +let _ = add_internal_name (is_tagged evar_tag) +let mk_evar_name n = Name (mk_tagged_id evar_tag n) +let nb_evar_deps = function + | Name id -> + let s = string_of_id id in + if not (is_tagged evar_tag s) then 0 else + let m = String.length evar_tag in + (try int_of_string (String.sub s m (String.length s - 1 - m)) with _ -> 0) + | _ -> 0 + +let discharged_tag = "_discharged_" +let mk_discharged_id id = + id_of_string (sprintf "%s%s_" discharged_tag (string_of_id id)) +let has_discharged_tag s = + let m = String.length discharged_tag and n = String.length s - 1 in + m < n && s.[n] = '_' && same_prefix s discharged_tag m +let _ = add_internal_name has_discharged_tag +let is_discharged_id id = has_discharged_tag (string_of_id id) + +let wildcard_tag = "_the_" +let wildcard_post = "_wildcard_" +let mk_wildcard_id i = + id_of_string (sprintf "%s%s%s" wildcard_tag (ordinal i) wildcard_post) +let has_wildcard_tag s = + let n = String.length s in let m = String.length wildcard_tag in + let m' = String.length wildcard_post in + n < m + m' + 2 && same_prefix s wildcard_tag m && + String.sub s (n - m') m' = wildcard_post && + skip_digits s m = n - m' - 2 +let _ = add_internal_name has_wildcard_tag + +let max_suffix m (t, j0 as tj0) id = + let s = string_of_id id in let n = String.length s - 1 in + let dn = String.length t - 1 - n in let i0 = j0 - dn in + if not (i0 >= m && s.[n] = '_' && same_prefix s t m) then tj0 else + let rec loop i = + if i < i0 && s.[i] = '0' then loop (i + 1) else + if (if i < i0 then skip_digits s i = n else le_s_t i) then s, i else tj0 + and le_s_t i = + let ds = s.[i] and dt = t.[i + dn] in + if ds = dt then i = n || le_s_t (i + 1) else + dt < ds && skip_digits s i = n in + loop m + +let mk_anon_id t gl = + let m, si0, id0 = + let s = ref (sprintf "_%s_" t) in + if is_internal_name !s then s := "_" ^ !s; + let n = String.length !s - 1 in + let rec loop i j = + let d = !s.[i] in if not (is_digit d) then i + 1, j else + loop (i - 1) (if d = '0' then j else i) in + let m, j = loop (n - 1) n in m, (!s, j), id_of_string !s in + let gl_ids = pf_ids_of_hyps gl in + if not (List.mem id0 gl_ids) then id0 else + let s, i = List.fold_left (max_suffix m) si0 gl_ids in + let n = String.length s - 1 in + let rec loop i = + if s.[i] = '9' then (s.[i] <- '0'; loop (i - 1)) else + if i < m then (s.[n] <- '0'; s.[m] <- '1'; s ^ "_") else + (s.[i] <- Char.chr (Char.code s.[i] + 1); s) in + id_of_string (loop (n - 1)) + +(* We must not anonymize context names discharged by the "in" tactical. *) + +let ssr_anon_hyp = "Hyp" + +let anontac (x, _, _) gl = + let id = match x with + | Name id -> + if is_discharged_id id then id else mk_anon_id (string_of_id id) gl + | _ -> mk_anon_id ssr_anon_hyp gl in + introid id gl + +let rec constr_name c = match kind_of_term c with + | Var id -> Name id + | Cast (c', _, _) -> constr_name c' + | Const cn -> Name (id_of_label (con_label cn)) + | App (c', _) -> constr_name c' + | _ -> Anonymous + + +(* }}} *) + +(** Open term to lambda-term coercion {{{ ************************************) + +(* This operation takes a goal gl and an open term (sigma, t), and *) +(* returns a term t' where all the new evars in sigma are abstracted *) +(* with the mkAbs argument, i.e., for mkAbs = mkLambda then there is *) +(* some duplicate-free array args of evars of sigma such that the *) +(* term mkApp (t', args) is convertible to t. *) +(* This makes a useful shorthand for local definitions in proofs, *) +(* i.e., pose succ := _ + 1 means pose succ := fun n : nat => n + 1, *) +(* and, in context of the the 4CT library, pose mid := maps id means *) +(* pose mid := fun d : detaSet => @maps d d (@id (datum d)) *) +(* Note that this facility does not extend to set, which tries *) +(* instead to fill holes by matching a goal subterm. *) +(* The argument to "have" et al. uses product abstraction, e.g. *) +(* have Hmid: forall s, (maps id s) = s. *) +(* stands for *) +(* have Hmid: forall (d : dataSet) (s : seq d), (maps id s) = s. *) +(* We also use this feature for rewrite rules, so that, e.g., *) +(* rewrite: (plus_assoc _ 3). *) +(* will execute as *) +(* rewrite (fun n => plus_assoc n 3) *) +(* i.e., it will rewrite some subterm .. + (3 + ..) to .. + 3 + ... *) +(* The convention is also used for the argument of the congr tactic, *) +(* e.g., congr (x + _ * 1). *) + +(* Replace new evars with lambda variables, retaining local dependencies *) +(* but stripping global ones. We use the variable names to encode the *) +(* the number of dependencies, so that the transformation is reversible. *) + +let pf_abs_evars gl (sigma, c0) = + let sigma0 = project gl in + let nenv = env_size (pf_env gl) in + let abs_evar n k = + let evi = Evd.find sigma k in + let dc = list_firstn n (evar_filtered_context evi) in + let abs_dc c = function + | x, Some b, t -> mkNamedLetIn x b t (mkArrow t c) + | x, None, t -> mkNamedProd x t c in + let t = Sign.fold_named_context_reverse abs_dc ~init:evi.evar_concl dc in + Evarutil.nf_evar sigma t in + let rec put evlist c = match kind_of_term c with + | Evar (k, a) -> + if List.mem_assoc k evlist || Evd.mem sigma0 k then evlist else + let n = max 0 (Array.length a - nenv) in + let t = abs_evar n k in (k, (n, t)) :: put evlist t + | _ -> fold_constr put evlist c in + let evlist = put [] c0 in + if evlist = [] then 0, c0,[] else + let rec lookup k i = function + | [] -> 0, 0 + | (k', (n, _)) :: evl -> if k = k' then i, n else lookup k (i + 1) evl in + let rec get i c = match kind_of_term c with + | Evar (ev, a) -> + let j, n = lookup ev i evlist in + if j = 0 then map_constr (get i) c else if n = 0 then mkRel j else + mkApp (mkRel j, Array.init n (fun k -> get i a.(n - 1 - k))) + | _ -> map_constr_with_binders ((+) 1) get i c in + let rec loop c i = function + | (_, (n, t)) :: evl -> + loop (mkLambda (mk_evar_name n, get (i - 1) t, c)) (i - 1) evl + | [] -> c in + List.length evlist, loop (get 1 c0) 1 evlist, List.map fst evlist + + + +(* As before but if (?i : T(?j)) and (?j : P : Prop), then the lambda for i + * looks like (fun evar_i : (forall pi : P. T(pi))) thanks to "loopP" and all + * occurrences of evar_i are replaced by (evar_i evar_j) thanks to "app". + * + * If P can be solved by ssrautoprop (that defaults to trivial), then + * the corresponding lambda looks like (fun evar_i : T(c)) where c is + * the solution found by ssrautoprop. + *) +let ssrautoprop_tac = ref (fun gl -> assert false) + +(* Thanks to Arnaud Spiwack for this snippet *) +let call_on_evar tac e s = + let { it = gs ; sigma = s } = tac { it = Goal.build e ; sigma = s } in gs, s + +let pf_abs_evars_pirrel gl (sigma, c0) = + pp(lazy(str"==PF_ABS_EVARS_PIRREL==")); + pp(lazy(str"c0= " ++ pr_constr c0)); + let sigma0 = project gl in + let nenv = env_size (pf_env gl) in + let abs_evar n k = + let evi = Evd.find sigma k in + let dc = list_firstn n (evar_filtered_context evi) in + let abs_dc c = function + | x, Some b, t -> mkNamedLetIn x b t (mkArrow t c) + | x, None, t -> mkNamedProd x t c in + let t = Sign.fold_named_context_reverse abs_dc ~init:evi.evar_concl dc in + Evarutil.nf_evar sigma t in + let rec put evlist c = match kind_of_term c with + | Evar (k, a) -> + if List.mem_assoc k evlist || Evd.mem sigma0 k then evlist else + let n = max 0 (Array.length a - nenv) in + let k_ty = + Retyping.get_sort_family_of + (pf_env gl) sigma (Evd.evar_concl (Evd.find sigma k)) in + let is_prop = k_ty = InProp in + let t = abs_evar n k in (k, (n, t, is_prop)) :: put evlist t + | _ -> fold_constr put evlist c in + let evlist = put [] c0 in + if evlist = [] then 0, c0 else + let pr_constr t = pr_constr (Reductionops.nf_beta (project gl) t) in + let evplist = + let depev = List.fold_left (fun evs (_,(_,t,_)) -> + Intset.union evs (Evarutil.evars_of_term t)) Intset.empty evlist in + List.filter (fun i,(_,_,b) -> b && Intset.mem i depev) evlist in + let evlist, evplist, sigma = + if evplist = [] then evlist, [], sigma else + List.fold_left (fun (ev, evp, sigma) (i, (_,t,_) as p) -> + try + let ng, sigma = call_on_evar !ssrautoprop_tac i sigma in + if (ng <> []) then errorstrm (str "Should we tell the user?"); + List.filter (fun (j,_) -> j <> i) ev, evp, sigma + with _ -> ev, p::evp, sigma) (evlist, [], sigma) (List.rev evplist) in + let c0 = Evarutil.nf_evar sigma c0 in + let evlist = + List.map (fun (x,(y,t,z)) -> x,(y,Evarutil.nf_evar sigma t,z)) evlist in + let evplist = + List.map (fun (x,(y,t,z)) -> x,(y,Evarutil.nf_evar sigma t,z)) evplist in + pp(lazy(str"c0= " ++ pr_constr c0)); + let rec lookup k i = function + | [] -> 0, 0 + | (k', (n,_,_)) :: evl -> if k = k' then i,n else lookup k (i + 1) evl in + let rec get evlist i c = match kind_of_term c with + | Evar (ev, a) -> + let j, n = lookup ev i evlist in + if j = 0 then map_constr (get evlist i) c else if n = 0 then mkRel j else + mkApp (mkRel j, Array.init n (fun k -> get evlist i a.(n - 1 - k))) + | _ -> map_constr_with_binders ((+) 1) (get evlist) i c in + let rec app extra_args i c = match decompose_app c with + | hd, args when isRel hd && destRel hd = i -> + let j = destRel hd in + mkApp (mkRel j, Array.of_list (List.map (lift (i-1)) extra_args @ args)) + | _ -> map_constr_with_binders ((+) 1) (app extra_args) i c in + let rec loopP evlist c i = function + | (_, (n, t, _)) :: evl -> + let t = get evlist (i - 1) t in + let n = Name (id_of_string (ssr_anon_hyp ^ string_of_int n)) in + loopP evlist (mkProd (n, t, c)) (i - 1) evl + | [] -> c in + let rec loop c i = function + | (_, (n, t, _)) :: evl -> + let evs = Evarutil.evars_of_term t in + let t_evplist = List.filter (fun (k,_) -> Intset.mem k evs) evplist in + let t = loopP t_evplist (get t_evplist 1 t) 1 t_evplist in + let t = get evlist (i - 1) t in + let extra_args = + List.map (fun (k,_) -> mkRel (fst (lookup k i evlist))) + (List.rev t_evplist) in + let c = if extra_args = [] then c else app extra_args 1 c in + loop (mkLambda (mk_evar_name n, t, c)) (i - 1) evl + | [] -> c in + let res = loop (get evlist 1 c0) 1 evlist in + pp(lazy(str"res= " ++ pr_constr res)); + List.length evlist, res + +(* Strip all non-essential dependencies from an abstracted term, generating *) +(* standard names for the abstracted holes. *) + +let pf_abs_cterm gl n c0 = + if n <= 0 then c0 else + let noargs = [|0|] in + let eva = Array.make n noargs in + let rec strip i c = match kind_of_term c with + | App (f, a) when isRel f -> + let j = i - destRel f in + if j >= n || eva.(j) = noargs then mkApp (f, Array.map (strip i) a) else + let dp = eva.(j) in + let nd = Array.length dp - 1 in + let mkarg k = strip i a.(if k < nd then dp.(k + 1) - j else k + dp.(0)) in + mkApp (f, Array.init (Array.length a - dp.(0)) mkarg) + | _ -> map_constr_with_binders ((+) 1) strip i c in + let rec strip_ndeps j i c = match kind_of_term c with + | Prod (x, t, c1) when i < j -> + let dl, c2 = strip_ndeps j (i + 1) c1 in + if noccurn 1 c2 then dl, lift (-1) c2 else + i :: dl, mkProd (x, strip i t, c2) + | LetIn (x, b, t, c1) when i < j -> + let _, _, c1' = destProd c1 in + let dl, c2 = strip_ndeps j (i + 1) c1' in + if noccurn 1 c2 then dl, lift (-1) c2 else + i :: dl, mkLetIn (x, strip i b, strip i t, c2) + | _ -> [], strip i c in + let rec strip_evars i c = match kind_of_term c with + | Lambda (x, t1, c1) when i < n -> + let na = nb_evar_deps x in + let dl, t2 = strip_ndeps (i + na) i t1 in + let na' = List.length dl in + eva.(i) <- Array.of_list (na - na' :: dl); + let x' = + if na' = 0 then Name (pf_type_id gl t2) else mk_evar_name na' in + mkLambda (x', t2, strip_evars (i + 1) c1) +(* if noccurn 1 c2 then lift (-1) c2 else + mkLambda (Name (pf_type_id gl t2), t2, c2) *) + | _ -> strip i c in + strip_evars 0 c0 + +(* Undo the evar abstractions. Also works for non-evar variables. *) + +let pf_unabs_evars gl ise n c0 = + if n = 0 then c0 else + let evv = Array.make n mkProp in + let nev = ref 0 in + let env0 = pf_env gl in + let nenv0 = env_size env0 in + let rec unabs i c = match kind_of_term c with + | Rel j when i - j < !nev -> evv.(i - j) + | App (f, a0) when isRel f -> + let a = Array.map (unabs i) a0 in + let j = i - destRel f in + if j >= !nev then mkApp (f, a) else + let ev, eva = destEvar evv.(j) in + let nd = Array.length eva - nenv0 in + if nd = 0 then mkApp (evv.(j), a) else + let evarg k = if k < nd then a.(nd - 1 - k) else eva.(k) in + let c' = mkEvar (ev, Array.init (nd + nenv0) evarg) in + let na = Array.length a - nd in + if na = 0 then c' else mkApp (c', Array.sub a nd na) + | _ -> map_constr_with_binders ((+) 1) unabs i c in + let push_rel = Environ.push_rel in + let rec mk_evar j env i c = match kind_of_term c with + | Prod (x, t, c1) when i < j -> + mk_evar j (push_rel (x, None, unabs i t) env) (i + 1) c1 + | LetIn (x, b, t, c1) when i < j -> + let _, _, c2 = destProd c1 in + mk_evar j (push_rel (x, Some (unabs i b), unabs i t) env) (i + 1) c2 + | _ -> Evarutil.e_new_evar ise env (unabs i c) in + let rec unabs_evars c = + if !nev = n then unabs n c else match kind_of_term c with + | Lambda (x, t, c1) when !nev < n -> + let i = !nev in + evv.(i) <- mk_evar (i + nb_evar_deps x) env0 i t; + incr nev; unabs_evars c1 + | _ -> unabs !nev c in + unabs_evars c0 + +(* }}} *) + +(** Tactical extensions. {{{ **************************************************) + +(* The TACTIC EXTEND facility can't be used for defining new user *) +(* tacticals, because: *) +(* - the concrete syntax must start with a fixed string *) +(* - the lexical Ltac environment is NOT used to interpret tactic *) +(* arguments *) +(* The second limitation means that the extended tacticals will *) +(* exhibit run-time scope errors if used inside Ltac functions or *) +(* pattern-matching constructs. *) +(* We use the following workaround: *) +(* - We use the (unparsable) "Qed" token for tacticals that *) +(* don't start with a token, then redefine the grammar and *) +(* printer using GEXTEND and set_pr_ssrtac, respectively. *) +(* - We use a global stack and side effects to pass the lexical *) +(* Ltac evaluation context to the extended tactical. The context *) +(* is grabbed by interpreting an (empty) ltacctx argument, *) +(* which should appear last in the grammar rules; the *) +(* get_ltacctx function pops the stack and returns the context. *) +(* For additional safety, the push returns an integer key that *) +(* is checked by the pop; since arguments are interpreted *) +(* left-to-right, this checks that only one tactic argument *) +(* pushes a context. *) +(* - To avoid a spurrious option type, we don't push the context *) +(* for a null tag. *) + +type ssrargfmt = ArgSsr of string | ArgCoq of argument_type | ArgSep of string + +let set_pr_ssrtac name prec afmt = + let fmt = List.map (function ArgSep s -> Some s | _ -> None) afmt in + let rec mk_akey = function + | ArgSsr s :: afmt' -> ExtraArgType ("ssr" ^ s) :: mk_akey afmt' + | ArgCoq a :: afmt' -> a :: mk_akey afmt' + | ArgSep _ :: afmt' -> mk_akey afmt' + | [] -> [] in + let tacname = "ssr" ^ name in + Pptactic.declare_extra_tactic_pprule (tacname, mk_akey afmt, (prec, fmt)) + +let ssrtac_atom loc name args = TacExtend (loc, "ssr" ^ name, args) +let ssrtac_expr loc name args = TacAtom (loc, ssrtac_atom loc name args) + +let ssrevaltac ist gtac = + interp_tac_gen ist.lfun [] ist.debug (globTacticIn (fun _ -> gtac)) + +(* fun gl -> let lfun = [tacarg_id, val_interp ist gl gtac] in + interp_tac_gen lfun [] ist.debug tacarg_expr gl *) + +(** Generic argument-based globbing/typing utilities *) + + +let interp_wit globwit wit ist gl x = + let globarg = in_gen globwit x in + let sigma, arg = interp_genarg ist gl globarg in + sigma, out_gen wit arg + +let interp_intro_pattern = interp_wit globwit_intro_pattern wit_intro_pattern + +let interp_constr = interp_wit globwit_constr wit_constr + +let interp_open_constr ist gl gc = + interp_wit globwit_open_constr wit_open_constr ist gl ((), gc) + +let interp_refine ist gl rc = + let roc = (), (rc, None) in + interp_wit globwit_casted_open_constr wit_casted_open_constr ist gl roc + +let pf_match = pf_apply (fun e s c t -> understand_tcc s e ~expected_type:t c) + +(* Estimate a bound on the number of arguments of a raw constr. *) +(* This is not perfect, because the unifier may fail to *) +(* typecheck the partial application, so we use a minimum of 5. *) +(* Also, we don't handle delayed or iterated coercions to *) +(* FUNCLASS, which is probably just as well since these can *) +(* lead to infinite arities. *) + +let splay_open_constr gl (sigma, c) = + let env = pf_env gl in let t = Retyping.get_type_of env sigma c in + Reductionops.splay_prod env sigma t + +let nbargs_open_constr gl oc = + let pl, _ = splay_open_constr gl oc in List.length pl + +let interp_nbargs ist gl rc = + try + let rc6 = mkRApp rc (mkRHoles 6) in + let sigma, t = interp_open_constr ist gl (rc6, None) in + let si = sig_it gl in let gl = re_sig si sigma in + 6 + nbargs_open_constr gl t + with _ -> 5 + +let pf_nbargs gl c = nbargs_open_constr gl (project gl, c) + +let isAppInd gl c = + try ignore (pf_reduce_to_atomic_ind gl c); true with _ -> false + +let interp_view_nbimps ist gl rc = + try + let sigma, t = interp_open_constr ist gl (rc, None) in + let si = sig_it gl in let gl = re_sig si sigma in + let pl, c = splay_open_constr gl t in + if isAppInd gl c then List.length pl else ~-(List.length pl) + with _ -> 0 + +(* }}} *) + +(** Vernacular commands: Prenex Implicits and Search {{{ **********************) + +(* This should really be implemented as an extension to the implicit *) +(* arguments feature, but unfortuately that API is sealed. The current *) +(* workaround uses a combination of notations that works reasonably, *) +(* with the following caveats: *) +(* - The pretty-printing always elides prenex implicits, even when *) +(* they are obviously needed. *) +(* - Prenex Implicits are NEVER exported from a module, because this *) +(* would lead to faulty pretty-printing and scoping errors. *) +(* - The command "Import Prenex Implicits" can be used to reassert *) +(* Prenex Implicits for all the visible constants that had been *) +(* declared as Prenex Implicits. *) + +let declare_one_prenex_implicit locality f = + let fref = + try Smartlocate.global_with_alias f + with _ -> errorstrm (pr_reference f ++ str " is not declared") in + let rec loop = function + | a :: args' when Impargs.is_status_implicit a -> + (ExplByName (Impargs.name_of_implicit a), (true, true, true)) :: loop args' + | args' when List.exists Impargs.is_status_implicit args' -> + errorstrm (str "Expected prenex implicits for " ++ pr_reference f) + | _ -> [] in + let impls = + match Impargs.implicits_of_global fref with + | [cond,impls] -> impls + | [] -> errorstrm (str "Expected some implicits for " ++ pr_reference f) + | _ -> errorstrm (str "Multiple implicits not supported") in + match loop impls with + | [] -> + errorstrm (str "Expected some implicits for " ++ pr_reference f) + | impls -> + Impargs.declare_manual_implicits locality fref ~enriching:false [impls] + +VERNAC COMMAND EXTEND Ssrpreneximplicits + | [ "Prenex" "Implicits" ne_global_list(fl) ] + -> [ let locality = Vernacexpr.use_section_locality () in + List.iter (declare_one_prenex_implicit locality) fl ] +END + +(* Vernac grammar visibility patch *) + +GEXTEND Gram + GLOBAL: gallina_ext; + gallina_ext: + [ [ IDENT "Import"; IDENT "Prenex"; IDENT "Implicits" -> + Vernacexpr.VernacUnsetOption (None, ["Printing"; "Implicit"; "Defensive"]) + ] ] + ; +END + +(* Remove the silly restriction that forces coercion classes to be precise *) +(* aliases, e.g., allowing notations that specify some class parameters. *) + +let qualify_ref clref = + let loc, qid = qualid_of_reference clref in + try match Nametab.locate_extended qid with + | TrueGlobal _ -> clref + | SynDef kn -> + let rec head_of = function + | ARef gref -> + Qualid (loc, Nametab.shortest_qualid_of_global Idset.empty gref) + | AApp (rc, _) -> head_of rc + | ACast (rc, _) -> head_of rc + | ALetIn (_, _, rc) -> head_of rc + | rc -> + user_err_loc (loc, "qualify_ref", + str "The definition of " ++ Ppconstr.pr_qualid qid + ++ str " does not have a head constant") in + head_of (snd (Syntax_def.search_syntactic_definition kn)) + with _ -> clref + +let class_rawexpr = G_vernac.class_rawexpr in +GEXTEND Gram + GLOBAL: class_rawexpr; + ssrqref: [[ gref = global -> qualify_ref gref ]]; + class_rawexpr: [[ class_ref = ssrqref -> Vernacexpr.RefClass (Genarg.AN class_ref) ]]; +END + +(** Extend Search to subsume SearchAbout, also adding hidden Type coercions. *) + +(* Main prefilter *) + +let pr_search_item = function + | Search.GlobSearchString s -> str s + | Search.GlobSearchSubPattern p -> pr_constr_pattern p + +let wit_ssr_searchitem, globwit_ssr_searchitem, rawwit_ssr_searchitem = + add_genarg "ssr_searchitem" pr_search_item + +let interp_search_notation loc s opt_scope = + try + let interp = Notation.interp_notation_as_global_reference loc in + let ref = interp (fun _ -> true) s opt_scope in + Search.GlobSearchSubPattern (Pattern.PRef ref) + with _ -> + let diagnosis = + try + let ntns = Notation.locate_notation pr_glob_constr s opt_scope in + let ambig = "This string refers to a complex or ambiguous notation." in + str ambig ++ str "\nTry searching with one of\n" ++ ntns + with _ -> str "This string is not part of an identifier or notation." in + user_err_loc (loc, "interp_search_notation", diagnosis) + +let pr_ssr_search_item _ _ _ = pr_search_item + +(* Workaround the notation API that can only print notations *) + +let is_ident s = try Lexer.check_ident s; true with _ -> false + +let is_ident_part s = is_ident ("H" ^ s) + +let interp_search_notation loc tag okey = + let err msg = user_err_loc (loc, "interp_search_notation", msg) in + let mk_pntn s for_key = + let n = String.length s in + let s' = String.make (n + 2) ' ' in + let rec loop i i' = + if i >= n then s', i' - 2 else if s.[i] = ' ' then loop (i + 1) i' else + let j = try String.index_from s (i + 1) ' ' with _ -> n in + let m = j - i in + if s.[i] = '\'' && i < j - 2 && s.[j - 1] = '\'' then + (String.blit s (i + 1) s' i' (m - 2); loop (j + 1) (i' + m - 1)) + else if for_key && is_ident (String.sub s i m) then + (s'.[i'] <- '_'; loop (j + 1) (i' + 2)) + else (String.blit s i s' i' m; loop (j + 1) (i' + m + 1)) in + loop 0 1 in + let trim_ntn (pntn, m) = String.sub pntn 1 (max 0 m) in + let pr_ntn ntn = str "(" ++ str ntn ++ str ")" in + let pr_and_list pr = function + | [x] -> pr x + | x :: lx -> pr_list pr_comma pr lx ++ pr_comma () ++ str "and " ++ pr x + | [] -> mt () in + let pr_sc sc = str (if sc = "" then "independently" else sc) in + let pr_scs = function + | [""] -> pr_sc "" + | scs -> str "in " ++ pr_and_list pr_sc scs in + let generator, pr_tag_sc = + let ign _ = mt () in match okey with + | Some key -> + let sc = Notation.find_delimiters_scope loc key in + let pr_sc s_in = str s_in ++ spc() ++ str sc ++ pr_comma() in + Notation.pr_scope ign sc, pr_sc + | None -> Notation.pr_scopes ign, ign in + let qtag s_in = pr_tag_sc s_in ++ qstring tag ++ spc()in + let ptag, ttag = + let ptag, m = mk_pntn tag false in + if m <= 0 then err (str "empty notation fragment"); + ptag, trim_ntn (ptag, m) in + let last = ref "" and last_sc = ref "" in + let scs = ref [] and ntns = ref [] in + let push_sc sc = match !scs with + | "" :: scs' -> scs := "" :: sc :: scs' + | scs' -> scs := sc :: scs' in + let get s _ _ = match !last with + | "Scope " -> last_sc := s; last := "" + | "Lonely notation" -> last_sc := ""; last := "" + | "\"" -> + let pntn, m = mk_pntn s true in + if string_string_contains pntn ptag then begin + let ntn = trim_ntn (pntn, m) in + match !ntns with + | [] -> ntns := [ntn]; scs := [!last_sc] + | ntn' :: _ when ntn' = ntn -> push_sc !last_sc + | _ when ntn = ttag -> ntns := ntn :: !ntns; scs := [!last_sc] + | _ :: ntns' when List.mem ntn ntns' -> () + | ntn' :: ntns' -> ntns := ntn' :: ntn :: ntns' + end; + last := "" + | _ -> last := s in + pp_with (Format.make_formatter get (fun _ -> ())) generator; + let ntn = match !ntns with + | [] -> + err (hov 0 (qtag "in" ++ str "does not occur in any notation")) + | ntn :: ntns' when ntn = ttag -> + if ntns' <> [] then begin + let pr_ntns' = pr_and_list pr_ntn ntns' in + msg_warning (hov 4 (qtag "In" ++ str "also occurs in " ++ pr_ntns')) + end; ntn + | [ntn] -> + msgnl (hov 4 (qtag "In" ++ str "is part of notation " ++ pr_ntn ntn)); ntn + | ntns' -> + let e = str "occurs in" ++ spc() ++ pr_and_list pr_ntn ntns' in + err (hov 4 (str "ambiguous: " ++ qtag "in" ++ e)) in + let (nvars, body), ((_, pat), osc) = match !scs with + | [sc] -> Notation.interp_notation loc ntn (None, [sc]) + | scs' -> + try Notation.interp_notation loc ntn (None, []) with _ -> + let e = pr_ntn ntn ++ spc() ++ str "is defined " ++ pr_scs scs' in + err (hov 4 (str "ambiguous: " ++ pr_tag_sc "in" ++ e)) in + let sc = Option.default "" osc in + let _ = + let m_sc = + if osc <> None then str "In " ++ str sc ++ pr_comma() else mt() in + let ntn_pat = trim_ntn (mk_pntn pat false) in + let rbody = glob_constr_of_aconstr loc body in + let m_body = hov 0 (Constrextern.without_symbols prl_glob_constr rbody) in + let m = m_sc ++ pr_ntn ntn_pat ++ spc () ++ str "denotes " ++ m_body in + msgnl (hov 0 m) in + if List.length !scs > 1 then + let scs' = list_remove sc !scs in + let w = pr_ntn ntn ++ str " is also defined " ++ pr_scs scs' in + msg_warning (hov 4 w) + else if string_string_contains ntn " .. " then + err (pr_ntn ntn ++ str " is an n-ary notation"); + let nvars = List.filter (fun (_,(_,typ)) -> typ = NtnTypeConstr) nvars in + let rec sub () = function + | AVar x when List.mem_assoc x nvars -> GPatVar (loc, (false, x)) + | c -> + glob_constr_of_aconstr_with_binders loc (fun _ x -> (), x) sub () c in + let _, npat = Pattern.pattern_of_glob_constr (sub () body) in + Search.GlobSearchSubPattern npat + +ARGUMENT EXTEND ssr_search_item TYPED AS ssr_searchitem + PRINTED BY pr_ssr_search_item + | [ string(s) ] -> + [ if is_ident_part s then Search.GlobSearchString s else + interp_search_notation loc s None ] + | [ string(s) "%" preident(key) ] -> + [ interp_search_notation loc s (Some key) ] + | [ constr_pattern(p) ] -> + [ try + let intern = Constrintern.intern_constr_pattern Evd.empty in + Search.GlobSearchSubPattern (snd (intern (Global.env()) p)) + with e -> raise (Cerrors.process_vernac_interp_error e) + ] +END + +let pr_ssr_search_arg _ _ _ = + let pr_item (b, p) = str (if b then "-" else "") ++ pr_search_item p in + pr_list spc pr_item + +ARGUMENT EXTEND ssr_search_arg TYPED AS (bool * ssr_searchitem) list + PRINTED BY pr_ssr_search_arg + | [ "-" ssr_search_item(p) ssr_search_arg(a) ] -> [ (false, p) :: a ] + | [ ssr_search_item(p) ssr_search_arg(a) ] -> [ (true, p) :: a ] + | [ ] -> [ [] ] +END + +(* Main type conclusion pattern filter *) + +let rec splay_search_pattern na = function + | Pattern.PApp (fp, args) -> splay_search_pattern (na + Array.length args) fp + | Pattern.PLetIn (_, _, bp) -> splay_search_pattern na bp + | Pattern.PRef hr -> hr, na + | _ -> error "no head constant in head search pattern" + +let coerce_search_pattern_to_sort hpat = + let env = Global.env () and sigma = Evd.empty in + let mkPApp fp n_imps args = + let args' = Array.append (Array.make n_imps (Pattern.PMeta None)) args in + Pattern.PApp (fp, args') in + let hr, na = splay_search_pattern 0 hpat in + let dc, ht = Reductionops.splay_prod env sigma (Global.type_of_global hr) in + let np = List.length dc in + if np < na then error "too many arguments in head search pattern" else + let hpat' = if np = na then hpat else mkPApp hpat (np - na) [||] in + let warn () = + msg_warning (str "Listing only lemmas with conclusion matching " ++ + pr_constr_pattern hpat') in + if isSort ht then begin warn (); true, hpat' end else + let filter_head, coe_path = + try + let _, cp = + Classops.lookup_path_to_sort_from (push_rels_assum dc env) sigma ht in + warn (); + true, cp + with _ -> false, [] in + let coerce hp coe_index = + let coe = Classops.get_coercion_value coe_index in + try + let coe_ref = reference_of_constr coe in + let n_imps = Option.get (Classops.hide_coercion coe_ref) in + mkPApp (Pattern.PRef coe_ref) n_imps [|hp|] + with _ -> + errorstrm (str "need explicit coercion " ++ pr_constr coe ++ spc () + ++ str "to interpret head search pattern as type") in + filter_head, List.fold_left coerce hpat' coe_path + +let rec interp_head_pat hpat = + let filter_head, p = coerce_search_pattern_to_sort hpat in + let rec loop c = match kind_of_term c with + | Cast (c', _, _) -> loop c' + | Prod (_, _, c') -> loop c' + | LetIn (_, _, _, c') -> loop c' + | _ -> Matching.is_matching p c in + filter_head, loop + +let all_true _ = true + +let interp_search_arg a = + let hpat, a1 = match a with + | (_, Search.GlobSearchSubPattern (Pattern.PMeta _)) :: a' -> all_true, a' + | (true, Search.GlobSearchSubPattern p) :: a' -> + let filter_head, p = interp_head_pat p in + if filter_head then p, a' else all_true, a + | _ -> all_true, a in + let is_string = + function (_, Search.GlobSearchString _) -> true | _ -> false in + let a2, a3 = List.partition is_string a1 in + hpat, a2 @ a3 + +(* Module path postfilter *) + +let pr_modloc (b, m) = if b then str "-" ++ pr_reference m else pr_reference m + +let wit_ssrmodloc, globwit_ssrmodloc, rawwit_ssrmodloc = + add_genarg "ssrmodloc" pr_modloc + +let pr_ssr_modlocs _ _ _ ml = + if ml = [] then str "" else spc () ++ str "in " ++ pr_list spc pr_modloc ml + +ARGUMENT EXTEND ssr_modlocs TYPED AS ssrmodloc list PRINTED BY pr_ssr_modlocs + | [ ] -> [ [] ] +END + +GEXTEND Gram + GLOBAL: ssr_modlocs; + modloc: [[ "-"; m = global -> true, m | m = global -> false, m]]; + ssr_modlocs: [[ "in"; ml = LIST1 modloc -> ml ]]; +END + +let interp_modloc mr = + let interp_mod (_, mr) = + let (loc, qid) = qualid_of_reference mr in + try Nametab.full_name_module qid with Not_found -> + user_err_loc (loc, "interp_modloc", str "No Module " ++ pr_qualid qid) in + let mr_out, mr_in = List.partition fst mr in + let interp_bmod b rmods = + if rmods = [] then fun _ _ _ -> true else + Search.filter_by_module_from_list (List.map interp_mod rmods, b) in + let is_in = interp_bmod false mr_in and is_out = interp_bmod true mr_out in + fun gr env -> is_in gr env () && is_out gr env () + +(* The unified, extended vernacular "Search" command *) + +let ssrdisplaysearch gr env t = + let pr_res = pr_global gr ++ spc () ++ str " " ++ pr_lconstr_env env t in + msg (hov 2 pr_res ++ fnl ()) + +VERNAC COMMAND EXTEND SsrSearchPattern +| [ "Search" ssr_search_arg(a) ssr_modlocs(mr) ] -> + [ let hpat, a' = interp_search_arg a in + let in_mod = interp_modloc mr in + let post_filter gr env typ = in_mod gr env && hpat typ in + Search.raw_search_about post_filter ssrdisplaysearch a' ] +END + +(* }}} *) + +(** Alternative notations for "match" and anonymous arguments. {{{ ************) + +(* Syntax: *) +(* if <term> is <pattern> then ... else ... *) +(* if <term> is <pattern> [in ..] return ... then ... else ... *) +(* let: <pattern> := <term> in ... *) +(* let: <pattern> [in ...] := <term> return ... in ... *) +(* The scope of a top-level 'as' in the pattern extends over the *) +(* 'return' type (dependent if/let). *) +(* Note that the optional "in ..." appears next to the <pattern> *) +(* rather than the <term> in then "let:" syntax. The alternative *) +(* would lead to ambiguities in, e.g., *) +(* let: p1 := (*v---INNER LET:---v *) *) +(* let: p2 := let: p3 := e3 in k return t in k2 in k1 return t' *) +(* in b (*^--ALTERNATIVE INNER LET--------^ *) *) + +(* Caveat : There is no pretty-printing support, since this would *) +(* require a modification to the Coq kernel (adding a new match *) +(* display style -- why aren't these strings?); also, the v8.1 *) +(* pretty-printer only allows extension hooks for printing *) +(* integer or string literals. *) +(* Also note that in the v8 grammar "is" needs to be a keyword; *) +(* as this can't be done from an ML extension file, the new *) +(* syntax will only work when ssreflect.v is imported. *) + + +let no_ct = None, None and no_rt = None in +let aliasvar = function + | [_, [CPatAlias (loc, _, id)]] -> Some (loc,Name id) + | _ -> None in +let mk_cnotype mp = aliasvar mp, None in +let mk_ctype mp t = aliasvar mp, Some t in +let mk_rtype t = Some t in +let mk_dthen loc (mp, ct, rt) c = (loc, mp, c), ct, rt in +let mk_let loc rt ct mp c1 = + CCases (loc, LetPatternStyle, rt, ct, [loc, mp, c1]) in +GEXTEND Gram + GLOBAL: binder_constr; + ssr_rtype: [[ "return"; t = operconstr LEVEL "100" -> mk_rtype t ]]; + ssr_mpat: [[ p = pattern -> [loc, [p]] ]]; + ssr_dpat: [ + [ mp = ssr_mpat; "in"; t = lconstr; rt = ssr_rtype -> mp, mk_ctype mp t, rt + | mp = ssr_mpat; rt = ssr_rtype -> mp, mk_cnotype mp, rt + | mp = ssr_mpat -> mp, no_ct, no_rt + ] ]; + ssr_dthen: [[ dp = ssr_dpat; "then"; c = lconstr -> mk_dthen loc dp c ]]; + ssr_elsepat: [[ "else" -> [loc, [CPatAtom (loc, None)]] ]]; + ssr_else: [[ mp = ssr_elsepat; c = lconstr -> loc, mp, c ]]; + binder_constr: [ + [ "if"; c = operconstr LEVEL "200"; "is"; db1 = ssr_dthen; b2 = ssr_else -> + let b1, ct, rt = db1 in CCases (loc, MatchStyle, rt, [c, ct], [b1; b2]) + | "if"; c = operconstr LEVEL "200";"isn't";db1 = ssr_dthen; b2 = ssr_else -> + let b1, ct, rt = db1 in + let b1, b2 = + let (l1, p1, r1), (l2, p2, r2) = b1, b2 in (l1, p1, r2), (l2, p2, r1) in + CCases (loc, MatchStyle, rt, [c, ct], [b1; b2]) + | "let"; ":"; mp = ssr_mpat; ":="; c = lconstr; "in"; c1 = lconstr -> + mk_let loc no_rt [c, no_ct] mp c1 + | "let"; ":"; mp = ssr_mpat; ":="; c = lconstr; + rt = ssr_rtype; "in"; c1 = lconstr -> + mk_let loc rt [c, mk_cnotype mp] mp c1 + | "let"; ":"; mp = ssr_mpat; "in"; t = lconstr; ":="; c = lconstr; + rt = ssr_rtype; "in"; c1 = lconstr -> + mk_let loc rt [c, mk_ctype mp t] mp c1 + ] ]; +END + +GEXTEND Gram + GLOBAL: closed_binder; + closed_binder: [ + [ ["of" | "&"]; c = operconstr LEVEL "99" -> + [LocalRawAssum ([loc, Anonymous], Default Explicit, c)] + ] ]; +END +(* }}} *) + +(** Tacticals (+, -, *, done, by, do, =>, first, and last). {{{ ***************) + +(** Bracketing tactical *) + +(* The tactic pretty-printer doesn't know that some extended tactics *) +(* are actually tacticals. To prevent it from improperly removing *) +(* parentheses we override the parsing rule for bracketed tactic *) +(* expressions so that the pretty-print always reflects the input. *) +(* (Removing user-specified parentheses is dubious anyway). *) + +GEXTEND Gram + GLOBAL: tactic_expr; + ssrparentacarg: [[ "("; tac = tactic_expr; ")" -> loc, Tacexp tac ]]; + tactic_expr: LEVEL "0" [[ arg = ssrparentacarg -> TacArg arg ]]; +END + +(** The internal "done" and "ssrautoprop" tactics. *) + +(* For additional flexibility, "done" and "ssrautoprop" are *) +(* defined in Ltac. *) +(* Although we provide a default definition in ssreflect, *) +(* we look up the definition dynamically at each call point, *) +(* to allow for user extensions. "ssrautoprop" defaults to *) +(* trivial. *) + +let donetac gl = + let tacname = + try Nametab.locate_tactic (qualid_of_ident (id_of_string "done")) + with Not_found -> try Nametab.locate_tactic (ssrqid "done") + with Not_found -> Util.error "The ssreflect library was not loaded" in + let tacexpr = dummy_loc, Tacexpr.Reference (ArgArg (dummy_loc, tacname)) in + eval_tactic (Tacexpr.TacArg tacexpr) gl + +let prof_donetac = mk_profiler "donetac";; +let donetac gl = prof_donetac.profile donetac gl;; + +let ssrautoprop gl = + try + let tacname = + try Nametab.locate_tactic (qualid_of_ident (id_of_string "ssrautoprop")) + with Not_found -> Nametab.locate_tactic (ssrqid "ssrautoprop") in + let tacexpr = dummy_loc, Tacexpr.Reference (ArgArg (dummy_loc, tacname)) in + eval_tactic (Tacexpr.TacArg tacexpr) gl + with Not_found -> Auto.full_trivial [] gl + +let () = ssrautoprop_tac := ssrautoprop + +let tclBY tac = tclTHEN tac donetac + +(** Tactical arguments. *) + +(* We have four kinds: simple tactics, [|]-bracketed lists, hints, and swaps *) +(* The latter two are used in forward-chaining tactics (have, suffice, wlog) *) +(* and subgoal reordering tacticals (; first & ; last), respectively. *) + +(* Force use of the tactic_expr parsing entry, to rule out tick marks. *) +let pr_ssrtacarg _ _ prt = prt tacltop +ARGUMENT EXTEND ssrtacarg TYPED AS tactic PRINTED BY pr_ssrtacarg +| [ "Qed" ] -> [ Util.anomaly "Grammar placeholder match" ] +END +GEXTEND Gram + GLOBAL: ssrtacarg; + ssrtacarg: [[ tac = tactic_expr LEVEL "5" -> tac ]]; +END + +(* Lexically closed tactic for tacticals. *) +let pr_ssrtclarg _ _ prt (tac, _) = prt tacltop tac +ARGUMENT EXTEND ssrtclarg TYPED AS ssrtacarg * ltacctx + PRINTED BY pr_ssrtclarg +| [ ssrtacarg(tac) ] -> [ tac, rawltacctx ] +END +let eval_tclarg (tac, ctx) = ssrevaltac (get_ltacctx ctx) tac + +let pr_ortacs prt = + let rec pr_rec = function + | [None] -> spc() ++ str "|" ++ spc() + | None :: tacs -> spc() ++ str "|" ++ pr_rec tacs + | Some tac :: tacs -> spc() ++ str "| " ++ prt tacltop tac ++ pr_rec tacs + | [] -> mt() in + function + | [None] -> spc() + | None :: tacs -> pr_rec tacs + | Some tac :: tacs -> prt tacltop tac ++ pr_rec tacs + | [] -> mt() +let pr_ssrortacs _ _ = pr_ortacs + +ARGUMENT EXTEND ssrortacs TYPED AS tactic option list PRINTED BY pr_ssrortacs +| [ ssrtacarg(tac) "|" ssrortacs(tacs) ] -> [ Some tac :: tacs ] +| [ ssrtacarg(tac) "|" ] -> [ [Some tac; None] ] +| [ ssrtacarg(tac) ] -> [ [Some tac] ] +| [ "|" ssrortacs(tacs) ] -> [ None :: tacs ] +| [ "|" ] -> [ [None; None] ] +END + +let pr_hintarg prt = function + | true, tacs -> hv 0 (str "[ " ++ pr_ortacs prt tacs ++ str " ]") + | false, [Some tac] -> prt tacltop tac + | _, _ -> mt() + +let pr_ssrhintarg _ _ = pr_hintarg + +let mk_hint tac = false, [Some tac] +let mk_orhint tacs = true, tacs +let nullhint = true, [] +let nohint = false, [] + +ARGUMENT EXTEND ssrhintarg TYPED AS bool * ssrortacs PRINTED BY pr_ssrhintarg +| [ "[" "]" ] -> [ nullhint ] +| [ "[" ssrortacs(tacs) "]" ] -> [ mk_orhint tacs ] +| [ ssrtacarg(arg) ] -> [ mk_hint arg ] +END + +ARGUMENT EXTEND ssrortacarg TYPED AS ssrhintarg PRINTED BY pr_ssrhintarg +| [ "[" ssrortacs(tacs) "]" ] -> [ mk_orhint tacs ] +END + +let hinttac ist is_by (is_or, atacs) = + let dtac = if is_by then donetac else tclIDTAC in + let mktac = function + | Some atac -> tclTHEN (ssrevaltac ist atac) dtac + | _ -> dtac in + match List.map mktac atacs with + | [] -> if is_or then dtac else tclIDTAC + | [tac] -> tac + | tacs -> tclFIRST tacs + +(** The "-"/"+"/"*" tacticals. *) + +(* These are just visual cues to flag the beginning of the script for *) +(* new subgoals, when indentation is not appropriate (typically after *) +(* tactics that generate more than two subgoals). *) + +TACTIC EXTEND ssrtclplus +| [ "Qed" "+" ssrtclarg(arg) ] -> [ eval_tclarg arg ] +END +set_pr_ssrtac "tclplus" 5 [ArgSep "+ "; ArgSsr "tclarg"] + +TACTIC EXTEND ssrtclminus +| [ "Qed" "-" ssrtclarg(arg) ] -> [ eval_tclarg arg ] +END +set_pr_ssrtac "tclminus" 5 [ArgSep "- "; ArgSsr "tclarg"] + +TACTIC EXTEND ssrtclstar +| [ "Qed" "*" ssrtclarg(arg) ] -> [ eval_tclarg arg ] +END +set_pr_ssrtac "tclstar" 5 [ArgSep "- "; ArgSsr "tclarg"] + +let gen_tclarg = in_gen rawwit_ssrtclarg + +GEXTEND Gram + GLOBAL: tactic tactic_mode; + tactic: [ + [ "+"; tac = ssrtclarg -> ssrtac_expr loc "tclplus" [gen_tclarg tac] + | "-"; tac = ssrtclarg -> ssrtac_expr loc "tclminus" [gen_tclarg tac] + | "*"; tac = ssrtclarg -> ssrtac_expr loc "tclstar" [gen_tclarg tac] + ] ]; + tactic_mode: [ + [ "+"; tac = G_vernac.subgoal_command -> tac None + | "-"; tac = G_vernac.subgoal_command -> tac None + | "*"; tac = G_vernac.subgoal_command -> tac None + ] ]; +END + +(** The "by" tactical. *) + +let pr_hint prt arg = + if arg = nohint then mt() else str "by " ++ pr_hintarg prt arg +let pr_ssrhint _ _ = pr_hint + +ARGUMENT EXTEND ssrhint TYPED AS ssrhintarg PRINTED BY pr_ssrhint +| [ ] -> [ nohint ] +END + +TACTIC EXTEND ssrtclby +| [ "Qed" ssrhint(tac) ltacctx(ctx)] -> + [ hinttac (get_ltacctx ctx) true tac ] +END +set_pr_ssrtac "tclby" 0 [ArgSsr "hint"; ArgSsr "ltacctx"] + +(* We can't parse "by" in ARGUMENT EXTEND because it will only be made *) +(* into a keyword in ssreflect.v; so we anticipate this in GEXTEND. *) + +GEXTEND Gram + GLOBAL: ssrhint simple_tactic; + ssrhint: [[ "by"; arg = ssrhintarg -> arg ]]; + simple_tactic: [ + [ "by"; arg = ssrhintarg -> + let garg = in_gen rawwit_ssrhint arg in + let gctx = in_gen rawwit_ltacctx rawltacctx in + ssrtac_atom loc "tclby" [garg; gctx] + ] ]; +END +(* }}} *) + +(** Bound assumption argument *) + +(* The Ltac API does have a type for assumptions but it is level-dependent *) +(* and therefore impratical to use for complex arguments, so we substitute *) +(* our own to have a uniform representation. Also, we refuse to intern *) +(* idents that match global/section constants, since this would lead to *) +(* fragile Ltac scripts. *) + +type ssrhyp = SsrHyp of loc * identifier + +let hyp_id (SsrHyp (_, id)) = id +let pr_hyp (SsrHyp (_, id)) = pr_id id +let pr_ssrhyp _ _ _ = pr_hyp + +let wit_ssrhyprep, globwit_ssrhyprep, rawwit_ssrhyprep = + add_genarg "ssrhyprep" pr_hyp + +let hyp_err loc msg id = + user_err_loc (loc, "ssrhyp", str msg ++ pr_id id) + +let intern_hyp ist (SsrHyp (loc, id) as hyp) = + let _ = intern_genarg ist (in_gen rawwit_var (loc, id)) in + if not_section_id id then hyp else + hyp_err loc "Can't clear section hypothesis " id + +let interp_hyp ist gl (SsrHyp (loc, id)) = + let s, id' = interp_wit globwit_var wit_var ist gl (loc, id) in + if not_section_id id' then s, SsrHyp (loc, id') else + hyp_err loc "Can't clear section hypothesis " id' + +ARGUMENT EXTEND ssrhyp TYPED AS ssrhyprep PRINTED BY pr_ssrhyp + INTERPRETED BY interp_hyp + GLOBALIZED BY intern_hyp + | [ ident(id) ] -> [ SsrHyp (loc, id) ] +END + +type ssrhyp_or_id = Hyp of ssrhyp | Id of ssrhyp + +let hoik f = function Hyp x -> f x | Id x -> f x +let hoi_id = hoik hyp_id +let pr_hoi = hoik pr_hyp +let pr_ssrhoi _ _ _ = pr_hoi + +let wit_ssrhoirep, globwit_ssrhoirep, rawwit_ssrhoirep = + add_genarg "ssrhoirep" pr_hoi + +let intern_ssrhoi ist = function + | Hyp h -> Hyp (intern_hyp ist h) + | Id (SsrHyp (_, id)) as hyp -> + let _ = intern_genarg ist (in_gen rawwit_ident id) in + hyp + +let interp_ssrhoi ist gl = function + | Hyp h -> let s, h' = interp_hyp ist gl h in s, Hyp h' + | Id (SsrHyp (loc, id)) -> + let s, id' = interp_wit globwit_ident wit_ident ist gl id in + s, Id (SsrHyp (loc, id')) + +ARGUMENT EXTEND ssrhoi_hyp TYPED AS ssrhoirep PRINTED BY pr_ssrhoi + INTERPRETED BY interp_ssrhoi + GLOBALIZED BY intern_ssrhoi + | [ ident(id) ] -> [ Hyp (SsrHyp(loc, id)) ] +END +ARGUMENT EXTEND ssrhoi_id TYPED AS ssrhoirep PRINTED BY pr_ssrhoi + INTERPRETED BY interp_ssrhoi + GLOBALIZED BY intern_ssrhoi + | [ ident(id) ] -> [ Id (SsrHyp(loc, id)) ] +END + +type ssrhyps = ssrhyp list + +let pr_hyps = pr_list pr_spc pr_hyp +let pr_ssrhyps _ _ _ = pr_hyps +let hyps_ids = List.map hyp_id + +let rec check_hyps_uniq ids = function + | SsrHyp (loc, id) :: _ when List.mem id ids -> + hyp_err loc "Duplicate assumption " id + | SsrHyp (_, id) :: hyps -> check_hyps_uniq (id :: ids) hyps + | [] -> () + +let check_hyp_exists hyps (SsrHyp(_, id)) = + try ignore(Sign.lookup_named id hyps) + with Not_found -> errorstrm (str"No assumption is named " ++ pr_id id) + +let interp_hyps ist gl ghyps = + let hyps = List.map snd (List.map (interp_hyp ist gl) ghyps) in + check_hyps_uniq [] hyps; Tacmach.project gl, hyps + +ARGUMENT EXTEND ssrhyps TYPED AS ssrhyp list PRINTED BY pr_ssrhyps + INTERPRETED BY interp_hyps + | [ ssrhyp_list(hyps) ] -> [ check_hyps_uniq [] hyps; hyps ] +END + +(** Terms parsing. {{{ ********************************************************) + +(* Because we allow wildcards in term references, we need to stage the *) +(* interpretation of terms so that it occurs at the right time during *) +(* the execution of the tactic (e.g., so that we don't report an error *) +(* for a term that isn't actually used in the execution). *) +(* The term representation tracks whether the concrete initial term *) +(* started with an opening paren, which might avoid a conflict between *) +(* the ssrreflect term syntax and Gallina notation. *) + +(* kinds of terms *) + +type ssrtermkind = char (* print flag *) + +let input_ssrtermkind strm = match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "(" -> '(' + | Tok.KEYWORD "@" -> '@' + | _ -> ' ' + +let ssrtermkind = Gram.Entry.of_parser "ssrtermkind" input_ssrtermkind + +let id_of_Cterm t = match id_of_cpattern t with + | Some x -> x + | None -> loc_error (loc_of_cpattern t) "Only identifiers are allowed here" +let ssrhyp_of_ssrterm = function + | k, (_, Some c) as o -> + SsrHyp (constr_loc c, id_of_Cterm (cpattern_of_term o)), String.make 1 k + | _, (_, None) -> assert false + +(* terms *) +let pr_ssrterm _ _ _ = pr_term +let pf_intern_term ist gl (_, c) = glob_constr ist (project gl) (pf_env gl) c +let intern_term ist sigma env (_, c) = glob_constr ist sigma env c +let interp_term ist gl (_, c) = snd (interp_open_constr ist gl c) +let force_term ist gl (_, c) = interp_constr ist gl c +let glob_ssrterm gs = function + | k, (_, Some c) -> k, Tacinterp.intern_constr gs c + | ct -> ct +let subst_ssrterm s (k, c) = k, Tacinterp.subst_glob_constr_and_expr s c +let interp_ssrterm _ gl t = Tacmach.project gl, t + +ARGUMENT EXTEND ssrterm + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_ssrterm SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "Qed" constr(c) ] -> [ mk_lterm c ] +END + +GEXTEND Gram + GLOBAL: ssrterm; + ssrterm: [[ k = ssrtermkind; c = constr -> mk_term k c ]]; +END +(* }}} *) + +(** The "in" pseudo-tactical {{{ **********************************************) + +(* We can't make "in" into a general tactical because this would create a *) +(* crippling conflict with the ltac let .. in construct. Hence, we add *) +(* explicitly an "in" suffix to all the extended tactics for which it is *) +(* relevant (including move, case, elim) and to the extended do tactical *) +(* below, which yields a general-purpose "in" of the form do [...] in ... *) + +(* This tactical needs to come before the intro tactics because the latter *) +(* must take precautions in order not to interfere with the discharged *) +(* assumptions. This is especially difficult for discharged "let"s, which *) +(* the default simpl and unfold tactics would erase blindly. *) + +(** Clear switch *) + +type ssrclear = ssrhyps + +let pr_clear_ne clr = str "{" ++ pr_hyps clr ++ str "}" +let pr_clear sep clr = if clr = [] then mt () else sep () ++ pr_clear_ne clr + +let pr_ssrclear _ _ _ = pr_clear mt + +ARGUMENT EXTEND ssrclear_ne TYPED AS ssrhyps PRINTED BY pr_ssrclear +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ check_hyps_uniq [] clr; clr ] +END + +ARGUMENT EXTEND ssrclear TYPED AS ssrclear_ne PRINTED BY pr_ssrclear +| [ ssrclear_ne(clr) ] -> [ clr ] +| [ ] -> [ [] ] +END + +let cleartac clr = check_hyps_uniq [] clr; clear (hyps_ids clr) + +(* type ssrwgen = ssrclear * ssrhyp * string *) + +let pr_wgen = function + | (clr, Some((id,k),None)) -> spc() ++ pr_clear mt clr ++ str k ++ pr_hoi id + | (clr, Some((id,k),Some p)) -> + spc() ++ pr_clear mt clr ++ str"(" ++ str k ++ pr_hoi id ++ str ":=" ++ + pr_cpattern p ++ str ")" + | (clr, None) -> spc () ++ pr_clear mt clr +let pr_ssrwgen _ _ _ = pr_wgen + +(* no globwith for char *) +ARGUMENT EXTEND ssrwgen + TYPED AS ssrclear * ((ssrhoi_hyp * string) * cpattern option) option + PRINTED BY pr_ssrwgen +| [ ssrclear_ne(clr) ] -> [ clr, None ] +| [ ssrhoi_hyp(hyp) ] -> [ [], Some((hyp, " "), None) ] +| [ "@" ssrhoi_hyp(hyp) ] -> [ [], Some((hyp, "@"), None) ] +| [ "(" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id," "),Some p) ] +| [ "(" ssrhoi_id(id) ")" ] -> [ [], Some ((id,"("), None) ] +| [ "(@" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id,"@"),Some p) ] +| [ "(" "@" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id,"@"),Some p) ] +END + +type ssrclseq = InGoal | InHyps + | InHypsGoal | InHypsSeqGoal | InSeqGoal | InHypsSeq | InAll | InAllHyps + +let pr_clseq = function + | InGoal | InHyps -> mt () + | InSeqGoal -> str "|- *" + | InHypsSeqGoal -> str " |- *" + | InHypsGoal -> str " *" + | InAll -> str "*" + | InHypsSeq -> str " |-" + | InAllHyps -> str "* |-" + +let wit_ssrclseq, globwit_ssrclseq, rawwit_ssrclseq = + add_genarg "ssrclseq" pr_clseq + +let pr_clausehyps = pr_list pr_spc pr_wgen +let pr_ssrclausehyps _ _ _ = pr_clausehyps + +ARGUMENT EXTEND ssrclausehyps +TYPED AS ssrwgen list PRINTED BY pr_ssrclausehyps +| [ ssrwgen(hyp) "," ssrclausehyps(hyps) ] -> [ hyp :: hyps ] +| [ ssrwgen(hyp) ssrclausehyps(hyps) ] -> [ hyp :: hyps ] +| [ ssrwgen(hyp) ] -> [ [hyp] ] +END + +(* type ssrclauses = ssrahyps * ssrclseq *) + +let pr_clauses (hyps, clseq) = + if clseq = InGoal then mt () + else str "in " ++ pr_clausehyps hyps ++ pr_clseq clseq +let pr_ssrclauses _ _ _ = pr_clauses + +ARGUMENT EXTEND ssrclauses TYPED AS ssrwgen list * ssrclseq + PRINTED BY pr_ssrclauses + | [ "in" ssrclausehyps(hyps) "|-" "*" ] -> [ hyps, InHypsSeqGoal ] + | [ "in" ssrclausehyps(hyps) "|-" ] -> [ hyps, InHypsSeq ] + | [ "in" ssrclausehyps(hyps) "*" ] -> [ hyps, InHypsGoal ] + | [ "in" ssrclausehyps(hyps) ] -> [ hyps, InHyps ] + | [ "in" "|-" "*" ] -> [ [], InSeqGoal ] + | [ "in" "*" ] -> [ [], InAll ] + | [ "in" "*" "|-" ] -> [ [], InAllHyps ] + | [ ] -> [ [], InGoal ] +END + +let nohide = mkRel 0 +let hidden_goal_tag = "the_hidden_goal" + +(* Reduction that preserves the Prod/Let spine of the "in" tactical. *) + +let inc_safe n = if n = 0 then n else n + 1 +let rec safe_depth c = match kind_of_term c with +| LetIn (Name x, _, _, c') when is_discharged_id x -> safe_depth c' + 1 +| LetIn (_, _, _, c') | Prod (_, _, c') -> inc_safe (safe_depth c') +| _ -> 0 + +let red_safe r e s c0 = + let rec red_to e c n = match kind_of_term c with + | Prod (x, t, c') when n > 0 -> + let t' = r e s t in let e' = Environ.push_rel (x, None, t') e in + mkProd (x, t', red_to e' c' (n - 1)) + | LetIn (x, b, t, c') when n > 0 -> + let t' = r e s t in let e' = Environ.push_rel (x, None, t') e in + mkLetIn (x, r e s b, t', red_to e' c' (n - 1)) + | _ -> r e s c in + red_to e c0 (safe_depth c0) + +let check_wgen_uniq gens = + let clears = List.flatten (List.map fst gens) in + check_hyps_uniq [] clears; + let ids = Util.list_map_filter + (function (_,Some ((id,_),_)) -> Some (hoi_id id) | _ -> None) gens in + let rec check ids = function + | id :: _ when List.mem id ids -> + errorstrm (str"Duplicate generalization " ++ pr_id id) + | id :: hyps -> check (id :: ids) hyps + | [] -> () in + check [] ids + +let pf_clauseids gl gens clseq = + let keep_clears = List.map (fun x, _ -> x, None) in + if gens <> [] then (check_wgen_uniq gens; gens) else + if clseq <> InAll && clseq <> InAllHyps then keep_clears gens else + Util.error "assumptions should be named explicitly" + +let hidden_clseq = function InHyps | InHypsSeq | InAllHyps -> true | _ -> false + +let hidetacs clseq idhide cl0 = + if not (hidden_clseq clseq) then [] else + [posetac idhide cl0; convert_concl_no_check (mkVar idhide)] + +let discharge_hyp (id', (id, mode)) gl = + let cl' = subst_var id (pf_concl gl) in + match pf_get_hyp gl id, mode with + | (_, None, t), _ | (_, Some _, t), "(" -> + apply_type (mkProd (Name id', t, cl')) [mkVar id] gl + | (_, Some v, t), _ -> convert_concl (mkLetIn (Name id', v, t, cl')) gl + +let endclausestac id_map clseq gl_id cl0 gl = + let not_hyp' id = not (List.mem_assoc id id_map) in + let orig_id id = try List.assoc id id_map with _ -> id in + let dc, c = Term.decompose_prod_assum (pf_concl gl) in + let hide_goal = hidden_clseq clseq in + let c_hidden = hide_goal && c = mkVar gl_id in + let rec fits forced = function + | (id, _) :: ids, (Name id', _, _) :: dc' when id' = id -> + fits true (ids, dc') + | ids, dc' -> + forced && ids = [] && (not hide_goal || dc' = [] && c_hidden) in + let rec unmark c = match kind_of_term c with + | Var id when hidden_clseq clseq && id = gl_id -> cl0 + | Prod (Name id, t, c') when List.mem_assoc id id_map -> + mkProd (Name (orig_id id), unmark t, unmark c') + | LetIn (Name id, v, t, c') when List.mem_assoc id id_map -> + mkLetIn (Name (orig_id id), unmark v, unmark t, unmark c') + | _ -> map_constr unmark c in + let utac hyp = convert_hyp_no_check (map_named_declaration unmark hyp) in + let utacs = List.map utac (pf_hyps gl) in + let ugtac gl' = convert_concl_no_check (unmark (pf_concl gl')) gl' in + let ctacs = if hide_goal then [clear [gl_id]] else [] in + let mktac itacs = tclTHENLIST (itacs @ utacs @ ugtac :: ctacs) in + let itac (_, id) = introduction id in + if fits false (id_map, List.rev dc) then mktac (List.map itac id_map) gl else + let all_ids = ids_of_rel_context dc @ pf_ids_of_hyps gl in + if List.for_all not_hyp' all_ids && not c_hidden then mktac [] gl else + Util.error "tampering with discharged assumptions of \"in\" tactical" + +let is_id_constr c = match kind_of_term c with + | Lambda(_,_,c) when isRel c -> 1 = destRel c + | _ -> false + +let red_product_skip_id env sigma c = match kind_of_term c with + | App(hd,args) when Array.length args = 1 && is_id_constr hd -> args.(0) + | _ -> try Tacred.red_product env sigma c with _ -> c + +let abs_wgen keep_let ist gl f gen (args,c) = + let sigma, env = project gl, pf_env gl in + let evar_closed t p = + if occur_existential t then + Util.user_err_loc (loc_of_cpattern p,"ssreflect", + pr_constr_pat t ++ + str" contains holes and matches no subterm of the goal") in + match gen with + | _, Some ((x, mode), None) when mode = "@" || (mode = " " && keep_let) -> + let x = hoi_id x in + let _, bo, ty = pf_get_hyp gl x in + (if bo <> None then args else mkVar x :: args), + mkProd_or_LetIn (Name (f x),bo,ty) (subst_var x c) + | _, Some ((x, _), None) -> + let x = hoi_id x in + mkVar x :: args, mkProd (Name (f x), pf_get_hyp_typ gl x, subst_var x c) + | _, Some ((x, "@"), Some p) -> + let x = hoi_id x in + let cp = interp_cpattern ist gl p None in + let t, c = + try fill_occ_pattern ~raise_NoMatch:true env sigma c cp None 1 + with NoMatch -> redex_of_pattern env cp, c in + evar_closed t p; + let ut = red_product_skip_id env sigma t in + args, mkLetIn(Name (f x), ut, pf_type_of gl t, c) + | _, Some ((x, _), Some p) -> + let x = hoi_id x in + let cp = interp_cpattern ist gl p None in + let t, c = + try fill_occ_pattern ~raise_NoMatch:true env sigma c cp None 1 + with NoMatch -> redex_of_pattern env cp, c in + evar_closed t p; + t :: args, mkProd(Name (f x), pf_type_of gl t, c) + | _ -> args, c + +let clr_of_wgen gen clrs = match gen with + | clr, Some ((x, _), None) -> + let x = hoi_id x in + cleartac clr :: cleartac [SsrHyp(Util.dummy_loc,x)] :: clrs + | clr, _ -> cleartac clr :: clrs + +let tclCLAUSES ist tac (gens, clseq) gl = + if clseq = InGoal || clseq = InSeqGoal then tac gl else + let clr_gens = pf_clauseids gl gens clseq in + let clear = tclTHENLIST (List.rev(List.fold_right clr_of_wgen clr_gens [])) in + let gl_id = mk_anon_id hidden_goal_tag gl in + let cl0 = pf_concl gl in + let dtac gl = + let c = pf_concl gl in + let args, c = + List.fold_right (abs_wgen true ist gl mk_discharged_id) gens ([], c) in + apply_type c args gl in + let endtac = + let id_map = Util.list_map_filter (function + | _, Some ((x,_),_) -> let id = hoi_id x in Some (mk_discharged_id id, id) + | _, None -> None) gens in + endclausestac id_map clseq gl_id cl0 in + tclTHENLIST (hidetacs clseq gl_id cl0 @ [dtac; clear; tac; endtac]) gl +(* }}} *) + +(** Simpl switch *) + +type ssrsimpl = Simpl | Cut | SimplCut | Nop + +let pr_simpl = function + | Simpl -> str "/=" + | Cut -> str "//" + | SimplCut -> str "//=" + | Nop -> mt () + +let pr_ssrsimpl _ _ _ = pr_simpl + +let wit_ssrsimplrep, globwit_ssrsimplrep, rawwit_ssrsimplrep = + add_genarg "ssrsimplrep" pr_simpl + +ARGUMENT EXTEND ssrsimpl_ne TYPED AS ssrsimplrep PRINTED BY pr_ssrsimpl +| [ "/=" ] -> [ Simpl ] +| [ "//" ] -> [ Cut ] +| [ "//=" ] -> [ SimplCut ] +END + +ARGUMENT EXTEND ssrsimpl TYPED AS ssrsimplrep PRINTED BY pr_ssrsimpl +| [ ssrsimpl_ne(sim) ] -> [ sim ] +| [ ] -> [ Nop ] +END + +(* We must avoid zeta-converting any "let"s created by the "in" tactical. *) + +let safe_simpltac gl = + let cl' = red_safe Tacred.simpl (pf_env gl) (project gl) (pf_concl gl) in + convert_concl_no_check cl' gl + +let simpltac = function + | Simpl -> safe_simpltac + | Cut -> tclTRY donetac + | SimplCut -> tclTHEN safe_simpltac (tclTRY donetac) + | Nop -> tclIDTAC + +(** Rewriting direction *) + +let pr_dir = function L2R -> str "->" | R2L -> str "<-" +let pr_rwdir = function L2R -> mt() | R2L -> str "-" + +let rewritetac dir c = + (* Due to the new optional arg ?tac, application shouldn't be too partial *) + Equality.general_rewrite (dir = L2R) all_occurrences true false c + +let wit_ssrdir, globwit_ssrdir, rawwit_ssrdir = + add_genarg "ssrdir" pr_dir + +let dir_org = function L2R -> 1 | R2L -> 2 + +(** Indexes *) + +(* Since SSR indexes are always positive numbers, we use the 0 value *) +(* to encode an omitted index. We reuse the in or_var type, but we *) +(* supply our own interpretation function, which checks for non *) +(* positive values, and allows the use of constr numerals, so that *) +(* e.g., "let n := eval compute in (1 + 3) in (do n!clear)" works. *) + +type ssrindex = int or_var + +let pr_index = function + | ArgVar (_, id) -> pr_id id + | ArgArg n when n > 0 -> pr_int n + | _ -> mt () +let pr_ssrindex _ _ _ = pr_index + +let noindex = ArgArg 0 +let allocc = Some(false,[]) + +let check_index loc i = + if i > 0 then i else loc_error loc "Index not positive" +let mk_index loc = function ArgArg i -> ArgArg (check_index loc i) | iv -> iv + +let interp_index ist gl idx = + Tacmach.project gl, + match idx with + | ArgArg _ -> idx + | ArgVar (loc, id) -> + let i = try match List.assoc id ist.lfun with + | VInteger i -> i + | VConstr ([],c) -> + let rc = Detyping.detype false [] [] c in + begin match Notation.uninterp_prim_token rc with + | _, Numeral bigi -> int_of_string (Bigint.to_string bigi) + | _ -> raise Not_found + end + | _ -> raise Not_found + with _ -> loc_error loc "Index not a number" in + ArgArg (check_index loc i) + +ARGUMENT EXTEND ssrindex TYPED AS int_or_var PRINTED BY pr_ssrindex + INTERPRETED BY interp_index +| [ int_or_var(i) ] -> [ mk_index loc i ] +END + +(** Occurrence switch *) + +(* The standard syntax of complemented occurrence lists involves a single *) +(* initial "-", e.g., {-1 3 5}. An initial *) +(* "+" may be used to indicate positive occurrences (the default). The *) +(* "+" is optional, except if the list of occurrences starts with a *) +(* variable or is empty (to avoid confusion with a clear switch). The *) +(* empty positive switch "{+}" selects no occurrences, while the empty *) +(* negative switch "{-}" selects all occurrences explicitly; this is the *) +(* default, but "{-}" prevents the implicit clear, and can be used to *) +(* force dependent elimination -- see ndefectelimtac below. *) + +type ssrocc = occ + +let pr_occ = function + | Some (true, occ) -> str "{-" ++ pr_list pr_spc int occ ++ str "}" + | Some (false, occ) -> str "{+" ++ pr_list pr_spc int occ ++ str "}" + | None -> str "{}" + +let pr_ssrocc _ _ _ = pr_occ + +ARGUMENT EXTEND ssrocc TYPED AS (bool * int list) option PRINTED BY pr_ssrocc +| [ natural(n) natural_list(occ) ] -> [ + Some (false, List.map (check_index loc) (n::occ)) ] +| [ "-" natural_list(occ) ] -> [ Some (true, occ) ] +| [ "+" natural_list(occ) ] -> [ Some (false, occ) ] +END + +let pf_mkprod gl c ?(name=constr_name c) cl = + let t = pf_type_of gl c in + if name <> Anonymous || noccurn 1 cl then mkProd (name, t, cl) else + mkProd (Name (pf_type_id gl t), t, cl) + +let pf_abs_prod name gl c cl = pf_mkprod gl c ~name (subst_term c cl) + +(** Discharge occ switch (combined occurrence / clear switch *) + +type ssrdocc = ssrclear option * ssrocc option + +let mkocc occ = None, occ +let noclr = mkocc None +let mkclr clr = Some clr, None +let nodocc = mkclr [] + +let pr_docc = function + | None, occ -> pr_occ occ + | Some clr, _ -> pr_clear mt clr + +let pr_ssrdocc _ _ _ = pr_docc + +ARGUMENT EXTEND ssrdocc TYPED AS ssrclear option * ssrocc PRINTED BY pr_ssrdocc +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ mkclr clr ] +| [ "{" ssrocc(occ) "}" ] -> [ mkocc occ ] +END + +(** View hint database and View application. {{{ ******************************) + +(* There are three databases of lemmas used to mediate the application *) +(* of reflection lemmas: one for forward chaining, one for backward *) +(* chaining, and one for secondary backward chaining. *) + +(* View hints *) + +let rec isCxHoles = function (CHole _, None) :: ch -> isCxHoles ch | _ -> false + +let pr_raw_ssrhintref prc _ _ = function + | CAppExpl (_, (None, r), args) when isCHoles args -> + prc (CRef r) ++ str "|" ++ int (List.length args) + | CApp (_, (_, CRef _), _) as c -> prc c + | CApp (_, (_, c), args) when isCxHoles args -> + prc c ++ str "|" ++ int (List.length args) + | c -> prc c + +let pr_rawhintref = function + | GApp (_, f, args) when isRHoles args -> + pr_glob_constr f ++ str "|" ++ int (List.length args) + | c -> pr_glob_constr c + +let pr_glob_ssrhintref _ _ _ (c, _) = pr_rawhintref c + +let pr_ssrhintref prc _ _ = prc + +let mkhintref loc c n = match c with + | CRef r -> CAppExpl (loc, (None, r), mkCHoles loc n) + | _ -> mkAppC (c, mkCHoles loc n) + +ARGUMENT EXTEND ssrhintref + PRINTED BY pr_ssrhintref + RAW_TYPED AS constr RAW_PRINTED BY pr_raw_ssrhintref + GLOB_TYPED AS constr GLOB_PRINTED BY pr_glob_ssrhintref + | [ constr(c) ] -> [ c ] + | [ constr(c) "|" natural(n) ] -> [ mkhintref loc c n ] +END + +(* View purpose *) + +let pr_viewpos = function + | 0 -> str " for move/" + | 1 -> str " for apply/" + | 2 -> str " for apply//" + | _ -> mt () + +let pr_ssrviewpos _ _ _ = pr_viewpos + +ARGUMENT EXTEND ssrviewpos TYPED AS int PRINTED BY pr_ssrviewpos + | [ "for" "move" "/" ] -> [ 0 ] + | [ "for" "apply" "/" ] -> [ 1 ] + | [ "for" "apply" "/" "/" ] -> [ 2 ] + | [ "for" "apply" "//" ] -> [ 2 ] + | [ ] -> [ 3 ] +END + +let pr_ssrviewposspc _ _ _ i = pr_viewpos i ++ spc () + +ARGUMENT EXTEND ssrviewposspc TYPED AS ssrviewpos PRINTED BY pr_ssrviewposspc + | [ ssrviewpos(i) ] -> [ i ] +END + +(* The table and its display command *) + +let viewtab : glob_constr list array = Array.make 3 [] + +let _ = + let init () = Array.fill viewtab 0 3 [] in + let freeze () = Array.copy viewtab in + let unfreeze vt = Array.blit vt 0 viewtab 0 3 in + Summary.declare_summary "ssrview" + { Summary.freeze_function = freeze; + Summary.unfreeze_function = unfreeze; + Summary.init_function = init } + +let mapviewpos f n k = if n < 3 then f n else for i = 0 to k - 1 do f i done + +let print_view_hints i = + let pp_viewname = str "Hint View" ++ pr_viewpos i ++ str " " in + let pp_hints = pr_list spc pr_rawhintref viewtab.(i) in + ppnl (pp_viewname ++ hov 0 pp_hints ++ Pp.cut ()) + +VERNAC COMMAND EXTEND PrintView +| [ "Print" "Hint" "View" ssrviewpos(i) ] -> [ mapviewpos print_view_hints i 3 ] +END + +(* Populating the table *) + +let cache_viewhint (_, (i, lvh)) = + let mem_raw h = List.exists (Topconstr.eq_glob_constr h) in + let add_hint h hdb = if mem_raw h hdb then hdb else h :: hdb in + viewtab.(i) <- List.fold_right add_hint lvh viewtab.(i) + +let subst_viewhint ( subst, (i, lvh as ilvh)) = + let lvh' = list_smartmap (Detyping.subst_glob_constr subst) lvh in + if lvh' == lvh then ilvh else i, lvh' + +let classify_viewhint x = Libobject.Substitute x + +let in_viewhint = + Libobject.declare_object {(Libobject.default_object "VIEW_HINTS") with + Libobject.open_function = (fun i o -> if i = 1 then cache_viewhint o); + Libobject.cache_function = cache_viewhint; + Libobject.subst_function = subst_viewhint; + Libobject.classify_function = classify_viewhint } + +let glob_view_hints lvh = + List.map (Constrintern.intern_constr Evd.empty (Global.env ())) lvh + +let add_view_hints lvh i = Lib.add_anonymous_leaf (in_viewhint (i, lvh)) + +VERNAC COMMAND EXTEND HintView + | [ "Hint" "View" ssrviewposspc(n) ne_ssrhintref_list(lvh) ] -> + [ mapviewpos (add_view_hints (glob_view_hints lvh)) n 2 ] +END + +(** Views *) + +(* Views for the "move" and "case" commands are actually open *) +(* terms, but this is handled by interp_view, which is called *) +(* by interp_casearg. We use lists, to support the *) +(* "double-view" feature of the apply command. *) + +(* type ssrview = ssrterm list *) + +let pr_view = pr_list mt (fun c -> str "/" ++ pr_term c) + +let pr_ssrview _ _ _ = pr_view + +ARGUMENT EXTEND ssrview TYPED AS ssrterm list + PRINTED BY pr_ssrview +| [ "/" constr(c) ] -> [ [mk_term ' ' c] ] +| [ "/" constr(c) ssrview(w) ] -> [ (mk_term ' ' c) :: w ] +END + +(* There are two ways of "applying" a view to term: *) +(* 1- using a view hint if the view is an instance of some *) +(* (reflection) inductive predicate. *) +(* 2- applying the view if it coerces to a function, adding *) +(* implicit arguments. *) +(* They require guessing the view hints and the number of *) +(* implicits, respectively, which we do by brute force. *) + +let view_error s gv = + errorstrm (str ("Cannot " ^ s ^ " view ") ++ pr_term gv) + +let interp_view ist si env sigma gv rid = + match intern_term ist sigma env gv with + | GApp (loc, GHole _, rargs) -> + let rv = GApp (loc, rid, rargs) in + snd (interp_open_constr ist (re_sig si sigma) (rv, None)) + | rv -> + let interp rc rargs = + interp_open_constr ist (re_sig si sigma) (mkRApp rc rargs, None) in + let rec simple_view rargs n = + if n < 0 then view_error "use" gv else + try interp rv rargs with _ -> simple_view (mkRHole :: rargs) (n - 1) in + let view_nbimps = interp_view_nbimps ist (re_sig si sigma) rv in + let view_args = [mkRApp rv (mkRHoles view_nbimps); rid] in + let rec view_with = function + | [] -> simple_view [rid] (interp_nbargs ist (re_sig si sigma) rv) + | hint :: hints -> try interp hint view_args with _ -> view_with hints in + snd (view_with (if view_nbimps < 0 then [] else viewtab.(0))) + +let top_id = mk_internal_id "top assumption" + +let with_view ist si env gl0 c name cl prune = + let c2r ist x = { ist with lfun = (top_id, VConstr ([], x)) :: ist.lfun } in + let rec loop (sigma, c') = function + | f :: view -> + let rid, ist = match kind_of_term c' with + | Var id -> mkRVar id, ist + | _ -> mkRltacVar top_id, c2r ist c' in + loop (interp_view ist si env sigma f rid) view + | [] -> + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + let c' = Reductionops.nf_evar sigma c' in + let n, c', _ = pf_abs_evars gl0 (sigma, c') in + let c' = if not prune then c' else pf_abs_cterm gl0 n c' in + pf_abs_prod name gl0 c' (prod_applist cl [c]), c' + in loop + +let pf_with_view ist gl (prune, view) cl c = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + with_view ist si env gl c (constr_name c) cl prune (sigma, c) view +(* }}} *) + +(** Extended intro patterns {{{ ***********************************************) + +type ssrtermrep = char * glob_constr_and_expr +type ssripat = + | IpatSimpl of ssrclear * ssrsimpl + | IpatId of identifier + | IpatWild + | IpatCase of ssripats list + | IpatRw of ssrocc * ssrdir + | IpatAll + | IpatAnon + | IpatView of ssrtermrep list + | IpatNoop + | IpatNewHidden of identifier list +and ssripats = ssripat list + +let remove_loc = snd + +let rec ipat_of_intro_pattern = function + | IntroIdentifier id -> IpatId id + | IntroWildcard -> IpatWild + | IntroOrAndPattern iorpat -> + IpatCase + (List.map (List.map ipat_of_intro_pattern) + (List.map (List.map remove_loc) iorpat)) + | IntroAnonymous -> IpatAnon + | IntroRewrite b -> IpatRw (allocc, if b then L2R else R2L) + | IntroFresh id -> IpatAnon + | IntroForthcoming _ -> (* Unable to determine which kind of ipat interp_introid could return [HH] *) + assert false + +let rec pr_ipat = function + | IpatId id -> pr_id id + | IpatSimpl (clr, sim) -> pr_clear mt clr ++ pr_simpl sim + | IpatCase iorpat -> hov 1 (str "[" ++ pr_iorpat iorpat ++ str "]") + | IpatRw (occ, dir) -> pr_occ occ ++ pr_dir dir + | IpatAll -> str "*" + | IpatWild -> str "_" + | IpatAnon -> str "?" + | IpatView v -> pr_view v + | IpatNoop -> str "-" + | IpatNewHidden l -> str "[:" ++ pr_list spc pr_id l ++ str "]" +and pr_iorpat iorpat = pr_list pr_bar pr_ipats iorpat +and pr_ipats ipats = pr_list spc pr_ipat ipats + +let wit_ssripatrep, globwit_ssripatrep, rawwit_ssripatrep = + add_genarg "ssripatrep" pr_ipat + +let pr_ssripat _ _ _ = pr_ipat +let pr_ssripats _ _ _ = pr_ipats +let pr_ssriorpat _ _ _ = pr_iorpat + +let intern_ipat ist ipat = + let rec check_pat = function + | IpatSimpl (clr, _) -> ignore (List.map (intern_hyp ist) clr) + | IpatCase iorpat -> List.iter (List.iter check_pat) iorpat + | _ -> () in + check_pat ipat; ipat + +let intern_ipats ist = List.map (intern_ipat ist) + +let interp_introid ist gl id = + try IntroIdentifier (hyp_id (snd (interp_hyp ist gl (SsrHyp (dummy_loc, id))))) + with _ -> snd(snd (interp_intro_pattern ist gl (dummy_loc,IntroIdentifier id))) + +let rec add_intro_pattern_hyps (loc, ipat) hyps = match ipat with + | IntroIdentifier id -> + if not_section_id id then SsrHyp (loc, id) :: hyps else + hyp_err loc "Can't delete section hypothesis " id + | IntroWildcard -> hyps + | IntroOrAndPattern iorpat -> + List.fold_right (List.fold_right add_intro_pattern_hyps) iorpat hyps + | IntroAnonymous -> [] + | IntroFresh _ -> [] + | IntroRewrite _ -> hyps + | IntroForthcoming _ -> + (* As in ipat_of_intro_pattern, was unable to determine which kind + of ipat interp_introid could return [HH] *) assert false + +let rec interp_ipat ist gl = + let ltacvar id = List.mem_assoc id ist.lfun in + let rec interp = function + | IpatId id when ltacvar id -> + ipat_of_intro_pattern (interp_introid ist gl id) + | IpatSimpl (clr, sim) -> + let add_hyps (SsrHyp (loc, id) as hyp) hyps = + if not (ltacvar id) then hyp :: hyps else + add_intro_pattern_hyps (loc, (interp_introid ist gl id)) hyps in + let clr' = List.fold_right add_hyps clr [] in + check_hyps_uniq [] clr'; IpatSimpl (clr', sim) + | IpatCase iorpat -> IpatCase (List.map (List.map interp) iorpat) + | IpatNewHidden l -> + IpatNewHidden + (List.map (function + | IntroIdentifier id -> id + | _ -> assert false) + (List.map (interp_introid ist gl) l)) + | ipat -> ipat in + interp + +let interp_ipats ist gl l = project gl, List.map (interp_ipat ist gl) l + +let pushIpatRw = function + | pats :: orpat -> (IpatRw (allocc, L2R) :: pats) :: orpat + | [] -> [] + +let pushIpatNoop = function + | pats :: orpat -> (IpatNoop :: pats) :: orpat + | [] -> [] + +ARGUMENT EXTEND ssripat TYPED AS ssripatrep list PRINTED BY pr_ssripats + INTERPRETED BY interp_ipats + GLOBALIZED BY intern_ipats + | [ "_" ] -> [ [IpatWild] ] + | [ "*" ] -> [ [IpatAll] ] + | [ ident(id) ] -> [ [IpatId id] ] + | [ "?" ] -> [ [IpatAnon] ] + | [ ssrsimpl_ne(sim) ] -> [ [IpatSimpl ([], sim)] ] + | [ ssrdocc(occ) "->" ] -> [ match occ with + | None, occ -> [IpatRw (occ, L2R)] + | Some clr, _ -> [IpatSimpl (clr, Nop); IpatRw (allocc, L2R)]] + | [ ssrdocc(occ) "<-" ] -> [ match occ with + | None, occ -> [IpatRw (occ, R2L)] + | Some clr, _ -> [IpatSimpl (clr, Nop); IpatRw (allocc, R2L)]] + | [ ssrdocc(occ) ] -> [ match occ with + | Some cl, _ -> check_hyps_uniq [] cl; [IpatSimpl (cl, Nop)] + | _ -> loc_error loc "Only identifiers are allowed here"] + | [ "->" ] -> [ [IpatRw (allocc, L2R)] ] + | [ "<-" ] -> [ [IpatRw (allocc, R2L)] ] + | [ "-" ] -> [ [IpatNoop] ] + | [ "-/" "=" ] -> [ [IpatNoop;IpatSimpl([],Simpl)] ] + | [ "-/=" ] -> [ [IpatNoop;IpatSimpl([],Simpl)] ] + | [ "-/" "/" ] -> [ [IpatNoop;IpatSimpl([],Cut)] ] + | [ "-//" ] -> [ [IpatNoop;IpatSimpl([],Cut)] ] + | [ "-/" "/=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ "-//" "=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ "-//=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ ssrview(v) ] -> [ [IpatView v] ] + | [ "[" ":" ident_list(idl) "]" ] -> [ [IpatNewHidden idl] ] +END + +ARGUMENT EXTEND ssripats TYPED AS ssripat PRINTED BY pr_ssripats + | [ ssripat(i) ssripats(tl) ] -> [ i @ tl ] + | [ ] -> [ [] ] +END + +ARGUMENT EXTEND ssriorpat TYPED AS ssripat list PRINTED BY pr_ssriorpat +| [ ssripats(pats) "|" ssriorpat(orpat) ] -> [ pats :: orpat ] +| [ ssripats(pats) "|-" ">" ssriorpat(orpat) ] -> [ pats :: pushIpatRw orpat ] +| [ ssripats(pats) "|-" ssriorpat(orpat) ] -> [ pats :: pushIpatNoop orpat ] +| [ ssripats(pats) "|->" ssriorpat(orpat) ] -> [ pats :: pushIpatRw orpat ] +| [ ssripats(pats) "||" ssriorpat(orpat) ] -> [ pats :: [] :: orpat ] +| [ ssripats(pats) "|||" ssriorpat(orpat) ] -> [ pats :: [] :: [] :: orpat ] +| [ ssripats(pats) "||||" ssriorpat(orpat) ] -> [ [pats; []; []; []] @ orpat ] +| [ ssripats(pats) ] -> [ [pats] ] +END + +let reject_ssrhid strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "[" -> + (match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD ":" -> raise Stream.Failure + | _ -> ()) + | _ -> () + +let test_nohidden = Gram.Entry.of_parser "test_ssrhid" reject_ssrhid + +ARGUMENT EXTEND ssrcpat TYPED AS ssripatrep PRINTED BY pr_ssripat + | [ "Qed" ssriorpat(x) ] -> [ IpatCase x ] +END + +GEXTEND Gram + GLOBAL: ssrcpat; + ssrcpat: [[ test_nohidden; "["; iorpat = ssriorpat; "]" -> IpatCase iorpat ]]; +END + +GEXTEND Gram + GLOBAL: ssripat; + ssripat: [[ pat = ssrcpat -> [pat] ]]; +END + +ARGUMENT EXTEND ssripats_ne TYPED AS ssripat PRINTED BY pr_ssripats + | [ ssripat(i) ssripats(tl) ] -> [ i @ tl ] +END + +(* subsets of patterns *) +let check_ssrhpats loc w_binders ipats = + let err_loc s = user_err_loc (loc, "ssreflect", s) in + let clr, ipats = + let rec aux clr = function + | IpatSimpl (cl, Nop) :: tl -> aux (clr @ cl) tl + | IpatSimpl (cl, sim) :: tl -> clr @ cl, IpatSimpl ([], sim) :: tl + | tl -> clr, tl + in aux [] ipats in + let simpl, ipats = + match List.rev ipats with + | IpatSimpl ([],_) as s :: tl -> [s], List.rev tl + | _ -> [], ipats in + if simpl <> [] && not w_binders then + err_loc (str "No s-item allowed here: " ++ pr_ipats simpl); + let ipat, binders = + let rec loop ipat = function + | [] -> ipat, [] + | ( IpatId _| IpatAnon| IpatCase _| IpatRw _ as i) :: tl -> + if w_binders then + if simpl <> [] && tl <> [] then + err_loc(str"binders XOR s-item allowed here: "++pr_ipats(tl@simpl)) + else if not (List.for_all (function IpatId _ -> true | _ -> false) tl) + then err_loc (str "Only binders allowed here: " ++ pr_ipats tl) + else ipat @ [i], tl + else + if tl = [] then ipat @ [i], [] + else err_loc (str "No binder or s-item allowed here: " ++ pr_ipats tl) + | hd :: tl -> loop (ipat @ [hd]) tl + in loop [] ipats in + ((clr, ipat), binders), simpl + +let single loc = + function [x] -> x | _ -> loc_error loc "Only one intro pattern is allowed" + +let pr_hpats (((clr, ipat), binders), simpl) = + pr_clear mt clr ++ pr_ipats ipat ++ pr_ipats binders ++ pr_ipats simpl +let pr_ssrhpats _ _ _ = pr_hpats +let pr_ssrhpats_wtransp _ _ _ (_, x) = pr_hpats x + +ARGUMENT EXTEND ssrhpats TYPED AS ((ssrclear * ssripat) * ssripat) * ssripat +PRINTED BY pr_ssrhpats + | [ ssripats(i) ] -> [ check_ssrhpats loc true i ] +END + +ARGUMENT EXTEND ssrhpats_wtransp + TYPED AS bool * (((ssrclear * ssripat) * ssripat) * ssripat) + PRINTED BY pr_ssrhpats_wtransp + | [ ssripats(i) ] -> [ false,check_ssrhpats loc true i ] + | [ ssripats(i) "@" ssripats(j) ] -> [ true,check_ssrhpats loc true (i @ j) ] +END + +ARGUMENT EXTEND ssrhpats_nobs +TYPED AS ((ssrclear * ssripat) * ssripat) * ssripat PRINTED BY pr_ssrhpats + | [ ssripats(i) ] -> [ check_ssrhpats loc false i ] +END + +ARGUMENT EXTEND ssrrpat TYPED AS ssripatrep PRINTED BY pr_ssripat + | [ "->" ] -> [ IpatRw (allocc, L2R) ] + | [ "<-" ] -> [ IpatRw (allocc, R2L) ] +END + +type ssrintros = ssripats * ltacctx + +let pr_intros sep (intrs, _) = + if intrs = [] then mt() else sep () ++ str "=> " ++ pr_ipats intrs +let pr_ssrintros _ _ _ = pr_intros mt + +ARGUMENT EXTEND ssrintros_ne TYPED AS ssripat * ltacctx + PRINTED BY pr_ssrintros + | [ "=>" ssripats_ne(pats) ] -> [ pats, rawltacctx ] +END + +ARGUMENT EXTEND ssrintros TYPED AS ssrintros_ne PRINTED BY pr_ssrintros + | [ ssrintros_ne(intrs) ] -> [ intrs ] + | [ ] -> [ [], rawltacctx ] +END + +let injecteq_id = mk_internal_id "injection equation" + +let pf_nb_prod gl = nb_prod (pf_concl gl) + +let rev_id = mk_internal_id "rev concl" + +let revtoptac n0 gl = + let n = pf_nb_prod gl - n0 in + let dc, cl = decompose_prod_n n (pf_concl gl) in + let dc' = dc @ [Name rev_id, compose_prod (List.rev dc) cl] in + let f = compose_lam dc' (mkEtaApp (mkRel (n + 1)) (-n) 1) in + refine (mkApp (f, [|Evarutil.mk_new_meta ()|])) gl + +let equality_inj l b id c gl = + let msg = ref "" in + try Equality.inj l b c gl + with Compat.Loc.Exc_located(_,UserError (_,s)) | UserError (_,s) + when msg := Pp.string_of_ppcmds s; + !msg = "Not a projectable equality but a discriminable one." || + !msg = "Nothing to inject." -> + msg_warning (str !msg); + discharge_hyp (id, (id, "")) gl + +let injectidl2rtac id c gl = + tclTHEN (equality_inj [] true id c) (revtoptac (pf_nb_prod gl)) gl + +let injectl2rtac c = match kind_of_term c with +| Var id -> injectidl2rtac id (mkVar id, NoBindings) +| _ -> + let id = injecteq_id in + tclTHENLIST [havetac id c; injectidl2rtac id (mkVar id, NoBindings); clear [id]] + +let is_injection_case c gl = + let mind, _ = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in + mkInd mind = build_coq_eq () + +let perform_injection c gl = + let mind, t = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in + let dc, eqt = decompose_prod t in + if dc = [] then injectl2rtac c gl else + if not (closed0 eqt) then error "can't decompose a quantified equality" else + let cl = pf_concl gl in let n = List.length dc in + let c_eq = mkEtaApp c n 2 in + let cl1 = mkLambda (Anonymous, mkArrow eqt cl, mkApp (mkRel 1, [|c_eq|])) in + let id = injecteq_id in + let id_with_ebind = (mkVar id, NoBindings) in + let injtac = tclTHEN (introid id) (injectidl2rtac id id_with_ebind) in + tclTHENLAST (apply (compose_lam dc cl1)) injtac gl + +let simplest_newcase_ref = ref (fun t gl -> assert false) +let simplest_newcase x gl = !simplest_newcase_ref x gl + +let ssrscasetac c gl = + if is_injection_case c gl then perform_injection c gl + else simplest_newcase c gl + +let intro_all gl = + let dc, _ = Term.decompose_prod_assum (pf_concl gl) in + tclTHENLIST (List.map anontac (List.rev dc)) gl + +let rec intro_anon gl = + try anontac (List.hd (fst (Term.decompose_prod_n_assum 1 (pf_concl gl)))) gl + with err0 -> try tclTHEN red_in_concl intro_anon gl with _ -> raise err0 + (* with _ -> error "No product even after reduction" *) + +let with_top tac = + tclTHENLIST [introid top_id; tac (mkVar top_id); clear [top_id]] + +let rec mapLR f = function [] -> [] | x :: s -> let y = f x in y :: mapLR f s + +let wild_ids = ref [] + +let new_wild_id () = + let i = 1 + List.length !wild_ids in + let id = mk_wildcard_id i in + wild_ids := id :: !wild_ids; + id + +let clear_wilds wilds gl = + clear (List.filter (fun id -> List.mem id wilds) (pf_ids_of_hyps gl)) gl + +let clear_with_wilds wilds clr0 gl = + let extend_clr clr (id, _, _ as nd) = + if List.mem id clr || not (List.mem id wilds) then clr else + let vars = global_vars_set_of_decl (pf_env gl) nd in + let occurs id' = Idset.mem id' vars in + if List.exists occurs clr then id :: clr else clr in + clear (Sign.fold_named_context_reverse extend_clr ~init:clr0 (pf_hyps gl)) gl + +let tclTHENS_nonstrict tac tacl taclname gl = + let tacres = tac gl in + let n_gls = List.length (sig_it tacres) in + let n_tac = List.length tacl in + if n_gls = n_tac then tclTHENS (fun _ -> tacres) tacl gl else + if n_gls = 0 then tacres else + let pr_only n1 n2 = if n1 < n2 then str "only " else mt () in + let pr_nb n1 n2 name = + pr_only n1 n2 ++ int n1 ++ str (" " ^ plural n1 name) in + errorstrm (pr_nb n_tac n_gls taclname ++ spc () + ++ str "for " ++ pr_nb n_gls n_tac "subgoal") + +(* Forward reference to extended rewrite *) +let ipat_rewritetac = ref (fun _ -> rewritetac) + +let rec is_name_in_ipats name = function + | IpatSimpl(clr,_) :: tl -> + List.exists (function SsrHyp(_,id) -> id = name) clr + || is_name_in_ipats name tl + | IpatId id :: tl -> id = name || is_name_in_ipats name tl + | IpatCase l :: tl -> is_name_in_ipats name (List.flatten l @ tl) + | _ :: tl -> is_name_in_ipats name tl + | [] -> false + +let move_top_with_view = ref (fun _ -> assert false) + +let rec nat_of_n n = + if n = 0 then mkConstruct path_of_O + else mkApp (mkConstruct path_of_S, [|nat_of_n (n-1)|]) + +let ssr_abstract_id = ref 0 +let _ = + Summary.declare_summary "SSR:abstractid" + { Summary.freeze_function = (fun _ -> !ssr_abstract_id); + Summary.unfreeze_function = (fun x -> ssr_abstract_id := x); + Summary.init_function = (fun () -> ssr_abstract_id := 0) } + +let mk_abstract_id () = incr ssr_abstract_id; nat_of_n !ssr_abstract_id + +let ssrmkabs id gl = + let env, concl = pf_env gl, pf_concl gl in + let sigma, abstract_proof, abstract_ty = + let sigma, ty = Evarutil.new_type_evar Evd.empty env in + let sigma,lock = Evarutil.new_evar sigma env (mkSsrConst "abstract_lock") in + let abstract_ty = + mkApp(mkSsrConst "abstract", [|ty;mk_abstract_id ();lock|]) in + let sigma, m = Evarutil.new_evar sigma env abstract_ty in + sigma, m, abstract_ty in + let sigma, kont = + let rd = Name id, None, abstract_ty in + Evarutil.new_evar sigma (Environ.push_rel rd env) concl in + pp(lazy(pr_constr concl)); + let step = mkApp (mkLambda(Name id,abstract_ty,kont) ,[|abstract_proof|]) in + Refine.refine (sigma, step) gl + +let ssrmkabstac ids = + List.fold_right (fun id tac -> tclTHENFIRST (ssrmkabs id) tac) ids tclIDTAC + +(* introstac: for "move" and "clear", tclEQINTROS: for "case" and "elim" *) +(* This block hides the spaghetti-code needed to implement the only two *) +(* tactics that should be used to process intro patters. *) +(* The difficulty is that we don't want to always rename, but we can *) +(* compute needeed renamings only at runtime, so we theread a tree like *) +(* imperativestructure so that outer renamigs are inherited by inner *) +(* ipts and that the cler performed at the end of ipatstac clears hyps *) +(* eventually renamed at runtime. *) +(* TODO: hide wild_ids in this block too *) +let introstac, tclEQINTROS = + let rec map_acc_k f k = function + | [] -> (* tricky: we save wilds now, we get to_cler (aka k) later *) + let clear_ww = clear_with_wilds !wild_ids in + [fun gl -> clear_ww (hyps_ids (List.flatten (List.map (!) k))) gl] + | x :: xs -> let k, x = f k xs x in x :: map_acc_k f k xs in + let rename force to_clr rest clr gl = + let hyps = pf_hyps gl in + pp(lazy(str"rename " ++ pr_clear spc clr)); + let () = if not force then List.iter (check_hyp_exists hyps) clr in + if List.exists (fun x -> force || is_name_in_ipats (hyp_id x) rest) clr then + let ren_clr, ren = + List.split (List.map (fun x -> let x = hyp_id x in + let x' = mk_anon_id (string_of_id x) gl in + SsrHyp (dummy_loc, x'), (x, x')) clr) in + let () = to_clr := ren_clr in + rename_hyp ren gl + else + let () = to_clr := clr in + tclIDTAC gl in + let rec ipattac ?ist k rest = function + | IpatWild -> k, introid (new_wild_id ()) + | IpatCase iorpat -> k, tclIORPAT ?ist k (with_top ssrscasetac) iorpat + | IpatRw (occ, dir) -> k, with_top (!ipat_rewritetac occ dir) + | IpatId id -> k, introid id + | IpatNewHidden idl -> k, ssrmkabstac idl + | IpatSimpl (clr, sim) -> + let to_clr = ref [] in + to_clr :: k, tclTHEN (rename false to_clr rest clr) (simpltac sim) + | IpatAll -> k, intro_all + | IpatAnon -> k, intro_anon + | IpatNoop -> k, tclIDTAC + | IpatView v -> match ist with + | None -> anomaly "ipattac with no ist but view" + | Some ist -> match rest with + | (IpatCase _ | IpatRw _)::_ -> + let to_clr = ref [] in let top_id = ref top_id in + to_clr :: k, + tclTHEN + (!move_top_with_view false top_id (false,v) ist) + (fun gl -> + rename true to_clr rest [SsrHyp (dummy_loc, !top_id)]gl) + | _ -> k, !move_top_with_view true (ref top_id) (true,v) ist + and tclIORPAT ?ist k tac = function + | [[]] -> tac + | orp -> + tclTHENS_nonstrict tac (mapLR (ipatstac ?ist k) orp) "intro pattern" + and ipatstac ?ist k ipats = + tclTHENLIST (map_acc_k (ipattac ?ist) k ipats) in + let introstac ?ist ipats = + wild_ids := []; + let tac = ipatstac ?ist [] ipats in + tclTHENLIST [tac; clear_wilds !wild_ids] in + let tclEQINTROS ?ist tac eqtac ipats = + wild_ids := []; + let rec split_itacs to_clr tac' = function + | (IpatSimpl _ as spat) :: ipats' -> + let to_clr, tac = ipattac ?ist to_clr ipats' spat in + split_itacs to_clr (tclTHEN tac' tac) ipats' + | IpatCase iorpat :: ipats' -> + to_clr, tclIORPAT ?ist to_clr tac' iorpat, ipats' + | ipats' -> to_clr, tac', ipats' in + let to_clr, tac1, ipats' = split_itacs [] tac ipats in + let tac2 = ipatstac ?ist to_clr ipats' in + tclTHENLIST [tac1; eqtac; tac2; clear_wilds !wild_ids] in + introstac, tclEQINTROS +;; + +let rec eqmoveipats eqpat = function + | (IpatSimpl _ as ipat) :: ipats -> ipat :: eqmoveipats eqpat ipats + | (IpatAll :: _ | []) as ipats -> IpatAnon :: eqpat :: ipats + | ipat :: ipats -> ipat :: eqpat :: ipats + +(* General case *) +let tclINTROS tac (ipats, ctx) = + let ist = get_ltacctx ctx in + tclEQINTROS ~ist (tac ist) tclIDTAC ipats + +(** The "=>" tactical *) + +let ssrintros_sep = + let atom_sep = function + | TacSplit (_,_, [NoBindings]) -> mt + | TacLeft (_, NoBindings) -> mt + | TacRight (_, NoBindings) -> mt + (* | TacExtend (_, "ssrapply", []) -> mt *) + | _ -> spc in + function + | TacId [] -> mt + | TacArg (_, Tacexp _) -> mt + | TacArg (_, Reference _) -> mt + | TacAtom (_, atom) -> atom_sep atom + | _ -> spc + +let pr_ssrintrosarg _ _ prt (tac, ipats) = + prt tacltop tac ++ pr_intros (ssrintros_sep tac) ipats + +ARGUMENT EXTEND ssrintrosarg TYPED AS tactic * ssrintros + PRINTED BY pr_ssrintrosarg +| [ "Qed" ssrtacarg(arg) ssrintros_ne(ipats) ] -> [ arg, ipats ] +END + +TACTIC EXTEND ssrtclintros +| [ "Qed" ssrintrosarg(arg) ] -> + [ let tac, intros = arg in + tclINTROS (fun ist -> ssrevaltac ist tac) intros ] +END +set_pr_ssrtac "tclintros" 0 [ArgSsr "introsarg"] + +let tclintros_expr loc tac ipats = + let args = [in_gen rawwit_ssrintrosarg (tac, ipats)] in + ssrtac_expr loc "tclintros" args + +GEXTEND Gram + GLOBAL: tactic_expr; + tactic_expr: LEVEL "1" [ RIGHTA + [ tac = tactic_expr; intros = ssrintros_ne -> tclintros_expr loc tac intros + ] ]; +END +(* }}} *) + +(** Multipliers {{{ ***********************************************************) + +(* modality *) + +type ssrmmod = May | Must | Once + +let pr_mmod = function May -> str "?" | Must -> str "!" | Once -> mt () + +let wit_ssrmmod, globwit_ssrmmod, rawwit_ssrmmod = add_genarg "ssrmmod" pr_mmod +let ssrmmod = Gram.Entry.create "ssrmmod" +GEXTEND Gram + GLOBAL: ssrmmod; + ssrmmod: [[ "!" -> Must | LEFTQMARK -> May | "?" -> May]]; +END + +(* tactical *) + +let tclID tac = tac + +let tclDOTRY n tac = + if n <= 0 then tclIDTAC else + let rec loop i gl = + if i = n then tclTRY tac gl else + tclTRY (tclTHEN tac (loop (i + 1))) gl in + loop 1 + +let tclDO n tac = + let prefix i = str"At iteration " ++ int i ++ str": " in + let tac_err_at i gl = + try tac gl + with + | UserError (l, s) -> raise (UserError (l, prefix i ++ s)) + | Compat.Loc.Exc_located(loc, UserError (l, s)) -> + raise (Compat.Loc.Exc_located(loc, UserError (l, prefix i ++ s))) in + let rec loop i gl = + if i = n then tac_err_at i gl else + (tclTHEN (tac_err_at i) (loop (i + 1))) gl in + loop 1 + +let tclMULT = function + | 0, May -> tclREPEAT + | 1, May -> tclTRY + | n, May -> tclDOTRY n + | 0, Must -> tclAT_LEAST_ONCE + | n, Must when n > 1 -> tclDO n + | _ -> tclID + +(** The "do" tactical. ********************************************************) + +(* +type ssrdoarg = ((ssrindex * ssrmmod) * (ssrhint * ltacctx)) * ssrclauses +*) + +let pr_ssrdoarg prc _ prt (((n, m), (tac, _)), clauses) = + pr_index n ++ pr_mmod m ++ pr_hintarg prt tac ++ pr_clauses clauses + +ARGUMENT EXTEND ssrdoarg + TYPED AS ((ssrindex * ssrmmod) * (ssrhintarg * ltacctx)) * ssrclauses + PRINTED BY pr_ssrdoarg +| [ "Qed" ] -> [ anomaly "Grammar placeholder match" ] +END + +let ssrdotac (((n, m), (tac, ctx)), clauses) = + let mul = get_index n, m in + let ist = get_ltacctx ctx in + tclCLAUSES ist (tclMULT mul (hinttac ist false tac)) clauses + +TACTIC EXTEND ssrtcldo +| [ "Qed" "do" ssrdoarg(arg) ] -> [ ssrdotac arg ] +END +set_pr_ssrtac "tcldo" 3 [ArgSep "do "; ArgSsr "doarg"] + +let ssrdotac_expr loc n m tac clauses = + let arg = ((n, m), (tac, rawltacctx)), clauses in + ssrtac_expr loc "tcldo" [in_gen rawwit_ssrdoarg arg] + +GEXTEND Gram + GLOBAL: tactic_expr; + ssrdotac: [ + [ tac = tactic_expr LEVEL "3" -> mk_hint tac + | tacs = ssrortacarg -> tacs + ] ]; + tactic_expr: LEVEL "3" [ RIGHTA + [ IDENT "do"; m = ssrmmod; tac = ssrdotac; clauses = ssrclauses -> + ssrdotac_expr loc noindex m tac clauses + | IDENT "do"; tac = ssrortacarg; clauses = ssrclauses -> + ssrdotac_expr loc noindex Once tac clauses + | IDENT "do"; n = int_or_var; m = ssrmmod; + tac = ssrdotac; clauses = ssrclauses -> + ssrdotac_expr loc (mk_index loc n) m tac clauses + ] ]; +END +(* }}} *) + +(** The "first" and "last" tacticals. {{{ *************************************) + +(* type ssrseqarg = ssrindex * (ssrtacarg * ssrtac option) *) + +let pr_seqtacarg prt = function + | (is_first, []), _ -> str (if is_first then "first" else "last") + | tac, Some dtac -> + hv 0 (pr_hintarg prt tac ++ spc() ++ str "|| " ++ prt tacltop dtac) + | tac, _ -> pr_hintarg prt tac + +let pr_ssrseqarg _ _ prt = function + | ArgArg 0, tac -> pr_seqtacarg prt tac + | i, tac -> pr_index i ++ str " " ++ pr_seqtacarg prt tac + +(* We must parse the index separately to resolve the conflict with *) +(* an unindexed tactic. *) +ARGUMENT EXTEND ssrseqarg TYPED AS ssrindex * (ssrhintarg * tactic option) + PRINTED BY pr_ssrseqarg +| [ "Qed" ] -> [ anomaly "Grammar placeholder match" ] +END + +let sq_brace_tacnames = + ["first"; "solve"; "do"; "rewrite"; "have"; "suffices"; "wlog"] + (* "by" is a keyword *) +let accept_ssrseqvar strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT id when not (List.mem id sq_brace_tacnames) -> + accept_before_syms_or_ids ["["] ["first";"last"] strm + | _ -> raise Stream.Failure + +let test_ssrseqvar = Gram.Entry.of_parser "test_ssrseqvar" accept_ssrseqvar + +let swaptacarg (loc, b) = (b, []), Some (TacAtom (loc, TacRevert [])) + +let check_seqtacarg dir arg = match snd arg, dir with + | ((true, []), Some (TacAtom (loc, _))), L2R -> + loc_error loc "expected \"last\"" + | ((false, []), Some (TacAtom (loc, _))), R2L -> + loc_error loc "expected \"first\"" + | _, _ -> arg + +let ssrorelse = Gram.Entry.create "ssrorelse" +GEXTEND Gram + GLOBAL: ssrorelse ssrseqarg; + ssrseqidx: [ + [ test_ssrseqvar; id = Prim.ident -> ArgVar (loc, id) + | n = Prim.natural -> ArgArg (check_index loc n) + ] ]; + ssrswap: [[ IDENT "first" -> loc, true | IDENT "last" -> loc, false ]]; + ssrorelse: [[ "||"; tac = tactic_expr LEVEL "2" -> tac ]]; + ssrseqarg: [ + [ arg = ssrswap -> noindex, swaptacarg arg + | i = ssrseqidx; tac = ssrortacarg; def = OPT ssrorelse -> i, (tac, def) + | i = ssrseqidx; arg = ssrswap -> i, swaptacarg arg + | tac = tactic_expr LEVEL "3" -> noindex, (mk_hint tac, None) + ] ]; +END + +let tclPERM perm tac gls = + let subgls = tac gls in + let sigma, subgll = Refiner.unpackage subgls in + let subgll' = perm subgll in + Refiner.repackage sigma subgll' +(* +let tclPERM perm tac gls = + let mkpft n g r = + {Proof_type.open_subgoals = n; Proof_type.goal = g; Proof_type.ref = r} in + let mkleaf g = mkpft 0 g None in + let mkprpft n g pr a = mkpft n g (Some (Proof_type.Prim pr, a)) in + let mkrpft n g c = mkprpft n g (Proof_type.Refine c) in + let mkipft n g = + let mki pft (id, _, _ as d) = + let g' = {g with evar_concl = mkNamedProd_or_LetIn d g.evar_concl} in + mkprpft n g' (Proof_type.Intro id) [pft] in + List.fold_left mki in + let gl = Refiner.sig_it gls in + let mkhyp subgl = + let rec chop_section = function + | (x, _, _ as d) :: e when not_section_id x -> d :: chop_section e + | _ -> [] in + let lhyps = Environ.named_context_of_val subgl.evar_hyps in + mk_perm_id (), subgl, chop_section lhyps in + let mkpfvar (hyp, subgl, lhyps) = + let mkarg args (lhyp, body, _) = + if body = None then mkVar lhyp :: args else args in + mkrpft 0 subgl (applist (mkVar hyp, List.fold_left mkarg [] lhyps)) [] in + let mkpfleaf (_, subgl, lhyps) = mkipft 1 gl (mkleaf subgl) lhyps in + let mkmeta _ = Evarutil.mk_new_meta () in + let mkhypdecl (hyp, subgl, lhyps) = + hyp, None, it_mkNamedProd_or_LetIn subgl.evar_concl lhyps in + let subgls, v as res0 = tac gls in + let sigma, subgll = Refiner.unpackage subgls in + let n = List.length subgll in if n = 0 then res0 else + let hyps = List.map mkhyp subgll in + let hyp_decls = List.map mkhypdecl (List.rev (perm hyps)) in + let c = applist (mkmeta (), List.map mkmeta subgll) in + let pft0 = mkipft 0 gl (v (List.map mkpfvar hyps)) hyp_decls in + let pft1 = mkrpft n gl c (pft0 :: List.map mkpfleaf (perm hyps)) in + let subgll', v' = Refiner.frontier pft1 in + Refiner.repackage sigma subgll', v' +*) + +let tclREV tac gl = tclPERM List.rev tac gl + +let rot_hyps dir i hyps = + let n = List.length hyps in + if i = 0 then List.rev hyps else + if i > n then error "Not enough subgoals" else + let rec rot i l_hyps = function + | hyp :: hyps' when i > 0 -> rot (i - 1) (hyp :: l_hyps) hyps' + | hyps' -> hyps' @ (List.rev l_hyps) in + rot (match dir with L2R -> i | R2L -> n - i) [] hyps + +let tclSEQAT (atac1, ctx) dir (ivar, ((_, atacs2), atac3)) = + let i = get_index ivar in + let evtac = ssrevaltac (get_ltacctx ctx) in + let tac1 = evtac atac1 in + if atacs2 = [] && atac3 <> None then tclPERM (rot_hyps dir i) tac1 else + let evotac = function Some atac -> evtac atac | _ -> tclIDTAC in + let tac3 = evotac atac3 in + let rec mk_pad n = if n > 0 then tac3 :: mk_pad (n - 1) else [] in + match dir, mk_pad (i - 1), List.map evotac atacs2 with + | L2R, [], [tac2] when atac3 = None -> tclTHENFIRST tac1 tac2 + | L2R, [], [tac2] when atac3 = None -> tclTHENLAST tac1 tac2 + | L2R, pad, tacs2 -> tclTHENSFIRSTn tac1 (Array.of_list (pad @ tacs2)) tac3 + | R2L, pad, tacs2 -> tclTHENSLASTn tac1 tac3 (Array.of_list (tacs2 @ pad)) + +(* We can't actually parse the direction separately because this *) +(* would introduce conflicts with the basic ltac syntax. *) +let pr_ssrseqdir _ _ _ = function + | L2R -> str ";" ++ spc () ++ str "first " + | R2L -> str ";" ++ spc () ++ str "last " + +ARGUMENT EXTEND ssrseqdir TYPED AS ssrdir PRINTED BY pr_ssrseqdir +| [ "Qed" ] -> [ anomaly "Grammar placeholder match" ] +END + +TACTIC EXTEND ssrtclseq +| [ "Qed" ssrtclarg(tac) ssrseqdir(dir) ssrseqarg(arg) ] -> + [ tclSEQAT tac dir arg ] +END +set_pr_ssrtac "tclseq" 5 [ArgSsr "tclarg"; ArgSsr "seqdir"; ArgSsr "seqarg"] + +let tclseq_expr loc tac dir arg = + let arg1 = in_gen rawwit_ssrtclarg (tac, rawltacctx) in + let arg2 = in_gen rawwit_ssrseqdir dir in + let arg3 = in_gen rawwit_ssrseqarg (check_seqtacarg dir arg) in + ssrtac_expr loc "tclseq" [arg1; arg2; arg3] + +GEXTEND Gram + GLOBAL: tactic_expr; + ssr_first: [ + [ tac = ssr_first; ipats = ssrintros_ne -> tclintros_expr loc tac ipats + | "["; tacl = LIST0 tactic_expr SEP "|"; "]" -> TacFirst tacl + ] ]; + ssr_first_else: [ + [ tac1 = ssr_first; tac2 = ssrorelse -> TacOrelse (tac1, tac2) + | tac = ssr_first -> tac ]]; + tactic_expr: LEVEL "4" [ LEFTA + [ tac1 = tactic_expr; ";"; IDENT "first"; tac2 = ssr_first_else -> + TacThen (tac1,[||], tac2,[||]) + | tac = tactic_expr; ";"; IDENT "first"; arg = ssrseqarg -> + tclseq_expr loc tac L2R arg + | tac = tactic_expr; ";"; IDENT "last"; arg = ssrseqarg -> + tclseq_expr loc tac R2L arg + ] ]; +END +(* }}} *) + +(** 5. Bookkeeping tactics (clear, move, case, elim) *) + +(* post-interpretation of terms *) +let all_ok _ _ = true + +let pf_abs_ssrterm ?(resolve_typeclasses=false) ist gl t = + let sigma, ct as t = interp_term ist gl t in + let sigma, _ as t = + let env = pf_env gl in + if not resolve_typeclasses then t + else + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + sigma, Evarutil.nf_evar sigma ct in + let n, c, abstracted_away = pf_abs_evars gl t in + List.fold_left Evd.remove sigma abstracted_away, pf_abs_cterm gl n c + +let pf_interp_ty ?(resolve_typeclasses=false) ist gl ty = + let n_binders = ref 0 in + let ty = match ty with + | a, (t, None) -> + let rec force_type = function + | GProd (l, x, k, s, t) -> incr n_binders; GProd (l, x, k, s, force_type t) + | GLetIn (l, x, v, t) -> incr n_binders; GLetIn (l, x, v, force_type t) + | ty -> mkRCast ty mkRType in + a, (force_type t, None) + | _, (_, Some ty) -> + let rec force_type = function + | CProdN (l, abs, t) -> + n_binders := !n_binders + List.length (List.flatten (List.map pi1 abs)); + CProdN (l, abs, force_type t) + | CLetIn (l, n, v, t) -> incr n_binders; CLetIn (l, n, v, force_type t) + | ty -> mkCCast dummy_loc ty (mkCType dummy_loc) in + mk_term ' ' (force_type ty) in + let strip_cast (sigma, t) = + let rec aux t = match kind_of_type t with + | CastType (t, ty) when !n_binders = 0 && isSort ty -> t + | ProdType(n,s,t) -> decr n_binders; mkProd (n, s, aux t) + | LetInType(n,v,ty,t) -> decr n_binders; mkLetIn (n, v, ty, aux t) + | _ -> anomaly "pf_interp_ty: ssr Type cast deleted by typecheck" in + sigma, aux t in + let sigma, cty as ty = strip_cast (interp_term ist gl ty) in + let ty = + let env = pf_env gl in + if not resolve_typeclasses then ty + else + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + sigma, Evarutil.nf_evar sigma cty in + let n, c, _ = pf_abs_evars gl ty in + let lam_c = pf_abs_cterm gl n c in + let ctx, c = decompose_lam_n n lam_c in + n, compose_prod ctx c, lam_c +;; + +let whd_app f args = Reductionops.whd_betaiota Evd.empty (mkApp (f, args)) + +let pr_cargs a = + str "[" ++ pr_list pr_spc pr_constr (Array.to_list a) ++ str "]" + +let pp_term gl t = + let t = Reductionops.nf_evar (project gl) t in pr_constr t +let pp_concat hd ?(sep=str", ") = function [] -> hd | x :: xs -> + hd ++ List.fold_left (fun acc x -> acc ++ sep ++ x) x xs + +let fake_pmatcher_end () = mkProp, L2R, (Evd.empty, mkProp) + +(* TASSI: given (c : ty), generates (c ??? : ty[???/...]) with m evars *) +exception NotEnoughProducts +let prof_saturate_whd = mk_profiler "saturate.whd";; +let saturate ?(beta=false) ?(bi_types=false) env sigma c ?(ty=Retyping.get_type_of env sigma c) m += + let rec loop ty args sigma n = + if n = 0 then + let args = List.rev args in + (if beta then Reductionops.whd_beta sigma else fun x -> x) + (mkApp (c, Array.of_list (List.map snd args))), ty, args, sigma + else match kind_of_type ty with + | ProdType (_, src, tgt) -> + let sigma, x = + Evarutil.new_evar (create_evar_defs sigma) env + (if bi_types then Reductionops.nf_betaiota sigma src else src) in + loop (subst1 x tgt) ((m - n,x) :: args) sigma (n-1) + | CastType (t, _) -> loop t args sigma n + | LetInType (_, v, _, t) -> loop (subst1 v t) args sigma n + | SortType _ -> assert false + | AtomicType _ -> + let ty = + prof_saturate_whd.profile + (Reductionops.whd_betadeltaiota env sigma) ty in + match kind_of_type ty with + | ProdType _ -> loop ty args sigma n + | _ -> anomaly "saturate did not find enough products" + in + loop ty [] sigma m + +let pf_saturate ?beta ?bi_types gl c ?ty m = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + let t, ty, args, sigma = saturate ?beta ?bi_types env sigma c ?ty m in + t, ty, args, re_sig si sigma + +(** Rewrite redex switch *) + +(** Generalization (discharge) item *) + +(* An item is a switch + term pair. *) + +(* type ssrgen = ssrdocc * ssrterm *) + +let pr_gen (docc, dt) = pr_docc docc ++ pr_cpattern dt + +let pr_ssrgen _ _ _ = pr_gen + +ARGUMENT EXTEND ssrgen TYPED AS ssrdocc * cpattern PRINTED BY pr_ssrgen +| [ ssrdocc(docc) cpattern(dt) ] -> [ docc, dt ] +| [ cpattern(dt) ] -> [ nodocc, dt ] +END + +let has_occ ((_, occ), _) = occ <> None +let hyp_of_var v = SsrHyp (dummy_loc, destVar v) + +let interp_clr = function +| Some clr, (k, c) + when (k = ' ' || k = '@') && is_pf_var c -> hyp_of_var c :: clr +| Some clr, _ -> clr +| None, _ -> [] + +(* XXX the k of the redex should percolate out *) +let pf_interp_gen_aux ist gl to_ind ((oclr, occ), t) = + let pat = interp_cpattern ist gl t None in (* UGLY API *) + let cl, env, sigma = pf_concl gl, pf_env gl, project gl in + let c = redex_of_pattern env pat in + let c, cl = + try fill_occ_pattern ~raise_NoMatch:true env sigma cl pat occ 1 + with NoMatch -> c, cl in + let clr = interp_clr (oclr, (tag_of_cpattern t, c)) in + if not(occur_existential c) then + if tag_of_cpattern t = '@' then + if not (isVar c) then errorstrm (str "@ can be used with variables only") + else match pf_get_hyp gl (destVar c) with + | _, None, _ -> errorstrm (str "@ can be used with let-ins only") + | name, Some bo, ty -> true, pat, mkLetIn (Name name, bo, ty, cl), c, clr + else false, pat, pf_mkprod gl c cl, c, clr + else if to_ind && occ = None then + let nv, p, _ = pf_abs_evars gl (fst pat, c) in + if nv = 0 then anomaly "occur_existential but no evars" else + false, pat, mkProd (constr_name c, pf_type_of gl p, pf_concl gl), p, clr + else loc_error (loc_of_cpattern t) "generalized term didn't match" + +let genclrtac cl cs clr = + let tclmyORELSE tac1 tac2 gl = + try tac1 gl + with e when Errors.noncritical e -> tac2 e gl in + (* apply_type may give a type error, but the useful message is + * the one of clear. You type "move: x" and you get + * "x is used in hyp H" instead of + * "The term H has type T x but is expected to have type T x0". *) + tclTHEN + (tclmyORELSE + (apply_type cl cs) + (fun type_err gl -> + tclTHEN + (tclTHEN (elim_type (build_coq_False ())) (cleartac clr)) + (fun gl -> raise type_err) + gl)) + (cleartac clr) + +let gentac ist gen gl = + let conv, _, cl, c, clr = pf_interp_gen_aux ist gl false gen in + if conv then tclTHEN (convert_concl cl) (cleartac clr) gl + else genclrtac cl [c] clr gl + +let pf_interp_gen ist gl to_ind gen = + let _, _, a, b ,c = pf_interp_gen_aux ist gl to_ind gen in a, b ,c + +(** Generalization (discharge) sequence *) + +(* A discharge sequence is represented as a list of up to two *) +(* lists of d-items, plus an ident list set (the possibly empty *) +(* final clear switch). The main list is empty iff the command *) +(* is defective, and has length two if there is a sequence of *) +(* dependent terms (and in that case it is the first of the two *) +(* lists). Thus, the first of the two lists is never empty. *) + +(* type ssrgens = ssrgen list *) +(* type ssrdgens = ssrgens list * ssrclear *) + +let gens_sep = function [], [] -> mt | _ -> spc + +let pr_dgens pr_gen (gensl, clr) = + let prgens s gens = str s ++ pr_list spc pr_gen gens in + let prdeps deps = prgens ": " deps ++ spc () ++ str "/" in + match gensl with + | [deps; []] -> prdeps deps ++ pr_clear pr_spc clr + | [deps; gens] -> prdeps deps ++ prgens " " gens ++ pr_clear spc clr + | [gens] -> prgens ": " gens ++ pr_clear spc clr + | _ -> pr_clear pr_spc clr + +let pr_ssrdgens _ _ _ = pr_dgens pr_gen + +let cons_gen gen = function + | gens :: gensl, clr -> (gen :: gens) :: gensl, clr + | _ -> anomaly "missing gen list" + +let cons_dep (gensl, clr) = + if List.length gensl = 1 then ([] :: gensl, clr) else + error "multiple dependents switches '/'" + +ARGUMENT EXTEND ssrdgens_tl TYPED AS ssrgen list list * ssrclear + PRINTED BY pr_ssrdgens +| [ "{" ne_ssrhyp_list(clr) "}" cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (mkclr clr, dt) dgens ] +| [ "{" ne_ssrhyp_list(clr) "}" ] -> + [ [[]], clr ] +| [ "{" ssrocc(occ) "}" cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (mkocc occ, dt) dgens ] +| [ "/" ssrdgens_tl(dgens) ] -> + [ cons_dep dgens ] +| [ cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (nodocc, dt) dgens ] +| [ ] -> + [ [[]], [] ] +END + +ARGUMENT EXTEND ssrdgens TYPED AS ssrdgens_tl PRINTED BY pr_ssrdgens +| [ ":" ssrgen(gen) ssrdgens_tl(dgens) ] -> [ cons_gen gen dgens ] +END + +let genstac (gens, clr) ist = + tclTHENLIST (cleartac clr :: List.rev_map (gentac ist) gens) + +(* Common code to handle generalization lists along with the defective case *) + +let with_defective maintac deps clr ist gl = + let top_id = + match kind_of_type (pf_concl gl) with + | ProdType (Name id, _, _) + when has_discharged_tag (string_of_id id) -> id + | _ -> top_id in + let top_gen = mkclr clr, cpattern_of_id top_id in + tclTHEN (introid top_id) (maintac deps top_gen ist) gl + +let with_dgens (gensl, clr) maintac ist = match gensl with + | [deps; []] -> with_defective maintac deps clr ist + | [deps; gen :: gens] -> + tclTHEN (genstac (gens, clr) ist) (maintac deps gen ist) + | [gen :: gens] -> tclTHEN (genstac (gens, clr) ist) (maintac [] gen ist) + | _ -> with_defective maintac [] clr ist + +let with_deps deps0 maintac cl0 cs0 clr0 ist gl0 = + let rec loop gl cl cs clr args clrs = function + | [] -> + let n = List.length args in + maintac (if n > 0 then applist (to_lambda n cl, args) else cl) clrs ist gl0 + | dep :: deps -> + let gl' = first_goal (genclrtac cl cs clr gl) in + let cl', c', clr' = pf_interp_gen ist gl' false dep in + loop gl' cl' [c'] clr' (c' :: args) (clr' :: clrs) deps in + loop gl0 cl0 cs0 clr0 cs0 [clr0] (List.rev deps0) + +(** Equations *) + +(* argument *) + +type ssreqid = ssripat option + +let pr_eqid = function Some pat -> str " " ++ pr_ipat pat | None -> mt () +let pr_ssreqid _ _ _ = pr_eqid + +(* We must use primitive parsing here to avoid conflicts with the *) +(* basic move, case, and elim tactics. *) +ARGUMENT EXTEND ssreqid TYPED AS ssripatrep option PRINTED BY pr_ssreqid +| [ "Qed" ] -> [ Util.anomaly "Grammar placeholder match" ] +END + +let accept_ssreqid strm = + match Compat.get_tok (Util.stream_nth 0 strm) with + | Tok.IDENT _ -> accept_before_syms [":"] strm + | Tok.KEYWORD ":" -> () + | Tok.KEYWORD pat when List.mem pat ["_"; "?"; "->"; "<-"] -> + accept_before_syms [":"] strm + | _ -> raise Stream.Failure + +let test_ssreqid = Gram.Entry.of_parser "test_ssreqid" accept_ssreqid + +GEXTEND Gram + GLOBAL: ssreqid; + ssreqpat: [ + [ id = Prim.ident -> IpatId id + | "_" -> IpatWild + | "?" -> IpatAnon + | occ = ssrdocc; "->" -> (match occ with + | None, occ -> IpatRw (occ, L2R) + | _ -> loc_error loc "Only occurrences are allowed here") + | occ = ssrdocc; "<-" -> (match occ with + | None, occ -> IpatRw (occ, R2L) + | _ -> loc_error loc "Only occurrences are allowed here") + | "->" -> IpatRw (allocc, L2R) + | "<-" -> IpatRw (allocc, R2L) + ]]; + ssreqid: [ + [ test_ssreqid; pat = ssreqpat -> Some pat + | test_ssreqid -> None + ]]; +END + +(* creation *) + +let mkEq dir cl c t n = + let eqargs = [|t; c; c|] in eqargs.(dir_org dir) <- mkRel n; + mkArrow (mkApp (build_coq_eq(), eqargs)) (lift 1 cl), mkRefl t c + +let pushmoveeqtac cl c = + let x, t, cl1 = destProd cl in + let cl2, eqc = mkEq R2L cl1 c t 1 in + apply_type (mkProd (x, t, cl2)) [c; eqc] + +let pushcaseeqtac cl gl = + let cl1, args = destApplication cl in + let n = Array.length args in + let dc, cl2 = decompose_lam_n n cl1 in + let _, t = List.nth dc (n - 1) in + let cl3, eqc = mkEq R2L cl2 args.(0) t n in + let cl4 = mkApp (compose_lam dc (mkProt (pf_type_of gl cl) cl3), args) in + tclTHEN (apply_type cl4 [eqc]) (convert_concl cl4) gl + +let pushelimeqtac gl = + let _, args = destApplication (pf_concl gl) in + let x, t, _ = destLambda args.(1) in + let cl1 = mkApp (args.(1), Array.sub args 2 (Array.length args - 2)) in + let cl2, eqc = mkEq L2R cl1 args.(2) t 1 in + tclTHEN (apply_type (mkProd (x, t, cl2)) [args.(2); eqc]) intro gl + +(** Bookkeeping (discharge-intro) argument *) + +(* Since all bookkeeping ssr commands have the same discharge-intro *) +(* argument format we use a single grammar entry point to parse them. *) +(* the entry point parses only non-empty arguments to avoid conflicts *) +(* with the basic Coq tactics. *) + +(* type ssrarg = ssrview * (ssreqid * (ssrdgens * ssripats)) *) + +let pr_ssrarg _ _ _ (view, (eqid, (dgens, ipats))) = + let pri = pr_intros (gens_sep dgens) in + pr_view view ++ pr_eqid eqid ++ pr_dgens pr_gen dgens ++ pri ipats + +ARGUMENT EXTEND ssrarg TYPED AS ssrview * (ssreqid * (ssrdgens * ssrintros)) + PRINTED BY pr_ssrarg +| [ ssrview(view) ssreqid(eqid) ssrdgens(dgens) ssrintros(ipats) ] -> + [ view, (eqid, (dgens, ipats)) ] +| [ ssrview(view) ssrclear(clr) ssrintros(ipats) ] -> + [ view, (None, (([], clr), ipats)) ] +| [ ssreqid(eqid) ssrdgens(dgens) ssrintros(ipats) ] -> + [ [], (eqid, (dgens, ipats)) ] +| [ ssrclear_ne(clr) ssrintros(ipats) ] -> + [ [], (None, (([], clr), ipats)) ] +| [ ssrintros_ne(ipats) ] -> + [ [], (None, (([], []), ipats)) ] +END + +(** The "clear" tactic *) + +(* We just add a numeric version that clears the n top assumptions. *) + +let poptac ?ist n = introstac ?ist (list_tabulate (fun _ -> IpatWild) n) + +TACTIC EXTEND ssrclear + | [ "clear" natural(n) ltacctx(ctx) ] -> [poptac ~ist:(get_ltacctx ctx) n] +END + +(** The "move" tactic *) + +let rec improper_intros = function + | IpatSimpl _ :: ipats -> improper_intros ipats + | (IpatId _ | IpatAnon | IpatCase _ | IpatAll) :: _ -> false + | _ -> true + +let check_movearg = function + | view, (eqid, _) when view <> [] && eqid <> None -> + Util.error "incompatible view and equation in move tactic" + | view, (_, (([gen :: _], _), _)) when view <> [] && has_occ gen -> + Util.error "incompatible view and occurrence switch in move tactic" + | _, (_, ((dgens, _), _)) when List.length dgens > 1 -> + Util.error "dependents switch `/' in move tactic" + | _, (eqid, (_, (ipats, _))) when eqid <> None && improper_intros ipats -> + Util.error "no proper intro pattern for equation in move tactic" + | arg -> arg + +ARGUMENT EXTEND ssrmovearg TYPED AS ssrarg PRINTED BY pr_ssrarg +| [ ssrarg(arg) ] -> [ check_movearg arg ] +END + +let viewmovetac_aux clear name_ref (_, vl as v) _ gen ist gl = + let cl, c, clr = pf_interp_gen ist gl false gen in + let cl, c = if vl = [] then cl, c else pf_with_view ist gl v cl c in + let clr = if clear then clr else [] in + name_ref := (match id_of_cpattern (snd gen) with Some id -> id | _ -> top_id); + genclrtac cl [c] clr gl + +let () = move_top_with_view := + (fun c r v -> with_defective (viewmovetac_aux c r v) [] []) + +let viewmovetac v deps gen ist gl = + viewmovetac_aux true (ref top_id) v deps gen ist gl + +let eqmovetac _ gen ist gl = + let cl, c, _ = pf_interp_gen ist gl false gen in pushmoveeqtac cl c gl + +let movehnftac gl = match kind_of_term (pf_concl gl) with + | Prod _ | LetIn _ -> tclIDTAC gl + | _ -> hnf_in_concl gl + +let ssrmovetac ist = function + | _::_ as view, (_, (dgens, (ipats, _))) -> + let dgentac = with_dgens dgens (viewmovetac (true, view)) ist in + tclTHEN dgentac (introstac ~ist ipats) + | _, (Some pat, (dgens, (ipats, _))) -> + let dgentac = with_dgens dgens eqmovetac ist in + tclTHEN dgentac (introstac ~ist (eqmoveipats pat ipats)) + | _, (_, (([gens], clr), (ipats, _))) -> + let gentac = genstac (gens, clr) ist in + tclTHEN gentac (introstac ~ist ipats) + | _, (_, ((_, clr), (ipats, _))) -> + tclTHENLIST [movehnftac; cleartac clr; introstac ~ist ipats] + +let ist_of_arg (_, (_, (_, (_, ctx)))) = get_ltacctx ctx + +TACTIC EXTEND ssrmove +| [ "move" ssrmovearg(arg) ssrrpat(pat) ] -> + [ let ist = ist_of_arg arg in + tclTHEN (ssrmovetac ist arg) (introstac ~ist [pat]) ] +| [ "move" ssrmovearg(arg) ssrclauses(clauses) ] -> + [ let ist = ist_of_arg arg in tclCLAUSES ist (ssrmovetac ist arg) clauses ] +| [ "move" ssrrpat(pat) ltacctx(ctx) ] -> + [ introstac ~ist:(get_ltacctx ctx) [pat] ] +| [ "move" ] -> [ movehnftac ] +END + +(* TASSI: given the type of an elimination principle, it finds the higher order + * argument (index), it computes it's arity and the arity of the eliminator and + * checks if the eliminator is recursive or not *) +let analyze_eliminator elimty env sigma = + let rec loop ctx t = match kind_of_type t with + | AtomicType (hd, args) when isRel hd -> + ctx, destRel hd, not (noccurn 1 t), Array.length args + | CastType (t, _) -> loop ctx t + | ProdType (x, ty, t) -> loop ((x,None,ty) :: ctx) t + | LetInType (x,b,ty,t) -> loop ((x,Some b,ty) :: ctx) (subst1 b t) + | _ -> + let env' = Environ.push_rel_context ctx env in + let t' = Reductionops.whd_betadeltaiota env' sigma t in + if not (eq_constr t t') then loop ctx t' else + errorstrm (str"The eliminator has the wrong shape."++spc()++ + str"A (applied) bound variable was expected as the conclusion of "++ + str"the eliminator's"++Pp.cut()++str"type:"++spc()++pr_constr elimty) in + let ctx, pred_id, elim_is_dep, n_pred_args = loop [] elimty in + let n_elim_args = rel_context_nhyps ctx in + let is_rec_elim = + let count_occurn n term = + let count = ref 0 in + let rec occur_rec n c = match kind_of_term c with + | Rel m -> if m = n then incr count + | _ -> iter_constr_with_binders succ occur_rec n c + in + occur_rec n term; !count in + let occurr2 n t = count_occurn n t > 1 in + not (list_for_all_i + (fun i (_,rd) -> pred_id <= i || not (occurr2 (pred_id - i) rd)) + 1 (assums_of_rel_context ctx)) + in + n_elim_args - pred_id, n_elim_args, is_rec_elim, elim_is_dep, n_pred_args + +(* TASSI: This version of unprotects inlines the unfold tactic definition, + * since we don't want to wipe out let-ins, and it seems there is no flag + * to change that behaviour in the standard unfold code *) +let unprotecttac gl = + let prot = destConst (mkSsrConst "protect_term") in + onClause (fun idopt -> + let hyploc = Option.map (fun id -> id, InHyp) idopt in + reduct_option + (Reductionops.clos_norm_flags + (Closure.RedFlags.mkflags + [Closure.RedFlags.fBETA; + Closure.RedFlags.fCONST prot; + Closure.RedFlags.fIOTA]), DEFAULTcast) hyploc) + allHypsAndConcl gl + +let dependent_apply_error = + try Util.error "Could not fill dependent hole in \"apply\"" with err -> err + +(* TASSI: Sometimes Coq's apply fails. According to my experience it may be + * related to goals that are products and with beta redexes. In that case it + * guesses the wrong number of implicit arguments for your lemma. What follows + * is just like apply, but with a user-provided number n of implicits. + * + * Refine.refine function that handles type classes and evars but fails to + * handle "dependently typed higher order evars". + * + * Refiner.refiner that does not handle metas with a non ground type but works + * with dependently typed higher order metas. *) +let applyn ~with_evars ?beta n t gl = + if with_evars then + let t, ty, args, gl = pf_saturate ?beta ~bi_types:true gl t n in + let gl = pf_unify_HO gl ty (pf_concl gl) in + let gs = list_map_filter (fun (_, e) -> + if isEvar (pf_nf_evar gl e) then Some e else None) + args in + pf_partial_solution ~shelve:true gl t gs + else + let t, gl = if n = 0 then t, gl else + let sigma, si = project gl, sig_it gl in + let rec loop sigma bo args = function (* saturate with metas *) + | 0 -> mkApp (t, Array.of_list (List.rev args)), re_sig si sigma + | n -> match kind_of_term bo with + | Lambda (_, ty, bo) -> + if not (closed0 ty) then raise dependent_apply_error; + let m = Evarutil.new_meta () in + loop (meta_declare m ty sigma) bo ((mkMeta m)::args) (n-1) + | _ -> assert false + in loop sigma t [] n in + pp(lazy(str"Refiner.refiner " ++ pr_constr t)); + Refiner.refiner (Proof_type.Prim (Proof_type.Refine t)) gl + +let refine_with ?(first_goes_last=false) ?beta ?(with_evars=true) oc gl = + let rec mkRels = function 1 -> [] | n -> mkRel n :: mkRels (n-1) in + let n, oc = pf_abs_evars_pirrel gl oc in + let oc = if not first_goes_last || n <= 1 then oc else + let l, c = decompose_lam oc in + if not (list_for_all_i (fun i (_,t) -> closedn ~-i t) (1-n) l) then oc else + compose_lam (let xs,y = list_chop (n-1) l in y @ xs) + (mkApp (compose_lam l c, Array.of_list (mkRel 1 :: mkRels n))) + in + pp(lazy(str"after: " ++ pr_constr oc)); + try applyn ~with_evars ?beta n oc gl with _ -> raise dependent_apply_error + +(** The "case" and "elim" tactic *) + +(* A case without explicit dependent terms but with both a view and an *) +(* occurrence switch and/or an equation is treated as dependent, with the *) +(* viewed term as the dependent term (the occurrence switch would be *) +(* meaningless otherwise). When both a view and explicit dependents are *) +(* present, it is forbidden to put a (meaningless) occurrence switch on *) +(* the viewed term. *) + +(* This is both elim and case (defaulting to the former). If ~elim is omitted + * the standard eliminator is chosen. The code is made of 4 parts: + * 1. find the eliminator if not given as ~elim and analyze it + * 2. build the patterns to be matched against the conclusion, looking at + * (occ, c), deps and the pattern inferred from the type of the eliminator + * 3. build the new predicate matching the patterns, and the tactic to + * generalize the equality in case eqid is not None + * 4. build the tactic handle intructions and clears as required in ipats and + * by eqid *) +let ssrelim ?(is_case=false) ?ist deps what ?elim eqid ipats gl = + (* some sanity checks *) + let oc, orig_clr, occ, c_gen = match what with + | `EConstr(_,_,t) when isEvar t -> anomaly "elim called on a constr evar" + | `EGen _ when ist = None -> anomaly "no ist and non simple elimination" + | `EGen (_, g) when elim = None && is_wildcard g -> + errorstrm(str"Indeterminate pattern and no eliminator") + | `EGen ((Some clr,occ), g) when is_wildcard g -> + None, clr, occ, None + | `EGen ((None, occ), g) when is_wildcard g -> None,[],occ,None + | `EGen ((_, occ), p as gen) -> + let _, c, clr = pf_interp_gen (Option.get ist) gl true gen in + Some c, clr, occ, Some p + | `EConstr (clr, occ, c) -> Some c, clr, occ, None in + let orig_gl, concl, env = gl, pf_concl gl, pf_env gl in + pp(lazy(str(if is_case then "==CASE==" else "==ELIM=="))); + (* Utils of local interest only *) + let iD s ?t gl = let t = match t with None -> pf_concl gl | Some x -> x in + pp(lazy(str s ++ pr_constr t)); tclIDTAC gl in + let eq, protectC = build_coq_eq (), mkSsrConst "protect_term" in + let fire_subst gl t = Reductionops.nf_evar (project gl) t in + let fire_sigma sigma t = Reductionops.nf_evar sigma t in + let is_undef_pat = function + | sigma, T t -> + (match kind_of_term (fire_sigma sigma t) with Evar _ -> true | _ -> false) + | _ -> false in + let match_pat env p occ h cl = + let sigma0 = project orig_gl in + pp(lazy(str"matching: " ++ pp_pattern p)); + let c, cl = fill_occ_pattern ~raise_NoMatch:true env sigma0 cl p occ h in + pp(lazy(str" got: " ++ pr_constr c)); + c, cl in + let mkTpat gl t = (* takes a term, refreshes it and makes a T pattern *) + let n, t, _ = pf_abs_evars orig_gl (project gl, fire_subst gl t) in + let t, _, _, sigma = saturate ~beta:true env (project gl) t n in + sigma, T t in + let unif_redex gl (sigma, r as p) t = (* t is a hint for the redex of p *) + let n, t, _ = pf_abs_evars orig_gl (project gl, fire_subst gl t) in + let t, _, _, sigma = saturate ~beta:true env sigma t n in + match r with + | X_In_T (e, p) -> sigma, E_As_X_In_T (t, e, p) + | _ -> try unify_HO env sigma t (redex_of_pattern env p), r with _ -> p in + (* finds the eliminator applies it to evars and c saturated as needed *) + (* obtaining "elim ??? (c ???)". pred is the higher order evar *) + let cty, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl = + match elim with + | Some elim -> + let elimty = pf_type_of gl elim in + let pred_id, n_elim_args, is_rec, elim_is_dep, n_pred_args = + analyze_eliminator elimty env (project gl) in + let elim, elimty, elim_args, gl = + pf_saturate ~beta:is_case gl elim ~ty:elimty n_elim_args in + let pred = List.assoc pred_id elim_args in + let elimty = Reductionops.whd_betadeltaiota env (project gl) elimty in + None, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl + | None -> + let c = Option.get oc in let c_ty = pf_type_of gl c in + let (kn, i) as ind, unfolded_c_ty = pf_reduce_to_quantified_ind gl c_ty in + let sort = elimination_sort_of_goal gl in + let elim = if not is_case then Indrec.lookup_eliminator ind sort + else pf_apply Indrec.build_case_analysis_scheme gl ind true sort in + let elimty = pf_type_of gl elim in + let pred_id,n_elim_args,is_rec,elim_is_dep,n_pred_args = + analyze_eliminator elimty env (project gl) in + let rctx = fst (decompose_prod_assum unfolded_c_ty) in + let n_c_args = rel_context_length rctx in + let c, c_ty, t_args, gl = pf_saturate gl c ~ty:c_ty n_c_args in + let elim, elimty, elim_args, gl = + pf_saturate ~beta:is_case gl elim ~ty:elimty n_elim_args in + let pred = List.assoc pred_id elim_args in + let pc = match n_c_args, c_gen with + | 0, Some p -> interp_cpattern (Option.get ist) orig_gl p None + | _ -> mkTpat gl c in + let cty = Some (c, c_ty, pc) in + let elimty = Reductionops.whd_betadeltaiota env (project gl) elimty in + cty, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl + in + pp(lazy(str"elim= "++ pr_constr_pat elim)); + pp(lazy(str"elimty= "++ pr_constr elimty)); + let inf_deps_r = match kind_of_type elimty with + | AtomicType (_, args) -> List.rev (Array.to_list args) + | _ -> assert false in + let saturate_until gl c c_ty f = + let rec loop n = try + let c, c_ty, _, gl = pf_saturate gl c ~ty:c_ty n in + let gl' = f c c_ty gl in + Some (c, c_ty, gl, gl') + with NotEnoughProducts -> None | _ -> loop (n+1) in loop 0 in + let elim_is_dep, gl = match cty with + | None -> true, gl + | Some (c, c_ty, _) -> + let res = + if elim_is_dep then None else + let arg = List.assoc (n_elim_args - 1) elim_args in + let arg_ty = pf_type_of gl arg in + match saturate_until gl c c_ty (fun c c_ty gl -> + pf_unify_HO (pf_unify_HO gl c_ty arg_ty) arg c) with + | Some (c, _, _, gl) -> Some (false, gl) + | None -> None in + match res with + | Some x -> x + | None -> + let inf_arg = List.hd inf_deps_r in + let inf_arg_ty = pf_type_of gl inf_arg in + match saturate_until gl c c_ty (fun _ c_ty gl -> + pf_unify_HO gl c_ty inf_arg_ty) with + | Some (c, _, _,gl) -> true, gl + | None -> errorstrm (str"Unable to apply the eliminator to the term"++ + spc()++pr_constr c++spc()++str"or to unify it's type with"++ + pr_constr inf_arg_ty) in + pp(lazy(str"elim_is_dep= " ++ bool elim_is_dep)); + let predty = pf_type_of gl pred in + (* Patterns for the inductive types indexes to be bound in pred are computed + * looking at the ones provided by the user and the inferred ones looking at + * the type of the elimination principle *) + let pp_pat (_,p,_,_) = pp_pattern p in + let pp_inf_pat gl (_,_,t,_) = pr_constr_pat (fire_subst gl t) in + let patterns, clr, gl = + let rec loop patterns clr i = function + | [],[] -> patterns, clr, gl + | ((oclr, occ), t):: deps, inf_t :: inf_deps -> + let ist = match ist with Some x -> x | None -> assert false in + let p = interp_cpattern ist orig_gl t None in + let clr_t = + interp_clr (oclr,(tag_of_cpattern t,redex_of_pattern env p))in + (* if we are the index for the equation we do not clear *) + let clr_t = if deps = [] && eqid <> None then [] else clr_t in + let p = if is_undef_pat p then mkTpat gl inf_t else p in + loop (patterns @ [i, p, inf_t, occ]) + (clr_t @ clr) (i+1) (deps, inf_deps) + | [], c :: inf_deps -> + pp(lazy(str"adding inferred pattern " ++ pr_constr_pat c)); + loop (patterns @ [i, mkTpat gl c, c, allocc]) + clr (i+1) ([], inf_deps) + | _::_, [] -> errorstrm (str "Too many dependent abstractions") in + let deps, head_p, inf_deps_r = match what, elim_is_dep, cty with + | `EConstr _, _, None -> anomaly "Simple welim with no term" + | _, false, _ -> deps, [], inf_deps_r + | `EGen gen, true, None -> deps @ [gen], [], inf_deps_r + | _, true, Some (c, _, pc) -> + let occ = if occ = None then allocc else occ in + let inf_p, inf_deps_r = List.hd inf_deps_r, List.tl inf_deps_r in + deps, [1, pc, inf_p, occ], inf_deps_r in + let patterns, clr, gl = + loop [] orig_clr (List.length head_p+1) (List.rev deps, inf_deps_r) in + head_p @ patterns, Util.list_uniquize clr, gl + in + pp(lazy(pp_concat (str"patterns=") (List.map pp_pat patterns))); + pp(lazy(pp_concat (str"inf. patterns=") (List.map (pp_inf_pat gl) patterns))); + (* Predicate generation, and (if necessary) tactic to generalize the + * equation asked by the user *) + let elim_pred, gen_eq_tac, clr, gl = + let error gl t inf_t = errorstrm (str"The given pattern matches the term"++ + spc()++pp_term gl t++spc()++str"while the inferred pattern"++ + spc()++pr_constr_pat (fire_subst gl inf_t)++spc()++ str"doesn't") in + let match_or_postpone (cl, gl, post) (h, p, inf_t, occ) = + let p = unif_redex gl p inf_t in + if is_undef_pat p then + let () = pp(lazy(str"postponing " ++ pp_pattern p)) in + cl, gl, post @ [h, p, inf_t, occ] + else try + let c, cl = match_pat env p occ h cl in + let gl = try pf_unify_HO gl inf_t c with _ -> error gl c inf_t in + cl, gl, post + with + | NoMatch | NoProgress -> + let e = redex_of_pattern env p in + let n, e, _ = pf_abs_evars gl (fst p, e) in + let e, _, _, gl = pf_saturate ~beta:true gl e n in + let gl = try pf_unify_HO gl inf_t e with _ -> error gl e inf_t in + cl, gl, post + in + let rec match_all concl gl patterns = + let concl, gl, postponed = + List.fold_left match_or_postpone (concl, gl, []) patterns in + if postponed = [] then concl, gl + else if List.length postponed = List.length patterns then + errorstrm (str "Some patterns are undefined even after all"++spc()++ + str"the defined ones matched") + else match_all concl gl postponed in + let concl, gl = match_all concl gl patterns in + let pred_rctx, _ = decompose_prod_assum (fire_subst gl predty) in + let concl, gen_eq_tac, clr = match eqid with + | Some (IpatId _) when not is_rec -> + let k = List.length deps in + let c = fire_subst gl (List.assoc (n_elim_args - k - 1) elim_args) in + let t = pf_type_of gl c in + let gen_eq_tac = + let refl = mkApp (eq, [|t; c; c|]) in + let new_concl = mkArrow refl (lift 1 (pf_concl orig_gl)) in + let new_concl = fire_subst gl new_concl in + let erefl = fire_subst gl (mkRefl t c) in + apply_type new_concl [erefl] in + let rel = k + if elim_is_dep then 1 else 0 in + let src = mkProt mkProp (mkApp (eq,[|t; c; mkRel rel|])) in + let concl = mkArrow src (lift 1 concl) in + let clr = if deps <> [] then clr else [] in + concl, gen_eq_tac, clr + | _ -> concl, tclIDTAC, clr in + let mk_lam t r = mkLambda_or_LetIn r t in + let concl = List.fold_left mk_lam concl pred_rctx in + let concl = + if eqid <> None && is_rec then mkProt (pf_type_of gl concl) concl + else concl in + concl, gen_eq_tac, clr, gl in + pp(lazy(str"elim_pred=" ++ pp_term gl elim_pred)); + let pty = Typing.type_of env (project gl) elim_pred in + pp(lazy(str"elim_pred_ty=" ++ pp_term gl pty)); + let gl = pf_unify_HO gl pred elim_pred in + (* check that the patterns do not contain non instantiated dependent metas *) + let () = + let evars_of_term = Evarutil.evars_of_term in + let patterns = List.map (fun (_,_,t,_) -> fire_subst gl t) patterns in + let patterns_ev = List.map evars_of_term patterns in + let ev = List.fold_left Intset.union Intset.empty patterns_ev in + let ty_ev = Intset.fold (fun i e -> + let ex = Evd.existential_of_int i in + let i_ty = Evd.evar_concl (Evd.find (project gl) ex) in + Intset.union e (evars_of_term i_ty)) + ev Intset.empty in + let inter = Intset.inter ev ty_ev in + if not (Intset.is_empty inter) then begin + let i = Intset.choose inter in + let pat = List.find (fun t -> Intset.mem i (evars_of_term t)) patterns in + errorstrm(str"Pattern"++spc()++pr_constr_pat pat++spc()++ + str"was not completely instantiated and one of its variables"++spc()++ + str"occurs in the type of another non instantieted pattern variable"); + end + in + (* the elim tactic, with the eliminator and the predicated we computed *) + let elim = project gl, fire_subst gl elim in + let elim_tac gl = + tclTHENLIST [refine_with ~with_evars:false elim; cleartac clr] gl in + (* handling of following intro patterns and equation introduction if any *) + let elim_intro_tac gl = + let intro_eq = + match eqid with + | Some (IpatId ipat) when not is_rec -> + let rec intro_eq gl = match kind_of_type (pf_concl gl) with + | ProdType (_, src, tgt) -> + (match kind_of_type src with + | AtomicType (hd, _) when eq_constr hd protectC -> + tclTHENLIST [unprotecttac; introid ipat] gl + | _ -> tclTHENLIST [ iD "IA"; introstac [IpatAnon]; intro_eq] gl) + |_ -> errorstrm (str "Too many names in intro pattern") in + intro_eq + | Some (IpatId ipat) -> + let name gl = mk_anon_id "K" gl in + let intro_lhs gl = + let elim_name = match orig_clr, what with + | [SsrHyp(_, x)], _ -> x + | _, `EConstr(_,_,t) when isVar t -> destVar t + | _ -> name gl in + if is_name_in_ipats elim_name ipats then introid (name gl) gl + else introid elim_name gl + in + let rec gen_eq_tac gl = + let concl = pf_concl gl in + let ctx, last = decompose_prod_assum concl in + let args = match kind_of_type last with + | AtomicType (hd, args) -> assert(eq_constr hd protectC); args + | _ -> assert false in + let case = args.(Array.length args-1) in + if not(closed0 case) then tclTHEN (introstac [IpatAnon]) gen_eq_tac gl + else + let case_ty = pf_type_of gl case in + let refl = mkApp (eq, [|lift 1 case_ty; mkRel 1; lift 1 case|]) in + let new_concl = fire_subst gl + (mkProd (Name (name gl), case_ty, mkArrow refl (lift 2 concl))) in + let erefl = fire_subst gl (mkRefl case_ty case) in + apply_type new_concl [case;erefl] gl in + tclTHENLIST [gen_eq_tac; intro_lhs; introid ipat] + | _ -> tclIDTAC in + let unprot = if eqid <> None && is_rec then unprotecttac else tclIDTAC in + tclEQINTROS ?ist elim_tac (tclTHENLIST [intro_eq; unprot]) ipats gl + in + tclTHENLIST [gen_eq_tac; elim_intro_tac] orig_gl +;; + +let simplest_newelim x= ssrelim ~is_case:false [] (`EConstr ([],None,x)) None [] +let simplest_newcase x= ssrelim ~is_case:true [] (`EConstr ([],None,x)) None [] +let _ = simplest_newcase_ref := simplest_newcase + +let check_casearg = function +| view, (_, (([_; gen :: _], _), _)) when view <> [] && has_occ gen -> + Util.error "incompatible view and occurrence switch in dependent case tactic" +| arg -> arg + +ARGUMENT EXTEND ssrcasearg TYPED AS ssrarg PRINTED BY pr_ssrarg +| [ ssrarg(arg) ] -> [ check_casearg arg ] +END + +let ssrcasetac ist (view, (eqid, (dgens, (ipats, _)))) = + let ndefectcasetac view eqid ipats deps ((_, occ), _ as gen) ist gl = + let simple = (eqid = None && deps = [] && occ = None) in + let cl, c, clr = pf_interp_gen ist gl true gen in + let vc = + if view = [] then c else snd(pf_with_view ist gl (false, view) cl c) in + if simple && is_injection_case vc gl then + tclTHENLIST [perform_injection vc; cleartac clr; introstac ~ist ipats] gl + else + (* macro for "case/v E: x" ---> "case E: x / (v x)" *) + let deps, clr, occ = + if view <> [] && eqid <> None && deps = [] then [gen], [], None + else deps, clr, occ in + ssrelim ~is_case:true ~ist deps (`EConstr (clr,occ, vc)) eqid ipats gl + in + with_dgens dgens (ndefectcasetac view eqid ipats) ist + +TACTIC EXTEND ssrcase +| [ "case" ssrcasearg(arg) ssrclauses(clauses) ] -> + [ let _, (_, (_, (_, ctx))) = arg in + let ist = get_ltacctx ctx in + tclCLAUSES ist (ssrcasetac ist arg) clauses ] +| [ "case" ] -> [ with_top ssrscasetac ] +END + +(** The "elim" tactic *) + +(* Elim views are elimination lemmas, so the eliminated term is not addded *) +(* to the dependent terms as for "case", unless it actually occurs in the *) +(* goal, the "all occurrences" {+} switch is used, or the equation switch *) +(* is used and there are no dependents. *) + +let ssrelimtac ist (view, (eqid, (dgens, (ipats, _)))) = + let ndefectelimtac view eqid ipats deps gen ist gl = + let elim = match view with [v] -> Some (snd(force_term ist gl v)) | _ -> None in + ssrelim ~ist deps (`EGen gen) ?elim eqid ipats gl + in + with_dgens dgens (ndefectelimtac view eqid ipats) ist + +TACTIC EXTEND ssrelim +| [ "elim" ssrarg(arg) ssrclauses(clauses) ] -> + [ let _, (_, (_, (_, ctx))) = arg in + let ist = get_ltacctx ctx in + tclCLAUSES ist (ssrelimtac ist arg) clauses ] +| [ "elim" ] -> [ with_top simplest_newelim ] +END + +(** 6. Backward chaining tactics: apply, exact, congr. *) + +(** The "apply" tactic *) + +let pr_agen (docc, dt) = pr_docc docc ++ pr_term dt +let pr_ssragen _ _ _ = pr_agen +let pr_ssragens _ _ _ = pr_dgens pr_agen + +ARGUMENT EXTEND ssragen TYPED AS ssrdocc * ssrterm PRINTED BY pr_ssragen +| [ "{" ne_ssrhyp_list(clr) "}" ssrterm(dt) ] -> [ mkclr clr, dt ] +| [ ssrterm(dt) ] -> [ nodocc, dt ] +END + +ARGUMENT EXTEND ssragens TYPED AS ssragen list list * ssrclear +PRINTED BY pr_ssragens +| [ "{" ne_ssrhyp_list(clr) "}" ssrterm(dt) ssragens(agens) ] -> + [ cons_gen (mkclr clr, dt) agens ] +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ [[]], clr] +| [ ssrterm(dt) ssragens(agens) ] -> + [ cons_gen (nodocc, dt) agens ] +| [ ] -> [ [[]], [] ] +END + +let mk_applyarg views agens intros = views, (None, (agens, intros)) + +let pr_ssraarg _ _ _ (view, (eqid, (dgens, ipats))) = + let pri = pr_intros (gens_sep dgens) in + pr_view view ++ pr_eqid eqid ++ pr_dgens pr_agen dgens ++ pri ipats + +ARGUMENT EXTEND ssrapplyarg +TYPED AS ssrview * (ssreqid * (ssragens * ssrintros)) +PRINTED BY pr_ssraarg +| [ ":" ssragen(gen) ssragens(dgens) ssrintros(intros) ] -> + [ mk_applyarg [] (cons_gen gen dgens) intros ] +| [ ssrclear_ne(clr) ssrintros(intros) ] -> + [ mk_applyarg [] ([], clr) intros ] +| [ ssrintros_ne(intros) ] -> + [ mk_applyarg [] ([], []) intros ] +| [ ssrview(view) ":" ssragen(gen) ssragens(dgens) ssrintros(intros) ] -> + [ mk_applyarg view (cons_gen gen dgens) intros ] +| [ ssrview(view) ssrclear(clr) ssrintros(intros) ] -> + [ mk_applyarg view ([], clr) intros ] +END + +let interp_agen ist gl ((goclr, _), (k, gc)) (clr, rcs) = + let rc = glob_constr ist (project gl) (pf_env gl) gc in + let rcs' = rc :: rcs in + match goclr with + | None -> clr, rcs' + | Some ghyps -> + let clr' = snd (interp_hyps ist gl ghyps) @ clr in + if k <> ' ' then clr', rcs' else + match rc with + | GVar (loc, id) when not_section_id id -> SsrHyp (loc, id) :: clr', rcs' + | GRef (loc, VarRef id) when not_section_id id -> + SsrHyp (loc, id) :: clr', rcs' + | _ -> clr', rcs' + +let interp_agens ist gl gagens = + match List.fold_right (interp_agen ist gl) gagens ([], []) with + | clr, rlemma :: args -> + let n = interp_nbargs ist gl rlemma - List.length args in + let rec loop i = + if i > n then + errorstrm (str "Cannot apply lemma " ++ pf_pr_glob_constr gl rlemma) + else + try interp_refine ist gl (mkRApp rlemma (mkRHoles i @ args)) + with _ -> loop (i + 1) in + clr, loop 0 + | _ -> assert false + +let apply_rconstr ?ist t gl = + let n = match ist, t with + | None, (GVar (_, id) | GRef (_, VarRef id)) -> pf_nbargs gl (mkVar id) + | Some ist, _ -> interp_nbargs ist gl t + | _ -> anomaly "apply_rconstr without ist and not RVar" in + let mkRlemma i = mkRApp t (mkRHoles i) in + let cl = pf_concl gl in + let rec loop i = + if i > n then errorstrm (str"Cannot apply lemma "++pf_pr_glob_constr gl t) + else try pf_match gl (mkRlemma i) cl with _ -> loop (i + 1) in + refine_with (loop 0) gl + +let mkRAppView ist gl rv gv = + let nb_view_imps = interp_view_nbimps ist gl rv in + mkRApp rv (mkRHoles (abs nb_view_imps)) + +let prof_apply_interp_with = mk_profiler "ssrapplytac.interp_with";; + +let refine_interp_apply_view i ist gl gv = + let pair i = List.map (fun x -> i, x) in + let rv = pf_intern_term ist gl gv in + let v = mkRAppView ist gl rv gv in + let interp_with (i, hint) = + interp_refine ist gl (mkRApp hint (v :: mkRHoles i)) in + let interp_with x = prof_apply_interp_with.profile interp_with x in + let rec loop = function + | [] -> (try apply_rconstr ~ist rv gl with _ -> view_error "apply" gv) + | h :: hs -> (try refine_with (snd (interp_with h)) gl with _ -> loop hs) in + loop (pair i viewtab.(i) @ if i = 2 then pair 1 viewtab.(1) else []) + +let apply_top_tac gl = + tclTHENLIST [introid top_id; apply_rconstr (mkRVar top_id); clear [top_id]] gl + +let inner_ssrapplytac gviews ggenl gclr ist gl = + let _, clr = interp_hyps ist gl gclr in + let vtac gv i gl' = refine_interp_apply_view i ist gl' gv in + let ggenl, tclGENTAC = + if gviews <> [] && ggenl <> [] then + let ggenl= List.map (fun (x,g) -> x, cpattern_of_term g) (List.hd ggenl) in + [], tclTHEN (genstac (ggenl,[]) ist) + else ggenl, tclTHEN tclIDTAC in + tclGENTAC (fun gl -> + match gviews, ggenl with + | v :: tl, [] -> + let dbl = if List.length tl = 1 then 2 else 1 in + tclTHEN + (List.fold_left (fun acc v -> tclTHENLAST acc (vtac v dbl)) (vtac v 1) tl) + (cleartac clr) gl + | [], [agens] -> + let clr', (_, lemma) = interp_agens ist gl agens in + tclTHENLIST [cleartac clr; refine_with ~beta:true lemma; cleartac clr'] gl + | _, _ -> tclTHEN apply_top_tac (cleartac clr) gl) gl + +let ssrapplytac (views, (_, ((gens, clr), intros))) = + tclINTROS (inner_ssrapplytac views gens clr) intros + +let prof_ssrapplytac = mk_profiler "ssrapplytac";; +let ssrapplytac arg gl = prof_ssrapplytac.profile (ssrapplytac arg) gl;; + +TACTIC EXTEND ssrapply +| [ "apply" ssrapplyarg(arg) ] -> [ ssrapplytac arg ] +| [ "apply" ] -> [ apply_top_tac ] +END + +(** The "exact" tactic *) + +let mk_exactarg views dgens = mk_applyarg views dgens ([], rawltacctx) + +ARGUMENT EXTEND ssrexactarg TYPED AS ssrapplyarg PRINTED BY pr_ssraarg +| [ ":" ssragen(gen) ssragens(dgens) ] -> + [ mk_exactarg [] (cons_gen gen dgens) ] +| [ ssrview(view) ssrclear(clr) ] -> + [ mk_exactarg view ([], clr) ] +| [ ssrclear_ne(clr) ] -> + [ mk_exactarg [] ([], clr) ] +END + +let vmexacttac pf gl = exact_no_check (mkCast (pf, VMcast, pf_concl gl)) gl + +TACTIC EXTEND ssrexact +| [ "exact" ssrexactarg(arg) ] -> [ tclBY (ssrapplytac arg) ] +| [ "exact" ] -> [ tclORELSE donetac (tclBY apply_top_tac) ] +| [ "exact" "<:" lconstr(pf) ] -> [ vmexacttac pf ] +END + +(** The "congr" tactic *) + +type ssrcongrarg = open_constr * (int * constr) + +let pr_ssrcongrarg _ _ _ (_, ((n, f), dgens)) = + (if n <= 0 then mt () else str " " ++ pr_int n) ++ + str " " ++ pr_term f ++ pr_dgens pr_gen dgens + +ARGUMENT EXTEND ssrcongrarg TYPED AS ltacctx * ((int * ssrterm) * ssrdgens) + PRINTED BY pr_ssrcongrarg +| [ natural(n) constr(c) ssrdgens(dgens) ] -> + [ rawltacctx, ((n, mk_term ' ' c), dgens) ] +| [ natural(n) constr(c) ] -> + [ rawltacctx, ((n, mk_term ' ' c),([[]],[])) ] +| [ constr(c) ssrdgens(dgens) ] -> [ rawltacctx, ((0, mk_term ' ' c), dgens) ] +| [ constr(c) ] -> [ rawltacctx, ((0, mk_term ' ' c), ([[]],[])) ] +END + +let rec mkRnat n = + if n <= 0 then GRef (dummy_loc, glob_O) else + mkRApp (GRef (dummy_loc, glob_S)) [mkRnat (n - 1)] + +let interp_congrarg_at ist gl n rf ty m = + pp(lazy(str"===interp_congrarg_at===")); + let congrn = mkSsrRRef "nary_congruence" in + let args1 = mkRnat n :: mkRHoles n @ [ty] in + let args2 = mkRHoles (3 * n) in + let rec loop i = + if i + n > m then None else + try + let rt = mkRApp congrn (args1 @ mkRApp rf (mkRHoles i) :: args2) in + pp(lazy(str"rt=" ++ pr_glob_constr rt)); + Some (interp_refine ist gl rt) + with _ -> loop (i + 1) in + loop 0 + +let pattern_id = mk_internal_id "pattern value" + +let congrtac ((n, t), ty) ist gl = + pp(lazy(str"===congr===")); + pp(lazy(str"concl=" ++ pr_constr (pf_concl gl))); + let _, f, _ = pf_abs_evars gl (interp_term ist gl t) in + let ist' = {ist with lfun = [pattern_id, VConstr ([],f)]} in + let rf = mkRltacVar pattern_id in + let m = pf_nbargs gl f in + let _, cf = if n > 0 then + match interp_congrarg_at ist' gl n rf ty m with + | Some cf -> cf + | None -> errorstrm (str "No " ++ pr_int n ++ str "-congruence with " + ++ pr_term t) + else let rec loop i = + if i > m then errorstrm (str "No congruence with " ++ pr_term t) + else match interp_congrarg_at ist' gl i rf ty m with + | Some cf -> cf + | None -> loop (i + 1) in + loop 1 in + tclTHEN (refine_with cf) (tclTRY reflexivity) gl + +let newssrcongrtac arg ist gl = + pp(lazy(str"===newcongr===")); + pp(lazy(str"concl=" ++ pr_constr (pf_concl gl))); + (* utils *) + let fs gl t = Reductionops.nf_evar (project gl) t in + let tclMATCH_GOAL (c, gl_c) proj t_ok t_fail gl = + match try Some (pf_unify_HO gl_c (pf_concl gl) c) with _ -> None with + | Some gl_c -> tclTHEN (convert_concl (fs gl_c c)) (t_ok (proj gl_c)) gl + | None -> t_fail () gl in + let mk_evar gl ty = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + let sigma, x = Evarutil.new_evar (create_evar_defs sigma) env ty in + x, re_sig si sigma in + let ssr_congr lr = mkApp (mkSsrConst "ssr_congr_arrow",lr) in + (* here thw two cases: simple equality or arrow *) + let equality, _, eq_args, gl' = pf_saturate gl (build_coq_eq ()) 3 in + tclMATCH_GOAL (equality, gl') (fun gl' -> fs gl' (List.assoc 0 eq_args)) + (fun ty -> congrtac (arg, Detyping.detype false [] [] ty) ist) + (fun () -> + let lhs, gl' = mk_evar gl mkProp in let rhs, gl' = mk_evar gl' mkProp in + let arrow = mkArrow lhs (lift 1 rhs) in + tclMATCH_GOAL (arrow, gl') (fun gl' -> [|fs gl' lhs;fs gl' rhs|]) + (fun lr -> tclTHEN (apply (ssr_congr lr)) (congrtac (arg, mkRType) ist)) + (fun _ _ -> errorstrm (str"Conclusion is not an equality nor an arrow"))) + gl +;; + +TACTIC EXTEND ssrcongr +| [ "congr" ssrcongrarg(arg) ] -> +[ let ctx, (arg, dgens) = arg in + let ist = get_ltacctx ctx in + match dgens with + | [gens], clr -> tclTHEN (genstac (gens,clr) ist) (newssrcongrtac arg ist) + | _ -> errorstrm (str"Dependent family abstractions not allowed in congr")] +END + +(** 7. Rewriting tactics (rewrite, unlock) *) + +(** Coq rewrite compatibility flag *) + +let ssr_strict_match = ref false + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "strict redex matching"; + Goptions.optkey = ["Match"; "Strict"]; + Goptions.optread = (fun () -> !ssr_strict_match); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssr_strict_match := b) } + +(** Rewrite multiplier *) + +type ssrmult = int * ssrmmod + +let notimes = 0 +let nomult = 1, Once + +let pr_mult (n, m) = + if n > 0 && m <> Once then pr_int n ++ pr_mmod m else pr_mmod m + +let pr_ssrmult _ _ _ = pr_mult + +ARGUMENT EXTEND ssrmult_ne TYPED AS int * ssrmmod PRINTED BY pr_ssrmult + | [ natural(n) ssrmmod(m) ] -> [ check_index loc n, m ] + | [ ssrmmod(m) ] -> [ notimes, m ] +END + +ARGUMENT EXTEND ssrmult TYPED AS ssrmult_ne PRINTED BY pr_ssrmult + | [ ssrmult_ne(m) ] -> [ m ] + | [ ] -> [ nomult ] +END + +(** Rewrite clear/occ switches *) + +let pr_rwocc = function + | None, None -> mt () + | None, occ -> pr_occ occ + | Some clr, _ -> pr_clear_ne clr + +let pr_ssrrwocc _ _ _ = pr_rwocc + +ARGUMENT EXTEND ssrrwocc TYPED AS ssrdocc PRINTED BY pr_ssrrwocc +| [ "{" ssrhyp_list(clr) "}" ] -> [ mkclr clr ] +| [ "{" ssrocc(occ) "}" ] -> [ mkocc occ ] +| [ ] -> [ noclr ] +END + +(** Rewrite rules *) + +type ssrwkind = RWred of ssrsimpl | RWdef | RWeq +(* type ssrrule = ssrwkind * ssrterm *) + +let pr_rwkind = function + | RWred s -> pr_simpl s + | RWdef -> str "/" + | RWeq -> mt () + +let wit_ssrrwkind, globwit_ssrrwkind, rawwit_ssrrwkind = + add_genarg "ssrrwkind" pr_rwkind + +let pr_rule = function + | RWred s, _ -> pr_simpl s + | RWdef, r-> str "/" ++ pr_term r + | RWeq, r -> pr_term r + +let pr_ssrrule _ _ _ = pr_rule + +let noruleterm loc = mk_term ' ' (mkCProp loc) + +ARGUMENT EXTEND ssrrule_ne TYPED AS ssrrwkind * ssrterm PRINTED BY pr_ssrrule + | [ ssrsimpl_ne(s) ] -> [ RWred s, noruleterm loc ] + | [ "/" ssrterm(t) ] -> [ RWdef, t ] + | [ ssrterm(t) ] -> [ RWeq, t ] +END + +ARGUMENT EXTEND ssrrule TYPED AS ssrrule_ne PRINTED BY pr_ssrrule + | [ ssrrule_ne(r) ] -> [ r ] + | [ ] -> [ RWred Nop, noruleterm loc ] +END + +(** Rewrite arguments *) + +(* type ssrrwarg = (ssrdir * ssrmult) * ((ssrdocc * ssrpattern) * ssrrule) *) + +let pr_option f = function None -> mt() | Some x -> f x +let pr_pattern_squarep= pr_option (fun r -> str "[" ++ pr_rpattern r ++ str "]") +let pr_ssrpattern_squarep _ _ _ = pr_pattern_squarep +let pr_rwarg ((d, m), ((docc, rx), r)) = + pr_rwdir d ++ pr_mult m ++ pr_rwocc docc ++ pr_pattern_squarep rx ++ pr_rule r + +let pr_ssrrwarg _ _ _ = pr_rwarg + +let mk_rwarg (d, (n, _ as m)) ((clr, occ as docc), rx) (rt, _ as r) = + if rt <> RWeq then begin + if rt = RWred Nop && not (m = nomult && occ = None && rx = None) + && (clr = None || clr = Some []) then + anomaly "Improper rewrite clear switch"; + if d = R2L && rt <> RWdef then + error "Right-to-left switch on simplification"; + if n <> 1 && rt = RWred Cut then + error "Bad or useless multiplier"; + if occ <> None && rx = None && rt <> RWdef then + error "Missing redex for simplification occurrence" + end; (d, m), ((docc, rx), r) + +let norwmult = L2R, nomult +let norwocc = noclr, None + +(* +let pattern_ident = Prim.pattern_ident in +GEXTEND Gram +GLOBAL: pattern_ident; +pattern_ident: +[[c = pattern_ident -> (CRef (Ident (loc,c)), NoBindings)]]; +END +*) + +ARGUMENT EXTEND ssrpattern_squarep +TYPED AS rpattern option PRINTED BY pr_ssrpattern_squarep + | [ "[" rpattern(rdx) "]" ] -> [ Some rdx ] + | [ ] -> [ None ] +END + +ARGUMENT EXTEND ssrpattern_ne_squarep +TYPED AS rpattern option PRINTED BY pr_ssrpattern_squarep + | [ "[" rpattern(rdx) "]" ] -> [ Some rdx ] +END + + +ARGUMENT EXTEND ssrrwarg + TYPED AS (ssrdir * ssrmult) * ((ssrdocc * rpattern option) * ssrrule) + PRINTED BY pr_ssrrwarg + | [ "-" ssrmult(m) ssrrwocc(docc) ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg (R2L, m) (docc, rx) r ] + | [ "-/" ssrterm(t) ] -> (* just in case '-/' should become a token *) + [ mk_rwarg (R2L, nomult) norwocc (RWdef, t) ] + | [ ssrmult_ne(m) ssrrwocc(docc) ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg (L2R, m) (docc, rx) r ] + | [ "{" ne_ssrhyp_list(clr) "}" ssrpattern_ne_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (mkclr clr, rx) r ] + | [ "{" ne_ssrhyp_list(clr) "}" ssrrule(r) ] -> + [ mk_rwarg norwmult (mkclr clr, None) r ] + | [ "{" ssrocc(occ) "}" ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (mkocc occ, rx) r ] + | [ "{" "}" ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (nodocc, rx) r ] + | [ ssrpattern_ne_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (noclr, rx) r ] + | [ ssrrule_ne(r) ] -> + [ mk_rwarg norwmult norwocc r ] +END + +let simplintac occ rdx sim gl = + let simptac gl = + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let simp env c _ = red_safe Tacred.simpl env sigma0 c in + convert_concl_no_check (eval_pattern env0 sigma0 concl0 rdx occ simp) gl in + match sim with + | Simpl -> simptac gl + | SimplCut -> tclTHEN simptac (tclTRY donetac) gl + | _ -> simpltac sim gl + +let rec get_evalref c = match kind_of_term c with + | Var id -> EvalVarRef id + | Const k -> EvalConstRef k + | App (c', _) -> get_evalref c' + | Cast (c', _, _) -> get_evalref c' + | _ -> errorstrm (str "The term " ++ pr_constr c ++ str " is not unfoldable") + +(* Strip a pattern generated by a prenex implicit to its constant. *) +let strip_unfold_term ((sigma, t) as p) kt = match kind_of_term t with + | App (f, a) when kt = ' ' && array_for_all isEvar a && isConst f -> + (sigma, f), true + | Const _ | Var _ -> p, true + | _ -> p, false + +let unfoldintac occ rdx t (kt,_) gl = + let fs sigma x = Reductionops.nf_evar sigma x in + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let (sigma, t), const = strip_unfold_term t kt in + let body env t c = Tacred.unfoldn [(true, [1]), get_evalref t] env sigma0 c in + let easy = occ = None && rdx = None in + let red_flags = if easy then Closure.betaiotazeta else Closure.betaiota in + let beta env = Reductionops.clos_norm_flags red_flags env sigma0 in + let unfold, conclude = match rdx with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let ise = create_evar_defs sigma in + let ise, u = mk_tpattern env0 sigma0 (ise,t) all_ok L2R t in + let find_T, end_T = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[u]) in + (fun env c h -> + try find_T env c h (fun env c _ -> body env t c) + with NoMatch when easy -> c + | NoMatch | NoProgress -> errorstrm (str"No occurrence of " + ++ pr_constr_pat t ++ spc() ++ str "in " ++ pr_constr c)), + (fun () -> try end_T () with + | NoMatch when easy -> fake_pmatcher_end () + | NoMatch -> anomaly "unfoldintac") + | _ -> + (fun env (c as orig_c) h -> + if const then + let rec aux c = + match kind_of_term c with + | Const _ when eq_constr c t -> body env t t + | App (f,a) when eq_constr f t -> mkApp (body env f f,a) + | _ -> let c = Reductionops.whd_betaiotazeta sigma0 c in + match kind_of_term c with + | Const _ when eq_constr c t -> body env t t + | App (f,a) when eq_constr f t -> mkApp (body env f f,a) + | Const f -> aux (body env c c) + | App (f, a) -> aux (mkApp (body env f f, a)) + | _ -> errorstrm (str "The term "++pr_constr orig_c++ + str" contains no " ++ pr_constr t ++ str" even after unfolding") + in aux c + else + try body env t (fs (unify_HO env sigma c t) t) + with _ -> errorstrm (str "The term " ++ + pr_constr c ++spc()++ str "does not unify with " ++ pr_constr_pat t)), + fake_pmatcher_end in + let concl = + try beta env0 (eval_pattern env0 sigma0 concl0 rdx occ unfold) + with Option.IsNone -> errorstrm (str"Failed to unfold " ++ pr_constr_pat t) in + let _ = conclude () in + convert_concl concl gl +;; + +let foldtac occ rdx ft gl = + let fs sigma x = Reductionops.nf_evar sigma x in + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let sigma, t = ft in + let fold, conclude = match rdx with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let ise = create_evar_defs sigma in + let ut = red_product_skip_id env0 sigma t in + let ise, ut = mk_tpattern env0 sigma0 (ise,t) all_ok L2R ut in + let find_T, end_T = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[ut]) in + (fun env c h -> try find_T env c h (fun env t _ -> t) with NoMatch -> c), + (fun () -> try end_T () with NoMatch -> fake_pmatcher_end ()) + | _ -> + (fun env c h -> try let sigma = unify_HO env sigma c t in fs sigma t + with _ -> errorstrm (str "fold pattern " ++ pr_constr_pat t ++ spc () + ++ str "does not match redex " ++ pr_constr_pat c)), + fake_pmatcher_end in + let concl = eval_pattern env0 sigma0 concl0 rdx occ fold in + let _ = conclude () in + convert_concl concl gl +;; + +let converse_dir = function L2R -> R2L | R2L -> L2R + +let rw_progress rhs lhs ise = not (eq_constr lhs (Evarutil.nf_evar ise rhs)) + +(* Coq has a more general form of "equation" (any type with a single *) +(* constructor with no arguments with_rect_r elimination lemmas). *) +(* However there is no clear way of determining the LHS and RHS of *) +(* such a generic Leibnitz equation -- short of inspecting the type *) +(* of the elimination lemmas. *) + +let rec strip_prod_assum c = match kind_of_term c with + | Prod (_, _, c') -> strip_prod_assum c' + | LetIn (_, v, _, c') -> strip_prod_assum (subst1 v c) + | Cast (c', _, _) -> strip_prod_assum c' + | _ -> c + +let rule_id = mk_internal_id "rewrite rule" + +exception PRtype_error +exception PRindetermined_rhs of constr + +let pirrel_rewrite pred rdx rdx_ty new_rdx dir (sigma, c) c_ty gl = + let env = pf_env gl in + let beta = Reductionops.clos_norm_flags Closure.beta env sigma in + let sigma, p = + let sigma = create_evar_defs sigma in + Evarutil.new_evar sigma env (beta (subst1 new_rdx pred)) in + let pred = mkNamedLambda pattern_id rdx_ty pred in + let elim = + let (kn, i) as ind, unfolded_c_ty = pf_reduce_to_quantified_ind gl c_ty in + let sort = elimination_sort_of_goal gl in + let elim = Indrec.lookup_eliminator ind sort in + if dir = R2L then elim else (* taken from Coq's rewrite *) + let elim = destConst elim in + let mp,dp,l = repr_con (constant_of_kn (canonical_con elim)) in + let l' = label_of_id (Nameops.add_suffix (id_of_label l) "_r") in + let c1' = Global.constant_of_delta_kn (canonical_con (make_con mp dp l')) in + mkConst c1' in + let proof = mkApp (elim, [| rdx_ty; new_rdx; pred; p; rdx; c |]) in + (* We check the proof is well typed *) + let proof_ty = + try Typing.type_of env sigma proof with _ -> raise PRtype_error in + pp(lazy(str"pirrel_rewrite proof term of type: " ++ pr_constr proof_ty)); + try refine_with + ~first_goes_last:(not !ssroldreworder) ~with_evars:false (sigma, proof) gl + with _ -> + (* we generate a msg like: "Unable to find an instance for the variable" *) + let c = Reductionops.nf_evar sigma c in + let hd_ty, miss = match kind_of_term c with + | App (hd, args) -> + let hd_ty = Retyping.get_type_of env sigma hd in + let names = let rec aux t = function 0 -> [] | n -> + let t = Reductionops.whd_betadeltaiota env sigma t in + match kind_of_type t with + | ProdType (name, _, t) -> name :: aux t (n-1) + | _ -> assert false in aux hd_ty (Array.length args) in + hd_ty, Util.list_map_filter (fun (t, name) -> + let evs = Util.Intset.elements (Evarutil.evars_of_term t) in + let open_evs = List.filter (fun k -> + InProp <> Retyping.get_sort_family_of + env sigma (Evd.evar_concl (Evd.find sigma k))) + evs in + if open_evs <> [] then Some name else None) + (List.combine (Array.to_list args) names) + | _ -> anomaly "rewrite rule not an application" in + errorstrm (Himsg.explain_refiner_error (Logic.UnresolvedBindings miss) ++ + (Pp.fnl()++str"Rule's type:" ++ spc() ++ pr_constr hd_ty)) +;; + +let rwcltac cl rdx dir sr gl = + let n, r_n,_ = pf_abs_evars gl sr in + let r_n' = pf_abs_cterm gl n r_n in + let r' = subst_var pattern_id r_n' in + let rdxt = Retyping.get_type_of (pf_env gl) (fst sr) rdx in + let cvtac, rwtac = + if closed0 r' then + let env, sigma, c, c_eq = pf_env gl, fst sr, snd sr, build_coq_eq () in + let c_ty = Typing.type_of env sigma c in + match kind_of_type (Reductionops.whd_betadeltaiota env sigma c_ty) with + | AtomicType(e, a) when eq_constr e c_eq -> + let new_rdx = if dir = L2R then a.(2) else a.(1) in + pirrel_rewrite cl rdx rdxt new_rdx dir sr c_ty, tclIDTAC + | _ -> + let cl' = mkApp (mkNamedLambda pattern_id rdxt cl, [|rdx|]) in + convert_concl cl', rewritetac dir r' + else + let dc, r2 = decompose_lam_n n r' in + let r3, _, r3t = + try destCast r2 with _ -> + errorstrm (str "no cast from " ++ pr_constr_pat (snd sr) + ++ str " to " ++ pr_constr r2) in + let cl' = mkNamedProd rule_id (compose_prod dc r3t) (lift 1 cl) in + let cl'' = mkNamedProd pattern_id rdxt cl' in + let itacs = [introid pattern_id; introid rule_id] in + let cltac = clear [pattern_id; rule_id] in + let rwtacs = [rewritetac dir (mkVar rule_id); cltac] in + (apply_type cl'' [rdx; compose_lam dc r3], tclTHENLIST (itacs @ rwtacs)) + in + let cvtac' _ = + try cvtac gl with + | PRtype_error -> + if occur_existential (pf_concl gl) + then errorstrm (str "Rewriting impacts evars") + else errorstrm (str "Dependent type error in rewrite of " + ++ pf_pr_constr gl (mkNamedLambda pattern_id rdxt cl)) + | Util.UserError _ as e -> raise e + | e -> anomaly ("cvtac's exception: " ^ Printexc.to_string e); + in + tclTHEN cvtac' rwtac gl + +let prof_rwcltac = mk_profiler "rwrxtac.rwcltac";; +let rwcltac cl rdx dir sr gl = + prof_rwcltac.profile (rwcltac cl rdx dir sr) gl +;; + + +let lz_coq_prod = + let prod = lazy (build_prod ()) in fun () -> Lazy.force prod + +let lz_setoid_relation = + let sdir = ["Classes"; "RelationClasses"] in + let last_srel = ref (Environ.empty_env, None) in + fun env -> match !last_srel with + | env', srel when env' == env -> srel + | _ -> + let srel = + try Some (coq_constant "Class_setoid" sdir "RewriteRelation") + with _ -> None in + last_srel := (env, srel); srel + +let ssr_is_setoid env = + match lz_setoid_relation env with + | None -> fun _ _ _ -> false + | Some srel -> + fun sigma r args -> + Rewrite.is_applied_rewrite_relation env + sigma [] (mkApp (r, args)) <> None + +let prof_rwxrtac_find_rule = mk_profiler "rwrxtac.find_rule";; + +let closed0_check cl p gl = + if closed0 cl then errorstrm (str"No occurrence of redex "++pf_pr_constr gl p) + +let rwrxtac occ rdx_pat dir rule gl = + let env = pf_env gl in + let coq_prod = lz_coq_prod () in + let is_setoid = ssr_is_setoid env in + let r_sigma, rules = + let rec loop d sigma r t0 rs red = + let t = + if red = 1 then Tacred.hnf_constr env sigma t0 + else Reductionops.whd_betaiotazeta sigma t0 in + match kind_of_term t with + | Prod (_, xt, at) -> + let ise, x = Evarutil.new_evar (create_evar_defs sigma) env xt in + loop d ise (mkApp (r, [|x|])) (subst1 x at) rs 0 + | App (pr, a) when pr = coq_prod.Coqlib.typ -> + let sr = match kind_of_term (Tacred.hnf_constr env sigma r) with + | App (c, ra) when c = coq_prod.Coqlib.intro -> fun i -> ra.(i + 1) + | _ -> let ra = Array.append a [|r|] in + function 1 -> mkApp (coq_prod.Coqlib.proj1, ra) + | _ -> mkApp (coq_prod.Coqlib.proj2, ra) in + if a.(0) = build_coq_True () then + loop (converse_dir d) sigma (sr 2) a.(1) rs 0 + else + let sigma2, rs2 = loop d sigma (sr 2) a.(1) rs 0 in + loop d sigma2 (sr 1) a.(0) rs2 0 + | App (r_eq, a) when Hipattern.match_with_equality_type t != None -> + let ind = destInd r_eq and rhs = array_last a in + let np, ndep = Inductiveops.inductive_nargs env ind in + let ind_ct = Inductiveops.type_of_constructors env ind in + let lhs0 = last_arg (strip_prod_assum ind_ct.(0)) in + let rdesc = match kind_of_term lhs0 with + | Rel i -> + let lhs = a.(np - i) in + let lhs, rhs = if d = L2R then lhs, rhs else rhs, lhs in +(* msgnl (str "RW: " ++ pr_rwdir d ++ str " " ++ pr_constr_pat r ++ str " : " + ++ pr_constr_pat lhs ++ str " ~> " ++ pr_constr_pat rhs); *) + d, r, lhs, rhs +(* + let l_i, r_i = if d = L2R then i, 1 - ndep else 1 - ndep, i in + let lhs = a.(np - l_i) and rhs = a.(np - r_i) in + let a' = Array.copy a in let _ = a'.(np - l_i) <- mkVar pattern_id in + let r' = mkCast (r, DEFAULTcast, mkApp (r_eq, a')) in + (d, r', lhs, rhs) +*) + | _ -> + let lhs = substl (array_list_of_tl (Array.sub a 0 np)) lhs0 in + let lhs, rhs = if d = R2L then lhs, rhs else rhs, lhs in + let d' = if Array.length a = 1 then d else converse_dir d in + d', r, lhs, rhs in + sigma, rdesc :: rs + | App (s_eq, a) when is_setoid sigma s_eq a -> + let np = Array.length a and i = 3 - dir_org d in + let lhs = a.(np - i) and rhs = a.(np + i - 3) in + let a' = Array.copy a in let _ = a'.(np - i) <- mkVar pattern_id in + let r' = mkCast (r, DEFAULTcast, mkApp (s_eq, a')) in + sigma, (d, r', lhs, rhs) :: rs + | _ -> + if red = 0 then loop d sigma r t rs 1 + else errorstrm (str "not a rewritable relation: " ++ pr_constr_pat t + ++ spc() ++ str "in rule " ++ pr_constr_pat (snd rule)) + in + let sigma, r = rule in + let t = Retyping.get_type_of env sigma r in + loop dir sigma r t [] 0 in + let find_rule rdx = + let rec rwtac = function + | [] -> + errorstrm (str "pattern " ++ pr_constr_pat rdx ++ + str " does not match " ++ pr_dir_side dir ++ + str " of " ++ pr_constr_pat (snd rule)) + | (d, r, lhs, rhs) :: rs -> + try + let ise = unify_HO env (create_evar_defs r_sigma) lhs rdx in + if not (rw_progress rhs rdx ise) then raise NoMatch else + d, (ise, Reductionops.nf_evar ise r) + with _ -> rwtac rs in + rwtac rules in + let find_rule rdx = prof_rwxrtac_find_rule.profile find_rule rdx in + let sigma0, env0, concl0 = project gl, pf_env gl, pf_concl gl in + let find_R, conclude = match rdx_pat with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let upats_origin = dir, snd rule in + let rpat env sigma0 (sigma, pats) (d, r, lhs, rhs) = + let sigma, pat = + mk_tpattern env sigma0 (sigma,r) (rw_progress rhs) d lhs in + sigma, pats @ [pat] in + let rpats = List.fold_left (rpat env0 sigma0) (r_sigma,[]) rules in + let find_R, end_R = mk_tpattern_matcher sigma0 occ ~upats_origin rpats in + find_R ~k:(fun _ _ h -> mkRel h), + fun cl -> let rdx,d,r = end_R () in closed0_check cl rdx gl; (d,r),rdx + | Some(_, (T e | X_In_T (_,e) | E_As_X_In_T (e,_,_) | E_In_X_In_T (e,_,_))) -> + let r = ref None in + (fun env c h -> do_once r (fun () -> find_rule c, c); mkRel h), + (fun concl -> closed0_check concl e gl; assert_done r) in + let concl = eval_pattern env0 sigma0 concl0 rdx_pat occ find_R in + let (d, r), rdx = conclude concl in + rwcltac concl rdx d r gl +;; + +let prof_rwxrtac = mk_profiler "rwrxtac";; +let rwrxtac occ rdx_pat dir rule gl = + prof_rwxrtac.profile (rwrxtac occ rdx_pat dir rule) gl +;; + + +(* Resolve forward reference *) +let _ = + ipat_rewritetac := fun occ dir c gl -> rwrxtac occ None dir (project gl, c) gl + +let rwargtac ist ((dir, mult), (((oclr, occ), grx), (kind, gt))) gl = + let fail = ref false in + let interp_rpattern ist gl gc = + try interp_rpattern ist gl gc + with _ when snd mult = May -> fail := true; project gl, T mkProp in + let interp gc gl = + try interp_term ist gl gc + with _ when snd mult = May -> fail := true; (project gl, mkProp) in + let rwtac gl = + let rx = Option.map (interp_rpattern ist gl) grx in + let t = interp gt gl in + (match kind with + | RWred sim -> simplintac occ rx sim + | RWdef -> if dir = R2L then foldtac occ rx t else unfoldintac occ rx t gt + | RWeq -> rwrxtac occ rx dir t) gl in + let ctac = cleartac (interp_clr (oclr, (fst gt, snd (interp gt gl)))) in + if !fail then ctac gl else tclTHEN (tclMULT mult rwtac) ctac gl + +(** Rewrite argument sequence *) + +(* type ssrrwargs = ssrrwarg list * ltacctx *) + +let pr_ssrrwargs _ _ _ (rwargs, _) = pr_list spc pr_rwarg rwargs + +ARGUMENT EXTEND ssrrwargs TYPED AS ssrrwarg list * ltacctx + PRINTED BY pr_ssrrwargs + | [ "Qed" ] -> [ anomaly "Grammar placeholder match" ] +END + +let ssr_rw_syntax = ref true + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "ssreflect rewrite"; + Goptions.optkey = ["SsrRewrite"]; + Goptions.optread = (fun _ -> !ssr_rw_syntax); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssr_rw_syntax := b) } + +let test_ssr_rw_syntax = + let test strm = + if not !ssr_rw_syntax then raise Stream.Failure else + if ssr_loaded () then () else + match Compat.get_tok (Util.stream_nth 0 strm) with + | Tok.KEYWORD key when List.mem key.[0] ['{'; '['; '/'] -> () + | _ -> raise Stream.Failure in + Gram.Entry.of_parser "test_ssr_rw_syntax" test + +GEXTEND Gram + GLOBAL: ssrrwargs; + ssrrwargs: [[ test_ssr_rw_syntax; a = LIST1 ssrrwarg -> a, rawltacctx ]]; +END + +(** The "rewrite" tactic *) + +let ssrrewritetac ist rwargs = tclTHENLIST (List.map (rwargtac ist) rwargs) + +TACTIC EXTEND ssrrewrite + | [ "rewrite" ssrrwargs(args) ssrclauses(clauses) ] -> + [ let ist = get_ltacctx (snd args) in + tclCLAUSES ist (ssrrewritetac ist (fst args)) clauses ] +END + +(** The "unlock" tactic *) + +let pr_unlockarg (occ, t) = pr_occ occ ++ pr_term t +let pr_ssrunlockarg _ _ _ = pr_unlockarg + +ARGUMENT EXTEND ssrunlockarg TYPED AS ssrocc * ssrterm + PRINTED BY pr_ssrunlockarg + | [ "{" ssrocc(occ) "}" ssrterm(t) ] -> [ occ, t ] + | [ ssrterm(t) ] -> [ None, t ] +END + +let pr_ssrunlockargs _ _ _ (args, _) = pr_list spc pr_unlockarg args + +ARGUMENT EXTEND ssrunlockargs TYPED AS ssrunlockarg list * ltacctx + PRINTED BY pr_ssrunlockargs + | [ ssrunlockarg_list(args) ] -> [ args, rawltacctx ] +END + +let unfoldtac occ ko t kt gl = + let cl, c = pf_fill_occ_term gl occ (fst (strip_unfold_term t kt)) in + let cl' = subst1 (pf_unfoldn [(true, [1]), get_evalref c] gl c) cl in + let f = if ko = None then Closure.betaiotazeta else Closure.betaiota in + convert_concl (pf_reduce (Reductionops.clos_norm_flags f) gl cl') gl + +let unlocktac ist args = + let utac (occ, gt) gl = + unfoldtac occ occ (interp_term ist gl gt) (fst gt) gl in + let locked = mkSsrConst "locked" in + let key = mkSsrConst "master_key" in + let ktacs = [ + (fun gl -> unfoldtac None None (project gl,locked) '(' gl); + simplest_newcase key ] in + tclTHENLIST (List.map utac args @ ktacs) + +TACTIC EXTEND ssrunlock + | [ "unlock" ssrunlockargs(args) ssrclauses(clauses) ] -> + [ let ist = get_ltacctx (snd args) in + tclCLAUSES ist (unlocktac ist (fst args)) clauses ] +END + +(** 8. Forward chaining tactics (pose, set, have, suffice, wlog) *) + +(** Defined identifier *) + +type ssrfwdid = identifier + +let pr_ssrfwdid _ _ _ id = pr_spc () ++ pr_id id + +(* We use a primitive parser for the head identifier of forward *) +(* tactis to avoid syntactic conflicts with basic Coq tactics. *) +ARGUMENT EXTEND ssrfwdid TYPED AS ident PRINTED BY pr_ssrfwdid + | [ "Qed" ] -> [ Util.anomaly "Grammar placeholder match" ] +END + +let accept_ssrfwdid strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT id -> accept_before_syms_or_any_id [":"; ":="; "("] strm + | _ -> raise Stream.Failure + + +let test_ssrfwdid = Gram.Entry.of_parser "test_ssrfwdid" accept_ssrfwdid + +GEXTEND Gram + GLOBAL: ssrfwdid; + ssrfwdid: [[ test_ssrfwdid; id = Prim.ident -> id ]]; + END + + + +(** Definition value formatting *) + +(* We use an intermediate structure to correctly render the binder list *) +(* abbreviations. We use a list of hints to extract the binders and *) +(* base term from a term, for the two first levels of representation of *) +(* of constr terms. *) + +type 'term ssrbind = + | Bvar of name + | Bdecl of name list * 'term + | Bdef of name * 'term option * 'term + | Bstruct of name + | Bcast of 'term + +let pr_binder prl = function + | Bvar x -> + pr_name x + | Bdecl (xs, t) -> + str "(" ++ pr_list pr_spc pr_name xs ++ str " : " ++ prl t ++ str ")" + | Bdef (x, None, v) -> + str "(" ++ pr_name x ++ str " := " ++ prl v ++ str ")" + | Bdef (x, Some t, v) -> + str "(" ++ pr_name x ++ str " : " ++ prl t ++ + str " := " ++ prl v ++ str ")" + | Bstruct x -> + str "{struct " ++ pr_name x ++ str "}" + | Bcast t -> + str ": " ++ prl t + +type 'term ssrbindval = 'term ssrbind list * 'term + +type ssrbindfmt = + | BFvar + | BFdecl of int (* #xs *) + | BFcast (* final cast *) + | BFdef of bool (* has cast? *) + | BFrec of bool * bool (* has struct? * has cast? *) + +let rec mkBstruct i = function + | Bvar x :: b -> + if i = 0 then [Bstruct x] else mkBstruct (i - 1) b + | Bdecl (xs, _) :: b -> + let i' = i - List.length xs in + if i' < 0 then [Bstruct (List.nth xs i)] else mkBstruct i' b + | _ :: b -> mkBstruct i b + | [] -> [] + +let rec format_local_binders h0 bl0 = match h0, bl0 with + | BFvar :: h, LocalRawAssum ([_, x], _, _) :: bl -> + Bvar x :: format_local_binders h bl + | BFdecl _ :: h, LocalRawAssum (lxs, _, t) :: bl -> + Bdecl (List.map snd lxs, t) :: format_local_binders h bl + | BFdef false :: h, LocalRawDef ((_, x), v) :: bl -> + Bdef (x, None, v) :: format_local_binders h bl + | BFdef true :: h, + LocalRawDef ((_, x), CCast (_, v, CastConv (_, t))) :: bl -> + Bdef (x, Some t, v) :: format_local_binders h bl + | _ -> [] + +let rec format_constr_expr h0 c0 = match h0, c0 with + | BFvar :: h, CLambdaN (_, [[_, x], _, _], c) -> + let bs, c' = format_constr_expr h c in + Bvar x :: bs, c' + | BFdecl _:: h, CLambdaN (_, [lxs, _, t], c) -> + let bs, c' = format_constr_expr h c in + Bdecl (List.map snd lxs, t) :: bs, c' + | BFdef false :: h, CLetIn(_, (_, x), v, c) -> + let bs, c' = format_constr_expr h c in + Bdef (x, None, v) :: bs, c' + | BFdef true :: h, CLetIn(_, (_, x), CCast (_, v, CastConv (_, t)), c) -> + let bs, c' = format_constr_expr h c in + Bdef (x, Some t, v) :: bs, c' + | [BFcast], CCast (_, c, CastConv (_, t)) -> + [Bcast t], c + | BFrec (has_str, has_cast) :: h, + CFix (_, _, [_, (Some locn, CStructRec), bl, t, c]) -> + let bs = format_local_binders h bl in + let bstr = if has_str then [Bstruct (Name (snd locn))] else [] in + bs @ bstr @ (if has_cast then [Bcast t] else []), c + | BFrec (_, has_cast) :: h, CCoFix (_, _, [_, bl, t, c]) -> + format_local_binders h bl @ (if has_cast then [Bcast t] else []), c + | _, c -> + [], c + +let rec format_glob_decl h0 d0 = match h0, d0 with + | BFvar :: h, (x, _, None, _) :: d -> + Bvar x :: format_glob_decl h d + | BFdecl 1 :: h, (x, _, None, t) :: d -> + Bdecl ([x], t) :: format_glob_decl h d + | BFdecl n :: h, (x, _, None, t) :: d when n > 1 -> + begin match format_glob_decl (BFdecl (n - 1) :: h) d with + | Bdecl (xs, _) :: bs -> Bdecl (x :: xs, t) :: bs + | bs -> Bdecl ([x], t) :: bs + end + | BFdef false :: h, (x, _, Some v, _) :: d -> + Bdef (x, None, v) :: format_glob_decl h d + | BFdef true:: h, (x, _, Some (GCast (_, v, CastConv (_, t))), _) :: d -> + Bdef (x, Some t, v) :: format_glob_decl h d + | _, (x, _, None, t) :: d -> + Bdecl ([x], t) :: format_glob_decl [] d + | _, (x, _, Some v, _) :: d -> + Bdef (x, None, v) :: format_glob_decl [] d + | _, [] -> [] + +let rec format_glob_constr h0 c0 = match h0, c0 with + | BFvar :: h, GLambda (_, x, _, _, c) -> + let bs, c' = format_glob_constr h c in + Bvar x :: bs, c' + | BFdecl 1 :: h, GLambda (_, x, _, t, c) -> + let bs, c' = format_glob_constr h c in + Bdecl ([x], t) :: bs, c' + | BFdecl n :: h, GLambda (_, x, _, t, c) when n > 1 -> + begin match format_glob_constr (BFdecl (n - 1) :: h) c with + | Bdecl (xs, _) :: bs, c' -> Bdecl (x :: xs, t) :: bs, c' + | _ -> [Bdecl ([x], t)], c + end + | BFdef false :: h, GLetIn(_, x, v, c) -> + let bs, c' = format_glob_constr h c in + Bdef (x, None, v) :: bs, c' + | BFdef true :: h, GLetIn(_, x, GCast (_, v, CastConv (_, t)), c) -> + let bs, c' = format_glob_constr h c in + Bdef (x, Some t, v) :: bs, c' + | [BFcast], GCast (_, c, CastConv(_, t)) -> + [Bcast t], c + | BFrec (has_str, has_cast) :: h, GRec (_, f, _, bl, t, c) + when Array.length c = 1 -> + let bs = format_glob_decl h bl.(0) in + let bstr = match has_str, f with + | true, GFix ([|Some i, GStructRec|], _) -> mkBstruct i bs + | _ -> [] in + bs @ bstr @ (if has_cast then [Bcast t.(0)] else []), c.(0) + | _, c -> + [], c + +(** Forward chaining argument *) + +(* There are three kinds of forward definitions: *) +(* - Hint: type only, cast to Type, may have proof hint. *) +(* - Have: type option + value, no space before type *) +(* - Pose: binders + value, space before binders. *) + +type ssrfwdkind = FwdHint of string * bool | FwdHave | FwdPose + +type ssrfwdfmt = ssrfwdkind * ssrbindfmt list + +let pr_fwdkind = function + | FwdHint (s,_) -> str (s ^ " ") | _ -> str " :=" ++ spc () +let pr_fwdfmt (fk, _ : ssrfwdfmt) = pr_fwdkind fk + +let wit_ssrfwdfmt, globwit_ssrfwdfmt, rawwit_ssrfwdfmt = + add_genarg "ssrfwdfmt" pr_fwdfmt + +(* type ssrfwd = ssrfwdfmt * ssrterm *) + +let mkFwdVal fk c = ((fk, []), mk_term ' ' c), rawltacctx +let mkssrFwdVal fk c = ((fk, []), (c,None)), rawltacctx + +let mkFwdCast fk loc t c = + ((fk, [BFcast]), mk_term ' ' (CCast (loc, c, dC t))), rawltacctx +let mkssrFwdCast fk loc t c = ((fk, [BFcast]), (c, Some t)), rawltacctx + +let mkFwdHint s t = + let loc = constr_loc t in + mkFwdCast (FwdHint (s,false)) loc t (CHole (loc, None)) +let mkFwdHintNoTC s t = + let loc = constr_loc t in + mkFwdCast (FwdHint (s,true)) loc t (CHole (loc, None)) + +let pr_gen_fwd prval prc prlc fk (bs, c) = + let prc s = str s ++ spc () ++ prval prc prlc c in + match fk, bs with + | FwdHint (s,_), [Bcast t] -> str s ++ spc () ++ prlc t + | FwdHint (s,_), _ -> prc (s ^ "(* typeof *)") + | FwdHave, [Bcast t] -> str ":" ++ spc () ++ prlc t ++ prc " :=" + | _, [] -> prc " :=" + | _, _ -> spc () ++ pr_list spc (pr_binder prlc) bs ++ prc " :=" + +let pr_fwd_guarded prval prval' = function +| ((fk, h), (_, (_, Some c))), _ -> + pr_gen_fwd prval pr_constr_expr prl_constr_expr fk (format_constr_expr h c) +| ((fk, h), (_, (c, None))), _ -> + pr_gen_fwd prval' pr_glob_constr prl_glob_constr fk (format_glob_constr h c) + +let pr_unguarded prc prlc = prlc + +let pr_fwd = pr_fwd_guarded pr_unguarded pr_unguarded +let pr_ssrfwd _ _ _ = pr_fwd + +ARGUMENT EXTEND ssrfwd TYPED AS (ssrfwdfmt * ssrterm) * ltacctx + PRINTED BY pr_ssrfwd + | [ ":=" lconstr(c) ] -> [ mkFwdVal FwdPose c ] + | [ ":" lconstr (t) ":=" lconstr(c) ] -> [ mkFwdCast FwdPose loc t c ] +END + +(** Independent parsing for binders *) + +(* The pose, pose fix, and pose cofix tactics use these internally to *) +(* parse argument fragments. *) + +let pr_ssrbvar prc _ _ v = prc v + +ARGUMENT EXTEND ssrbvar TYPED AS constr PRINTED BY pr_ssrbvar +| [ ident(id) ] -> [ mkCVar loc id ] +| [ "_" ] -> [ CHole (loc, None) ] +END + +let bvar_lname = function + | CRef (Ident (loc, id)) -> loc, Name id + | c -> constr_loc c, Anonymous + +let pr_ssrbinder prc _ _ (_, c) = prc c + +ARGUMENT EXTEND ssrbinder TYPED AS ssrfwdfmt * constr PRINTED BY pr_ssrbinder + | [ ssrbvar(bv) ] -> + [ let xloc, _ as x = bvar_lname bv in + (FwdPose, [BFvar]), + CLambdaN (loc,[[x],Default Explicit,CHole (xloc,None)],CHole (loc,None)) ] + | [ "(" ssrbvar(bv) ")" ] -> + [ let xloc, _ as x = bvar_lname bv in + (FwdPose, [BFvar]), + CLambdaN (loc,[[x],Default Explicit,CHole (xloc,None)],CHole (loc,None)) ] + | [ "(" ssrbvar(bv) ":" lconstr(t) ")" ] -> + [ let x = bvar_lname bv in + (FwdPose, [BFdecl 1]), + CLambdaN (loc, [[x], Default Explicit, t], CHole (loc, None)) ] + | [ "(" ssrbvar(bv) ne_ssrbvar_list(bvs) ":" lconstr(t) ")" ] -> + [ let xs = List.map bvar_lname (bv :: bvs) in + let n = List.length xs in + (FwdPose, [BFdecl n]), + CLambdaN (loc, [xs, Default Explicit, t], CHole (loc, None)) ] + | [ "(" ssrbvar(id) ":" lconstr(t) ":=" lconstr(v) ")" ] -> + [ let loc' = Util.join_loc (constr_loc t) (constr_loc v) in + let v' = CCast (loc', v, dC t) in + (FwdPose,[BFdef true]), CLetIn (loc,bvar_lname id, v',CHole (loc,None)) ] + | [ "(" ssrbvar(id) ":=" lconstr(v) ")" ] -> + [ (FwdPose,[BFdef false]), CLetIn (loc,bvar_lname id, v,CHole (loc,None)) ] +END + +GEXTEND Gram + GLOBAL: ssrbinder; + ssrbinder: [ + [ ["of" | "&"]; c = operconstr LEVEL "99" -> + (FwdPose, [BFvar]), + CLambdaN (loc,[[loc,Anonymous],Default Explicit,c],CHole (loc,None)) ] + ]; +END + +let rec binders_fmts = function + | ((_, h), _) :: bs -> h @ binders_fmts bs + | _ -> [] + +let push_binders c2 bs = + let loc2 = constr_loc c2 in let mkloc loc1 = Util.join_loc loc1 loc2 in + let rec loop ty c = function + | (_, CLambdaN (loc1, b, _)) :: bs when ty -> + CProdN (mkloc loc1, b, loop ty c bs) + | (_, CLambdaN (loc1, b, _)) :: bs -> + CLambdaN (mkloc loc1, b, loop ty c bs) + | (_, CLetIn (loc1, x, v, _)) :: bs -> + CLetIn (mkloc loc1, x, v, loop ty c bs) + | [] -> c + | _ -> anomaly "binder not a lambda nor a let in" in + match c2 with + | CCast (x, ct, CastConv (y, cty)) -> + (CCast (x, loop false ct bs, CastConv (y, loop true cty bs))) + | ct -> loop false ct bs + +let rec fix_binders = function + | (_, CLambdaN (_, [xs, _, t], _)) :: bs -> + LocalRawAssum (xs, Default Explicit, t) :: fix_binders bs + | (_, CLetIn (_, x, v, _)) :: bs -> + LocalRawDef (x, v) :: fix_binders bs + | _ -> [] + +let pr_ssrstruct _ _ _ = function + | Some id -> str "{struct " ++ pr_id id ++ str "}" + | None -> mt () + +ARGUMENT EXTEND ssrstruct TYPED AS ident option PRINTED BY pr_ssrstruct +| [ "{" "struct" ident(id) "}" ] -> [ Some id ] +| [ ] -> [ None ] +END + +(** The "pose" tactic *) + +(* The plain pose form. *) + +let bind_fwd bs = function + | ((fk, h), (ck, (rc, Some c))), ctx -> + ((fk,binders_fmts bs @ h), (ck,(rc,Some (push_binders c bs)))), ctx + | fwd -> fwd + +ARGUMENT EXTEND ssrposefwd TYPED AS ssrfwd PRINTED BY pr_ssrfwd + | [ ssrbinder_list(bs) ssrfwd(fwd) ] -> [ bind_fwd bs fwd ] +END + +(* The pose fix form. *) + +let pr_ssrfixfwd _ _ _ (id, fwd) = str " fix " ++ pr_id id ++ pr_fwd fwd + +let bvar_locid = function + | CRef (Ident (loc, id)) -> loc, id + | _ -> Util.error "Missing identifier after \"(co)fix\"" + + +ARGUMENT EXTEND ssrfixfwd TYPED AS ident * ssrfwd PRINTED BY pr_ssrfixfwd + | [ "fix" ssrbvar(bv) ssrbinder_list(bs) ssrstruct(sid) ssrfwd(fwd) ] -> + [ let (_, id) as lid = bvar_locid bv in + let ((fk, h), (ck, (rc, oc))), ctx = fwd in + let c = Option.get oc in + let has_cast, t', c' = match format_constr_expr h c with + | [Bcast t'], c' -> true, t', c' + | _ -> false, CHole (constr_loc c, None), c in + let lb = fix_binders bs in + let has_struct, i = + let rec loop = function + (l', Name id') :: _ when sid = Some id' -> true, (l', id') + | [l', Name id'] when sid = None -> false, (l', id') + | _ :: bn -> loop bn + | [] -> Util.error "Bad structural argument" in + loop (names_of_local_assums lb) in + let h' = BFrec (has_struct, has_cast) :: binders_fmts bs in + let fix = CFix (loc, lid, [lid, (Some i, CStructRec), lb, t', c']) in + id, (((fk, h'), (ck, (rc, Some fix))), ctx) ] +END + + +(* The pose cofix form. *) + +let pr_ssrcofixfwd _ _ _ (id, fwd) = str " cofix " ++ pr_id id ++ pr_fwd fwd + +ARGUMENT EXTEND ssrcofixfwd TYPED AS ssrfixfwd PRINTED BY pr_ssrcofixfwd + | [ "cofix" ssrbvar(bv) ssrbinder_list(bs) ssrfwd(fwd) ] -> + [ let _, id as lid = bvar_locid bv in + let ((fk, h), (ck, (rc, oc))), ctx = fwd in + let c = Option.get oc in + let has_cast, t', c' = match format_constr_expr h c with + | [Bcast t'], c' -> true, t', c' + | _ -> false, CHole (constr_loc c, None), c in + let h' = BFrec (false, has_cast) :: binders_fmts bs in + let cofix = CCoFix (loc, lid, [lid, fix_binders bs, t', c']) in + id, (((fk, h'), (ck, (rc, Some cofix))), ctx) + ] +END + +let ssrposetac (id, ((_, t), ctx)) gl = + posetac id (snd(pf_abs_ssrterm (get_ltacctx ctx) gl t)) gl + + +let prof_ssrposetac = mk_profiler "ssrposetac";; +let ssrposetac arg gl = prof_ssrposetac.profile (ssrposetac arg) gl;; + +TACTIC EXTEND ssrpose +| [ "pose" ssrfixfwd(ffwd) ] -> [ ssrposetac ffwd ] +| [ "pose" ssrcofixfwd(ffwd) ] -> [ ssrposetac ffwd ] +| [ "pose" ssrfwdid(id) ssrposefwd(fwd) ] -> [ ssrposetac (id, fwd) ] +END + +(** The "set" tactic *) + +(* type ssrsetfwd = ssrfwd * ssrdocc *) + +let guard_setrhs s i = s.[i] = '{' + +let pr_setrhs occ prc prlc c = + if occ = nodocc then pr_guarded guard_setrhs prlc c else pr_docc occ ++ prc c + +let pr_fwd_guarded prval prval' = function +| ((fk, h), (_, (_, Some c))), _ -> + pr_gen_fwd prval pr_constr_expr prl_constr_expr fk (format_constr_expr h c) +| ((fk, h), (_, (c, None))), _ -> + pr_gen_fwd prval' pr_glob_constr prl_glob_constr fk (format_glob_constr h c) + +(* This does not print the type, it should be fixed... *) +let pr_ssrsetfwd _ _ _ ((((fk,_),(t,_)),ctx), docc) = + pr_gen_fwd (fun _ _ -> pr_cpattern) + (fun _ -> mt()) (fun _ -> mt()) fk ([Bcast ()],t) + +ARGUMENT EXTEND ssrsetfwd +TYPED AS ((ssrfwdfmt * (lcpattern * ssrterm option)) * ltacctx) * ssrdocc +PRINTED BY pr_ssrsetfwd +| [ ":" lconstr(t) ":=" "{" ssrocc(occ) "}" cpattern(c) ] -> + [ mkssrFwdCast FwdPose loc (mk_lterm t) c, mkocc occ ] +| [ ":" lconstr(t) ":=" lcpattern(c) ] -> + [ mkssrFwdCast FwdPose loc (mk_lterm t) c, nodocc ] +| [ ":=" "{" ssrocc(occ) "}" cpattern(c) ] -> + [ mkssrFwdVal FwdPose c, mkocc occ ] +| [ ":=" lcpattern(c) ] -> [ mkssrFwdVal FwdPose c, nodocc ] +END + +let ssrsettac ist id (((_, (pat, pty)), _), (_, occ)) gl = + let pat = interp_cpattern ist gl pat (Option.map snd pty) in + let cl, sigma, env = pf_concl gl, project gl, pf_env gl in + let c, cl = + try fill_occ_pattern ~raise_NoMatch:true env sigma cl pat occ 1 + with NoMatch -> redex_of_pattern ~resolve_typeclasses:true env pat, cl in + if occur_existential c then errorstrm(str"The pattern"++spc()++ + pr_constr_pat c++spc()++str"did not match and has holes."++spc()++ + str"Did you mean pose?") else + let c, cty = match kind_of_term c with + | Cast(t, DEFAULTcast, ty) -> t, ty + | _ -> c, pf_type_of gl c in + let cl' = mkLetIn (Name id, c, cty, cl) in + tclTHEN (convert_concl cl') (introid id) gl + +TACTIC EXTEND ssrset +| [ "set" ssrfwdid(id) ssrsetfwd(fwd) ssrclauses(clauses) ] -> + [ let (_, ctx), _ = fwd in + let ist = get_ltacctx ctx in + tclCLAUSES ist (ssrsettac ist id fwd) clauses ] +END + +(** The "have" tactic *) + +(* type ssrhavefwd = ssrfwd * ssrhint *) + +let pr_ssrhavefwd _ _ prt (fwd, hint) = pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrhavefwd TYPED AS ssrfwd * ssrhint PRINTED BY pr_ssrhavefwd +| [ ":" lconstr(t) ssrhint(hint) ] -> [ mkFwdHint ":" t, hint ] +| [ ":" lconstr(t) ":=" lconstr(c) ] -> [ mkFwdCast FwdHave loc t c, nohint ] +| [ ":" lconstr(t) ":=" ] -> [ mkFwdHintNoTC ":" t, nohint ] +| [ ":=" lconstr(c) ] -> [ mkFwdVal FwdHave c, nohint ] +END + +let intro_id_to_binder = List.map (function + | IpatId id -> + let xloc, _ as x = bvar_lname (mkCVar dummy_loc id) in + (FwdPose, [BFvar]), + CLambdaN (dummy_loc, [[x], Default Explicit, CHole (xloc, None)], + CHole (dummy_loc, None)) + | _ -> anomaly "non-id accepted as binder") + +let binder_to_intro_id = List.map (function + | (FwdPose, [BFvar]), CLambdaN (_,[ids,_,_],_) + | (FwdPose, [BFdecl _]), CLambdaN (_,[ids,_,_],_) -> + List.map (function (_, Name id) -> IpatId id | _ -> IpatAnon) ids + | (FwdPose, [BFdef _]), CLetIn (_,(_,Name id),_,_) -> [IpatId id] + | (FwdPose, [BFdef _]), CLetIn (_,(_,Anonymous),_,_) -> [IpatAnon] + | _ -> anomaly "ssrbinder is not a binder") + +let pr_ssrhavefwdwbinders _ _ prt (tr,(hpats, (fwd, hint))) = + pr_hpats hpats ++ pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrhavefwdwbinders + TYPED AS bool * (ssrhpats * (ssrfwd * ssrhint)) + PRINTED BY pr_ssrhavefwdwbinders +| [ ssrhpats_wtransp(trpats) ssrbinder_list(bs) ssrhavefwd(fwd) ] -> + [ let tr, pats = trpats in + let ((clr, pats), binders), simpl = pats in + let allbs = intro_id_to_binder binders @ bs in + let allbinders = binders @ List.flatten (binder_to_intro_id bs) in + let hint = bind_fwd allbs (fst fwd), snd fwd in + tr, ((((clr, pats), allbinders), simpl), hint) ] +END + +(* Tactic. *) + +let examine_abstract id gl = + let tid = pf_type_of gl id in + let abstract = mkSsrConst "abstract" in + if not (isApp tid) || not (eq_constr (fst(destApp tid)) abstract) then + errorstrm(strbrk"not an abstract constant: "++pr_constr id); + let _, args_id = destApp tid in + if Array.length args_id <> 3 then + errorstrm(strbrk"not a proper abstract constant: "++pr_constr id); + if not (isEvar args_id.(2)) then + errorstrm(strbrk"abstract constant "++pr_constr id++str" already used"); + args_id + +let pf_find_abstract_proof check_lock gl abstract_n = + let fire gl t = Reductionops.nf_evar (project gl) t in + let abstract = mkSsrConst "abstract" in + let l = Evd.fold_undefined (fun e ei l -> + match kind_of_term ei.Evd.evar_concl with + | App(hd, [|ty; n; lock|]) + when (not check_lock || + (occur_existential (fire gl ty) && + isEvar (fire gl lock))) && + eq_constr hd abstract && eq_constr n abstract_n -> e::l + | _ -> l) (project gl) [] in + match l with + | [e] -> e + | _ -> errorstrm(strbrk"abstract constant "++pr_constr abstract_n++ + strbrk" not found in the evar map exactly once. "++ + strbrk"Did you tamper with it?") + +let unfold cl = + let module R = Reductionops in let module F = Closure.RedFlags in + reduct_in_concl (R.clos_norm_flags (F.mkflags + (List.map (fun c -> F.fCONST (destConst c)) cl @ [F.fBETA; F.fIOTA]))) + +let havegentac ist t gl = + let _, c = pf_abs_ssrterm ist gl t in + apply_type (mkArrow (pf_type_of gl c) (pf_concl gl)) [c] gl + +let havetac + (transp,((((clr, pats), binders), simpl), ((((fk, _), t), ctx), hint))) + suff namefst gl += + let ist, concl = get_ltacctx ctx, pf_concl gl in + let skols, pats = + List.partition (function IpatNewHidden _ -> true | _ -> false) pats in + let itac_mkabs = introstac ~ist skols in + let itac_c = introstac ~ist (IpatSimpl(clr,Nop) :: pats) in + let itac, id, clr = introstac ~ist pats, tclIDTAC, cleartac clr in + let binderstac n = + let rec aux = function 0 -> [] | n -> IpatAnon :: aux (n-1) in + tclTHEN (if binders <> [] then introstac ~ist (aux n) else tclIDTAC) + (introstac ~ist binders) in + let simpltac = introstac ~ist simpl in + let fixtc = + not !ssrhaveNOtcresolution && + match fk with FwdHint(_,true) -> false | _ -> true in + let hint = hinttac ist true hint in + let cuttac t gl = + if transp then + (* The v8.4 refiner is too stupid to deal with a let-in, apparently. + * Let's do things by hand... *) + let concl = pf_concl gl in + let gl, extra = pf_new_evar gl t in + let gl, kont = pf_new_evar gl (mkLetIn (Anonymous, extra, t, concl)) in + let step = mkApp (mkSsrConst "ssr_have_let", [|concl; t; extra; kont|]) in + pf_partial_solution gl step [extra; kont] + else + basecuttac "ssr_have" t gl in + (* Introduce now abstract constants, so that everything sees them *) + let abstract_key = mkSsrConst "abstract_key" in + let unlock_abs args_id gl = pf_unify_HO gl args_id.(2) abstract_key in + tclTHENFIRST itac_mkabs (fun gl -> + let mkt t = mk_term ' ' t in + let mkl t = (' ', (t, None)) in + let interp gl rtc t = pf_abs_ssrterm ~resolve_typeclasses:rtc ist gl t in + let interp_ty gl rtc t = + let a,b,_ = pf_interp_ty ~resolve_typeclasses:rtc ist gl t in a, b in + let ct, cty, hole, loc = match t with + | _, (_, Some (CCast (loc, ct, CastConv (_, cty)))) -> + mkt ct, mkt cty, mkt (mkCHole dummy_loc), loc + | _, (_, Some ct) -> + mkt ct, mkt (mkCHole dummy_loc), mkt (mkCHole dummy_loc), dummy_loc + | _, (GCast (loc, ct, CastConv (_, cty)), None) -> + mkl ct, mkl cty, mkl mkRHole, loc + | _, (t, None) -> mkl t, mkl mkRHole, mkl mkRHole, dummy_loc in + let gl, cut, sol, itac1, itac2 = + match fk, namefst, suff with + | FwdHave, true, true -> + errorstrm (str"Suff have does not accept a proof term") + | FwdHave, false, true -> + let cty = combineCG cty hole (mkCArrow loc) mkRArrow in + let _, t = interp gl false (combineCG ct cty (mkCCast loc) mkRCast) in + let ty = pf_type_of gl t in + let ctx, _ = decompose_prod_n 1 ty in + let assert_is_conv gl = + try convert_concl (compose_prod ctx concl) gl + with _ -> errorstrm (str "Given proof term is not of type " ++ + pr_constr (mkArrow (mkVar (id_of_string "_")) concl)) in + gl, ty, tclTHEN assert_is_conv (apply t), id, itac_c + | FwdHave, false, false -> + let skols = List.flatten (List.map (function + | IpatNewHidden ids -> ids + | _ -> assert false) skols) in + let skols_args = + List.map (fun id -> examine_abstract (mkVar id) gl) skols in + let gl = List.fold_right unlock_abs skols_args gl in + let sigma, t = interp gl false (combineCG ct cty (mkCCast loc) mkRCast) in + let gl = re_sig (sig_it gl) sigma in + let gs = + List.map (fun a -> pf_find_abstract_proof false gl a.(1)) skols_args in + let tacopen_skols gl = + let stuff, g = Refiner.unpackage gl in + Refiner.repackage stuff ((List.map Goal.build gs) @ [g]) in + gl, pf_type_of gl t, apply t, id, + tclTHEN (tclTHEN itac_c simpltac) + (tclTHEN tacopen_skols (unfold [mkSsrConst "abstract";abstract_key])) + | _,true,true -> + gl, mkArrow (snd (interp_ty gl fixtc cty)) concl, hint, itac, clr + | _,false,true -> + gl, mkArrow (snd (interp_ty gl fixtc cty)) concl, hint, id, itac_c + | _, false, false -> + let n, cty = interp_ty gl fixtc cty in + gl, cty, tclTHEN (binderstac n) hint, id, tclTHEN itac_c simpltac + | _, true, false -> assert false in + tclTHENS (cuttac cut) [ tclTHEN sol itac1; itac2 ] gl) + gl +;; + +(* to extend the abstract value one needs: + Utility lemma to partially instantiate an abstract constant type. + Lemma use_abstract T n l (x : abstract T n l) : T. + Proof. by case: l x. Qed. +*) +let ssrabstract ctx gens (*last*) gl = + let ist = get_ltacctx ctx in + let main _ (_,cid) ist gl = +(* + let proj1, proj2, prod = + let pdata = build_prod () in + pdata.Coqlib.proj1, pdata.Coqlib.proj2, pdata.Coqlib.typ in +*) + let concl, env = pf_concl gl, pf_env gl in + let fire gl t = Reductionops.nf_evar (project gl) t in + let abstract = mkSsrConst "abstract" in + let abstract_key = mkSsrConst "abstract_key" in + let id = mkVar (Option.get (id_of_cpattern cid)) in + let args_id = examine_abstract id gl in + let abstract_n = args_id.(1) in + let abstract_proof = pf_find_abstract_proof true gl abstract_n in + let gl, proof = + let pf_unify_HO gl a b = + try pf_unify_HO gl a b + with _ -> errorstrm(strbrk"The abstract variable "++pr_constr id++ + strbrk" cannot abstract this goal. Did you generalize it?") in + let rec find_hole p t = + match kind_of_term t with + | Evar _ (*when last*) -> pf_unify_HO gl concl t, p +(* + | Evar _ -> + let sigma, it = project gl, sig_it gl in + let sigma, ty = Evarutil.new_type_evar sigma env in + let gl = re_sig it sigma in + let p = mkApp (proj2,[|ty;concl;p|]) in + let concl = mkApp(prod,[|ty; concl|]) in + pf_unify_HO gl concl t, p + | App(hd, [|left; right|]) when eq_constr hd prod -> + find_hole (mkApp (proj1,[|left;right;p|])) left +*) + | _ -> errorstrm(strbrk"abstract constant "++pr_constr abstract_n++ + strbrk" has an unexpected shape. Did you tamper with it?") + in + find_hole + ((*if last then*) id + (*else mkApp(mkSsrConst "use_abstract",Array.append args_id [|id|])*)) + (fire gl args_id.(0)) in + let gl = (*if last then*) pf_unify_HO gl abstract_key args_id.(2) (*else gl*) in + let proof = fire gl proof in +(* if last then *) + let tacopen gl = + let stuff, g = Refiner.unpackage gl in + Refiner.repackage stuff [ g; Goal.build abstract_proof ] in + tclTHENS tacopen [tclSOLVE [apply proof];unfold[abstract;abstract_key]] gl +(* else apply proof gl *) + in + let introback ist (gens, _) = + introstac ~ist + (List.map (fun (_,cp) -> match id_of_cpattern cp with + | None -> IpatAnon + | Some id -> IpatId id) + (List.tl (List.hd gens))) in + tclTHEN (with_dgens gens main ist) (introback ist gens) gl + +(* The standard TACTIC EXTEND does not work for abstract *) +GEXTEND Gram + GLOBAL: tactic_expr; + tactic_expr: LEVEL "3" + [ RIGHTA [ IDENT "abstract"; gens = ssrdgens; ctx = ltacctx -> + Tacexpr.TacAtom(loc,Tacexpr.TacExtend (loc, "ssrabstract", + [Genarg.in_gen rawwit_ssrdgens gens; + Genarg.in_gen rawwit_ltacctx ctx])) ]]; +END +TACTIC EXTEND ssrabstract +| [ "abstract" ssrdgens(gens) ltacctx(ctx) ] -> [ + if List.length (fst gens) <> 1 then + errorstrm (str"dependents switches '/' not allowed here"); + ssrabstract ctx gens ] +END + +let prof_havetac = mk_profiler "havetac";; +let havetac arg a b gl = prof_havetac.profile (havetac arg a b) gl;; + +TACTIC EXTEND ssrhave +| [ "have" ssrhavefwdwbinders(fwd) ] -> + [ havetac fwd false false ] +END + +TACTIC EXTEND ssrhavesuff +| [ "have" "suff" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ havetac (false,(pats,fwd)) true false ] +END + +TACTIC EXTEND ssrhavesuffices +| [ "have" "suffices" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ havetac (false,(pats,fwd)) true false ] +END + +TACTIC EXTEND ssrsuffhave +| [ "suff" "have" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ havetac (false,(pats,fwd)) true true ] +END + +TACTIC EXTEND ssrsufficeshave +| [ "suffices" "have" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ havetac (false,(pats,fwd)) true true ] +END + +(** The "suffice" tactic *) + +let pr_ssrsufffwdwbinders _ _ prt (hpats, (fwd, hint)) = + pr_hpats hpats ++ pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrsufffwd + TYPED AS ssrhpats * (ssrfwd * ssrhint) PRINTED BY pr_ssrsufffwdwbinders +| [ ssrhpats(pats) ssrbinder_list(bs) ":" lconstr(t) ssrhint(hint) ] -> + [ let ((clr, pats), binders), simpl = pats in + let allbs = intro_id_to_binder binders @ bs in + let allbinders = binders @ List.flatten (binder_to_intro_id bs) in + let fwd = mkFwdHint ":" t in + (((clr, pats), allbinders), simpl), (bind_fwd allbs fwd, hint) ] +END + +let sufftac ((((clr, pats),binders),simpl), (((_, c), ctx), hint)) = + let ist = get_ltacctx ctx in + let htac = tclTHEN (introstac ~ist pats) (hinttac ist true hint) in + let c = match c with + | (a, (b, Some (CCast (_, _, CastConv (_, cty))))) -> a, (b, Some cty) + | (a, (GCast (_, _, CastConv (_, cty)), None)) -> a, (cty, None) + | _ -> anomaly "suff: ssr cast hole deleted by typecheck" in + let ctac gl = basecuttac "ssr_suff" (pi2 (pf_interp_ty ist gl c)) gl in + tclTHENS ctac [htac; tclTHEN (cleartac clr) (introstac ~ist (binders@simpl))] + +TACTIC EXTEND ssrsuff +| [ "suff" ssrsufffwd(fwd) ] -> [ sufftac fwd ] +END + +TACTIC EXTEND ssrsuffices +| [ "suffices" ssrsufffwd(fwd) ] -> [ sufftac fwd ] +END + +(** The "wlog" (Without Loss Of Generality) tactic *) + +(* type ssrwlogfwd = ssrwgen list * ssrfwd *) + +let pr_ssrwlogfwd _ _ _ (gens, t) = + str ":" ++ pr_list mt pr_wgen gens ++ spc() ++ pr_fwd t + +ARGUMENT EXTEND ssrwlogfwd TYPED AS ssrwgen list * ssrfwd + PRINTED BY pr_ssrwlogfwd +| [ ":" ssrwgen_list(gens) "/" lconstr(t) ] -> [ gens, mkFwdHint "/" t] +END + +let destProd_or_LetIn c = + match kind_of_term c with + | Prod (n,ty,c) -> (n,None,ty), c + | LetIn (n,bo,ty,c) -> (n,Some bo,ty), c + | _ -> raise (Invalid_argument "destProd_or_LetIn") + +let wlogtac (((clr0, pats),_),_) (gens, ((_, ct), ctx)) hint suff ghave gl = + let ist = get_ltacctx ctx in + let mkabs gen = abs_wgen false ist gl (fun x -> x) gen in + let mkclr gen clrs = clr_of_wgen gen clrs in + let mkpats = function + | _, Some ((x, _), _) -> fun pats -> IpatId (hoi_id x) :: pats + | _ -> fun x -> x in + let ct = match ct with + | (a, (b, Some (CCast (_, _, CastConv (_, cty))))) -> a, (b, Some cty) + | (a, (GCast (_, _, CastConv (_, cty)), None)) -> a, (cty, None) + | _ -> anomaly "wlog: ssr cast hole deleted by typecheck" in + let cut_implies_goal = not (suff || ghave <> `NoGen) in + let c, args, ct = + let gens = List.filter (function _, Some _ -> true | _ -> false) gens in + let concl = pf_concl gl in + let c = mkProp in + let c = if cut_implies_goal then mkArrow c concl else c in + let args, c = List.fold_right mkabs gens ([],c) in + let env, _ = + List.fold_left (fun (env, c) _ -> + let rd, c = destProd_or_LetIn c in + Environ.push_rel rd env, c) (pf_env gl, c) gens in + let sigma, ev = Evarutil.new_evar (project gl) env Term.mkProp in + let k, _ = Term.destEvar ev in + let fake_gl = { Evd.it = Goal.build k; Evd.sigma = sigma } in + let _, ct, _ = pf_interp_ty ist fake_gl ct in + let rec subst c g s = match kind_of_term c, g with + | Prod(Anonymous,_,c), [] -> mkProd(Anonymous, subst_vars s ct, c) + | Sort _, [] -> subst_vars s ct + | LetIn(Name id as n,b,ty,c), _::g -> mkLetIn (n,b,ty,subst c g (id::s)) + | Prod(Name id as n,ty,c), _::g -> mkProd (n,ty,subst c g (id::s)) + | _ -> anomaly "SSR: wlog: subst" in + let c = subst c gens [] in + let rec pired c = function + | [] -> c + | t::ts as args -> match kind_of_term c with + | Prod(_,_,c) -> pired (subst1 t c) ts + | LetIn(id,b,ty,c) -> mkLetIn (id,b,ty,pired c args) + | _ -> anomaly "SSR: wlog: pired" in + c, args, pired c args in + let tacipat pats = introstac ~ist pats in + let tacigens = + tclTHEN + (tclTHENLIST(List.rev(List.fold_right mkclr gens [cleartac clr0]))) + (introstac ~ist (List.fold_right mkpats gens [])) in + let hinttac = hinttac ist true hint in + let cut_kind, fst_goal_tac, snd_goal_tac = + match suff, ghave with + | true, `NoGen -> "ssr_wlog", tclTHEN hinttac (tacipat pats), tacigens + | false, `NoGen -> "ssr_wlog", hinttac, tclTHEN tacigens (tacipat pats) + | true, `Gen _ -> assert false + | false, `Gen id -> + if gens = [] then errorstrm(str"gen have requires some generalizations"); + let clear0 = cleartac clr0 in + let id, name_general_hyp, cleanup, pats = match id, pats with + | None, (IpatId id as ip)::pats -> Some id, tacipat [ip], clear0, pats + | None, _ -> None, tclIDTAC, clear0, pats + | Some (Some id),_ -> Some id, introid id, clear0, pats + | Some _,_ -> + let id = mk_anon_id "tmp" gl in + Some id, introid id, tclTHEN clear0 (clear [id]), pats in + let tac_specialize = match id with + | None -> tclIDTAC + | Some id -> + if pats = [] then tclIDTAC else + let args = Array.of_list args in + pp(lazy(str"specialized="++pr_constr (mkApp (mkVar id,args)))); + pp(lazy(str"specialized_ty="++pr_constr ct)); + tclTHENS (basecuttac "ssr_have" ct) + [apply (mkApp (mkVar id,args)); tclIDTAC] in + "ssr_have", + (if hint = nohint then tacigens else hinttac), + tclTHENLIST [name_general_hyp; tac_specialize; tacipat pats; cleanup] + in + tclTHENS (basecuttac cut_kind c) [fst_goal_tac; snd_goal_tac] gl + +TACTIC EXTEND ssrwlog +| [ "wlog" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ wlogtac pats fwd hint false `NoGen ] +END + +TACTIC EXTEND ssrwlogs +| [ "wlog" "suff" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ wlogtac pats fwd hint true `NoGen ] +END + +TACTIC EXTEND ssrwlogss +| [ "wlog" "suffices" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ]-> + [ wlogtac pats fwd hint true `NoGen ] +END + +TACTIC EXTEND ssrwithoutloss +| [ "without" "loss" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ wlogtac pats fwd hint false `NoGen ] +END + +TACTIC EXTEND ssrwithoutlosss +| [ "without" "loss" "suff" + ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ wlogtac pats fwd hint true `NoGen ] +END + +TACTIC EXTEND ssrwithoutlossss +| [ "without" "loss" "suffices" + ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ]-> + [ wlogtac pats fwd hint true `NoGen ] +END + +(* Generally have *) +let pr_idcomma _ _ _ = function + | None -> mt() + | Some None -> str"_, " + | Some (Some id) -> pr_id id ++ str", " + +ARGUMENT EXTEND ssr_idcomma TYPED AS ident option option PRINTED BY pr_idcomma + | [ ] -> [ None ] +END + +let accept_idcomma strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT _ | Tok.KEYWORD "_" -> accept_before_syms [","] strm + | _ -> raise Stream.Failure + +let test_idcomma = Gram.Entry.of_parser "test_idcomma" accept_idcomma + +GEXTEND Gram + GLOBAL: ssr_idcomma; + ssr_idcomma: [ [ test_idcomma; + ip = [ id = IDENT -> Some (id_of_string id) | "_" -> None ]; "," -> + Some ip + ] ]; +END + +let augment_preclr clr1 (((clr0, x),y),z) = (((clr1 @ clr0, x),y),z) + +TACTIC EXTEND ssrgenhave +| [ "gen" "have" ssrclear(clr) + ssr_idcomma(id) ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ let pats = augment_preclr clr pats in + wlogtac pats fwd hint false (`Gen id) ] +END + +TACTIC EXTEND ssrgenhave2 +| [ "generally" "have" ssrclear(clr) + ssr_idcomma(id) ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ let pats = augment_preclr clr pats in + wlogtac pats fwd hint false (`Gen id) ] +END + +(** Canonical Structure alias *) + +let def_body : Vernacexpr.definition_expr Gram.Entry.e = Obj.magic + (Grammar.Entry.find (Obj.magic gallina_ext) "vernac:def_body") in + +GEXTEND Gram + GLOBAL: gallina_ext; + + gallina_ext: + (* Canonical structure *) + [[ IDENT "Canonical"; qid = Constr.global -> + Vernacexpr.VernacCanonical (AN qid) + | IDENT "Canonical"; ntn = Prim.by_notation -> + Vernacexpr.VernacCanonical (ByNotation ntn) + | IDENT "Canonical"; qid = Constr.global; + d = def_body -> + let s = coerce_reference_to_id qid in + Vernacexpr.VernacDefinition + ((Decl_kinds.Global,Decl_kinds.CanonicalStructure), + (dummy_loc,s),(d ), + (fun _ -> Recordops.declare_canonical_structure)) + ]]; +END + +(** 9. Keyword compatibility fixes. *) + +(* Coq v8.1 notation uses "by" and "of" quasi-keywords, i.e., reserved *) +(* identifiers used as keywords. This is incompatible with ssreflect.v *) +(* which makes "by" and "of" true keywords, because of technicalities *) +(* in the internal lexer-parser API of Coq. We patch this here by *) +(* adding new parsing rules that recognize the new keywords. *) +(* To make matters worse, the Coq grammar for tactics fails to *) +(* export the non-terminals we need to patch. Fortunately, the CamlP5 *) +(* API provides a backdoor access (with loads of Obj.magic trickery). *) + +(* Coq v8.3 defines "by" as a keyword, some hacks are not needed any *) +(* longer and thus comment out. Such comments are marked with v8.3 *) + +let tac_ent = List.fold_left Grammar.Entry.find (Obj.magic simple_tactic) in +let hypident_ent = + tac_ent ["clause_dft_all"; "in_clause"; "hypident_occ"; "hypident"] in +let id_or_meta : Obj.t Gram.Entry.e = Obj.magic + (Grammar.Entry.find hypident_ent "id_or_meta") in +let hypident : (Obj.t * hyp_location_flag) Gram.Entry.e = + Obj.magic hypident_ent in +GEXTEND Gram + GLOBAL: hypident; +hypident: [ + [ "("; IDENT "type"; "of"; id = id_or_meta; ")" -> id, InHypTypeOnly + | "("; IDENT "value"; "of"; id = id_or_meta; ")" -> id, InHypValueOnly + ] ]; +END + +GEXTEND Gram + GLOBAL: hloc; +hloc: [ + [ "in"; "("; "Type"; "of"; id = ident; ")" -> + HypLocation ((Util.dummy_loc,id), InHypTypeOnly) + | "in"; "("; IDENT "Value"; "of"; id = ident; ")" -> + HypLocation ((Util.dummy_loc,id), InHypValueOnly) + ] ]; +END + +let constr_eval + : (Topconstr.constr_expr,Obj.t,Obj.t) Glob_term.may_eval Gram.Entry.e + = Obj.magic (Grammar.Entry.find (Obj.magic constr_may_eval) "constr_eval") + +GEXTEND Gram + GLOBAL: constr_eval; + constr_eval: [ + [ IDENT "type"; "of"; c = Constr.constr -> ConstrTypeOf c ] + ]; +END + +(* We wipe out all the keywords generated by the grammar rules we defined. *) +(* The user is supposed to Require Import ssreflect or Require ssreflect *) +(* and Import ssreflect.SsrSyntax to obtain these keywords and as a *) +(* consequence the extended ssreflect grammar. *) +let () = Lexer.unfreeze frozen_lexer ;; + +(* vim: set filetype=ocaml foldmethod=marker: *) diff --git a/mathcomp/ssreflect/plugin/v8.4/ssreflect.mllib b/mathcomp/ssreflect/plugin/v8.4/ssreflect.mllib new file mode 100644 index 0000000..006b70f --- /dev/null +++ b/mathcomp/ssreflect/plugin/v8.4/ssreflect.mllib @@ -0,0 +1,2 @@ +Ssrmatching +Ssreflect diff --git a/mathcomp/ssreflect/plugin/v8.4/ssrmatching.ml4 b/mathcomp/ssreflect/plugin/v8.4/ssrmatching.ml4 new file mode 100644 index 0000000..b24b0a6 --- /dev/null +++ b/mathcomp/ssreflect/plugin/v8.4/ssrmatching.ml4 @@ -0,0 +1,1223 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +(* Defining grammar rules with "xx" in it automatically declares keywords too, + * we thus save the lexer to restore it at the end of the file *) +let frozen_lexer = Lexer.freeze () ;; + +(*i camlp4use: "pa_extend.cmo" i*) +(*i camlp4deps: "grammar/grammar.cma" i*) + +open Names +open Pp +open Pcoq +open Genarg +open Term +open Topconstr +open Libnames +open Tactics +open Tacticals +open Termops +open Namegen +open Recordops +open Tacmach +open Coqlib +open Glob_term +open Util +open Evd +open Extend +open Goptions +open Tacexpr +open Tacinterp +open Pretyping +open Constr +open Tactic +open Extraargs +open Ppconstr +open Printer + +type loc = Util.loc +let errorstrm = errorlabstrm "ssreflect" +let loc_error loc msg = user_err_loc (loc, msg, str msg) + +(* 0 cost pp function. Active only if env variable SSRDEBUG is set *) +(* or if SsrDebug is Set *) +let pp_ref = ref (fun _ -> ()) +let ssr_pp s = pperrnl (str"SSR: "++Lazy.force s) +let _ = try ignore(Sys.getenv "SSRDEBUG"); pp_ref := ssr_pp with Not_found -> () +let debug b = + if b then pp_ref := ssr_pp else pp_ref := fun _ -> () +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssrmatching debugging"; + Goptions.optkey = ["SsrMatchingDebug"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !pp_ref == ssr_pp); + Goptions.optwrite = debug } +let pp s = !pp_ref s + +(** Utils {{{ *****************************************************************) +let env_size env = List.length (Environ.named_context env) +let safeDestApp c = + match kind_of_term c with App (f, a) -> f, a | _ -> c, [| |] +let get_index = function ArgArg i -> i | _ -> + anomaly "Uninterpreted index" +(* Toplevel constr must be globalized twice ! *) +let glob_constr ist gsigma genv = function + | _, Some ce -> + let ltacvars = List.map fst ist.lfun, [] in + Constrintern.intern_gen false ~ltacvars:ltacvars gsigma genv ce + | rc, None -> rc + +(* Term printing utilities functions for deciding bracketing. *) +let pr_paren prx x = hov 1 (str "(" ++ prx x ++ str ")") +(* String lexing utilities *) +let skip_wschars s = + let rec loop i = match s.[i] with '\n'..' ' -> loop (i + 1) | _ -> i in loop +(* We also guard characters that might interfere with the ssreflect *) +(* tactic syntax. *) +let guard_term ch1 s i = match s.[i] with + | '(' -> false + | '{' | '/' | '=' -> true + | _ -> ch1 = '(' +(* The call 'guard s i' should return true if the contents of s *) +(* starting at i need bracketing to avoid ambiguities. *) +let pr_guarded guard prc c = + msg_with Format.str_formatter (prc c); + let s = Format.flush_str_formatter () ^ "$" in + if guard s (skip_wschars s 0) then pr_paren prc c else prc c +(* More sensible names for constr printers *) +let pr_constr = pr_constr +let prl_glob_constr c = pr_lglob_constr_env (Global.env ()) c +let pr_glob_constr c = pr_glob_constr_env (Global.env ()) c +let prl_constr_expr = pr_lconstr_expr +let pr_constr_expr = pr_constr_expr +let prl_glob_constr_and_expr = function + | _, Some c -> prl_constr_expr c + | c, None -> prl_glob_constr c +let pr_glob_constr_and_expr = function + | _, Some c -> pr_constr_expr c + | c, None -> pr_glob_constr c +let pr_term (k, c) = pr_guarded (guard_term k) pr_glob_constr_and_expr c +let prl_term (k, c) = pr_guarded (guard_term k) prl_glob_constr_and_expr c + +(** Adding a new uninterpreted generic argument type *) +let add_genarg tag pr = + let wit, globwit, rawwit as wits = create_arg None tag in + let glob _ rarg = in_gen globwit (out_gen rawwit rarg) in + let interp _ gl garg = Tacmach.project gl,in_gen wit (out_gen globwit garg) in + let subst _ garg = garg in + add_interp_genarg tag (glob, interp, subst); + let gen_pr _ _ _ = pr in + Pptactic.declare_extra_genarg_pprule + (rawwit, gen_pr) (globwit, gen_pr) (wit, gen_pr); + wits + +(** Constructors for cast type *) +let dC t = CastConv (DEFAULTcast, t) +(** Constructors for constr_expr *) +let isCVar = function CRef (Ident _) -> true | _ -> false +let destCVar = function CRef (Ident (_, id)) -> id | _ -> + anomaly "not a CRef" +let mkCHole loc = CHole (loc, None) +let mkCLambda loc name ty t = + CLambdaN (loc, [[loc, name], Default Explicit, ty], t) +let mkCLetIn loc name bo t = + CLetIn (loc, (loc, name), bo, t) +let mkCCast loc t ty = CCast (loc,t, dC ty) +(** Constructors for rawconstr *) +let mkRHole = GHole (dummy_loc, InternalHole) +let mkRApp f args = if args = [] then f else GApp (dummy_loc, f, args) +let mkRCast rc rt = GCast (dummy_loc, rc, dC rt) +let mkRLambda n s t = GLambda (dummy_loc, n, Explicit, s, t) + +(* ssrterm conbinators *) +let combineCG t1 t2 f g = match t1, t2 with + | (x, (t1, None)), (_, (t2, None)) -> x, (g t1 t2, None) + | (x, (_, Some t1)), (_, (_, Some t2)) -> x, (mkRHole, Some (f t1 t2)) + | _, (_, (_, None)) -> anomaly "have: mixed C-G constr" + | _ -> anomaly "have: mixed G-C constr" +let loc_ofCG = function + | (_, (s, None)) -> loc_of_glob_constr s + | (_, (_, Some s)) -> constr_loc s + +let mk_term k c = k, (mkRHole, Some c) +let mk_lterm = mk_term ' ' + +(* }}} *) + +(** Profiling {{{ *************************************************************) +type profiler = { + profile : 'a 'b. ('a -> 'b) -> 'a -> 'b; + reset : unit -> unit; + print : unit -> unit } +let profile_now = ref false +let something_profiled = ref false +let profilers = ref [] +let add_profiler f = profilers := f :: !profilers;; +let profile b = + profile_now := b; + if b then List.iter (fun f -> f.reset ()) !profilers; + if not b then List.iter (fun f -> f.print ()) !profilers +;; +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssrmatching profiling"; + Goptions.optkey = ["SsrMatchingProfiling"]; + Goptions.optread = (fun _ -> !profile_now); + Goptions.optdepr = false; + Goptions.optwrite = profile } +let () = + let prof_total = + let init = ref 0.0 in { + profile = (fun f x -> assert false); + reset = (fun () -> init := Unix.gettimeofday ()); + print = (fun () -> if !something_profiled then + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + "total" 0 (Unix.gettimeofday() -. !init) 0.0 0.0)) } in + let prof_legenda = { + profile = (fun f x -> assert false); + reset = (fun () -> ()); + print = (fun () -> if !something_profiled then begin + prerr_endline + (Printf.sprintf "!! %39s ---------- --------- --------- ---------" + (String.make 39 '-')); + prerr_endline + (Printf.sprintf "!! %-39s %10s %9s %9s %9s" + "function" "#calls" "total" "max" "average") end) } in + add_profiler prof_legenda; + add_profiler prof_total +;; + +let mk_profiler s = + let total, calls, max = ref 0.0, ref 0, ref 0.0 in + let reset () = total := 0.0; calls := 0; max := 0.0 in + let profile f x = + if not !profile_now then f x else + let before = Unix.gettimeofday () in + try + incr calls; + let res = f x in + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + res + with exc -> + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + raise exc in + let print () = + if !calls <> 0 then begin + something_profiled := true; + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + s !calls !total !max (!total /. (float_of_int !calls))) end in + let prof = { profile = profile; reset = reset; print = print } in + add_profiler prof; + prof +;; +(* }}} *) + +exception NoProgress + +(** Unification procedures. *) + +(* To enforce the rigidity of the rooted match we always split *) +(* top applications, so the unification procedures operate on *) +(* arrays of patterns and terms. *) +(* We perform three kinds of unification: *) +(* EQ: exact conversion check *) +(* FO: first-order unification of evars, without conversion *) +(* HO: higher-order unification with conversion *) +(* The subterm unification strategy is to find the first FO *) +(* match, if possible, and the first HO match otherwise, then *) +(* compute all the occurrences that are EQ matches for the *) +(* relevant subterm. *) +(* Additional twists: *) +(* - If FO/HO fails then we attempt to fill evars using *) +(* typeclasses before raising an outright error. We also *) +(* fill typeclasses even after a successful match, since *) +(* beta-reduction and canonical instances may leave *) +(* undefined evars. *) +(* - We do postchecks to rule out matches that are not *) +(* closed or that assign to a global evar; these can be *) +(* disabled for rewrite or dependent family matches. *) +(* - We do a full FO scan before turning to HO, as the FO *) +(* comparison can be much faster than the HO one. *) + +let unif_EQ env sigma p c = + let evars = existential_opt_value sigma in + try let _ = Reduction.conv env p ~evars c in true with _ -> false + +let unif_EQ_args env sigma pa a = + let n = Array.length pa in + let rec loop i = (i = n) || unif_EQ env sigma pa.(i) a.(i) && loop (i + 1) in + loop 0 + +let prof_unif_eq_args = mk_profiler "unif_EQ_args";; +let unif_EQ_args env sigma pa a = + prof_unif_eq_args.profile (unif_EQ_args env sigma pa) a +;; + +let unif_HO env ise p c = Evarconv.the_conv_x env p c ise + +let unif_HOtype env ise p c = Evarconv.the_conv_x_leq env p c ise + +let unif_HO_args env ise0 pa i ca = + let n = Array.length pa in + let rec loop ise j = + if j = n then ise else loop (unif_HO env ise pa.(j) ca.(i + j)) (j + 1) in + loop ise0 0 + +(* FO unification should boil down to calling w_unify with no_delta, but *) +(* alas things are not so simple: w_unify does partial type-checking, *) +(* which breaks down when the no-delta flag is on (as the Coq type system *) +(* requires full convertibility. The workaround here is to convert all *) +(* evars into metas, since 8.2 does not TC metas. This means some lossage *) +(* for HO evars, though hopefully Miller patterns can pick up some of *) +(* those cases, and HO matching will mop up the rest. *) +let flags_FO = {Unification.default_no_delta_unify_flags with + Unification.modulo_conv_on_closed_terms = None; + Unification.modulo_eta = true; + Unification.modulo_betaiota = true; + Unification.modulo_delta_types = full_transparent_state; + Unification.allow_K_in_toplevel_higher_order_unification=false} +let unif_FO env ise p c = + Unification.w_unify env ise Reduction.CONV ~flags:flags_FO p c + +(* Perform evar substitution in main term and prune substitution. *) +let nf_open_term sigma0 ise c = + let s = ise and s' = ref sigma0 in + let rec nf c' = match kind_of_term c' with + | Evar ex -> + begin try nf (existential_value s ex) with _ -> + let k, a = ex in let a' = Array.map nf a in + if not (Evd.mem !s' k) then + s' := Evd.add !s' k (Evarutil.nf_evar_info s (Evd.find s k)); + mkEvar (k, a') + end + | _ -> map_constr nf c' in + let copy_def k evi () = + if evar_body evi != Evd.Evar_empty then () else + match Evd.evar_body (Evd.find s k) with + | Evar_defined c' -> s' := Evd.define k (nf c') !s' + | _ -> () in + let c' = nf c in let _ = Evd.fold copy_def sigma0 () in !s', c' + +let unif_end env sigma0 ise0 pt ok = + let ise = Evarconv.consider_remaining_unif_problems env ise0 in + let s, t = nf_open_term sigma0 ise pt in + let ise1 = create_evar_defs s in + let ise2 = Typeclasses.resolve_typeclasses ~fail:true env ise1 in + if not (ok ise) then raise NoProgress else + if ise2 == ise1 then (s, t) else nf_open_term sigma0 ise2 t + +let pf_unif_HO gl sigma pt p c = + let env = pf_env gl in + let ise = unif_HO env (create_evar_defs sigma) p c in + unif_end env (project gl) ise pt (fun _ -> true) + +let unify_HO env sigma0 t1 t2 = + let sigma = unif_HO env sigma0 t1 t2 in + let sigma, _ = unif_end env sigma0 sigma t2 (fun _ -> true) in + sigma + +let pf_unify_HO gl t1 t2 = + let env, sigma0, si = pf_env gl, project gl, sig_it gl in + let sigma = unify_HO env sigma0 t1 t2 in + re_sig si sigma + +(* This is what the definition of iter_constr should be... *) +let iter_constr_LR f c = match kind_of_term c with + | Evar (k, a) -> Array.iter f a + | Cast (cc, _, t) -> f cc; f t + | Prod (_, t, b) | Lambda (_, t, b) -> f t; f b + | LetIn (_, v, t, b) -> f v; f t; f b + | App (cf, a) -> f cf; Array.iter f a + | Case (_, p, v, b) -> f v; f p; Array.iter f b + | Fix (_, (_, t, b)) | CoFix (_, (_, t, b)) -> + for i = 0 to Array.length t - 1 do f t.(i); f b.(i) done + | _ -> () + +(* The comparison used to determine which subterms matches is KEYED *) +(* CONVERSION. This looks for convertible terms that either have the same *) +(* same head constant as pat if pat is an application (after beta-iota), *) +(* or start with the same constr constructor (esp. for LetIn); this is *) +(* disregarded if the head term is let x := ... in x, and casts are always *) +(* ignored and removed). *) +(* Record projections get special treatment: in addition to the projection *) +(* constant itself, ssreflect also recognizes head constants of canonical *) +(* projections. *) + +exception NoMatch +type ssrdir = L2R | R2L +let pr_dir_side = function L2R -> str "LHS" | R2L -> str "RHS" +let inv_dir = function L2R -> R2L | R2L -> L2R + + +type pattern_class = + | KpatFixed + | KpatConst + | KpatEvar of existential_key + | KpatLet + | KpatLam + | KpatRigid + | KpatFlex + | KpatProj of constant + +type tpattern = { + up_k : pattern_class; + up_FO : constr; + up_f : constr; + up_a : constr array; + up_t : constr; (* equation proof term or matched term *) + up_dir : ssrdir; (* direction of the rule *) + up_ok : constr -> evar_map -> bool; (* progess test for rewrite *) + } + +let all_ok _ _ = true + +let proj_nparams c = + try 1 + Recordops.find_projection_nparams (ConstRef c) with _ -> 0 + +let isFixed c = match kind_of_term c with + | Var _ | Ind _ | Construct _ | Const _ -> true + | _ -> false + +let isRigid c = match kind_of_term c with + | Prod _ | Sort _ | Lambda _ | Case _ | Fix _ | CoFix _ -> true + | _ -> false + +exception UndefPat + +let hole_var = mkVar (id_of_string "_") +let pr_constr_pat c0 = + let rec wipe_evar c = + if isEvar c then hole_var else map_constr wipe_evar c in + pr_constr (wipe_evar c0) + +(* Turn (new) evars into metas *) +let evars_for_FO ~hack env sigma0 (ise0:evar_map) c0 = + let ise = ref ise0 in + let sigma = ref ise0 in + let nenv = env_size env + if hack then 1 else 0 in + let rec put c = match kind_of_term c with + | Evar (k, a as ex) -> + begin try put (existential_value !sigma ex) + with NotInstantiatedEvar -> + if Evd.mem sigma0 k then map_constr put c else + let evi = Evd.find !sigma k in + let dc = list_firstn (max 0 (Array.length a - nenv)) (evar_filtered_context evi) in + let abs_dc (d, c) = function + | x, Some b, t -> d, mkNamedLetIn x (put b) (put t) c + | x, None, t -> mkVar x :: d, mkNamedProd x (put t) c in + let a, t = + Sign.fold_named_context_reverse abs_dc ~init:([], (put evi.evar_concl)) dc in + let m = Evarutil.new_meta () in + ise := meta_declare m t !ise; + sigma := Evd.define k (applist (mkMeta m, a)) !sigma; + put (existential_value !sigma ex) + end + | _ -> map_constr put c in + let c1 = put c0 in !ise, c1 + +(* Compile a match pattern from a term; t is the term to fill. *) +(* p_origin can be passed to obtain a better error message *) +let mk_tpattern ?p_origin ?(hack=false) env sigma0 (ise, t) ok dir p = + let k, f, a = + let f, a = Reductionops.whd_betaiota_stack ise p in + match kind_of_term f with + | Const p -> + let np = proj_nparams p in + if np = 0 || np > List.length a then KpatConst, f, a else + let a1, a2 = list_chop np a in KpatProj p, applist(f, a1), a2 + | Var _ | Ind _ | Construct _ -> KpatFixed, f, a + | Evar (k, _) -> + if Evd.mem sigma0 k then KpatEvar k, f, a else + if a <> [] then KpatFlex, f, a else + (match p_origin with None -> error "indeterminate pattern" + | Some (dir, rule) -> + errorstrm (str "indeterminate " ++ pr_dir_side dir + ++ str " in " ++ pr_constr_pat rule)) + | LetIn (_, v, _, b) -> + if b <> mkRel 1 then KpatLet, f, a else KpatFlex, v, a + | Lambda _ -> KpatLam, f, a + | _ -> KpatRigid, f, a in + let aa = Array.of_list a in + let ise', p' = evars_for_FO ~hack env sigma0 ise (mkApp (f, aa)) in + ise', + { up_k = k; up_FO = p'; up_f = f; + up_a = aa; up_ok = ok; up_dir = dir; up_t = t} + +(* Specialize a pattern after a successful match: assign a precise head *) +(* kind and arity for Proj and Flex patterns. *) +let ungen_upat lhs (sigma, t) u = + let f, a = safeDestApp lhs in + let k = match kind_of_term f with + | Var _ | Ind _ | Construct _ -> KpatFixed + | Const _ -> KpatConst + | Evar (k, _) -> if is_defined sigma k then raise NoMatch else KpatEvar k + | LetIn _ -> KpatLet + | Lambda _ -> KpatLam + | _ -> KpatRigid in + sigma, {u with up_k = k; up_FO = lhs; up_f = f; up_a = a; up_t = t} + +let nb_cs_proj_args pc f u = + let na k = + List.length (lookup_canonical_conversion (ConstRef pc, k)).o_TCOMPS in + try match kind_of_term f with + | Prod _ -> na Prod_cs + | Sort s -> na (Sort_cs (family_of_sort s)) + | Const c' when c' = pc -> Array.length (snd (destApp u.up_f)) + | Var _ | Ind _ | Construct _ | Const _ -> na (Const_cs (global_of_constr f)) + | _ -> -1 + with Not_found -> -1 + +let isEvar_k k f = + match kind_of_term f with Evar (k', _) -> k = k' | _ -> false + +let nb_args c = + match kind_of_term c with App (_, a) -> Array.length a | _ -> 0 + +let mkSubArg i a = if i = Array.length a then a else Array.sub a 0 i +let mkSubApp f i a = if i = 0 then f else mkApp (f, mkSubArg i a) + +let splay_app ise = + let rec loop c a = match kind_of_term c with + | App (f, a') -> loop f (Array.append a' a) + | Cast (c', _, _) -> loop c' a + | Evar ex -> + (try loop (existential_value ise ex) a with _ -> c, a) + | _ -> c, a in + fun c -> match kind_of_term c with + | App (f, a) -> loop f a + | Cast _ | Evar _ -> loop c [| |] + | _ -> c, [| |] + +let filter_upat i0 f n u fpats = + let na = Array.length u.up_a in + if n < na then fpats else + let np = match u.up_k with + | KpatConst when eq_constr u.up_f f -> na + | KpatFixed when u.up_f = f -> na + | KpatEvar k when isEvar_k k f -> na + | KpatLet when isLetIn f -> na + | KpatLam when isLambda f -> na + | KpatRigid when isRigid f -> na + | KpatFlex -> na + | KpatProj pc -> + let np = na + nb_cs_proj_args pc f u in if n < np then -1 else np + | _ -> -1 in + if np < na then fpats else + let () = if !i0 < np then i0 := n in (u, np) :: fpats + +let filter_upat_FO i0 f n u fpats = + let np = nb_args u.up_FO in + if n < np then fpats else + let ok = match u.up_k with + | KpatConst -> eq_constr u.up_f f + | KpatFixed -> u.up_f = f + | KpatEvar k -> isEvar_k k f + | KpatLet -> isLetIn f + | KpatLam -> isLambda f + | KpatRigid -> isRigid f + | KpatProj pc -> f = mkConst pc + | KpatFlex -> i0 := n; true in + if ok then begin if !i0 < np then i0 := np; (u, np) :: fpats end else fpats + +exception FoundUnif of (evar_map * tpattern) +(* Note: we don't update env as we descend into the term, as the primitive *) +(* unification procedure always rejects subterms with bound variables. *) + +(* We are forced to duplicate code between the FO/HO matching because we *) +(* have to work around several kludges in unify.ml: *) +(* - w_unify drops into second-order unification when the pattern is an *) +(* application whose head is a meta. *) +(* - w_unify tries to unify types without subsumption when the pattern *) +(* head is an evar or meta (e.g., it fails on ?1 = nat when ?1 : Type). *) +(* - w_unify expands let-in (zeta conversion) eagerly, whereas we want to *) +(* match a head let rigidly. *) +let match_upats_FO upats env sigma0 ise = + let rec loop c = + let f, a = splay_app ise c in let i0 = ref (-1) in + let fpats = + List.fold_right (filter_upat_FO i0 f (Array.length a)) upats [] in + while !i0 >= 0 do + let i = !i0 in i0 := -1; + let c' = mkSubApp f i a in + let one_match (u, np) = + let skip = + if i <= np then i < np else + if u.up_k == KpatFlex then begin i0 := i - 1; false end else + begin if !i0 < np then i0 := np; true end in + if skip || not (closed0 c') then () else try + let _ = match u.up_k with + | KpatFlex -> + let kludge v = mkLambda (Anonymous, mkProp, v) in + unif_FO env ise (kludge u.up_FO) (kludge c') + | KpatLet -> + let kludge vla = + let vl, a = safeDestApp vla in + let x, v, t, b = destLetIn vl in + mkApp (mkLambda (x, t, b), array_cons v a) in + unif_FO env ise (kludge u.up_FO) (kludge c') + | _ -> unif_FO env ise u.up_FO c' in + let ise' = (* Unify again using HO to assign evars *) + let p = mkApp (u.up_f, u.up_a) in + try unif_HO env ise p c' with _ -> raise NoMatch in + let lhs = mkSubApp f i a in + let pt' = unif_end env sigma0 ise' u.up_t (u.up_ok lhs) in + raise (FoundUnif (ungen_upat lhs pt' u)) + with FoundUnif _ as sigma_u -> raise sigma_u + | Not_found -> anomaly "incomplete ise in match_upats_FO" + | _ -> () in + List.iter one_match fpats + done; + iter_constr_LR loop f; Array.iter loop a in + fun c -> try loop c with Invalid_argument _ -> anomaly "IN FO" + +let prof_FO = mk_profiler "match_upats_FO";; +let match_upats_FO upats env sigma0 ise c = + prof_FO.profile (match_upats_FO upats env sigma0) ise c +;; + + +let match_upats_HO upats env sigma0 ise c = + let it_did_match = ref false in + let rec aux upats env sigma0 ise c = + let f, a = splay_app ise c in let i0 = ref (-1) in + let fpats = List.fold_right (filter_upat i0 f (Array.length a)) upats [] in + while !i0 >= 0 do + let i = !i0 in i0 := -1; + let one_match (u, np) = + let skip = + if i <= np then i < np else + if u.up_k == KpatFlex then begin i0 := i - 1; false end else + begin if !i0 < np then i0 := np; true end in + if skip then () else try + let ise' = match u.up_k with + | KpatFixed | KpatConst -> ise + | KpatEvar _ -> + let _, pka = destEvar u.up_f and _, ka = destEvar f in + unif_HO_args env ise pka 0 ka + | KpatLet -> + let x, v, t, b = destLetIn f in + let _, pv, _, pb = destLetIn u.up_f in + let ise' = unif_HO env ise pv v in + unif_HO (Environ.push_rel (x, None, t) env) ise' pb b + | KpatFlex | KpatProj _ -> + unif_HO env ise u.up_f (mkSubApp f (i - Array.length u.up_a) a) + | _ -> unif_HO env ise u.up_f f in + let ise'' = unif_HO_args env ise' u.up_a (i - Array.length u.up_a) a in + let lhs = mkSubApp f i a in + let pt' = unif_end env sigma0 ise'' u.up_t (u.up_ok lhs) in + raise (FoundUnif (ungen_upat lhs pt' u)) + with FoundUnif _ as sigma_u -> raise sigma_u + | NoProgress -> it_did_match := true + | _ -> () in + List.iter one_match fpats + done; + iter_constr_LR (aux upats env sigma0 ise) f; + Array.iter (aux upats env sigma0 ise) a + in + aux upats env sigma0 ise c; + if !it_did_match then raise NoProgress + +let prof_HO = mk_profiler "match_upats_HO";; +let match_upats_HO upats env sigma0 ise c = + prof_HO.profile (match_upats_HO upats env sigma0) ise c +;; + + +let fixed_upat = function +| {up_k = KpatFlex | KpatEvar _ | KpatProj _} -> false +| {up_t = t} -> not (occur_existential t) + +let do_once r f = match !r with Some _ -> () | None -> r := Some (f ()) + +let assert_done r = + match !r with Some x -> x | None -> anomaly "do_once never called" + +type subst = Environ.env -> Term.constr -> int -> Term.constr +type find_P = + Environ.env -> Term.constr -> int -> + k:subst -> + Term.constr +type conclude = unit -> Term.constr * ssrdir * (Evd.evar_map * Term.constr) + +(* upats_origin makes a better error message only *) +let mk_tpattern_matcher + ?(raise_NoMatch=false) ?upats_origin sigma0 occ (ise, upats) += + let nocc = ref 0 and skip_occ = ref false in + let use_occ, occ_list = match occ with + | Some (true, ol) -> ol = [], ol + | Some (false, ol) -> ol <> [], ol + | None -> false, [] in + let max_occ = List.fold_right max occ_list 0 in + let subst_occ = + let occ_set = Array.make max_occ (not use_occ) in + let _ = List.iter (fun i -> occ_set.(i - 1) <- use_occ) occ_list in + let _ = if max_occ = 0 then skip_occ := use_occ in + fun () -> incr nocc; + if !nocc = max_occ then skip_occ := use_occ; + if !nocc <= max_occ then occ_set.(!nocc - 1) else not use_occ in + let upat_that_matched = ref None in + let match_EQ env sigma u = + match u.up_k with + | KpatLet -> + let x, pv, t, pb = destLetIn u.up_f in + let env' = Environ.push_rel (x, None, t) env in + let match_let f = match kind_of_term f with + | LetIn (_, v, _, b) -> unif_EQ env sigma pv v && unif_EQ env' sigma pb b + | _ -> false in match_let + | KpatFixed -> (=) u.up_f + | KpatConst -> eq_constr u.up_f + | KpatLam -> fun c -> + (match kind_of_term c with + | Lambda _ -> unif_EQ env sigma u.up_f c + | _ -> false) + | _ -> unif_EQ env sigma u.up_f in +let p2t p = mkApp(p.up_f,p.up_a) in +let source () = match upats_origin, upats with + | None, [p] -> + (if fixed_upat p then str"term " else str"partial term ") ++ + pr_constr_pat (p2t p) ++ spc() + | Some (dir,rule), [p] -> str"The " ++ pr_dir_side dir ++ str" of " ++ + pr_constr_pat rule ++ fnl() ++ ws 4 ++ pr_constr_pat (p2t p) ++ fnl() + | Some (dir,rule), _ -> str"The " ++ pr_dir_side dir ++ str" of " ++ + pr_constr_pat rule ++ spc() + | _, [] | None, _::_::_ -> + anomaly "mk_tpattern_matcher with no upats_origin" in +((fun env c h ~k -> + do_once upat_that_matched (fun () -> + try + match_upats_FO upats env sigma0 ise c; + match_upats_HO upats env sigma0 ise c; + raise NoMatch + with FoundUnif sigma_u -> sigma_u + | NoMatch when (not raise_NoMatch) -> + errorstrm (source () ++ str "does not match any subterm of the goal") + | NoProgress when (not raise_NoMatch) -> + let dir = match upats_origin with Some (d,_) -> d | _ -> + anomaly "mk_tpattern_matcher with no upats_origin" in + errorstrm (str"all matches of "++source()++ + str"are equal to the " ++ pr_dir_side (inv_dir dir)) + | NoProgress -> raise NoMatch); + let sigma, ({up_f = pf; up_a = pa} as u) = assert_done upat_that_matched in + if !skip_occ then (ignore(k env u.up_t 0); c) else + let match_EQ = match_EQ env sigma u in + let pn = Array.length pa in + let rec subst_loop (env,h as acc) c' = + if !skip_occ then c' else + let f, a = splay_app sigma c' in + if Array.length a >= pn && match_EQ f && unif_EQ_args env sigma pa a then + let a1, a2 = array_chop (Array.length pa) a in + let fa1 = mkApp (f, a1) in + let f' = if subst_occ () then k env u.up_t h else fa1 in + mkApp (f', array_map_left (subst_loop acc) a2) + else + (* TASSI: clear letin values to avoid unfolding *) + let inc_h (n,_,ty) (env,h') = Environ.push_rel (n,None,ty) env, h' + 1 in + let f' = map_constr_with_binders_left_to_right inc_h subst_loop acc f in + mkApp (f', array_map_left (subst_loop acc) a) in + subst_loop (env,h) c) : find_P), +((fun () -> + let sigma, ({up_f = pf; up_a = pa} as u) = + match !upat_that_matched with + | Some x -> x | None when raise_NoMatch -> raise NoMatch + | None -> anomaly "companion function never called" in + let p' = mkApp (pf, pa) in + if max_occ <= !nocc then p', u.up_dir, (sigma, u.up_t) + else errorstrm (str"Only " ++ int !nocc ++ str" < " ++ int max_occ ++ + str(plural !nocc " occurence") ++ match upats_origin with + | None -> str" of" ++ spc() ++ pr_constr_pat p' + | Some (dir,rule) -> str" of the " ++ pr_dir_side dir ++ fnl() ++ + ws 4 ++ pr_constr_pat p' ++ fnl () ++ + str"of " ++ pr_constr_pat rule)) : conclude) + +type ('ident, 'term) ssrpattern = + | T of 'term + | In_T of 'term + | X_In_T of 'ident * 'term + | In_X_In_T of 'ident * 'term + | E_In_X_In_T of 'term * 'ident * 'term + | E_As_X_In_T of 'term * 'ident * 'term + +let pr_pattern = function + | T t -> prl_term t + | In_T t -> str "in " ++ prl_term t + | X_In_T (x,t) -> prl_term x ++ str " in " ++ prl_term t + | In_X_In_T (x,t) -> str "in " ++ prl_term x ++ str " in " ++ prl_term t + | E_In_X_In_T (e,x,t) -> + prl_term e ++ str " in " ++ prl_term x ++ str " in " ++ prl_term t + | E_As_X_In_T (e,x,t) -> + prl_term e ++ str " as " ++ prl_term x ++ str " in " ++ prl_term t + +let pr_pattern_w_ids = function + | T t -> prl_term t + | In_T t -> str "in " ++ prl_term t + | X_In_T (x,t) -> pr_id x ++ str " in " ++ prl_term t + | In_X_In_T (x,t) -> str "in " ++ pr_id x ++ str " in " ++ prl_term t + | E_In_X_In_T (e,x,t) -> + prl_term e ++ str " in " ++ pr_id x ++ str " in " ++ prl_term t + | E_As_X_In_T (e,x,t) -> + prl_term e ++ str " as " ++ pr_id x ++ str " in " ++ prl_term t + +let pr_pattern_aux pr_constr = function + | T t -> pr_constr t + | In_T t -> str "in " ++ pr_constr t + | X_In_T (x,t) -> pr_constr x ++ str " in " ++ pr_constr t + | In_X_In_T (x,t) -> str "in " ++ pr_constr x ++ str " in " ++ pr_constr t + | E_In_X_In_T (e,x,t) -> + pr_constr e ++ str " in " ++ pr_constr x ++ str " in " ++ pr_constr t + | E_As_X_In_T (e,x,t) -> + pr_constr e ++ str " as " ++ pr_constr x ++ str " in " ++ pr_constr t +let pp_pattern (sigma, p) = + pr_pattern_aux (fun t -> pr_constr (snd (nf_open_term sigma sigma t))) p +let pr_cpattern = pr_term +let pr_rpattern _ _ _ = pr_pattern + +let pr_option f = function None -> mt() | Some x -> f x +let pr_ssrpattern _ _ _ = pr_option pr_pattern +let pr_pattern_squarep = pr_option (fun r -> str "[" ++ pr_pattern r ++ str "]") +let pr_ssrpattern_squarep _ _ _ = pr_pattern_squarep +let pr_pattern_roundp = pr_option (fun r -> str "(" ++ pr_pattern r ++ str ")") +let pr_ssrpattern_roundp _ _ _ = pr_pattern_roundp + +let wit_rpatternty, globwit_rpatternty, rawwit_rpatternty = + add_genarg "rpatternty" pr_pattern + +ARGUMENT EXTEND rpattern TYPED AS rpatternty PRINTED BY pr_rpattern + | [ lconstr(c) ] -> [ T (mk_lterm c) ] + | [ "in" lconstr(c) ] -> [ In_T (mk_lterm c) ] + | [ lconstr(x) "in" lconstr(c) ] -> + [ X_In_T (mk_lterm x, mk_lterm c) ] + | [ "in" lconstr(x) "in" lconstr(c) ] -> + [ In_X_In_T (mk_lterm x, mk_lterm c) ] + | [ lconstr(e) "in" lconstr(x) "in" lconstr(c) ] -> + [ E_In_X_In_T (mk_lterm e, mk_lterm x, mk_lterm c) ] + | [ lconstr(e) "as" lconstr(x) "in" lconstr(c) ] -> + [ E_As_X_In_T (mk_lterm e, mk_lterm x, mk_lterm c) ] +END + +type cpattern = char * glob_constr_and_expr +let tag_of_cpattern = fst +let loc_of_cpattern = loc_ofCG +let cpattern_of_term t = t +type occ = (bool * int list) option + +type rpattern = (cpattern, cpattern) ssrpattern +let pr_rpattern = pr_pattern + +type pattern = Evd.evar_map * (Term.constr, Term.constr) ssrpattern + + +let id_of_cpattern = function + | _,(_,Some (CRef (Ident (_, x)))) -> Some x + | _,(_,Some (CAppExpl (_, (_, Ident (_, x)), []))) -> Some x + | _,(GRef (_, VarRef x) ,None) -> Some x + | _ -> None +let id_of_Cterm t = match id_of_cpattern t with + | Some x -> x + | None -> loc_error (loc_of_cpattern t) "Only identifiers are allowed here" + +let interp_wit globwit wit ist gl x = + let globarg = in_gen globwit x in + let sigma, arg = interp_genarg ist gl globarg in + sigma, out_gen wit arg +let interp_constr = interp_wit globwit_constr wit_constr +let interp_open_constr ist gl gc = + interp_wit globwit_open_constr wit_open_constr ist gl ((), gc) +let pf_intern_term ist gl (_, c) = glob_constr ist (project gl) (pf_env gl) c +let interp_term ist gl (_, c) = snd (interp_open_constr ist gl c) +let glob_ssrterm gs = function + | k, (_, Some c) -> k, Tacinterp.intern_constr gs c + | ct -> ct +let subst_ssrterm s (k, c) = k, Tacinterp.subst_glob_constr_and_expr s c +let pr_ssrterm _ _ _ = pr_term +let input_ssrtermkind strm = match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "(" -> '(' + | Tok.KEYWORD "@" -> '@' + | _ -> ' ' +let ssrtermkind = Gram.Entry.of_parser "ssrtermkind" input_ssrtermkind + +(* This piece of code asserts the following notations are reserved *) +(* Reserved Notation "( a 'in' b )" (at level 0). *) +(* Reserved Notation "( a 'as' b )" (at level 0). *) +(* Reserved Notation "( a 'in' b 'in' c )" (at level 0). *) +(* Reserved Notation "( a 'as' b 'in' c )" (at level 0). *) +let glob_cpattern gs p = + pp(lazy(str"globbing pattern: " ++ pr_term p)); + let glob x = snd (glob_ssrterm gs (mk_lterm x)) in + let encode k s l = + let name = Name (id_of_string ("_ssrpat_" ^ s)) in + k, (mkRCast mkRHole (mkRLambda name mkRHole (mkRApp mkRHole l)), None) in + let bind_in t1 t2 = + let d = dummy_loc in let n = Name (destCVar t1) in + fst (glob (mkCCast d (mkCHole d) (mkCLambda d n (mkCHole d) t2))) in + let check_var t2 = if not (isCVar t2) then + loc_error (constr_loc t2) "Only identifiers are allowed here" in + match p with + | _, (_, None) as x -> x + | k, (v, Some t) as orig -> + if k = 'x' then glob_ssrterm gs ('(', (v, Some t)) else + match t with + | CNotation(_, "( _ in _ )", ([t1; t2], [], [])) -> + (try match glob t1, glob t2 with + | (r1, None), (r2, None) -> encode k "In" [r1;r2] + | (r1, Some _), (r2, Some _) when isCVar t1 -> + encode k "In" [r1; r2; bind_in t1 t2] + | (r1, Some _), (r2, Some _) -> encode k "In" [r1; r2] + | _ -> anomaly "where are we?" + with _ when isCVar t1 -> encode k "In" [bind_in t1 t2]) + | CNotation(_, "( _ in _ in _ )", ([t1; t2; t3], [], [])) -> + check_var t2; encode k "In" [fst (glob t1); bind_in t2 t3] + | CNotation(_, "( _ as _ )", ([t1; t2], [], [])) -> + encode k "As" [fst (glob t1); fst (glob t2)] + | CNotation(_, "( _ as _ in _ )", ([t1; t2; t3], [], [])) -> + check_var t2; encode k "As" [fst (glob t1); bind_in t2 t3] + | _ -> glob_ssrterm gs orig +;; + +let interp_ssrterm _ gl t = Tacmach.project gl, t + +ARGUMENT EXTEND cpattern + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_cpattern SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "Qed" constr(c) ] -> [ mk_lterm c ] +END + +GEXTEND Gram + GLOBAL: cpattern; + cpattern: [[ k = ssrtermkind; c = constr -> + let pattern = mk_term k c in + if loc_ofCG pattern <> loc && k = '(' then mk_term 'x' c else pattern ]]; +END + +ARGUMENT EXTEND lcpattern + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_cpattern SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "Qed" lconstr(c) ] -> [ mk_lterm c ] +END + +GEXTEND Gram + GLOBAL: lcpattern; + lcpattern: [[ k = ssrtermkind; c = lconstr -> + let pattern = mk_term k c in + if loc_ofCG pattern <> loc && k = '(' then mk_term 'x' c else pattern ]]; +END + +let interp_pattern ist gl red redty = + pp(lazy(str"interpreting: " ++ pr_pattern red)); + let xInT x y = X_In_T(x,y) and inXInT x y = In_X_In_T(x,y) in + let inT x = In_T x and eInXInT e x t = E_In_X_In_T(e,x,t) in + let eAsXInT e x t = E_As_X_In_T(e,x,t) in + let mkG ?(k=' ') x = k,(x,None) in + let decode t f g = + try match (pf_intern_term ist gl t) with + | GCast(_,GHole _,CastConv(_,GLambda(_,Name x,_,_,c))) -> f x (' ',(c,None)) + | it -> g t with _ -> g t in + let decodeG t f g = decode (mkG t) f g in + let bad_enc id _ = anomaly ("bad encoding for pattern " ^ id) in + let cleanup_XinE h x rp sigma = + let h_k = match kind_of_term h with Evar (k,_) -> k | _ -> assert false in + let to_clean, update = (* handle rename if x is already used *) + let ctx = pf_hyps gl in + let len = Sign.named_context_length ctx in + let name = ref None in + try ignore(Sign.lookup_named x ctx); (name, fun k -> + if !name = None then + let nctx = Evd.evar_context (Evd.find sigma k) in + let nlen = Sign.named_context_length nctx in + if nlen > len then begin + name := Some (pi1 (List.nth nctx (nlen - len - 1))) + end) + with Not_found -> ref (Some x), fun _ -> () in + let sigma0 = project gl in + let new_evars = + let rec aux acc t = match kind_of_term t with + | Evar (k,_) -> + if k = h_k || List.mem k acc || Evd.mem sigma0 k then acc else + (update k; k::acc) + | _ -> fold_constr aux acc t in + aux [] (Evarutil.nf_evar sigma rp) in + let sigma = + List.fold_left (fun sigma e -> + if Evd.is_defined sigma e then sigma else (* clear may be recursiv *) + let name = Option.get !to_clean in + let g = Goal.build e in + pp(lazy(pr_id name)); + try snd(Logic.prim_refiner (Proof_type.Thin [name]) sigma g) + with Evarutil.ClearDependencyError _ -> sigma) + sigma new_evars in + sigma in + let red = match red with + | T(k,(GCast (_,GHole _,(CastConv(_,GLambda (_,Name id,_,_,t)))),None)) + when let id = string_of_id id in let len = String.length id in + (len > 8 && String.sub id 0 8 = "_ssrpat_") -> + let id = string_of_id id in let len = String.length id in + (match String.sub id 8 (len - 8), t with + | "In", GApp(_, _, [t]) -> decodeG t xInT (fun x -> T x) + | "In", GApp(_, _, [e; t]) -> decodeG t (eInXInT (mkG e)) (bad_enc id) + | "In", GApp(_, _, [e; t; e_in_t]) -> + decodeG t (eInXInT (mkG e)) + (fun _ -> decodeG e_in_t xInT (fun _ -> assert false)) + | "As", GApp(_, _, [e; t]) -> decodeG t (eAsXInT (mkG e)) (bad_enc id) + | _ -> bad_enc id ()) + | T t -> decode t xInT (fun x -> T x) + | In_T t -> decode t inXInT inT + | X_In_T (e,t) -> decode t (eInXInT e) (fun x -> xInT (id_of_Cterm e) x) + | In_X_In_T (e,t) -> inXInT (id_of_Cterm e) t + | E_In_X_In_T (e,x,rp) -> eInXInT e (id_of_Cterm x) rp + | E_As_X_In_T (e,x,rp) -> eAsXInT e (id_of_Cterm x) rp in + pp(lazy(str"decoded as: " ++ pr_pattern_w_ids red)); + let red = match redty with None -> red | Some ty -> let ty = ' ', ty in + match red with + | T t -> T (combineCG t ty (mkCCast (loc_ofCG t)) mkRCast) + | X_In_T (x,t) -> + let ty = pf_intern_term ist gl ty in + E_As_X_In_T (mkG (mkRCast mkRHole ty), x, t) + | E_In_X_In_T (e,x,t) -> + let ty = mkG (pf_intern_term ist gl ty) in + E_In_X_In_T (combineCG e ty (mkCCast (loc_ofCG t)) mkRCast, x, t) + | E_As_X_In_T (e,x,t) -> + let ty = mkG (pf_intern_term ist gl ty) in + E_As_X_In_T (combineCG e ty (mkCCast (loc_ofCG t)) mkRCast, x, t) + | red -> red in + pp(lazy(str"typed as: " ++ pr_pattern_w_ids red)); + let mkXLetIn loc x (a,(g,c)) = match c with + | Some b -> a,(g,Some (mkCLetIn loc x (CHole(loc,None)) b)) + | None -> a,(GLetIn (loc,x,(GHole (loc, BinderType x)), g), None) in + match red with + | T t -> let sigma, t = interp_term ist gl t in sigma, T t + | In_T t -> let sigma, t = interp_term ist gl t in sigma, In_T t + | X_In_T (x, rp) | In_X_In_T (x, rp) -> + let mk x p = match red with X_In_T _ -> X_In_T(x,p) | _ -> In_X_In_T(x,p) in + let rp = mkXLetIn dummy_loc (Name x) rp in + let sigma, rp = interp_term ist gl rp in + let _, h, _, rp = destLetIn rp in + let sigma = cleanup_XinE h x rp sigma in + let rp = subst1 h (Evarutil.nf_evar sigma rp) in + sigma, mk h rp + | E_In_X_In_T(e, x, rp) | E_As_X_In_T (e, x, rp) -> + let mk e x p = + match red with E_In_X_In_T _ ->E_In_X_In_T(e,x,p)|_->E_As_X_In_T(e,x,p) in + let rp = mkXLetIn dummy_loc (Name x) rp in + let sigma, rp = interp_term ist gl rp in + let _, h, _, rp = destLetIn rp in + let sigma = cleanup_XinE h x rp sigma in + let rp = subst1 h (Evarutil.nf_evar sigma rp) in + let sigma, e = interp_term ist (re_sig (sig_it gl) sigma) e in + sigma, mk e h rp +;; +let interp_cpattern ist gl red redty = interp_pattern ist gl (T red) redty;; +let interp_rpattern ist gl red = interp_pattern ist gl red None;; + +(* The full occurrence set *) +let noindex = Some(false,[]) + +(* calls do_subst on every sub-term identified by (pattern,occ) *) +let eval_pattern ?raise_NoMatch env0 sigma0 concl0 pattern occ do_subst = + let fs sigma x = Reductionops.nf_evar sigma x in + let pop_evar sigma e p = + let { Evd.evar_body = e_body } as e_def = Evd.find sigma e in + let e_body = match e_body with Evar_defined c -> c + | _ -> errorstrm (str "Matching the pattern " ++ pr_constr p ++ + str " did not instantiate ?" ++ int e ++ spc () ++ + str "Does the variable bound by the \"in\" construct occur "++ + str "in the pattern?") in + let sigma = + Evd.add (Evd.remove sigma e) e {e_def with Evd.evar_body = Evar_empty} in + sigma, e_body in + let ex_value hole = + match kind_of_term hole with Evar (e,_) -> e | _ -> assert false in + let mk_upat_for ?hack env sigma0 (sigma, t) ?(p=t) ok = + let sigma,pat= mk_tpattern ?hack env sigma0 (sigma,p) ok L2R (fs sigma t) in + sigma, [pat] in + match pattern with + | None -> do_subst env0 concl0 1 + | Some (sigma, (T rp | In_T rp)) -> + let rp = fs sigma rp in + let ise = create_evar_defs sigma in + let occ = match pattern with Some (_, T _) -> occ | _ -> noindex in + let rp = mk_upat_for env0 sigma0 (ise, rp) all_ok in + let find_T, end_T = mk_tpattern_matcher ?raise_NoMatch sigma0 occ rp in + let concl = find_T env0 concl0 1 do_subst in + let _ = end_T () in + concl + | Some (sigma, (X_In_T (hole, p) | In_X_In_T (hole, p))) -> + let p = fs sigma p in + let occ = match pattern with Some (_, X_In_T _) -> occ | _ -> noindex in + let ex = ex_value hole in + let rp = mk_upat_for ~hack:true env0 sigma0 (sigma, p) all_ok in + let find_T, end_T = mk_tpattern_matcher sigma0 noindex rp in + (* we start from sigma, so hole is considered a rigid head *) + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher ?raise_NoMatch sigma occ holep in + let concl = find_T env0 concl0 1 (fun env c h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h + (fun env _ -> do_subst env e_body))) in + let _ = end_X () in let _ = end_T () in + concl + | Some (sigma, E_In_X_In_T (e, hole, p)) -> + let p, e = fs sigma p, fs sigma e in + let ex = ex_value hole in + let rp = mk_upat_for ~hack:true env0 sigma0 (sigma, p) all_ok in + let find_T, end_T = mk_tpattern_matcher sigma0 noindex rp in + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher sigma noindex holep in + let re = mk_upat_for env0 sigma0 (sigma, e) all_ok in + let find_E, end_E = mk_tpattern_matcher ?raise_NoMatch sigma0 occ re in + let concl = find_T env0 concl0 1 (fun env c h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h (fun env c h -> + find_E env e_body h do_subst))) in + let _ = end_E () in let _ = end_X () in let _ = end_T () in + concl + | Some (sigma, E_As_X_In_T (e, hole, p)) -> + let p, e = fs sigma p, fs sigma e in + let ex = ex_value hole in + let rp = + let e_sigma = unify_HO env0 sigma hole e in + e_sigma, fs e_sigma p in + let rp = mk_upat_for ~hack:true env0 sigma0 rp all_ok in + let find_TE, end_TE = mk_tpattern_matcher sigma0 noindex rp in + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher sigma occ holep in + let concl = find_TE env0 concl0 1 (fun env c h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h (fun env c h -> + let e_sigma = unify_HO env sigma e_body e in + let e_body = fs e_sigma e in + do_subst env e_body h))) in + let _ = end_X () in let _ = end_TE () in + concl +;; + +let redex_of_pattern ?(resolve_typeclasses=false) env (sigma, p) = + let e = match p with + | In_T _ | In_X_In_T _ -> anomaly "pattern without redex" + | T e | X_In_T (e, _) | E_As_X_In_T (e, _, _) | E_In_X_In_T (e, _, _) -> e in + let sigma = + if not resolve_typeclasses then sigma + else Typeclasses.resolve_typeclasses ~fail:false env sigma in + Reductionops.nf_evar sigma e + +let fill_occ_pattern ?raise_NoMatch env sigma cl pat occ h = + let find_R, conclude = let r = ref None in + (fun env c h' -> do_once r (fun () -> c); mkRel (h'+h-1)), + (fun _ -> if !r = None then redex_of_pattern env pat else assert_done r) in + let cl = eval_pattern ?raise_NoMatch env sigma cl (Some pat) occ find_R in + let e = conclude cl in + e, cl +;; + +(* clenup interface for external use *) +let mk_tpattern ?p_origin env sigma0 sigma_t f dir c = + mk_tpattern ?p_origin env sigma0 sigma_t f dir c +;; + +let pf_fill_occ env concl occ sigma0 p (sigma, t) ok h = + let ise = create_evar_defs sigma in + let ise, u = mk_tpattern env sigma0 (ise,t) ok L2R p in + let find_U, end_U = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[u]) in + let concl = find_U env concl h (fun _ _ -> mkRel) in + let rdx, _, (sigma, p) = end_U () in + sigma, p, concl, rdx + +let fill_occ_term env cl occ sigma0 (sigma, t) = + try + let sigma',t',cl,_ = pf_fill_occ env cl occ sigma0 t (sigma, t) all_ok 1 in + if sigma' != sigma0 then error "matching impacts evars" else cl, (sigma',t') with NoMatch -> try + let sigma', t' = + unif_end env sigma0 (create_evar_defs sigma) t (fun _ -> true) in + if sigma' != sigma0 then raise NoMatch else cl, (sigma', t') + with _ -> + errorstrm (str "partial term " ++ pr_constr_pat t + ++ str " does not match any subterm of the goal") + +let pf_fill_occ_term gl occ t = + let sigma0 = project gl and env = pf_env gl and concl = pf_concl gl in + let cl,(_,t) = fill_occ_term env concl occ sigma0 t in + cl, t + +let cpattern_of_id id = ' ', (GRef (dummy_loc, VarRef id), None) + +let is_wildcard = function + | _,(_,Some (CHole _)|GHole _,None) -> true + | _ -> false + +type ltacctx = int + +let pr_ltacctx _ _ _ _ = mt () + +let ltacctxs = ref (1, []) + +let interp_ltacctx ist gl n0 = + Tacmach.project gl, + if n0 = 0 then 0 else + let n, s = !ltacctxs in + let n' = if n >= max_int then 1 else n + 1 in + ltacctxs := (n', (n, ist) :: s); n + +let noltacctx = 0 +let rawltacctx = 1 + +ARGUMENT EXTEND ltacctx TYPED AS int PRINTED BY pr_ltacctx + INTERPRETED BY interp_ltacctx +| [ ] -> [ rawltacctx ] +END + +let get_ltacctx i = match !ltacctxs with +| _ when i = noltacctx -> anomaly "Missing Ltac context" +| n, (i', ist) :: s when i' = i -> ltacctxs := (n, s); ist +| _ -> anomaly "Bad scope in SSR tactical" + +(* "ssrpattern" *) +let pr_ssrpatternarg _ _ _ cpat = pr_rpattern cpat + +ARGUMENT EXTEND ssrpatternarg + TYPED AS rpattern + PRINTED BY pr_ssrpatternarg +| [ "[" rpattern(pat) "]" ] -> [ pat ] +END + +let ssrpatterntac ctx arg gl = + let ist = get_ltacctx ctx in + let pat = interp_rpattern ist gl arg in + let sigma0 = project gl in + let concl0 = pf_concl gl in + let t, concl_x = fill_occ_pattern (Global.env()) sigma0 concl0 pat noindex 1 in + let tty = pf_type_of gl t in + let concl = mkLetIn (Name (id_of_string "toto"), t, tty, concl_x) in + convert_concl concl DEFAULTcast gl + +TACTIC EXTEND ssrat +| [ "ssrpattern" ssrpatternarg(arg) ltacctx(ctx) ] -> [ ssrpatterntac ctx arg ] +END + + +(* We wipe out all the keywords generated by the grammar rules we defined. *) +(* The user is supposed to Require Import ssreflect or Require ssreflect *) +(* and Import ssreflect.SsrSyntax to obtain these keywords and as a *) +(* consequence the extended ssreflect grammar. *) +let () = Lexer.unfreeze frozen_lexer ;; + +(* vim: set filetype=ocaml foldmethod=marker: *) diff --git a/mathcomp/ssreflect/plugin/v8.4/ssrmatching.mli b/mathcomp/ssreflect/plugin/v8.4/ssrmatching.mli new file mode 100644 index 0000000..4c6bfac --- /dev/null +++ b/mathcomp/ssreflect/plugin/v8.4/ssrmatching.mli @@ -0,0 +1,256 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +open Genarg +open Environ +open Evd +open Proof_type +open Term + +(** ******** Small Scale Reflection pattern matching facilities ************* *) + +(** Pattern parsing *) + +(** The type of context patterns, the patterns of the [set] tactic and + [:] tactical. These are patterns that identify a precise subterm. *) +type cpattern +val pr_cpattern : cpattern -> Pp.std_ppcmds + +(** CS cpattern: (f _), (X in t), (t in X in t), (t as X in t) *) +val cpattern : cpattern Pcoq.Gram.entry +val globwit_cpattern : (cpattern, glevel) abstract_argument_type +val rawwit_cpattern : (cpattern, rlevel) abstract_argument_type +val wit_cpattern : (cpattern, tlevel) abstract_argument_type + +(** OS cpattern: f _, (X in t), (t in X in t), (t as X in t) *) +val lcpattern : cpattern Pcoq.Gram.entry +val globwit_lcpattern : (cpattern, glevel) abstract_argument_type +val rawwit_lcpattern : (cpattern, rlevel) abstract_argument_type +val wit_lcpattern : (cpattern, tlevel) abstract_argument_type + +(** The type of rewrite patterns, the patterns of the [rewrite] tactic. + These patterns also include patterns that identify all the subterms + of a context (i.e. "in" prefix) *) +type rpattern +val pr_rpattern : rpattern -> Pp.std_ppcmds + +(** OS rpattern: f _, in t, X in t, in X in t, t in X in t, t as X in t *) +val rpattern : rpattern Pcoq.Gram.entry +val globwit_rpattern : (rpattern, glevel) abstract_argument_type +val rawwit_rpattern : (rpattern, rlevel) abstract_argument_type +val wit_rpattern : (rpattern, tlevel) abstract_argument_type + +(** Pattern interpretation and matching *) + +exception NoMatch +exception NoProgress + +(** AST for [rpattern] (and consequently [cpattern]) *) +type ('ident, 'term) ssrpattern = + | T of 'term + | In_T of 'term + | X_In_T of 'ident * 'term + | In_X_In_T of 'ident * 'term + | E_In_X_In_T of 'term * 'ident * 'term + | E_As_X_In_T of 'term * 'ident * 'term + +type pattern = evar_map * (constr, constr) ssrpattern +val pp_pattern : pattern -> Pp.std_ppcmds + +(** Extracts the redex and applies to it the substitution part of the pattern. + @raise Anomaly if called on [In_T] or [In_X_In_T] *) +val redex_of_pattern : ?resolve_typeclasses:bool -> env -> pattern -> constr + +(** [interp_rpattern ise gl rpat] "internalizes" and "interprets" [rpat] + in the current [Ltac] interpretation signature [ise] and tactic input [gl]*) +val interp_rpattern : + Tacinterp.interp_sign -> goal Tacmach.sigma -> + rpattern -> + pattern + +(** [interp_cpattern ise gl cpat ty] "internalizes" and "interprets" [cpat] + in the current [Ltac] interpretation signature [ise] and tactic input [gl]. + [ty] is an optional type for the redex of [cpat] *) +val interp_cpattern : + Tacinterp.interp_sign -> goal Tacmach.sigma -> + cpattern -> glob_constr_and_expr option -> + pattern + +(** The set of occurrences to be matched. The boolean is set to true + * to signal the complement of this set (i.e. {-1 3}) *) +type occ = (bool * int list) option + +(** Substitution function. The [int] argument is the number of binders + traversed so far *) +type subst = env -> constr -> int -> constr + +(** [eval_pattern b env sigma t pat occ subst] maps [t] calling [subst] on every + [occ] occurrence of [pat]. The [int] argument is the number of + binders traversed. If [pat] is [None] then then subst is called on [t]. + [t] must live in [env] and [sigma], [pat] must have been interpreted in + (an extension of) [sigma]. + @raise NoMatch if [pat] has no occurrence and [b] is [true] (default [false]) + @return [t] where all [occ] occurrences of [pat] have been mapped using + [subst] *) +val eval_pattern : + ?raise_NoMatch:bool -> + env -> evar_map -> constr -> + pattern option -> occ -> subst -> + constr + +(** [fill_occ_pattern b env sigma t pat occ h] is a simplified version of + [eval_pattern]. + It replaces all [occ] occurrences of [pat] in [t] with Rel [h]. + [t] must live in [env] and [sigma], [pat] must have been interpreted in + (an extension of) [sigma]. + @raise NoMatch if [pat] has no occurrence and [b] is [true] (default [false]) + @return the instance of the redex of [pat] that was matched and [t] + transformed as described above. *) +val fill_occ_pattern : + ?raise_NoMatch:bool -> + env -> evar_map -> constr -> + pattern -> occ -> int -> + constr * constr + +(** Grammar entry to grab [Ltac] context, needed for the functions above. + It must appear after the last tactic argument and only once. + For example {[ + TACTIC EXTEND ssrclear + | [ "clear" natural(n) ltacctx(ctx) ] -> [poptac ~ist:(get_ltacctx ctx) n] + END ]} *) +type ltacctx + +val get_ltacctx : ltacctx -> Tacinterp.interp_sign + +val ltacctx : ltacctx Pcoq.Gram.entry +val globwit_ltacctx : (ltacctx, glevel) abstract_argument_type +val rawwit_ltacctx : (ltacctx, rlevel) abstract_argument_type +val wit_ltacctx : (ltacctx, tlevel) abstract_argument_type + +(** *************************** Low level APIs ****************************** *) + +(* The primitive matching facility. It matches of a term with holes, like + the T pattern above, and calls a continuation on its occurrences. *) + +type ssrdir = L2R | R2L +val pr_dir_side : ssrdir -> Pp.std_ppcmds + +(** a pattern for a term with wildcards *) +type tpattern + +(** [mk_tpattern env sigma0 sigma_t ok p_origin dir p] compiles a term [t] + living in [env] [sigma] (an extension of [sigma0]) intro a [tpattern]. + The [tpattern] can hold a (proof) term [p] and a diction [dir]. The [ok] + callback is used to filter occurrences. + @return the compiled [tpattern] and its [evar_map] + @raise UserEerror is the pattern is a wildcard *) +val mk_tpattern : + ?p_origin:ssrdir * constr -> + env -> evar_map -> + evar_map * constr -> + (constr -> evar_map -> bool) -> + ssrdir -> constr -> + evar_map * tpattern + +(** [findP env t i k] is a stateful function that finds the next occurrence + of a tpattern and calls the callback [k] to map the subterm matched. + The [int] argument passed to [k] is the number of binders traversed so far + plus the initial value [i]. + @return [t] where the subterms identified by the selected occurrences of + the patter have been mapped using [k] + @raise NoMatch if the raise_NoMatch flag given to [mk_tpattern_matcher] is + [true] and if the pattern did not match + @raise UserEerror if the raise_NoMatch flag given to [mk_tpattern_matcher] is + [false] and if the pattern did not match *) +type find_P = + env -> constr -> int -> k:subst -> constr + +(** [conclude ()] asserts that all mentioned ocurrences have been visited. + @return the instance of the pattern, the evarmap after the pattern + instantiation, the proof term and the ssrdit stored in the tpattern + @raise UserEerror if too many occurrences were specified *) +type conclude = unit -> constr * ssrdir * (evar_map * constr) + +(** [mk_tpattern_matcher b o sigma0 occ sigma_tplist] creates a pair + a function [find_P] and [conclude] with the behaviour explained above. + The flag [b] (default [false]) changes the error reporting behaviour + of [find_P] if none of the [tpattern] matches. The argument [o] can + be passed to tune the [UserError] eventually raised (useful if the + pattern is coming from the LHS/RHS of an equation) *) +val mk_tpattern_matcher : + ?raise_NoMatch:bool -> + ?upats_origin:ssrdir * constr -> + evar_map -> occ -> evar_map * tpattern list -> + find_P * conclude + +(** Example of [mk_tpattern_matcher] to implement + [rewrite \{occ\}\[in t\]rules]. + It first matches "in t" (called [pat]), then in all matched subterms + it matches the LHS of the rules using [find_R]. + [concl0] is the initial goal, [concl] will be the goal where some terms + are replaced by a De Bruijn index. The [rw_progress] extra check + selects only occurrences that are not rewritten to themselves (e.g. + an occurrence "x + x" rewritten with the commutativity law of addition + is skipped) {[ + let find_R, conclude = match pat with + | Some (_, In_T _) -> + let aux (sigma, pats) (d, r, lhs, rhs) = + let sigma, pat = + mk_tpattern env0 sigma0 (sigma, r) (rw_progress rhs) d lhs in + sigma, pats @ [pat] in + let rpats = List.fold_left aux (r_sigma, []) rules in + let find_R, end_R = mk_tpattern_matcher sigma0 occ rpats in + find_R ~k:(fun _ _ h -> mkRel h), + fun cl -> let rdx, d, r = end_R () in (d,r),rdx + | _ -> ... in + let concl = eval_pattern env0 sigma0 concl0 pat occ find_R in + let (d, r), rdx = conclude concl in ]} *) + +(* convenience shortcut: [pf_fill_occ_term gl occ (sigma,t)] returns + * the conclusion of [gl] where [occ] occurrences of [t] have been replaced + * by [Rel 1] and the instance of [t] *) +val pf_fill_occ_term : + goal Tacmach.sigma -> occ -> evar_map * constr -> constr * constr + +(* It may be handy to inject a simple term into the first form of cpattern *) +val cpattern_of_term : char * glob_constr_and_expr -> cpattern + +(** Helpers to make stateful closures. Example: a [find_P] function may be + called many times, but the pattern instantiation phase is performed only the + first time. The corresponding [conclude] has to return the instantiated + pattern redex. Since it is up to [find_P] to raise [NoMatch] if the pattern + has no instance, [conclude] considers it an anomaly if the pattern did + not match *) + +(** [do_once r f] calls [f] and updates the ref only once *) +val do_once : 'a option ref -> (unit -> 'a) -> unit +(** [assert_done r] return the content of r. @raise Anomaly is r is [None] *) +val assert_done : 'a option ref -> 'a + +(** Very low level APIs. + these are calls to evarconv's [the_conv_x] followed by + [consider_remaining_unif_problems] and [resolve_typeclasses]. + In case of failure they raise [NoMatch] *) + +val unify_HO : env -> evar_map -> constr -> constr -> evar_map +val pf_unify_HO : goal Tacmach.sigma -> constr -> constr -> goal Tacmach.sigma + +(** Some more low level functions needed to implement the full SSR language + on top of the former APIs *) +val tag_of_cpattern : cpattern -> char +val loc_of_cpattern : cpattern -> Util.loc +val id_of_cpattern : cpattern -> Names.variable option +val is_wildcard : cpattern -> bool +val cpattern_of_id : Names.variable -> cpattern +val rawltacctx : ltacctx +val cpattern_of_id : Names.variable -> cpattern +val rawltacctx : ltacctx +val pr_constr_pat : constr -> Pp.std_ppcmds + +(* One can also "Set SsrMatchingDebug" from a .v *) +val debug : bool -> unit + +(* One should delimit a snippet with "Set SsrMatchingProfiling" and + * "Unset SsrMatchingProfiling" to get timings *) +val profile : bool -> unit + +(* eof *) diff --git a/mathcomp/ssreflect/plugin/v8.5/ssreflect.ml4 b/mathcomp/ssreflect/plugin/v8.5/ssreflect.ml4 new file mode 100644 index 0000000..f598c21 --- /dev/null +++ b/mathcomp/ssreflect/plugin/v8.5/ssreflect.ml4 @@ -0,0 +1,6164 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +(* This line is read by the Makefile's dist target: do not remove. *) +DECLARE PLUGIN "ssreflect" +let ssrversion = "1.5";; +let ssrAstVersion = 1;; +let () = Mltop.add_known_plugin (fun () -> + if Flags.is_verbose () && not !Flags.batch_mode then begin + Printf.printf "\nSmall Scale Reflection version %s loaded.\n" ssrversion; + Printf.printf "Copyright 2005-2014 Microsoft Corporation and INRIA.\n"; + Printf.printf "Distributed under the terms of the CeCILL-B license.\n\n" + end; + (* Disable any semantics associated with bullets *) + Goptions.set_string_option_value_gen + (Some false) ["Bullet";"Behavior"] "None") + "ssreflect" +;; + +(* Defining grammar rules with "xx" in it automatically declares keywords too, + * we thus save the lexer to restore it at the end of the file *) +let frozen_lexer = Lexer.freeze () ;; + +(*i camlp4use: "pa_extend.cmo" i*) +(*i camlp4deps: "grammar/grammar.cma" i*) + +open Names +open Pp +open Pcoq +open Genarg +open Stdarg +open Constrarg +open Term +open Vars +open Context +open Topconstr +open Libnames +open Tactics +open Tacticals +open Termops +open Namegen +open Recordops +open Tacmach +open Coqlib +open Glob_term +open Util +open Evd +open Extend +open Goptions +open Tacexpr +open Tacinterp +open Pretyping +open Constr +open Tactic +open Extraargs +open Ppconstr +open Printer + +open Globnames +open Misctypes +open Decl_kinds +open Evar_kinds +open Constrexpr +open Constrexpr_ops +open Notation_term +open Notation_ops +open Locus +open Locusops + +open Ssrmatching + + +(* Tentative patch from util.ml *) + +let array_fold_right_from n f v a = + let rec fold n = + if n >= Array.length v then a else f v.(n) (fold (succ n)) + in + fold n + +let array_app_tl v l = + if Array.length v = 0 then invalid_arg "array_app_tl"; + array_fold_right_from 1 (fun e l -> e::l) v l + +let array_list_of_tl v = + if Array.length v = 0 then invalid_arg "array_list_of_tl"; + array_fold_right_from 1 (fun e l -> e::l) v [] + +(* end patch *) + +module Intset = Evar.Set + +type loc = Loc.t +let dummy_loc = Loc.ghost +let errorstrm = Errors.errorlabstrm "ssreflect" +let loc_error loc msg = Errors.user_err_loc (loc, msg, str msg) +let anomaly s = Errors.anomaly (str s) + +(** look up a name in the ssreflect internals module *) +let ssrdirpath = make_dirpath [id_of_string "ssreflect"] +let ssrqid name = make_qualid ssrdirpath (id_of_string name) +let ssrtopqid name = make_short_qualid (id_of_string name) +let locate_reference qid = + Smartlocate.global_of_extended_global (Nametab.locate_extended qid) +let mkSsrRef name = + try locate_reference (ssrqid name) with Not_found -> + try locate_reference (ssrtopqid name) with Not_found -> + Errors.error "Small scale reflection library not loaded" +let mkSsrRRef name = GRef (dummy_loc, mkSsrRef name,None), None +let mkSsrConst name env sigma = + Evd.fresh_global env sigma (mkSsrRef name) +let pf_mkSsrConst name gl = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, t = mkSsrConst name env sigma in + t, re_sig it sigma +let pf_fresh_global name gl = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma,t = Evd.fresh_global env sigma name in + t, re_sig it sigma + +(** Ssreflect load check. *) + +(* To allow ssrcoq to be fully compatible with the "plain" Coq, we only *) +(* turn on its incompatible features (the new rewrite syntax, and the *) +(* reserved identifiers) when the theory library (ssreflect.v) has *) +(* has actually been required, or is being defined. Because this check *) +(* needs to be done often (for each identifier lookup), we implement *) +(* some caching, repeating the test only when the environment changes. *) +(* We check for protect_term because it is the first constant loaded; *) +(* ssr_have would ultimately be a better choice. *) +let ssr_loaded = Summary.ref ~name:"SSR:loaded" false +let is_ssr_loaded () = + !ssr_loaded || + (if Lexer.is_keyword "SsrSyntax_is_Imported" then ssr_loaded:=true; + !ssr_loaded) + +(* 0 cost pp function. Active only if env variable SSRDEBUG is set *) +(* or if SsrDebug is Set *) +let pp_ref = ref (fun _ -> ()) +let ssr_pp s = pperrnl (str"SSR: "++Lazy.force s) +let _ = try ignore(Sys.getenv "SSRDEBUG"); pp_ref := ssr_pp with Not_found -> () +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect debugging"; + Goptions.optkey = ["SsrDebug"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !pp_ref == ssr_pp); + Goptions.optwrite = (fun b -> + Ssrmatching.debug b; + if b then pp_ref := ssr_pp else pp_ref := fun _ -> ()) } +let pp s = !pp_ref s + +(** Utils {{{ *****************************************************************) +let env_size env = List.length (Environ.named_context env) +let safeDestApp c = + match kind_of_term c with App (f, a) -> f, a | _ -> c, [| |] +let get_index = function ArgArg i -> i | _ -> + anomaly "Uninterpreted index" +(* Toplevel constr must be globalized twice ! *) +let glob_constr ist genv = function + | _, Some ce -> + let vars = Id.Map.fold (fun x _ accu -> Id.Set.add x accu) ist.lfun Id.Set.empty in + let ltacvars = { + Constrintern.empty_ltac_sign with Constrintern.ltac_vars = vars } in + Constrintern.intern_gen WithoutTypeConstraint ~ltacvars genv ce + | rc, None -> rc + +(* Term printing utilities functions for deciding bracketing. *) +let pr_paren prx x = hov 1 (str "(" ++ prx x ++ str ")") +(* String lexing utilities *) +let skip_wschars s = + let rec loop i = match s.[i] with '\n'..' ' -> loop (i + 1) | _ -> i in loop +let skip_numchars s = + let rec loop i = match s.[i] with '0'..'9' -> loop (i + 1) | _ -> i in loop +(* We also guard characters that might interfere with the ssreflect *) +(* tactic syntax. *) +let guard_term ch1 s i = match s.[i] with + | '(' -> false + | '{' | '/' | '=' -> true + | _ -> ch1 = '(' +(* The call 'guard s i' should return true if the contents of s *) +(* starting at i need bracketing to avoid ambiguities. *) +let pr_guarded guard prc c = + msg_with Format.str_formatter (prc c); + let s = Format.flush_str_formatter () ^ "$" in + if guard s (skip_wschars s 0) then pr_paren prc c else prc c +(* More sensible names for constr printers *) +let prl_constr = pr_lconstr +let pr_constr = pr_constr +let prl_glob_constr c = pr_lglob_constr_env (Global.env ()) c +let pr_glob_constr c = pr_glob_constr_env (Global.env ()) c +let prl_constr_expr = pr_lconstr_expr +let pr_constr_expr = pr_constr_expr +let prl_glob_constr_and_expr = function + | _, Some c -> prl_constr_expr c + | c, None -> prl_glob_constr c +let pr_glob_constr_and_expr = function + | _, Some c -> pr_constr_expr c + | c, None -> pr_glob_constr c +let pr_term (k, c) = pr_guarded (guard_term k) pr_glob_constr_and_expr c +let prl_term (k, c) = pr_guarded (guard_term k) prl_glob_constr_and_expr c + +(** Adding a new uninterpreted generic argument type *) +let add_genarg tag pr = + let wit = Genarg.make0 None tag in + let glob ist x = (ist, x) in + let subst _ x = x in + let interp ist gl x = (gl.Evd.sigma, x) in + let gen_pr _ _ _ = pr in + let () = Genintern.register_intern0 wit glob in + let () = Genintern.register_subst0 wit subst in + let () = Geninterp.register_interp0 wit interp in + Pptactic.declare_extra_genarg_pprule wit gen_pr gen_pr gen_pr; + wit + +(** Constructors for cast type *) +let dC t = CastConv t + +(** Constructors for constr_expr *) +let mkCProp loc = CSort (loc, GProp) +let mkCType loc = CSort (loc, GType []) +let mkCVar loc id = CRef (Ident (loc, id),None) +let isCVar = function CRef (Ident _,_) -> true | _ -> false +let destCVar = function CRef (Ident (_, id),_) -> id | _ -> + anomaly "not a CRef" +let rec mkCHoles loc n = + if n <= 0 then [] else CHole (loc, None, IntroAnonymous, None) :: mkCHoles loc (n - 1) +let mkCHole loc = CHole (loc, None, IntroAnonymous, None) +let rec isCHoles = function CHole _ :: cl -> isCHoles cl | cl -> cl = [] +let mkCExplVar loc id n = + CAppExpl (loc, (None, Ident (loc, id), None), mkCHoles loc n) +let mkCLambda loc name ty t = + CLambdaN (loc, [[loc, name], Default Explicit, ty], t) +let mkCLetIn loc name bo t = + CLetIn (loc, (loc, name), bo, t) +let mkCArrow loc ty t = + CProdN (loc, [[dummy_loc,Anonymous], Default Explicit, ty], t) +let mkCCast loc t ty = CCast (loc,t, dC ty) +(** Constructors for rawconstr *) +let mkRHole = GHole (dummy_loc, InternalHole, IntroAnonymous, None) +let rec mkRHoles n = if n > 0 then mkRHole :: mkRHoles (n - 1) else [] +let rec isRHoles = function GHole _ :: cl -> isRHoles cl | cl -> cl = [] +let mkRApp f args = if args = [] then f else GApp (dummy_loc, f, args) +let mkRVar id = GRef (dummy_loc, VarRef id,None) +let mkRltacVar id = GVar (dummy_loc, id) +let mkRCast rc rt = GCast (dummy_loc, rc, dC rt) +let mkRType = GSort (dummy_loc, GType []) +let mkRProp = GSort (dummy_loc, GProp) +let mkRArrow rt1 rt2 = GProd (dummy_loc, Anonymous, Explicit, rt1, rt2) +let mkRConstruct c = GRef (dummy_loc, ConstructRef c,None) +let mkRInd mind = GRef (dummy_loc, IndRef mind,None) +let mkRLambda n s t = GLambda (dummy_loc, n, Explicit, s, t) + +(** Constructors for constr *) +let pf_e_type_of gl t = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, ty = Typing.e_type_of env sigma t in + re_sig it sigma, ty + +let mkAppRed f c = match kind_of_term f with +| Lambda (_, _, b) -> subst1 c b +| _ -> mkApp (f, [|c|]) + +let mkProt t c gl = + let prot, gl = pf_mkSsrConst "protect_term" gl in + mkApp (prot, [|t; c|]), gl + +let mkRefl t c gl = + let refl, gl = pf_fresh_global (build_coq_eq_data()).refl gl in + mkApp (refl, [|t; c|]), gl + + +(* Application to a sequence of n rels (for building eta-expansions). *) +(* The rel indices decrease down to imin (inclusive), unless n < 0, *) +(* in which case they're incresing (from imin). *) +let mkEtaApp c n imin = + if n = 0 then c else + let nargs, mkarg = + if n < 0 then -n, (fun i -> mkRel (imin + i)) else + let imax = imin + n - 1 in n, (fun i -> mkRel (imax - i)) in + mkApp (c, Array.init nargs mkarg) +(* Same, but optimizing head beta redexes *) +let rec whdEtaApp c n = + if n = 0 then c else match kind_of_term c with + | Lambda (_, _, c') -> whdEtaApp c' (n - 1) + | _ -> mkEtaApp (lift n c) n 1 +let mkType () = Universes.new_Type (Lib.cwd ()) + +(* ssrterm conbinators *) +let combineCG t1 t2 f g = match t1, t2 with + | (x, (t1, None)), (_, (t2, None)) -> x, (g t1 t2, None) + | (x, (_, Some t1)), (_, (_, Some t2)) -> x, (mkRHole, Some (f t1 t2)) + | _, (_, (_, None)) -> anomaly "have: mixed C-G constr" + | _ -> anomaly "have: mixed G-C constr" +let loc_ofCG = function + | (_, (s, None)) -> Glob_ops.loc_of_glob_constr s + | (_, (_, Some s)) -> Constrexpr_ops.constr_loc s + +let mk_term k c = k, (mkRHole, Some c) +let mk_lterm c = mk_term ' ' c + +let map_fold_constr g f ctx acc cstr = + let array_f ctx acc x = let x, acc = f ctx acc x in acc, x in + match kind_of_term cstr with + | (Rel _ | Meta _ | Var _ | Sort _ | Const _ | Ind _ | Construct _) -> + cstr, acc + | Proj (x,c) -> + let c', acc = f ctx acc c in + (if c == c' then cstr else mkProj (x,c')), acc + | Cast (c,k, t) -> + let c', acc = f ctx acc c in + let t', acc = f ctx acc t in + (if c==c' && t==t' then cstr else mkCast (c', k, t')), acc + | Prod (na,t,c) -> + let t', acc = f ctx acc t in + let c', acc = f (g (na,None,t) ctx) acc c in + (if t==t' && c==c' then cstr else mkProd (na, t', c')), acc + | Lambda (na,t,c) -> + let t', acc = f ctx acc t in + let c', acc = f (g (na,None,t) ctx) acc c in + (if t==t' && c==c' then cstr else mkLambda (na, t', c')), acc + | LetIn (na,b,t,c) -> + let b', acc = f ctx acc b in + let t', acc = f ctx acc t in + let c', acc = f (g (na,Some b,t) ctx) acc c in + (if b==b' && t==t' && c==c' then cstr else mkLetIn (na, b', t', c')), acc + | App (c,al) -> + let c', acc = f ctx acc c in + let acc, al' = CArray.smartfoldmap (array_f ctx) acc al in + (if c==c' && Array.for_all2 (==) al al' then cstr else mkApp (c', al')), + acc + | Evar (e,al) -> + let acc, al' = CArray.smartfoldmap (array_f ctx) acc al in + (if Array.for_all2 (==) al al' then cstr else mkEvar (e, al')), acc + | Case (ci,p,c,bl) -> + let p', acc = f ctx acc p in + let c', acc = f ctx acc c in + let acc, bl' = CArray.smartfoldmap (array_f ctx) acc bl in + (if p==p' && c==c' && Array.for_all2 (==) bl bl' then cstr else + mkCase (ci, p', c', bl')), + acc + | Fix (ln,(lna,tl,bl)) -> + let acc, tl' = CArray.smartfoldmap (array_f ctx) acc tl in + let ctx' = Array.fold_left2 (fun l na t -> g (na,None,t) l) ctx lna tl in + let acc, bl' = CArray.smartfoldmap (array_f ctx') acc bl in + (if Array.for_all2 (==) tl tl' && Array.for_all2 (==) bl bl' + then cstr + else mkFix (ln,(lna,tl',bl'))), acc + | CoFix(ln,(lna,tl,bl)) -> + let acc, tl' = CArray.smartfoldmap (array_f ctx) acc tl in + let ctx' = Array.fold_left2 (fun l na t -> g (na,None,t) l) ctx lna tl in + let acc,bl' = CArray.smartfoldmap (array_f ctx') acc bl in + (if Array.for_all2 (==) tl tl' && Array.for_all2 (==) bl bl' + then cstr + else mkCoFix (ln,(lna,tl',bl'))), acc + +let pf_merge_uc_of sigma gl = + let ucst = Evd.evar_universe_context sigma in + pf_merge_uc ucst gl + +(* }}} *) + +(** Profiling {{{ *************************************************************) +type profiler = { + profile : 'a 'b. ('a -> 'b) -> 'a -> 'b; + reset : unit -> unit; + print : unit -> unit } +let profile_now = ref false +let something_profiled = ref false +let profilers = ref [] +let add_profiler f = profilers := f :: !profilers;; +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect profiling"; + Goptions.optkey = ["SsrProfiling"]; + Goptions.optread = (fun _ -> !profile_now); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> + Ssrmatching.profile b; + profile_now := b; + if b then List.iter (fun f -> f.reset ()) !profilers; + if not b then List.iter (fun f -> f.print ()) !profilers) } +let () = + let prof_total = + let init = ref 0.0 in { + profile = (fun f x -> assert false); + reset = (fun () -> init := Unix.gettimeofday ()); + print = (fun () -> if !something_profiled then + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + "total" 0 (Unix.gettimeofday() -. !init) 0.0 0.0)) } in + let prof_legenda = { + profile = (fun f x -> assert false); + reset = (fun () -> ()); + print = (fun () -> if !something_profiled then begin + prerr_endline + (Printf.sprintf "!! %39s ---------- --------- --------- ---------" + (String.make 39 '-')); + prerr_endline + (Printf.sprintf "!! %-39s %10s %9s %9s %9s" + "function" "#calls" "total" "max" "average") end) } in + add_profiler prof_legenda; + add_profiler prof_total +;; + +let mk_profiler s = + let total, calls, max = ref 0.0, ref 0, ref 0.0 in + let reset () = total := 0.0; calls := 0; max := 0.0 in + let profile f x = + if not !profile_now then f x else + let before = Unix.gettimeofday () in + try + incr calls; + let res = f x in + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + res + with exc -> + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + raise exc in + let print () = + if !calls <> 0 then begin + something_profiled := true; + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + s !calls !total !max (!total /. (float_of_int !calls))) end in + let prof = { profile = profile; reset = reset; print = print } in + add_profiler prof; + prof +;; +(* }}} *) + +let inVersion = Libobject.declare_object { + (Libobject.default_object "SSRASTVERSION") with + Libobject.load_function = (fun _ (_,v) -> + if v <> ssrAstVersion then Errors.error "Please recompile your .vo files"); + Libobject.classify_function = (fun v -> Libobject.Keep v); +} + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect version"; + Goptions.optkey = ["SsrAstVersion"]; + Goptions.optread = (fun _ -> true); + Goptions.optdepr = false; + Goptions.optwrite = (fun _ -> + Lib.add_anonymous_leaf (inVersion ssrAstVersion)) } + +let tactic_expr = Tactic.tactic_expr +let gallina_ext = Vernac_.gallina_ext +let sprintf = Printf.sprintf +let tactic_mode = G_vernac.tactic_mode + +(** 1. Utilities *) + + +let ssroldreworder = Summary.ref ~name:"SSR:oldreworder" true +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssreflect 1.3 compatibility flag"; + Goptions.optkey = ["SsrOldRewriteGoalsOrder"]; + Goptions.optread = (fun _ -> !ssroldreworder); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssroldreworder := b) } + +let ssrhaveNOtcresolution = ref false + +let inHaveTCResolution = Libobject.declare_object { + (Libobject.default_object "SSRHAVETCRESOLUTION") with + Libobject.cache_function = (fun (_,v) -> ssrhaveNOtcresolution := v); + Libobject.load_function = (fun _ (_,v) -> ssrhaveNOtcresolution := v); + Libobject.classify_function = (fun v -> Libobject.Keep v); +} + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "have type classes"; + Goptions.optkey = ["SsrHave";"NoTCResolution"]; + Goptions.optread = (fun _ -> !ssrhaveNOtcresolution); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> + Lib.add_anonymous_leaf (inHaveTCResolution b)) } + + +(** Primitive parsing to avoid syntax conflicts with basic tactics. *) + +let accept_before_syms syms strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | _ -> raise Stream.Failure + +let accept_before_syms_or_any_id syms strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | Tok.IDENT _ -> () + | _ -> raise Stream.Failure + +let accept_before_syms_or_ids syms ids strm = + match Compat.get_tok (stream_nth 1 strm) with + | Tok.KEYWORD sym when List.mem sym syms -> () + | Tok.IDENT id when List.mem id ids -> () + | _ -> raise Stream.Failure + +(** Pretty-printing utilities *) + +let pr_id = Ppconstr.pr_id +let pr_name = function Name id -> pr_id id | Anonymous -> str "_" +let pr_spc () = str " " +let pr_bar () = Pp.cut() ++ str "|" +let pr_list = prlist_with_sep + +let tacltop = (5,Ppextend.E) + +(** Tactic-level diagnosis *) + +let pf_pr_constr gl = pr_constr_env (pf_env gl) + +let pf_pr_glob_constr gl = pr_glob_constr_env (pf_env gl) + +(* debug *) + +let pf_msg gl = + let ppgl = pr_lconstr_env (pf_env gl) (project gl) (pf_concl gl) in + msgnl (str "goal is " ++ ppgl) + +let msgtac gl = pf_msg gl; tclIDTAC gl + +(** Tactic utilities *) + +let last_goal gls = let sigma, gll = Refiner.unpackage gls in + Refiner.repackage sigma (List.nth gll (List.length gll - 1)) + +let pf_type_id gl t = id_of_string (hdchar (pf_env gl) t) + +let not_section_id id = not (is_section_variable id) + +let is_pf_var c = isVar c && not_section_id (destVar c) + +let pf_ids_of_proof_hyps gl = + let add_hyp (id, _, _) ids = if not_section_id id then id :: ids else ids in + Context.fold_named_context add_hyp (pf_hyps gl) ~init:[] + +let pf_nf_evar gl e = Reductionops.nf_evar (project gl) e + +let pf_partial_solution gl t evl = + let sigma, g = project gl, sig_it gl in + let sigma = Goal.V82.partial_solution sigma g t in + re_sig (List.map (fun x -> (fst (destEvar x))) evl) sigma + +let pf_new_evar gl ty = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, extra = Evarutil.new_evar env sigma ty in + re_sig it sigma, extra + +(* Basic tactics *) + +let convert_concl_no_check t = convert_concl_no_check t DEFAULTcast +let convert_concl t = convert_concl t DEFAULTcast +let reduct_in_concl t = reduct_in_concl (t, DEFAULTcast) +let havetac id c = Proofview.V82.of_tactic (pose_proof (Name id) c) +let settac id c = letin_tac None (Name id) c None +let posetac id cl = Proofview.V82.of_tactic (settac id cl nowhere) +let basecuttac name c gl = + let hd, gl = pf_mkSsrConst name gl in + let t = mkApp (hd, [|c|]) in + let gl, _ = pf_e_type_of gl t in + Proofview.V82.of_tactic (apply t) gl + +(* we reduce head beta redexes *) +let betared env = + Closure.create_clos_infos + (Closure.RedFlags.mkflags [Closure.RedFlags.fBETA]) + env +;; +let introid name = tclTHEN (fun gl -> + let g, env = pf_concl gl, pf_env gl in + match kind_of_term g with + | App (hd, _) when isLambda hd -> + let g = Closure.whd_val (betared env) (Closure.inject g) in + Proofview.V82.of_tactic (convert_concl_no_check g) gl + | _ -> tclIDTAC gl) + (Proofview.V82.of_tactic (intro_mustbe_force name)) +;; + + +(** Name generation {{{ *******************************************************) + +(* Since Coq now does repeated internal checks of its external lexical *) +(* rules, we now need to carve ssreflect reserved identifiers out of *) +(* out of the user namespace. We use identifiers of the form _id_ for *) +(* this purpose, e.g., we "anonymize" an identifier id as _id_, adding *) +(* an extra leading _ if this might clash with an internal identifier. *) +(* We check for ssreflect identifiers in the ident grammar rule; *) +(* when the ssreflect Module is present this is normally an error, *) +(* but we provide a compatibility flag to reduce this to a warning. *) + +let ssr_reserved_ids = Summary.ref ~name:"SSR:idents" true + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "ssreflect identifiers"; + Goptions.optkey = ["SsrIdents"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !ssr_reserved_ids); + Goptions.optwrite = (fun b -> ssr_reserved_ids := b) + } + +let is_ssr_reserved s = + let n = String.length s in n > 2 && s.[0] = '_' && s.[n - 1] = '_' + +let internal_names = ref [] +let add_internal_name pt = internal_names := pt :: !internal_names +let is_internal_name s = List.exists (fun p -> p s) !internal_names + +let ssr_id_of_string loc s = + if is_ssr_reserved s && is_ssr_loaded () then begin + if !ssr_reserved_ids then + loc_error loc ("The identifier " ^ s ^ " is reserved.") + else if is_internal_name s then + msg_warning (str ("Conflict between " ^ s ^ " and ssreflect internal names.")) + else msg_warning (str ( + "The name " ^ s ^ " fits the _xxx_ format used for anonymous variables.\n" + ^ "Scripts with explicit references to anonymous variables are fragile.")) + end; id_of_string s + +let ssr_null_entry = Gram.Entry.of_parser "ssr_null" (fun _ -> ()) + +let (!@) = Compat.to_coqloc + +GEXTEND Gram + GLOBAL: Prim.ident; + Prim.ident: [[ s = IDENT; ssr_null_entry -> ssr_id_of_string !@loc s ]]; +END + +let mk_internal_id s = + let s' = sprintf "_%s_" s in + for i = 1 to String.length s do if s'.[i] = ' ' then s'.[i] <- '_' done; + add_internal_name ((=) s'); id_of_string s' + +let same_prefix s t n = + let rec loop i = i = n || s.[i] = t.[i] && loop (i + 1) in loop 0 + +let skip_digits s = + let n = String.length s in + let rec loop i = if i < n && is_digit s.[i] then loop (i + 1) else i in loop + +let mk_tagged_id t i = id_of_string (sprintf "%s%d_" t i) +let is_tagged t s = + let n = String.length s - 1 and m = String.length t in + m < n && s.[n] = '_' && same_prefix s t m && skip_digits s m = n + +let perm_tag = "_perm_Hyp_" +let _ = add_internal_name (is_tagged perm_tag) +let mk_perm_id = + let salt = ref 1 in + fun () -> salt := !salt mod 10000 + 1; mk_tagged_id perm_tag !salt + +let evar_tag = "_evar_" +let _ = add_internal_name (is_tagged evar_tag) +let mk_evar_name n = Name (mk_tagged_id evar_tag n) +let nb_evar_deps = function + | Name id -> + let s = string_of_id id in + if not (is_tagged evar_tag s) then 0 else + let m = String.length evar_tag in + (try int_of_string (String.sub s m (String.length s - 1 - m)) with _ -> 0) + | _ -> 0 + +let discharged_tag = "_discharged_" +let mk_discharged_id id = + id_of_string (sprintf "%s%s_" discharged_tag (string_of_id id)) +let has_discharged_tag s = + let m = String.length discharged_tag and n = String.length s - 1 in + m < n && s.[n] = '_' && same_prefix s discharged_tag m +let _ = add_internal_name has_discharged_tag +let is_discharged_id id = has_discharged_tag (string_of_id id) + +let wildcard_tag = "_the_" +let wildcard_post = "_wildcard_" +let mk_wildcard_id i = + id_of_string (sprintf "%s%s%s" wildcard_tag (String.ordinal i) wildcard_post) +let has_wildcard_tag s = + let n = String.length s in let m = String.length wildcard_tag in + let m' = String.length wildcard_post in + n < m + m' + 2 && same_prefix s wildcard_tag m && + String.sub s (n - m') m' = wildcard_post && + skip_digits s m = n - m' - 2 +let _ = add_internal_name has_wildcard_tag + +let max_suffix m (t, j0 as tj0) id = + let s = string_of_id id in let n = String.length s - 1 in + let dn = String.length t - 1 - n in let i0 = j0 - dn in + if not (i0 >= m && s.[n] = '_' && same_prefix s t m) then tj0 else + let rec loop i = + if i < i0 && s.[i] = '0' then loop (i + 1) else + if (if i < i0 then skip_digits s i = n else le_s_t i) then s, i else tj0 + and le_s_t i = + let ds = s.[i] and dt = t.[i + dn] in + if ds = dt then i = n || le_s_t (i + 1) else + dt < ds && skip_digits s i = n in + loop m + +let mk_anon_id t gl = + let m, si0, id0 = + let s = ref (sprintf "_%s_" t) in + if is_internal_name !s then s := "_" ^ !s; + let n = String.length !s - 1 in + let rec loop i j = + let d = !s.[i] in if not (is_digit d) then i + 1, j else + loop (i - 1) (if d = '0' then j else i) in + let m, j = loop (n - 1) n in m, (!s, j), id_of_string !s in + let gl_ids = pf_ids_of_hyps gl in + if not (List.mem id0 gl_ids) then id0 else + let s, i = List.fold_left (max_suffix m) si0 gl_ids in + let n = String.length s - 1 in + let rec loop i = + if s.[i] = '9' then (s.[i] <- '0'; loop (i - 1)) else + if i < m then (s.[n] <- '0'; s.[m] <- '1'; s ^ "_") else + (s.[i] <- Char.chr (Char.code s.[i] + 1); s) in + id_of_string (loop (n - 1)) + +(* We must not anonymize context names discharged by the "in" tactical. *) + +let ssr_anon_hyp = "Hyp" + +let anontac (x, _, _) gl = + let id = match x with + | Name id -> + if is_discharged_id id then id else mk_anon_id (string_of_id id) gl + | _ -> mk_anon_id ssr_anon_hyp gl in + introid id gl + +let rec constr_name c = match kind_of_term c with + | Var id -> Name id + | Cast (c', _, _) -> constr_name c' + | Const (cn,_) -> Name (id_of_label (con_label cn)) + | App (c', _) -> constr_name c' + | _ -> Anonymous + +(* }}} *) + +(** Open term to lambda-term coercion {{{ ************************************) + +(* This operation takes a goal gl and an open term (sigma, t), and *) +(* returns a term t' where all the new evars in sigma are abstracted *) +(* with the mkAbs argument, i.e., for mkAbs = mkLambda then there is *) +(* some duplicate-free array args of evars of sigma such that the *) +(* term mkApp (t', args) is convertible to t. *) +(* This makes a useful shorthand for local definitions in proofs, *) +(* i.e., pose succ := _ + 1 means pose succ := fun n : nat => n + 1, *) +(* and, in context of the the 4CT library, pose mid := maps id means *) +(* pose mid := fun d : detaSet => @maps d d (@id (datum d)) *) +(* Note that this facility does not extend to set, which tries *) +(* instead to fill holes by matching a goal subterm. *) +(* The argument to "have" et al. uses product abstraction, e.g. *) +(* have Hmid: forall s, (maps id s) = s. *) +(* stands for *) +(* have Hmid: forall (d : dataSet) (s : seq d), (maps id s) = s. *) +(* We also use this feature for rewrite rules, so that, e.g., *) +(* rewrite: (plus_assoc _ 3). *) +(* will execute as *) +(* rewrite (fun n => plus_assoc n 3) *) +(* i.e., it will rewrite some subterm .. + (3 + ..) to .. + 3 + ... *) +(* The convention is also used for the argument of the congr tactic, *) +(* e.g., congr (x + _ * 1). *) + +(* Replace new evars with lambda variables, retaining local dependencies *) +(* but stripping global ones. We use the variable names to encode the *) +(* the number of dependencies, so that the transformation is reversible. *) + +let pf_abs_evars gl (sigma, c0) = + let sigma0, ucst = project gl, Evd.evar_universe_context sigma in + let nenv = env_size (pf_env gl) in + let abs_evar n k = + let evi = Evd.find sigma k in + let dc = List.firstn n (evar_filtered_context evi) in + let abs_dc c = function + | x, Some b, t -> mkNamedLetIn x b t (mkArrow t c) + | x, None, t -> mkNamedProd x t c in + let t = Context.fold_named_context_reverse abs_dc ~init:evi.evar_concl dc in + Evarutil.nf_evar sigma t in + let rec put evlist c = match kind_of_term c with + | Evar (k, a) -> + if List.mem_assoc k evlist || Evd.mem sigma0 k then evlist else + let n = max 0 (Array.length a - nenv) in + let t = abs_evar n k in (k, (n, t)) :: put evlist t + | _ -> fold_constr put evlist c in + let evlist = put [] c0 in + if evlist = [] then 0, c0,[], ucst else + let rec lookup k i = function + | [] -> 0, 0 + | (k', (n, _)) :: evl -> if k = k' then i, n else lookup k (i + 1) evl in + let rec get i c = match kind_of_term c with + | Evar (ev, a) -> + let j, n = lookup ev i evlist in + if j = 0 then map_constr (get i) c else if n = 0 then mkRel j else + mkApp (mkRel j, Array.init n (fun k -> get i a.(n - 1 - k))) + | _ -> map_constr_with_binders ((+) 1) get i c in + let rec loop c i = function + | (_, (n, t)) :: evl -> + loop (mkLambda (mk_evar_name n, get (i - 1) t, c)) (i - 1) evl + | [] -> c in + List.length evlist, loop (get 1 c0) 1 evlist, List.map fst evlist, ucst + + + +(* As before but if (?i : T(?j)) and (?j : P : Prop), then the lambda for i + * looks like (fun evar_i : (forall pi : P. T(pi))) thanks to "loopP" and all + * occurrences of evar_i are replaced by (evar_i evar_j) thanks to "app". + * + * If P can be solved by ssrautoprop (that defaults to trivial), then + * the corresponding lambda looks like (fun evar_i : T(c)) where c is + * the solution found by ssrautoprop. + *) +let ssrautoprop_tac = ref (fun gl -> assert false) + +(* Thanks to Arnaud Spiwack for this snippet *) +let call_on_evar tac e s = + let { it = gs ; sigma = s } = + tac { it = e ; sigma = s; } in + gs, s + +let pf_abs_evars_pirrel gl (sigma, c0) = + pp(lazy(str"==PF_ABS_EVARS_PIRREL==")); + pp(lazy(str"c0= " ++ pr_constr c0)); + let sigma0 = project gl in + let c0 = Evarutil.nf_evar sigma0 (Evarutil.nf_evar sigma c0) in + let nenv = env_size (pf_env gl) in + let abs_evar n k = + let evi = Evd.find sigma k in + let dc = List.firstn n (evar_filtered_context evi) in + let abs_dc c = function + | x, Some b, t -> mkNamedLetIn x b t (mkArrow t c) + | x, None, t -> mkNamedProd x t c in + let t = Context.fold_named_context_reverse abs_dc ~init:evi.evar_concl dc in + Evarutil.nf_evar sigma0 (Evarutil.nf_evar sigma t) in + let rec put evlist c = match kind_of_term c with + | Evar (k, a) -> + if List.mem_assoc k evlist || Evd.mem sigma0 k then evlist else + let n = max 0 (Array.length a - nenv) in + let k_ty = + Retyping.get_sort_family_of + (pf_env gl) sigma (Evd.evar_concl (Evd.find sigma k)) in + let is_prop = k_ty = InProp in + let t = abs_evar n k in (k, (n, t, is_prop)) :: put evlist t + | _ -> fold_constr put evlist c in + let evlist = put [] c0 in + if evlist = [] then 0, c0 else + let pr_constr t = pr_constr (Reductionops.nf_beta (project gl) t) in + pp(lazy(str"evlist=" ++ pr_list (fun () -> str";") + (fun (k,_) -> str(Evd.string_of_existential k)) evlist)); + let evplist = + let depev = List.fold_left (fun evs (_,(_,t,_)) -> + Intset.union evs (Evarutil.undefined_evars_of_term sigma t)) Intset.empty evlist in + List.filter (fun i,(_,_,b) -> b && Intset.mem i depev) evlist in + let evlist, evplist, sigma = + if evplist = [] then evlist, [], sigma else + List.fold_left (fun (ev, evp, sigma) (i, (_,t,_) as p) -> + try + let ng, sigma = call_on_evar !ssrautoprop_tac i sigma in + if (ng <> []) then errorstrm (str "Should we tell the user?"); + List.filter (fun (j,_) -> j <> i) ev, evp, sigma + with _ -> ev, p::evp, sigma) (evlist, [], sigma) (List.rev evplist) in + let c0 = Evarutil.nf_evar sigma c0 in + let evlist = + List.map (fun (x,(y,t,z)) -> x,(y,Evarutil.nf_evar sigma t,z)) evlist in + let evplist = + List.map (fun (x,(y,t,z)) -> x,(y,Evarutil.nf_evar sigma t,z)) evplist in + pp(lazy(str"c0= " ++ pr_constr c0)); + let rec lookup k i = function + | [] -> 0, 0 + | (k', (n,_,_)) :: evl -> if k = k' then i,n else lookup k (i + 1) evl in + let rec get evlist i c = match kind_of_term c with + | Evar (ev, a) -> + let j, n = lookup ev i evlist in + if j = 0 then map_constr (get evlist i) c else if n = 0 then mkRel j else + mkApp (mkRel j, Array.init n (fun k -> get evlist i a.(n - 1 - k))) + | _ -> map_constr_with_binders ((+) 1) (get evlist) i c in + let rec app extra_args i c = match decompose_app c with + | hd, args when isRel hd && destRel hd = i -> + let j = destRel hd in + mkApp (mkRel j, Array.of_list (List.map (lift (i-1)) extra_args @ args)) + | _ -> map_constr_with_binders ((+) 1) (app extra_args) i c in + let rec loopP evlist c i = function + | (_, (n, t, _)) :: evl -> + let t = get evlist (i - 1) t in + let n = Name (id_of_string (ssr_anon_hyp ^ string_of_int n)) in + loopP evlist (mkProd (n, t, c)) (i - 1) evl + | [] -> c in + let rec loop c i = function + | (_, (n, t, _)) :: evl -> + let evs = Evarutil.undefined_evars_of_term sigma t in + let t_evplist = List.filter (fun (k,_) -> Intset.mem k evs) evplist in + let t = loopP t_evplist (get t_evplist 1 t) 1 t_evplist in + let t = get evlist (i - 1) t in + let extra_args = + List.map (fun (k,_) -> mkRel (fst (lookup k i evlist))) + (List.rev t_evplist) in + let c = if extra_args = [] then c else app extra_args 1 c in + loop (mkLambda (mk_evar_name n, t, c)) (i - 1) evl + | [] -> c in + let res = loop (get evlist 1 c0) 1 evlist in + pp(lazy(str"res= " ++ pr_constr res)); + List.length evlist, res + +(* Strip all non-essential dependencies from an abstracted term, generating *) +(* standard names for the abstracted holes. *) + +let pf_abs_cterm gl n c0 = + if n <= 0 then c0 else + let noargs = [|0|] in + let eva = Array.make n noargs in + let rec strip i c = match kind_of_term c with + | App (f, a) when isRel f -> + let j = i - destRel f in + if j >= n || eva.(j) = noargs then mkApp (f, Array.map (strip i) a) else + let dp = eva.(j) in + let nd = Array.length dp - 1 in + let mkarg k = strip i a.(if k < nd then dp.(k + 1) - j else k + dp.(0)) in + mkApp (f, Array.init (Array.length a - dp.(0)) mkarg) + | _ -> map_constr_with_binders ((+) 1) strip i c in + let rec strip_ndeps j i c = match kind_of_term c with + | Prod (x, t, c1) when i < j -> + let dl, c2 = strip_ndeps j (i + 1) c1 in + if noccurn 1 c2 then dl, lift (-1) c2 else + i :: dl, mkProd (x, strip i t, c2) + | LetIn (x, b, t, c1) when i < j -> + let _, _, c1' = destProd c1 in + let dl, c2 = strip_ndeps j (i + 1) c1' in + if noccurn 1 c2 then dl, lift (-1) c2 else + i :: dl, mkLetIn (x, strip i b, strip i t, c2) + | _ -> [], strip i c in + let rec strip_evars i c = match kind_of_term c with + | Lambda (x, t1, c1) when i < n -> + let na = nb_evar_deps x in + let dl, t2 = strip_ndeps (i + na) i t1 in + let na' = List.length dl in + eva.(i) <- Array.of_list (na - na' :: dl); + let x' = + if na' = 0 then Name (pf_type_id gl t2) else mk_evar_name na' in + mkLambda (x', t2, strip_evars (i + 1) c1) +(* if noccurn 1 c2 then lift (-1) c2 else + mkLambda (Name (pf_type_id gl t2), t2, c2) *) + | _ -> strip i c in + strip_evars 0 c0 + +(* Undo the evar abstractions. Also works for non-evar variables. *) + +let pf_unabs_evars gl ise n c0 = + if n = 0 then c0 else + let evv = Array.make n mkProp in + let nev = ref 0 in + let env0 = pf_env gl in + let nenv0 = env_size env0 in + let rec unabs i c = match kind_of_term c with + | Rel j when i - j < !nev -> evv.(i - j) + | App (f, a0) when isRel f -> + let a = Array.map (unabs i) a0 in + let j = i - destRel f in + if j >= !nev then mkApp (f, a) else + let ev, eva = destEvar evv.(j) in + let nd = Array.length eva - nenv0 in + if nd = 0 then mkApp (evv.(j), a) else + let evarg k = if k < nd then a.(nd - 1 - k) else eva.(k) in + let c' = mkEvar (ev, Array.init (nd + nenv0) evarg) in + let na = Array.length a - nd in + if na = 0 then c' else mkApp (c', Array.sub a nd na) + | _ -> map_constr_with_binders ((+) 1) unabs i c in + let push_rel = Environ.push_rel in + let rec mk_evar j env i c = match kind_of_term c with + | Prod (x, t, c1) when i < j -> + mk_evar j (push_rel (x, None, unabs i t) env) (i + 1) c1 + | LetIn (x, b, t, c1) when i < j -> + let _, _, c2 = destProd c1 in + mk_evar j (push_rel (x, Some (unabs i b), unabs i t) env) (i + 1) c2 + | _ -> Evarutil.e_new_evar env ise (unabs i c) in + let rec unabs_evars c = + if !nev = n then unabs n c else match kind_of_term c with + | Lambda (x, t, c1) when !nev < n -> + let i = !nev in + evv.(i) <- mk_evar (i + nb_evar_deps x) env0 i t; + incr nev; unabs_evars c1 + | _ -> unabs !nev c in + unabs_evars c0 + +(* }}} *) + +(** Tactical extensions. {{{ **************************************************) + +(* The TACTIC EXTEND facility can't be used for defining new user *) +(* tacticals, because: *) +(* - the concrete syntax must start with a fixed string *) +(* We use the following workaround: *) +(* - We use the (unparsable) "YouShouldNotTypeThis" token for tacticals that *) +(* don't start with a token, then redefine the grammar and *) +(* printer using GEXTEND and set_pr_ssrtac, respectively. *) + +type ssrargfmt = ArgSsr of string | ArgCoq of argument_type | ArgSep of string + +let ssrtac_name name = { + mltac_plugin = "ssreflect"; + mltac_tactic = "ssr" ^ name; +} + +let set_pr_ssrtac name prec afmt = + let fmt = List.map (function ArgSep s -> Some s | _ -> None) afmt in + let rec mk_akey = function + | ArgSsr s :: afmt' -> ExtraArgType ("ssr" ^ s) :: mk_akey afmt' + | ArgCoq a :: afmt' -> a :: mk_akey afmt' + | ArgSep _ :: afmt' -> mk_akey afmt' + | [] -> [] in + let tacname = ssrtac_name name in + Pptactic.declare_ml_tactic_pprule tacname + { Pptactic.pptac_args = mk_akey afmt; + Pptactic.pptac_prods = (prec, fmt) } + +let ssrtac_atom loc name args = TacML (loc, ssrtac_name name, args) +let ssrtac_expr = ssrtac_atom + + +let ssrevaltac ist gtac = + let debug = match TacStore.get ist.extra f_debug with + | None -> Tactic_debug.DebugOff | Some level -> level + in + Proofview.V82.of_tactic (interp_tac_gen ist.lfun [] debug (globTacticIn (fun _ -> gtac))) + +(* fun gl -> let lfun = [tacarg_id, val_interp ist gl gtac] in + interp_tac_gen lfun [] ist.debug tacarg_expr gl *) + +(** Generic argument-based globbing/typing utilities *) + + +let interp_wit wit ist gl x = + let globarg = in_gen (glbwit wit) x in + let sigma, arg = interp_genarg ist (pf_env gl) (project gl) (pf_concl gl) gl.Evd.it globarg in + sigma, out_gen (topwit wit) arg + +let interp_intro_pattern = interp_wit wit_intro_pattern + +let interp_constr = interp_wit wit_constr + +let interp_open_constr ist gl gc = + interp_wit wit_open_constr ist gl ((), gc) + +let interp_refine ist gl rc = + let constrvars = extract_ltac_constr_values ist (pf_env gl) in + let vars = { Pretyping.empty_lvar with + Pretyping.ltac_constrs = constrvars; ltac_genargs = ist.lfun + } in + let kind = OfType (pf_concl gl) in + let flags = { + use_typeclasses = true; + use_unif_heuristics = true; + use_hook = None; + fail_evar = false; + expand_evars = true } + in + let sigma, c = understand_ltac flags (pf_env gl) (project gl) vars kind rc in +(* pp(lazy(str"sigma@interp_refine=" ++ pr_evar_map None sigma)); *) + pp(lazy(str"c@interp_refine=" ++ pr_constr c)); + (sigma, (sigma, c)) + +let pf_match = pf_apply (fun e s c t -> understand_tcc e s ~expected_type:t c) + +(* Estimate a bound on the number of arguments of a raw constr. *) +(* This is not perfect, because the unifier may fail to *) +(* typecheck the partial application, so we use a minimum of 5. *) +(* Also, we don't handle delayed or iterated coercions to *) +(* FUNCLASS, which is probably just as well since these can *) +(* lead to infinite arities. *) + +let splay_open_constr gl (sigma, c) = + let env = pf_env gl in let t = Retyping.get_type_of env sigma c in + Reductionops.splay_prod env sigma t + +let nbargs_open_constr gl oc = + let pl, _ = splay_open_constr gl oc in List.length pl + +let interp_nbargs ist gl rc = + try + let rc6 = mkRApp rc (mkRHoles 6) in + let sigma, t = interp_open_constr ist gl (rc6, None) in + let si = sig_it gl in + let gl = re_sig si sigma in + 6 + nbargs_open_constr gl t + with _ -> 5 + +let pf_nbargs gl c = nbargs_open_constr gl (project gl, c) + +let isAppInd gl c = + try ignore (pf_reduce_to_atomic_ind gl c); true with _ -> false + +let interp_view_nbimps ist gl rc = + try + let sigma, t = interp_open_constr ist gl (rc, None) in + let si = sig_it gl in + let gl = re_sig si sigma in + let pl, c = splay_open_constr gl t in + if isAppInd gl c then List.length pl else ~-(List.length pl) + with _ -> 0 + +(* }}} *) + +(** Vernacular commands: Prenex Implicits and Search {{{ **********************) + +(* This should really be implemented as an extension to the implicit *) +(* arguments feature, but unfortuately that API is sealed. The current *) +(* workaround uses a combination of notations that works reasonably, *) +(* with the following caveats: *) +(* - The pretty-printing always elides prenex implicits, even when *) +(* they are obviously needed. *) +(* - Prenex Implicits are NEVER exported from a module, because this *) +(* would lead to faulty pretty-printing and scoping errors. *) +(* - The command "Import Prenex Implicits" can be used to reassert *) +(* Prenex Implicits for all the visible constants that had been *) +(* declared as Prenex Implicits. *) + +let declare_one_prenex_implicit locality f = + let fref = + try Smartlocate.global_with_alias f + with _ -> errorstrm (pr_reference f ++ str " is not declared") in + let rec loop = function + | a :: args' when Impargs.is_status_implicit a -> + (ExplByName (Impargs.name_of_implicit a), (true, true, true)) :: loop args' + | args' when List.exists Impargs.is_status_implicit args' -> + errorstrm (str "Expected prenex implicits for " ++ pr_reference f) + | _ -> [] in + let impls = + match Impargs.implicits_of_global fref with + | [cond,impls] -> impls + | [] -> errorstrm (str "Expected some implicits for " ++ pr_reference f) + | _ -> errorstrm (str "Multiple implicits not supported") in + match loop impls with + | [] -> + errorstrm (str "Expected some implicits for " ++ pr_reference f) + | impls -> + Impargs.declare_manual_implicits locality fref ~enriching:false [impls] + +VERNAC COMMAND EXTEND Ssrpreneximplicits CLASSIFIED AS SIDEFF + | [ "Prenex" "Implicits" ne_global_list(fl) ] + -> [ let locality = + Locality.make_section_locality (Locality.LocalityFixme.consume ()) in + List.iter (declare_one_prenex_implicit locality) fl ] +END + +(* Vernac grammar visibility patch *) + +GEXTEND Gram + GLOBAL: gallina_ext; + gallina_ext: + [ [ IDENT "Import"; IDENT "Prenex"; IDENT "Implicits" -> + Vernacexpr.VernacUnsetOption (["Printing"; "Implicit"; "Defensive"]) + ] ] + ; +END + +(** Extend Search to subsume SearchAbout, also adding hidden Type coercions. *) + +(* Main prefilter *) + +type raw_glob_search_about_item = + | RGlobSearchSubPattern of constr_expr + | RGlobSearchString of Loc.t * string * string option + +let pr_search_item = function + | RGlobSearchString (_,s,_) -> str s + | RGlobSearchSubPattern p -> pr_constr_expr p + +let wit_ssr_searchitem = add_genarg "ssr_searchitem" pr_search_item + +let interp_search_notation loc s opt_scope = + try + let interp = Notation.interp_notation_as_global_reference loc in + let ref = interp (fun _ -> true) s opt_scope in + Search.GlobSearchSubPattern (Pattern.PRef ref) + with _ -> + let diagnosis = + try + let ntns = Notation.locate_notation pr_glob_constr s opt_scope in + let ambig = "This string refers to a complex or ambiguous notation." in + str ambig ++ str "\nTry searching with one of\n" ++ ntns + with _ -> str "This string is not part of an identifier or notation." in + Errors.user_err_loc (loc, "interp_search_notation", diagnosis) + +let pr_ssr_search_item _ _ _ = pr_search_item + +(* Workaround the notation API that can only print notations *) + +let is_ident s = try Lexer.check_ident s; true with _ -> false + +let is_ident_part s = is_ident ("H" ^ s) + +let interp_search_notation loc tag okey = + let err msg = Errors.user_err_loc (loc, "interp_search_notation", msg) in + let mk_pntn s for_key = + let n = String.length s in + let s' = String.make (n + 2) ' ' in + let rec loop i i' = + if i >= n then s', i' - 2 else if s.[i] = ' ' then loop (i + 1) i' else + let j = try String.index_from s (i + 1) ' ' with _ -> n in + let m = j - i in + if s.[i] = '\'' && i < j - 2 && s.[j - 1] = '\'' then + (String.blit s (i + 1) s' i' (m - 2); loop (j + 1) (i' + m - 1)) + else if for_key && is_ident (String.sub s i m) then + (s'.[i'] <- '_'; loop (j + 1) (i' + 2)) + else (String.blit s i s' i' m; loop (j + 1) (i' + m + 1)) in + loop 0 1 in + let trim_ntn (pntn, m) = String.sub pntn 1 (max 0 m) in + let pr_ntn ntn = str "(" ++ str ntn ++ str ")" in + let pr_and_list pr = function + | [x] -> pr x + | x :: lx -> pr_list pr_comma pr lx ++ pr_comma () ++ str "and " ++ pr x + | [] -> mt () in + let pr_sc sc = str (if sc = "" then "independently" else sc) in + let pr_scs = function + | [""] -> pr_sc "" + | scs -> str "in " ++ pr_and_list pr_sc scs in + let generator, pr_tag_sc = + let ign _ = mt () in match okey with + | Some key -> + let sc = Notation.find_delimiters_scope loc key in + let pr_sc s_in = str s_in ++ spc() ++ str sc ++ pr_comma() in + Notation.pr_scope ign sc, pr_sc + | None -> Notation.pr_scopes ign, ign in + let qtag s_in = pr_tag_sc s_in ++ qstring tag ++ spc()in + let ptag, ttag = + let ptag, m = mk_pntn tag false in + if m <= 0 then err (str "empty notation fragment"); + ptag, trim_ntn (ptag, m) in + let last = ref "" and last_sc = ref "" in + let scs = ref [] and ntns = ref [] in + let push_sc sc = match !scs with + | "" :: scs' -> scs := "" :: sc :: scs' + | scs' -> scs := sc :: scs' in + let get s _ _ = match !last with + | "Scope " -> last_sc := s; last := "" + | "Lonely notation" -> last_sc := ""; last := "" + | "\"" -> + let pntn, m = mk_pntn s true in + if String.string_contains pntn ptag then begin + let ntn = trim_ntn (pntn, m) in + match !ntns with + | [] -> ntns := [ntn]; scs := [!last_sc] + | ntn' :: _ when ntn' = ntn -> push_sc !last_sc + | _ when ntn = ttag -> ntns := ntn :: !ntns; scs := [!last_sc] + | _ :: ntns' when List.mem ntn ntns' -> () + | ntn' :: ntns' -> ntns := ntn' :: ntn :: ntns' + end; + last := "" + | _ -> last := s in + pp_with (Format.make_formatter get (fun _ -> ())) generator; + let ntn = match !ntns with + | [] -> + err (hov 0 (qtag "in" ++ str "does not occur in any notation")) + | ntn :: ntns' when ntn = ttag -> + if ntns' <> [] then begin + let pr_ntns' = pr_and_list pr_ntn ntns' in + msg_warning (hov 4 (qtag "In" ++ str "also occurs in " ++ pr_ntns')) + end; ntn + | [ntn] -> + msgnl (hov 4 (qtag "In" ++ str "is part of notation " ++ pr_ntn ntn)); ntn + | ntns' -> + let e = str "occurs in" ++ spc() ++ pr_and_list pr_ntn ntns' in + err (hov 4 (str "ambiguous: " ++ qtag "in" ++ e)) in + let (nvars, body), ((_, pat), osc) = match !scs with + | [sc] -> Notation.interp_notation loc ntn (None, [sc]) + | scs' -> + try Notation.interp_notation loc ntn (None, []) with _ -> + let e = pr_ntn ntn ++ spc() ++ str "is defined " ++ pr_scs scs' in + err (hov 4 (str "ambiguous: " ++ pr_tag_sc "in" ++ e)) in + let sc = Option.default "" osc in + let _ = + let m_sc = + if osc <> None then str "In " ++ str sc ++ pr_comma() else mt() in + let ntn_pat = trim_ntn (mk_pntn pat false) in + let rbody = glob_constr_of_notation_constr loc body in + let m_body = hov 0 (Constrextern.without_symbols prl_glob_constr rbody) in + let m = m_sc ++ pr_ntn ntn_pat ++ spc () ++ str "denotes " ++ m_body in + msgnl (hov 0 m) in + if List.length !scs > 1 then + let scs' = List.remove (=) sc !scs in + let w = pr_ntn ntn ++ str " is also defined " ++ pr_scs scs' in + msg_warning (hov 4 w) + else if String.string_contains ntn " .. " then + err (pr_ntn ntn ++ str " is an n-ary notation"); + let nvars = List.filter (fun (_,(_,typ)) -> typ = NtnTypeConstr) nvars in + let rec sub () = function + | NVar x when List.mem_assoc x nvars -> GPatVar (loc, (false, x)) + | c -> + glob_constr_of_notation_constr_with_binders loc (fun _ x -> (), x) sub () c in + let _, npat = Patternops.pattern_of_glob_constr (sub () body) in + Search.GlobSearchSubPattern npat + +ARGUMENT EXTEND ssr_search_item TYPED AS ssr_searchitem + PRINTED BY pr_ssr_search_item + | [ string(s) ] -> [ RGlobSearchString (loc,s,None) ] + | [ string(s) "%" preident(key) ] -> [ RGlobSearchString (loc,s,Some key) ] + | [ constr_pattern(p) ] -> [ RGlobSearchSubPattern p ] +END + +let pr_ssr_search_arg _ _ _ = + let pr_item (b, p) = str (if b then "-" else "") ++ pr_search_item p in + pr_list spc pr_item + +ARGUMENT EXTEND ssr_search_arg TYPED AS (bool * ssr_searchitem) list + PRINTED BY pr_ssr_search_arg + | [ "-" ssr_search_item(p) ssr_search_arg(a) ] -> [ (false, p) :: a ] + | [ ssr_search_item(p) ssr_search_arg(a) ] -> [ (true, p) :: a ] + | [ ] -> [ [] ] +END + +(* Main type conclusion pattern filter *) + +let rec splay_search_pattern na = function + | Pattern.PApp (fp, args) -> splay_search_pattern (na + Array.length args) fp + | Pattern.PLetIn (_, _, bp) -> splay_search_pattern na bp + | Pattern.PRef hr -> hr, na + | _ -> Errors.error "no head constant in head search pattern" + +let coerce_search_pattern_to_sort hpat = + let env = Global.env () and sigma = Evd.empty in + let mkPApp fp n_imps args = + let args' = Array.append (Array.make n_imps (Pattern.PMeta None)) args in + Pattern.PApp (fp, args') in + let hr, na = splay_search_pattern 0 hpat in + let dc, ht = + Reductionops.splay_prod env sigma (Universes.unsafe_type_of_global hr) in + let np = List.length dc in + if np < na then Errors.error "too many arguments in head search pattern" else + let hpat' = if np = na then hpat else mkPApp hpat (np - na) [||] in + let warn () = + msg_warning (str "Listing only lemmas with conclusion matching " ++ + pr_constr_pattern hpat') in + if isSort ht then begin warn (); true, hpat' end else + let filter_head, coe_path = + try + let _, cp = + Classops.lookup_path_to_sort_from (push_rels_assum dc env) sigma ht in + warn (); + true, cp + with _ -> false, [] in + let coerce hp coe_index = + let coe = Classops.get_coercion_value coe_index in + try + let coe_ref = reference_of_constr coe in + let n_imps = Option.get (Classops.hide_coercion coe_ref) in + mkPApp (Pattern.PRef coe_ref) n_imps [|hp|] + with _ -> + errorstrm (str "need explicit coercion " ++ pr_constr coe ++ spc () + ++ str "to interpret head search pattern as type") in + filter_head, List.fold_left coerce hpat' coe_path + +let rec interp_head_pat hpat = + let filter_head, p = coerce_search_pattern_to_sort hpat in + let rec loop c = match kind_of_term c with + | Cast (c', _, _) -> loop c' + | Prod (_, _, c') -> loop c' + | LetIn (_, _, _, c') -> loop c' + | _ -> Constr_matching.is_matching (Global.env()) Evd.empty p c in + filter_head, loop + +let all_true _ = true + +let rec interp_search_about args accu = match args with +| [] -> accu +| (flag, arg) :: rem -> + fun gr env typ -> + let ans = Search.search_about_filter arg gr env typ in + (if flag then ans else not ans) && interp_search_about rem accu gr env typ + +let interp_search_arg arg = + let arg = List.map (fun (x,arg) -> x, match arg with + | RGlobSearchString (loc,s,key) -> + if is_ident_part s then Search.GlobSearchString s else + interp_search_notation loc s key + | RGlobSearchSubPattern p -> + try + let intern = Constrintern.intern_constr_pattern in + Search.GlobSearchSubPattern (snd (intern (Global.env()) p)) + with e -> let e = Errors.push e in iraise (Cerrors.process_vernac_interp_error e)) arg in + let hpat, a1 = match arg with + | (_, Search.GlobSearchSubPattern (Pattern.PMeta _)) :: a' -> all_true, a' + | (true, Search.GlobSearchSubPattern p) :: a' -> + let filter_head, p = interp_head_pat p in + if filter_head then p, a' else all_true, arg + | _ -> all_true, arg in + let is_string = + function (_, Search.GlobSearchString _) -> true | _ -> false in + let a2, a3 = List.partition is_string a1 in + interp_search_about (a2 @ a3) (fun gr env typ -> hpat typ) + +(* Module path postfilter *) + +let pr_modloc (b, m) = if b then str "-" ++ pr_reference m else pr_reference m + +let wit_ssrmodloc = add_genarg "ssrmodloc" pr_modloc + +let pr_ssr_modlocs _ _ _ ml = + if ml = [] then str "" else spc () ++ str "in " ++ pr_list spc pr_modloc ml + +ARGUMENT EXTEND ssr_modlocs TYPED AS ssrmodloc list PRINTED BY pr_ssr_modlocs + | [ ] -> [ [] ] +END + +GEXTEND Gram + GLOBAL: ssr_modlocs; + modloc: [[ "-"; m = global -> true, m | m = global -> false, m]]; + ssr_modlocs: [[ "in"; ml = LIST1 modloc -> ml ]]; +END + +let interp_modloc mr = + let interp_mod (_, mr) = + let (loc, qid) = qualid_of_reference mr in + try Nametab.full_name_module qid with Not_found -> + Errors.user_err_loc (loc, "interp_modloc", str "No Module " ++ pr_qualid qid) in + let mr_out, mr_in = List.partition fst mr in + let interp_bmod b = function + | [] -> fun _ _ _ -> true + | rmods -> Search.module_filter (List.map interp_mod rmods, b) in + let is_in = interp_bmod false mr_in and is_out = interp_bmod true mr_out in + fun gr env typ -> is_in gr env typ && is_out gr env typ + +(* The unified, extended vernacular "Search" command *) + +let ssrdisplaysearch gr env t = + let pr_res = pr_global gr ++ spc () ++ str " " ++ pr_lconstr_env env Evd.empty t in + msg (hov 2 pr_res ++ fnl ()) + +VERNAC COMMAND EXTEND SsrSearchPattern CLASSIFIED AS QUERY +| [ "Search" ssr_search_arg(a) ssr_modlocs(mr) ] -> + [ let hpat = interp_search_arg a in + let in_mod = interp_modloc mr in + let post_filter gr env typ = in_mod gr env typ && hpat gr env typ in + let display gr env typ = + if post_filter gr env typ then ssrdisplaysearch gr env typ + in + Search.generic_search None display ] +END + +(* }}} *) + +(** Alternative notations for "match" and anonymous arguments. {{{ ************) + +(* Syntax: *) +(* if <term> is <pattern> then ... else ... *) +(* if <term> is <pattern> [in ..] return ... then ... else ... *) +(* let: <pattern> := <term> in ... *) +(* let: <pattern> [in ...] := <term> return ... in ... *) +(* The scope of a top-level 'as' in the pattern extends over the *) +(* 'return' type (dependent if/let). *) +(* Note that the optional "in ..." appears next to the <pattern> *) +(* rather than the <term> in then "let:" syntax. The alternative *) +(* would lead to ambiguities in, e.g., *) +(* let: p1 := (*v---INNER LET:---v *) *) +(* let: p2 := let: p3 := e3 in k return t in k2 in k1 return t' *) +(* in b (*^--ALTERNATIVE INNER LET--------^ *) *) + +(* Caveat : There is no pretty-printing support, since this would *) +(* require a modification to the Coq kernel (adding a new match *) +(* display style -- why aren't these strings?); also, the v8.1 *) +(* pretty-printer only allows extension hooks for printing *) +(* integer or string literals. *) +(* Also note that in the v8 grammar "is" needs to be a keyword; *) +(* as this can't be done from an ML extension file, the new *) +(* syntax will only work when ssreflect.v is imported. *) + +let no_ct = None, None and no_rt = None in +let aliasvar = function + | [_, [CPatAlias (loc, _, id)]] -> Some (loc,Name id) + | _ -> None in +let mk_cnotype mp = aliasvar mp, None in +let mk_ctype mp t = aliasvar mp, Some t in +let mk_rtype t = Some t in +let mk_dthen loc (mp, ct, rt) c = (loc, mp, c), ct, rt in +let mk_let loc rt ct mp c1 = + CCases (loc, LetPatternStyle, rt, ct, [loc, mp, c1]) in +GEXTEND Gram + GLOBAL: binder_constr; + ssr_rtype: [[ "return"; t = operconstr LEVEL "100" -> mk_rtype t ]]; + ssr_mpat: [[ p = pattern -> [!@loc, [p]] ]]; + ssr_dpat: [ + [ mp = ssr_mpat; "in"; t = pattern; rt = ssr_rtype -> mp, mk_ctype mp t, rt + | mp = ssr_mpat; rt = ssr_rtype -> mp, mk_cnotype mp, rt + | mp = ssr_mpat -> mp, no_ct, no_rt + ] ]; + ssr_dthen: [[ dp = ssr_dpat; "then"; c = lconstr -> mk_dthen !@loc dp c ]]; + ssr_elsepat: [[ "else" -> [!@loc, [CPatAtom (!@loc, None)]] ]]; + ssr_else: [[ mp = ssr_elsepat; c = lconstr -> !@loc, mp, c ]]; + binder_constr: [ + [ "if"; c = operconstr LEVEL "200"; "is"; db1 = ssr_dthen; b2 = ssr_else -> + let b1, ct, rt = db1 in CCases (!@loc, MatchStyle, rt, [c, ct], [b1; b2]) + | "if"; c = operconstr LEVEL "200";"isn't";db1 = ssr_dthen; b2 = ssr_else -> + let b1, ct, rt = db1 in + let b1, b2 = + let (l1, p1, r1), (l2, p2, r2) = b1, b2 in (l1, p1, r2), (l2, p2, r1) in + CCases (!@loc, MatchStyle, rt, [c, ct], [b1; b2]) + | "let"; ":"; mp = ssr_mpat; ":="; c = lconstr; "in"; c1 = lconstr -> + mk_let (!@loc) no_rt [c, no_ct] mp c1 + | "let"; ":"; mp = ssr_mpat; ":="; c = lconstr; + rt = ssr_rtype; "in"; c1 = lconstr -> + mk_let (!@loc) rt [c, mk_cnotype mp] mp c1 + | "let"; ":"; mp = ssr_mpat; "in"; t = pattern; ":="; c = lconstr; + rt = ssr_rtype; "in"; c1 = lconstr -> + mk_let (!@loc) rt [c, mk_ctype mp t] mp c1 + ] ]; +END + +GEXTEND Gram + GLOBAL: closed_binder; + closed_binder: [ + [ ["of" | "&"]; c = operconstr LEVEL "99" -> + [LocalRawAssum ([!@loc, Anonymous], Default Explicit, c)] + ] ]; +END +(* }}} *) + +(** Tacticals (+, -, *, done, by, do, =>, first, and last). {{{ ***************) + +(** Bracketing tactical *) + +(* The tactic pretty-printer doesn't know that some extended tactics *) +(* are actually tacticals. To prevent it from improperly removing *) +(* parentheses we override the parsing rule for bracketed tactic *) +(* expressions so that the pretty-print always reflects the input. *) +(* (Removing user-specified parentheses is dubious anyway). *) + +GEXTEND Gram + GLOBAL: tactic_expr; + ssrparentacarg: [[ "("; tac = tactic_expr; ")" -> !@loc, Tacexp tac ]]; + tactic_expr: LEVEL "0" [[ arg = ssrparentacarg -> TacArg arg ]]; +END + +(** The internal "done" and "ssrautoprop" tactics. *) + +(* For additional flexibility, "done" and "ssrautoprop" are *) +(* defined in Ltac. *) +(* Although we provide a default definition in ssreflect, *) +(* we look up the definition dynamically at each call point, *) +(* to allow for user extensions. "ssrautoprop" defaults to *) +(* trivial. *) + +let donetac gl = + let tacname = + try Nametab.locate_tactic (qualid_of_ident (id_of_string "done")) + with Not_found -> try Nametab.locate_tactic (ssrqid "done") + with Not_found -> Errors.error "The ssreflect library was not loaded" in + let tacexpr = dummy_loc, Tacexpr.Reference (ArgArg (dummy_loc, tacname)) in + Proofview.V82.of_tactic (eval_tactic (Tacexpr.TacArg tacexpr)) gl + +let prof_donetac = mk_profiler "donetac";; +let donetac gl = prof_donetac.profile donetac gl;; + +let ssrautoprop gl = + try + let tacname = + try Nametab.locate_tactic (qualid_of_ident (id_of_string "ssrautoprop")) + with Not_found -> Nametab.locate_tactic (ssrqid "ssrautoprop") in + let tacexpr = dummy_loc, Tacexpr.Reference (ArgArg (dummy_loc, tacname)) in + Proofview.V82.of_tactic (eval_tactic (Tacexpr.TacArg tacexpr)) gl + with Not_found -> Proofview.V82.of_tactic (Auto.full_trivial []) gl + +let () = ssrautoprop_tac := ssrautoprop + +let tclBY tac = tclTHEN tac donetac + +(** Tactical arguments. *) + +(* We have four kinds: simple tactics, [|]-bracketed lists, hints, and swaps *) +(* The latter two are used in forward-chaining tactics (have, suffice, wlog) *) +(* and subgoal reordering tacticals (; first & ; last), respectively. *) + +(* Force use of the tactic_expr parsing entry, to rule out tick marks. *) +let pr_ssrtacarg _ _ prt = prt tacltop +ARGUMENT EXTEND ssrtacarg TYPED AS tactic PRINTED BY pr_ssrtacarg +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END +GEXTEND Gram + GLOBAL: ssrtacarg; + ssrtacarg: [[ tac = tactic_expr LEVEL "5" -> tac ]]; +END + +(* Lexically closed tactic for tacticals. *) +let pr_ssrtclarg _ _ prt tac = prt tacltop tac +ARGUMENT EXTEND ssrtclarg TYPED AS ssrtacarg + PRINTED BY pr_ssrtclarg +| [ ssrtacarg(tac) ] -> [ tac ] +END +let eval_tclarg ist tac = ssrevaltac ist tac + +let pr_ortacs prt = + let rec pr_rec = function + | [None] -> spc() ++ str "|" ++ spc() + | None :: tacs -> spc() ++ str "|" ++ pr_rec tacs + | Some tac :: tacs -> spc() ++ str "| " ++ prt tacltop tac ++ pr_rec tacs + | [] -> mt() in + function + | [None] -> spc() + | None :: tacs -> pr_rec tacs + | Some tac :: tacs -> prt tacltop tac ++ pr_rec tacs + | [] -> mt() +let pr_ssrortacs _ _ = pr_ortacs + +ARGUMENT EXTEND ssrortacs TYPED AS tactic option list PRINTED BY pr_ssrortacs +| [ ssrtacarg(tac) "|" ssrortacs(tacs) ] -> [ Some tac :: tacs ] +| [ ssrtacarg(tac) "|" ] -> [ [Some tac; None] ] +| [ ssrtacarg(tac) ] -> [ [Some tac] ] +| [ "|" ssrortacs(tacs) ] -> [ None :: tacs ] +| [ "|" ] -> [ [None; None] ] +END + +let pr_hintarg prt = function + | true, tacs -> hv 0 (str "[ " ++ pr_ortacs prt tacs ++ str " ]") + | false, [Some tac] -> prt tacltop tac + | _, _ -> mt() + +let pr_ssrhintarg _ _ = pr_hintarg + +let mk_hint tac = false, [Some tac] +let mk_orhint tacs = true, tacs +let nullhint = true, [] +let nohint = false, [] + +ARGUMENT EXTEND ssrhintarg TYPED AS bool * ssrortacs PRINTED BY pr_ssrhintarg +| [ "[" "]" ] -> [ nullhint ] +| [ "[" ssrortacs(tacs) "]" ] -> [ mk_orhint tacs ] +| [ ssrtacarg(arg) ] -> [ mk_hint arg ] +END + +ARGUMENT EXTEND ssrortacarg TYPED AS ssrhintarg PRINTED BY pr_ssrhintarg +| [ "[" ssrortacs(tacs) "]" ] -> [ mk_orhint tacs ] +END + +let hinttac ist is_by (is_or, atacs) = + let dtac = if is_by then donetac else tclIDTAC in + let mktac = function + | Some atac -> tclTHEN (ssrevaltac ist atac) dtac + | _ -> dtac in + match List.map mktac atacs with + | [] -> if is_or then dtac else tclIDTAC + | [tac] -> tac + | tacs -> tclFIRST tacs + +(** The "-"/"+"/"*" tacticals. *) + +(* These are just visual cues to flag the beginning of the script for *) +(* new subgoals, when indentation is not appropriate (typically after *) +(* tactics that generate more than two subgoals). *) + +TACTIC EXTEND ssrtclplus +| [ "YouShouldNotTypeThis" "+" ssrtclarg(arg) ] -> [ Proofview.V82.tactic (eval_tclarg ist arg) ] +END +set_pr_ssrtac "tclplus" 5 [ArgSep "+ "; ArgSsr "tclarg"] + +TACTIC EXTEND ssrtclminus +| [ "YouShouldNotTypeThis" "-" ssrtclarg(arg) ] -> [ Proofview.V82.tactic (eval_tclarg ist arg) ] +END +set_pr_ssrtac "tclminus" 5 [ArgSep "- "; ArgSsr "tclarg"] + +TACTIC EXTEND ssrtclstar +| [ "YouShouldNotTypeThis" "*" ssrtclarg(arg) ] -> [ Proofview.V82.tactic (eval_tclarg ist arg) ] +END +set_pr_ssrtac "tclstar" 5 [ArgSep "- "; ArgSsr "tclarg"] + +let gen_tclarg = in_gen (rawwit wit_ssrtclarg) + +GEXTEND Gram + GLOBAL: tactic tactic_mode; + tactic: [ + [ "+"; tac = ssrtclarg -> ssrtac_expr !@loc "tclplus" [gen_tclarg tac] + | "-"; tac = ssrtclarg -> ssrtac_expr !@loc "tclminus" [gen_tclarg tac] + | "*"; tac = ssrtclarg -> ssrtac_expr !@loc "tclstar" [gen_tclarg tac] + ] ]; + tactic_mode: [ + [ "+"; tac = G_vernac.subgoal_command -> tac None + | "-"; tac = G_vernac.subgoal_command -> tac None + | "*"; tac = G_vernac.subgoal_command -> tac None + ] ]; +END + +(** The "by" tactical. *) + +let pr_hint prt arg = + if arg = nohint then mt() else str "by " ++ pr_hintarg prt arg +let pr_ssrhint _ _ = pr_hint + +ARGUMENT EXTEND ssrhint TYPED AS ssrhintarg PRINTED BY pr_ssrhint +| [ ] -> [ nohint ] +END + +TACTIC EXTEND ssrtclby +| [ "YouShouldNotTypeThis" ssrhint(tac) ] -> [ Proofview.V82.tactic (hinttac ist true tac) ] +END +set_pr_ssrtac "tclby" 0 [ArgSsr "hint"] + +(* We can't parse "by" in ARGUMENT EXTEND because it will only be made *) +(* into a keyword in ssreflect.v; so we anticipate this in GEXTEND. *) + +GEXTEND Gram + GLOBAL: ssrhint simple_tactic; + ssrhint: [[ "by"; arg = ssrhintarg -> arg ]]; + simple_tactic: [ + [ "by"; arg = ssrhintarg -> + let garg = in_gen (rawwit wit_ssrhint) arg in + ssrtac_atom !@loc "tclby" [garg] + ] ]; +END +(* }}} *) + +(** Bound assumption argument *) + +(* The Ltac API does have a type for assumptions but it is level-dependent *) +(* and therefore impratical to use for complex arguments, so we substitute *) +(* our own to have a uniform representation. Also, we refuse to intern *) +(* idents that match global/section constants, since this would lead to *) +(* fragile Ltac scripts. *) + +type ssrhyp = SsrHyp of loc * identifier + +let hyp_id (SsrHyp (_, id)) = id +let pr_hyp (SsrHyp (_, id)) = pr_id id +let pr_ssrhyp _ _ _ = pr_hyp + +let wit_ssrhyprep = add_genarg "ssrhyprep" pr_hyp + +let hyp_err loc msg id = + Errors.user_err_loc (loc, "ssrhyp", str msg ++ pr_id id) + +let intern_hyp ist (SsrHyp (loc, id) as hyp) = + let _ = Tacintern.intern_genarg ist (in_gen (rawwit wit_var) (loc, id)) in + if not_section_id id then hyp else + hyp_err loc "Can't clear section hypothesis " id + +let interp_hyp ist gl (SsrHyp (loc, id)) = + let s, id' = interp_wit wit_var ist gl (loc, id) in + if not_section_id id' then s, SsrHyp (loc, id') else + hyp_err loc "Can't clear section hypothesis " id' + +ARGUMENT EXTEND ssrhyp TYPED AS ssrhyprep PRINTED BY pr_ssrhyp + INTERPRETED BY interp_hyp + GLOBALIZED BY intern_hyp + | [ ident(id) ] -> [ SsrHyp (loc, id) ] +END + +type ssrhyp_or_id = Hyp of ssrhyp | Id of ssrhyp + +let hoik f = function Hyp x -> f x | Id x -> f x +let hoi_id = hoik hyp_id +let pr_hoi = hoik pr_hyp +let pr_ssrhoi _ _ _ = pr_hoi + +let wit_ssrhoirep = add_genarg "ssrhoirep" pr_hoi + +let intern_ssrhoi ist = function + | Hyp h -> Hyp (intern_hyp ist h) + | Id (SsrHyp (_, id)) as hyp -> + let _ = Tacintern.intern_genarg ist (in_gen (rawwit wit_ident) id) in + hyp + +let interp_ssrhoi ist gl = function + | Hyp h -> let s, h' = interp_hyp ist gl h in s, Hyp h' + | Id (SsrHyp (loc, id)) -> + let s, id' = interp_wit wit_ident ist gl id in + s, Id (SsrHyp (loc, id')) + +ARGUMENT EXTEND ssrhoi_hyp TYPED AS ssrhoirep PRINTED BY pr_ssrhoi + INTERPRETED BY interp_ssrhoi + GLOBALIZED BY intern_ssrhoi + | [ ident(id) ] -> [ Hyp (SsrHyp(loc, id)) ] +END +ARGUMENT EXTEND ssrhoi_id TYPED AS ssrhoirep PRINTED BY pr_ssrhoi + INTERPRETED BY interp_ssrhoi + GLOBALIZED BY intern_ssrhoi + | [ ident(id) ] -> [ Id (SsrHyp(loc, id)) ] +END + +type ssrhyps = ssrhyp list + +let pr_hyps = pr_list pr_spc pr_hyp +let pr_ssrhyps _ _ _ = pr_hyps +let hyps_ids = List.map hyp_id + +let rec check_hyps_uniq ids = function + | SsrHyp (loc, id) :: _ when List.mem id ids -> + hyp_err loc "Duplicate assumption " id + | SsrHyp (_, id) :: hyps -> check_hyps_uniq (id :: ids) hyps + | [] -> () + +let check_hyp_exists hyps (SsrHyp(_, id)) = + try ignore(Context.lookup_named id hyps) + with Not_found -> errorstrm (str"No assumption is named " ++ pr_id id) + +let interp_hyps ist gl ghyps = + let hyps = List.map snd (List.map (interp_hyp ist gl) ghyps) in + check_hyps_uniq [] hyps; Tacmach.project gl, hyps + +ARGUMENT EXTEND ssrhyps TYPED AS ssrhyp list PRINTED BY pr_ssrhyps + INTERPRETED BY interp_hyps + | [ ssrhyp_list(hyps) ] -> [ check_hyps_uniq [] hyps; hyps ] +END + +(** Terms parsing. {{{ ********************************************************) + +(* Because we allow wildcards in term references, we need to stage the *) +(* interpretation of terms so that it occurs at the right time during *) +(* the execution of the tactic (e.g., so that we don't report an error *) +(* for a term that isn't actually used in the execution). *) +(* The term representation tracks whether the concrete initial term *) +(* started with an opening paren, which might avoid a conflict between *) +(* the ssrreflect term syntax and Gallina notation. *) + +(* kinds of terms *) + +type ssrtermkind = char (* print flag *) + +let input_ssrtermkind strm = match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "(" -> '(' + | Tok.KEYWORD "@" -> '@' + | _ -> ' ' + +let ssrtermkind = Gram.Entry.of_parser "ssrtermkind" input_ssrtermkind + +let id_of_Cterm t = match id_of_cpattern t with + | Some x -> x + | None -> loc_error (loc_of_cpattern t) "Only identifiers are allowed here" +let ssrhyp_of_ssrterm = function + | k, (_, Some c) as o -> + SsrHyp (constr_loc c, id_of_Cterm (cpattern_of_term o)), String.make 1 k + | _, (_, None) -> assert false + +(* terms *) +let pr_ssrterm _ _ _ = pr_term +let pf_intern_term ist gl (_, c) = glob_constr ist (pf_env gl) c +let intern_term ist sigma env (_, c) = glob_constr ist env c +let interp_term ist gl (_, c) = snd (interp_open_constr ist gl c) +let force_term ist gl (_, c) = interp_constr ist gl c +let glob_ssrterm gs = function + | k, (_, Some c) -> k, Tacintern.intern_constr gs c + | ct -> ct +let subst_ssrterm s (k, c) = k, Tacsubst.subst_glob_constr_and_expr s c +let interp_ssrterm _ gl t = Tacmach.project gl, t + +ARGUMENT EXTEND ssrterm + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_ssrterm SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "YouShouldNotTypeThis" constr(c) ] -> [ mk_lterm c ] +END + +GEXTEND Gram + GLOBAL: ssrterm; + ssrterm: [[ k = ssrtermkind; c = constr -> mk_term k c ]]; +END +(* }}} *) + +(** The "in" pseudo-tactical {{{ **********************************************) + +(* We can't make "in" into a general tactical because this would create a *) +(* crippling conflict with the ltac let .. in construct. Hence, we add *) +(* explicitly an "in" suffix to all the extended tactics for which it is *) +(* relevant (including move, case, elim) and to the extended do tactical *) +(* below, which yields a general-purpose "in" of the form do [...] in ... *) + +(* This tactical needs to come before the intro tactics because the latter *) +(* must take precautions in order not to interfere with the discharged *) +(* assumptions. This is especially difficult for discharged "let"s, which *) +(* the default simpl and unfold tactics would erase blindly. *) + +(** Clear switch *) + +type ssrclear = ssrhyps + +let pr_clear_ne clr = str "{" ++ pr_hyps clr ++ str "}" +let pr_clear sep clr = if clr = [] then mt () else sep () ++ pr_clear_ne clr + +let pr_ssrclear _ _ _ = pr_clear mt + +ARGUMENT EXTEND ssrclear_ne TYPED AS ssrhyps PRINTED BY pr_ssrclear +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ check_hyps_uniq [] clr; clr ] +END + +ARGUMENT EXTEND ssrclear TYPED AS ssrclear_ne PRINTED BY pr_ssrclear +| [ ssrclear_ne(clr) ] -> [ clr ] +| [ ] -> [ [] ] +END + +let cleartac clr = check_hyps_uniq [] clr; clear (hyps_ids clr) + +(* type ssrwgen = ssrclear * ssrhyp * string *) + +let pr_wgen = function + | (clr, Some((id,k),None)) -> spc() ++ pr_clear mt clr ++ str k ++ pr_hoi id + | (clr, Some((id,k),Some p)) -> + spc() ++ pr_clear mt clr ++ str"(" ++ str k ++ pr_hoi id ++ str ":=" ++ + pr_cpattern p ++ str ")" + | (clr, None) -> spc () ++ pr_clear mt clr +let pr_ssrwgen _ _ _ = pr_wgen + +(* no globwith for char *) +ARGUMENT EXTEND ssrwgen + TYPED AS ssrclear * ((ssrhoi_hyp * string) * cpattern option) option + PRINTED BY pr_ssrwgen +| [ ssrclear_ne(clr) ] -> [ clr, None ] +| [ ssrhoi_hyp(hyp) ] -> [ [], Some((hyp, " "), None) ] +| [ "@" ssrhoi_hyp(hyp) ] -> [ [], Some((hyp, "@"), None) ] +| [ "(" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id," "),Some p) ] +| [ "(" ssrhoi_id(id) ")" ] -> [ [], Some ((id,"("), None) ] +| [ "(@" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id,"@"),Some p) ] +| [ "(" "@" ssrhoi_id(id) ":=" lcpattern(p) ")" ] -> + [ [], Some ((id,"@"),Some p) ] +END + +type ssrclseq = InGoal | InHyps + | InHypsGoal | InHypsSeqGoal | InSeqGoal | InHypsSeq | InAll | InAllHyps + +let pr_clseq = function + | InGoal | InHyps -> mt () + | InSeqGoal -> str "|- *" + | InHypsSeqGoal -> str " |- *" + | InHypsGoal -> str " *" + | InAll -> str "*" + | InHypsSeq -> str " |-" + | InAllHyps -> str "* |-" + +let wit_ssrclseq = add_genarg "ssrclseq" pr_clseq +let pr_clausehyps = pr_list pr_spc pr_wgen +let pr_ssrclausehyps _ _ _ = pr_clausehyps + +ARGUMENT EXTEND ssrclausehyps +TYPED AS ssrwgen list PRINTED BY pr_ssrclausehyps +| [ ssrwgen(hyp) "," ssrclausehyps(hyps) ] -> [ hyp :: hyps ] +| [ ssrwgen(hyp) ssrclausehyps(hyps) ] -> [ hyp :: hyps ] +| [ ssrwgen(hyp) ] -> [ [hyp] ] +END + +(* type ssrclauses = ssrahyps * ssrclseq *) + +let pr_clauses (hyps, clseq) = + if clseq = InGoal then mt () + else str "in " ++ pr_clausehyps hyps ++ pr_clseq clseq +let pr_ssrclauses _ _ _ = pr_clauses + +ARGUMENT EXTEND ssrclauses TYPED AS ssrwgen list * ssrclseq + PRINTED BY pr_ssrclauses + | [ "in" ssrclausehyps(hyps) "|-" "*" ] -> [ hyps, InHypsSeqGoal ] + | [ "in" ssrclausehyps(hyps) "|-" ] -> [ hyps, InHypsSeq ] + | [ "in" ssrclausehyps(hyps) "*" ] -> [ hyps, InHypsGoal ] + | [ "in" ssrclausehyps(hyps) ] -> [ hyps, InHyps ] + | [ "in" "|-" "*" ] -> [ [], InSeqGoal ] + | [ "in" "*" ] -> [ [], InAll ] + | [ "in" "*" "|-" ] -> [ [], InAllHyps ] + | [ ] -> [ [], InGoal ] +END + +let nohide = mkRel 0 +let hidden_goal_tag = "the_hidden_goal" + +(* Reduction that preserves the Prod/Let spine of the "in" tactical. *) + +let inc_safe n = if n = 0 then n else n + 1 +let rec safe_depth c = match kind_of_term c with +| LetIn (Name x, _, _, c') when is_discharged_id x -> safe_depth c' + 1 +| LetIn (_, _, _, c') | Prod (_, _, c') -> inc_safe (safe_depth c') +| _ -> 0 + +let red_safe r e s c0 = + let rec red_to e c n = match kind_of_term c with + | Prod (x, t, c') when n > 0 -> + let t' = r e s t in let e' = Environ.push_rel (x, None, t') e in + mkProd (x, t', red_to e' c' (n - 1)) + | LetIn (x, b, t, c') when n > 0 -> + let t' = r e s t in let e' = Environ.push_rel (x, None, t') e in + mkLetIn (x, r e s b, t', red_to e' c' (n - 1)) + | _ -> r e s c in + red_to e c0 (safe_depth c0) + +let check_wgen_uniq gens = + let clears = List.flatten (List.map fst gens) in + check_hyps_uniq [] clears; + let ids = CList.map_filter + (function (_,Some ((id,_),_)) -> Some (hoi_id id) | _ -> None) gens in + let rec check ids = function + | id :: _ when List.mem id ids -> + errorstrm (str"Duplicate generalization " ++ pr_id id) + | id :: hyps -> check (id :: ids) hyps + | [] -> () in + check [] ids + +let pf_clauseids gl gens clseq = + let keep_clears = List.map (fun x, _ -> x, None) in + if gens <> [] then (check_wgen_uniq gens; gens) else + if clseq <> InAll && clseq <> InAllHyps then keep_clears gens else + Errors.error "assumptions should be named explicitly" + +let hidden_clseq = function InHyps | InHypsSeq | InAllHyps -> true | _ -> false + +let hidetacs clseq idhide cl0 = + if not (hidden_clseq clseq) then [] else + [posetac idhide cl0; + Proofview.V82.of_tactic (convert_concl_no_check (mkVar idhide))] + +let discharge_hyp (id', (id, mode)) gl = + let cl' = subst_var id (pf_concl gl) in + match pf_get_hyp gl id, mode with + | (_, None, t), _ | (_, Some _, t), "(" -> + apply_type (mkProd (Name id', t, cl')) [mkVar id] gl + | (_, Some v, t), _ -> + Proofview.V82.of_tactic (convert_concl (mkLetIn (Name id', v, t, cl'))) gl + +let endclausestac id_map clseq gl_id cl0 gl = + let not_hyp' id = not (List.mem_assoc id id_map) in + let orig_id id = try List.assoc id id_map with _ -> id in + let dc, c = Term.decompose_prod_assum (pf_concl gl) in + let hide_goal = hidden_clseq clseq in + let c_hidden = hide_goal && c = mkVar gl_id in + let rec fits forced = function + | (id, _) :: ids, (Name id', _, _) :: dc' when id' = id -> + fits true (ids, dc') + | ids, dc' -> + forced && ids = [] && (not hide_goal || dc' = [] && c_hidden) in + let rec unmark c = match kind_of_term c with + | Var id when hidden_clseq clseq && id = gl_id -> cl0 + | Prod (Name id, t, c') when List.mem_assoc id id_map -> + mkProd (Name (orig_id id), unmark t, unmark c') + | LetIn (Name id, v, t, c') when List.mem_assoc id id_map -> + mkLetIn (Name (orig_id id), unmark v, unmark t, unmark c') + | _ -> map_constr unmark c in + let utac hyp = + Proofview.V82.of_tactic + (convert_hyp_no_check (map_named_declaration unmark hyp)) in + let utacs = List.map utac (pf_hyps gl) in + let ugtac gl' = + Proofview.V82.of_tactic + (convert_concl_no_check (unmark (pf_concl gl'))) gl' in + let ctacs = if hide_goal then [clear [gl_id]] else [] in + let mktac itacs = tclTHENLIST (itacs @ utacs @ ugtac :: ctacs) in + let itac (_, id) = Proofview.V82.of_tactic (introduction id) in + if fits false (id_map, List.rev dc) then mktac (List.map itac id_map) gl else + let all_ids = ids_of_rel_context dc @ pf_ids_of_hyps gl in + if List.for_all not_hyp' all_ids && not c_hidden then mktac [] gl else + Errors.error "tampering with discharged assumptions of \"in\" tactical" + +let is_id_constr c = match kind_of_term c with + | Lambda(_,_,c) when isRel c -> 1 = destRel c + | _ -> false + +let red_product_skip_id env sigma c = match kind_of_term c with + | App(hd,args) when Array.length args = 1 && is_id_constr hd -> args.(0) + | _ -> try Tacred.red_product env sigma c with _ -> c + +let abs_wgen keep_let ist f gen (gl,args,c) = + let sigma, env = project gl, pf_env gl in + let evar_closed t p = + if occur_existential t then + Errors.user_err_loc (loc_of_cpattern p,"ssreflect", + pr_constr_pat t ++ + str" contains holes and matches no subterm of the goal") in + match gen with + | _, Some ((x, mode), None) when mode = "@" || (mode = " " && keep_let) -> + let x = hoi_id x in + let _, bo, ty = pf_get_hyp gl x in + gl, + (if bo <> None then args else mkVar x :: args), + mkProd_or_LetIn (Name (f x),bo,ty) (subst_var x c) + | _, Some ((x, _), None) -> + let x = hoi_id x in + gl, mkVar x :: args, mkProd (Name (f x), pf_get_hyp_typ gl x, subst_var x c) + | _, Some ((x, "@"), Some p) -> + let x = hoi_id x in + let cp = interp_cpattern ist gl p None in + let (t, ucst), c = + try fill_occ_pattern ~raise_NoMatch:true env sigma c cp None 1 + with NoMatch -> redex_of_pattern env cp, c in + evar_closed t p; + let ut = red_product_skip_id env sigma t in + pf_merge_uc ucst gl, args, mkLetIn(Name (f x), ut, pf_type_of gl t, c) + | _, Some ((x, _), Some p) -> + let x = hoi_id x in + let cp = interp_cpattern ist gl p None in + let (t, ucst), c = + try fill_occ_pattern ~raise_NoMatch:true env sigma c cp None 1 + with NoMatch -> redex_of_pattern env cp, c in + evar_closed t p; + pf_merge_uc ucst gl, t :: args, mkProd(Name (f x), pf_type_of gl t, c) + | _ -> gl, args, c + +let clr_of_wgen gen clrs = match gen with + | clr, Some ((x, _), None) -> + let x = hoi_id x in + cleartac clr :: cleartac [SsrHyp(Loc.ghost,x)] :: clrs + | clr, _ -> cleartac clr :: clrs + +let tclCLAUSES ist tac (gens, clseq) gl = + if clseq = InGoal || clseq = InSeqGoal then tac gl else + let clr_gens = pf_clauseids gl gens clseq in + let clear = tclTHENLIST (List.rev(List.fold_right clr_of_wgen clr_gens [])) in + let gl_id = mk_anon_id hidden_goal_tag gl in + let cl0 = pf_concl gl in + let dtac gl = + let c = pf_concl gl in + let gl, args, c = + List.fold_right (abs_wgen true ist mk_discharged_id) gens (gl,[], c) in + apply_type c args gl in + let endtac = + let id_map = CList.map_filter (function + | _, Some ((x,_),_) -> let id = hoi_id x in Some (mk_discharged_id id, id) + | _, None -> None) gens in + endclausestac id_map clseq gl_id cl0 in + tclTHENLIST (hidetacs clseq gl_id cl0 @ [dtac; clear; tac; endtac]) gl +(* }}} *) + +(** Simpl switch *) + +type ssrsimpl = Simpl | Cut | SimplCut | Nop + +let pr_simpl = function + | Simpl -> str "/=" + | Cut -> str "//" + | SimplCut -> str "//=" + | Nop -> mt () + +let pr_ssrsimpl _ _ _ = pr_simpl + +let wit_ssrsimplrep = add_genarg "ssrsimplrep" pr_simpl + +ARGUMENT EXTEND ssrsimpl_ne TYPED AS ssrsimplrep PRINTED BY pr_ssrsimpl +| [ "/=" ] -> [ Simpl ] +| [ "//" ] -> [ Cut ] +| [ "//=" ] -> [ SimplCut ] +END + +ARGUMENT EXTEND ssrsimpl TYPED AS ssrsimplrep PRINTED BY pr_ssrsimpl +| [ ssrsimpl_ne(sim) ] -> [ sim ] +| [ ] -> [ Nop ] +END + +(* We must avoid zeta-converting any "let"s created by the "in" tactical. *) + +let safe_simpltac gl = + let cl' = red_safe Tacred.simpl (pf_env gl) (project gl) (pf_concl gl) in + Proofview.V82.of_tactic (convert_concl_no_check cl') gl + +let simpltac = function + | Simpl -> safe_simpltac + | Cut -> tclTRY donetac + | SimplCut -> tclTHEN safe_simpltac (tclTRY donetac) + | Nop -> tclIDTAC + +(** Rewriting direction *) + +let pr_dir = function L2R -> str "->" | R2L -> str "<-" +let pr_rwdir = function L2R -> mt() | R2L -> str "-" + +let rewritetac dir c = + (* Due to the new optional arg ?tac, application shouldn't be too partial *) + Proofview.V82.of_tactic begin + Equality.general_rewrite (dir = L2R) AllOccurrences true false c + end + +let wit_ssrdir = add_genarg "ssrdir" pr_dir + +let dir_org = function L2R -> 1 | R2L -> 2 + +(** Indexes *) + +(* Since SSR indexes are always positive numbers, we use the 0 value *) +(* to encode an omitted index. We reuse the in or_var type, but we *) +(* supply our own interpretation function, which checks for non *) +(* positive values, and allows the use of constr numerals, so that *) +(* e.g., "let n := eval compute in (1 + 3) in (do n!clear)" works. *) + +type ssrindex = int or_var + +let pr_index = function + | ArgVar (_, id) -> pr_id id + | ArgArg n when n > 0 -> int n + | _ -> mt () +let pr_ssrindex _ _ _ = pr_index + +let noindex = ArgArg 0 +let allocc = Some(false,[]) + +let check_index loc i = + if i > 0 then i else loc_error loc "Index not positive" +let mk_index loc = function ArgArg i -> ArgArg (check_index loc i) | iv -> iv + +let interp_index ist gl idx = + Tacmach.project gl, + match idx with + | ArgArg _ -> idx + | ArgVar (loc, id) -> + let i = + try + let v = Id.Map.find id ist.lfun in + begin match Value.to_int v with + | Some i -> i + | None -> + begin match Value.to_constr v with + | Some c -> + let rc = Detyping.detype false [] (pf_env gl) (project gl) c in + begin match Notation.uninterp_prim_token rc with + | _, Numeral bigi -> int_of_string (Bigint.to_string bigi) + | _ -> raise Not_found + end + | None -> raise Not_found + end end + with _ -> loc_error loc "Index not a number" in + ArgArg (check_index loc i) + +ARGUMENT EXTEND ssrindex TYPED AS int_or_var PRINTED BY pr_ssrindex + INTERPRETED BY interp_index +| [ int_or_var(i) ] -> [ mk_index loc i ] +END + +(** Occurrence switch *) + +(* The standard syntax of complemented occurrence lists involves a single *) +(* initial "-", e.g., {-1 3 5}. An initial *) +(* "+" may be used to indicate positive occurrences (the default). The *) +(* "+" is optional, except if the list of occurrences starts with a *) +(* variable or is empty (to avoid confusion with a clear switch). The *) +(* empty positive switch "{+}" selects no occurrences, while the empty *) +(* negative switch "{-}" selects all occurrences explicitly; this is the *) +(* default, but "{-}" prevents the implicit clear, and can be used to *) +(* force dependent elimination -- see ndefectelimtac below. *) + +type ssrocc = occ + +let pr_occ = function + | Some (true, occ) -> str "{-" ++ pr_list pr_spc int occ ++ str "}" + | Some (false, occ) -> str "{+" ++ pr_list pr_spc int occ ++ str "}" + | None -> str "{}" + +let pr_ssrocc _ _ _ = pr_occ + +ARGUMENT EXTEND ssrocc TYPED AS (bool * int list) option PRINTED BY pr_ssrocc +| [ natural(n) natural_list(occ) ] -> [ + Some (false, List.map (check_index loc) (n::occ)) ] +| [ "-" natural_list(occ) ] -> [ Some (true, occ) ] +| [ "+" natural_list(occ) ] -> [ Some (false, occ) ] +END + +let pf_mkprod gl c ?(name=constr_name c) cl = + let t = pf_type_of gl c in + if name <> Anonymous || noccurn 1 cl then mkProd (name, t, cl) else + mkProd (Name (pf_type_id gl t), t, cl) + +let pf_abs_prod name gl c cl = pf_mkprod gl c ~name (subst_term c cl) + +(** Discharge occ switch (combined occurrence / clear switch *) + +type ssrdocc = ssrclear option * ssrocc option + +let mkocc occ = None, occ +let noclr = mkocc None +let mkclr clr = Some clr, None +let nodocc = mkclr [] + +let pr_docc = function + | None, occ -> pr_occ occ + | Some clr, _ -> pr_clear mt clr + +let pr_ssrdocc _ _ _ = pr_docc + +ARGUMENT EXTEND ssrdocc TYPED AS ssrclear option * ssrocc PRINTED BY pr_ssrdocc +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ mkclr clr ] +| [ "{" ssrocc(occ) "}" ] -> [ mkocc occ ] +END + +(** View hint database and View application. {{{ ******************************) + +(* There are three databases of lemmas used to mediate the application *) +(* of reflection lemmas: one for forward chaining, one for backward *) +(* chaining, and one for secondary backward chaining. *) + +(* View hints *) + +let rec isCxHoles = function (CHole _, None) :: ch -> isCxHoles ch | _ -> false + +let pr_raw_ssrhintref prc _ _ = function + | CAppExpl (_, (None, r,x), args) when isCHoles args -> + prc (CRef (r,x)) ++ str "|" ++ int (List.length args) + | CApp (_, (_, CRef _), _) as c -> prc c + | CApp (_, (_, c), args) when isCxHoles args -> + prc c ++ str "|" ++ int (List.length args) + | c -> prc c + +let pr_rawhintref = function + | GApp (_, f, args) when isRHoles args -> + pr_glob_constr f ++ str "|" ++ int (List.length args) + | c -> pr_glob_constr c + +let pr_glob_ssrhintref _ _ _ (c, _) = pr_rawhintref c + +let pr_ssrhintref prc _ _ = prc + +let mkhintref loc c n = match c with + | CRef (r,x) -> CAppExpl (loc, (None, r, x), mkCHoles loc n) + | _ -> mkAppC (c, mkCHoles loc n) + +ARGUMENT EXTEND ssrhintref + PRINTED BY pr_ssrhintref + RAW_TYPED AS constr RAW_PRINTED BY pr_raw_ssrhintref + GLOB_TYPED AS constr GLOB_PRINTED BY pr_glob_ssrhintref + | [ constr(c) ] -> [ c ] + | [ constr(c) "|" natural(n) ] -> [ mkhintref loc c n ] +END + +(* View purpose *) + +let pr_viewpos = function + | 0 -> str " for move/" + | 1 -> str " for apply/" + | 2 -> str " for apply//" + | _ -> mt () + +let pr_ssrviewpos _ _ _ = pr_viewpos + +ARGUMENT EXTEND ssrviewpos TYPED AS int PRINTED BY pr_ssrviewpos + | [ "for" "move" "/" ] -> [ 0 ] + | [ "for" "apply" "/" ] -> [ 1 ] + | [ "for" "apply" "/" "/" ] -> [ 2 ] + | [ "for" "apply" "//" ] -> [ 2 ] + | [ ] -> [ 3 ] +END + +let pr_ssrviewposspc _ _ _ i = pr_viewpos i ++ spc () + +ARGUMENT EXTEND ssrviewposspc TYPED AS ssrviewpos PRINTED BY pr_ssrviewposspc + | [ ssrviewpos(i) ] -> [ i ] +END + +(* The table and its display command *) + +let viewtab : glob_constr list array = Array.make 3 [] + +let _ = + let init () = Array.fill viewtab 0 3 [] in + let freeze _ = Array.copy viewtab in + let unfreeze vt = Array.blit vt 0 viewtab 0 3 in + Summary.declare_summary "ssrview" + { Summary.freeze_function = freeze; + Summary.unfreeze_function = unfreeze; + Summary.init_function = init } + +let mapviewpos f n k = if n < 3 then f n else for i = 0 to k - 1 do f i done + +let print_view_hints i = + let pp_viewname = str "Hint View" ++ pr_viewpos i ++ str " " in + let pp_hints = pr_list spc pr_rawhintref viewtab.(i) in + ppnl (pp_viewname ++ hov 0 pp_hints ++ Pp.cut ()) + +VERNAC COMMAND EXTEND PrintView CLASSIFIED AS QUERY +| [ "Print" "Hint" "View" ssrviewpos(i) ] -> [ mapviewpos print_view_hints i 3 ] +END + +(* Populating the table *) + +let cache_viewhint (_, (i, lvh)) = + let mem_raw h = List.exists (Notation_ops.eq_glob_constr h) in + let add_hint h hdb = if mem_raw h hdb then hdb else h :: hdb in + viewtab.(i) <- List.fold_right add_hint lvh viewtab.(i) + +let subst_viewhint ( subst, (i, lvh as ilvh)) = + let lvh' = List.smartmap (Detyping.subst_glob_constr subst) lvh in + if lvh' == lvh then ilvh else i, lvh' + +let classify_viewhint x = Libobject.Substitute x + +let in_viewhint = + Libobject.declare_object {(Libobject.default_object "VIEW_HINTS") with + Libobject.open_function = (fun i o -> if i = 1 then cache_viewhint o); + Libobject.cache_function = cache_viewhint; + Libobject.subst_function = subst_viewhint; + Libobject.classify_function = classify_viewhint } + +let glob_view_hints lvh = + List.map (Constrintern.intern_constr (Global.env ())) lvh + +let add_view_hints lvh i = Lib.add_anonymous_leaf (in_viewhint (i, lvh)) + +VERNAC COMMAND EXTEND HintView CLASSIFIED AS QUERY + | [ "Hint" "View" ssrviewposspc(n) ne_ssrhintref_list(lvh) ] -> + [ mapviewpos (add_view_hints (glob_view_hints lvh)) n 2 ] +END + +(** Views *) + +(* Views for the "move" and "case" commands are actually open *) +(* terms, but this is handled by interp_view, which is called *) +(* by interp_casearg. We use lists, to support the *) +(* "double-view" feature of the apply command. *) + +(* type ssrview = ssrterm list *) + +let pr_view = pr_list mt (fun c -> str "/" ++ pr_term c) + +let pr_ssrview _ _ _ = pr_view + +ARGUMENT EXTEND ssrview TYPED AS ssrterm list + PRINTED BY pr_ssrview +| [ "/" constr(c) ] -> [ [mk_term ' ' c] ] +| [ "/" constr(c) ssrview(w) ] -> [ (mk_term ' ' c) :: w ] +END + +(* There are two ways of "applying" a view to term: *) +(* 1- using a view hint if the view is an instance of some *) +(* (reflection) inductive predicate. *) +(* 2- applying the view if it coerces to a function, adding *) +(* implicit arguments. *) +(* They require guessing the view hints and the number of *) +(* implicits, respectively, which we do by brute force. *) + +let view_error s gv = + errorstrm (str ("Cannot " ^ s ^ " view ") ++ pr_term gv) + +let interp_view ist si env sigma gv rid = + match intern_term ist sigma env gv with + | GApp (loc, GHole _, rargs) -> + let rv = GApp (loc, rid, rargs) in + snd (interp_open_constr ist (re_sig si sigma) (rv, None)) + | rv -> + let interp rc rargs = + interp_open_constr ist (re_sig si sigma) (mkRApp rc rargs, None) in + let rec simple_view rargs n = + if n < 0 then view_error "use" gv else + try interp rv rargs with _ -> simple_view (mkRHole :: rargs) (n - 1) in + let view_nbimps = interp_view_nbimps ist (re_sig si sigma) rv in + let view_args = [mkRApp rv (mkRHoles view_nbimps); rid] in + let rec view_with = function + | [] -> simple_view [rid] (interp_nbargs ist (re_sig si sigma) rv) + | hint :: hints -> try interp hint view_args with _ -> view_with hints in + snd (view_with (if view_nbimps < 0 then [] else viewtab.(0))) + +let top_id = mk_internal_id "top assumption" + +let with_view ist si env gl0 c name cl prune = + let c2r ist x = { ist with lfun = + Id.Map.add top_id (Value.of_constr x) ist.lfun } in + let rec loop (sigma, c') = function + | f :: view -> + let rid, ist = match kind_of_term c' with + | Var id -> mkRVar id, ist + | _ -> mkRltacVar top_id, c2r ist c' in + loop (interp_view ist si env sigma f rid) view + | [] -> + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + let c' = Reductionops.nf_evar sigma c' in + let n, c', _, ucst = pf_abs_evars gl0 (sigma, c') in + let c' = if not prune then c' else pf_abs_cterm gl0 n c' in + let gl0 = pf_merge_uc ucst gl0 in + pf_abs_prod name gl0 c' (prod_applist cl [c]), c', pf_merge_uc_of sigma gl0 + in loop + +let pf_with_view ist gl (prune, view) cl c = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + with_view ist si env gl c (constr_name c) cl prune (sigma, c) view +(* }}} *) + +(** Extended intro patterns {{{ ***********************************************) + +type ssrtermrep = char * glob_constr_and_expr +type ssripat = + | IpatSimpl of ssrclear * ssrsimpl + | IpatId of identifier + | IpatWild + | IpatCase of ssripats list + | IpatRw of ssrocc * ssrdir + | IpatAll + | IpatAnon + | IpatView of ssrtermrep list + | IpatNoop + | IpatNewHidden of identifier list +and ssripats = ssripat list + +let remove_loc = snd + +let rec ipat_of_intro_pattern = function + | IntroNaming (IntroIdentifier id) -> IpatId id + | IntroAction IntroWildcard -> IpatWild + | IntroAction (IntroOrAndPattern iorpat) -> + IpatCase + (List.map (List.map ipat_of_intro_pattern) + (List.map (List.map remove_loc) iorpat)) + | IntroNaming IntroAnonymous -> IpatAnon + | IntroAction (IntroRewrite b) -> IpatRw (allocc, if b then L2R else R2L) + | IntroNaming (IntroFresh id) -> IpatAnon + | IntroAction (IntroApplyOn _) -> (* to do *) Errors.error "TO DO" + | IntroAction (IntroInjection ips) -> + IpatCase [List.map ipat_of_intro_pattern (List.map remove_loc ips)] + | IntroForthcoming _ -> (* Unable to determine which kind of ipat interp_introid could return [HH] *) + assert false + +let rec pr_ipat = function + | IpatId id -> pr_id id + | IpatSimpl (clr, sim) -> pr_clear mt clr ++ pr_simpl sim + | IpatCase iorpat -> hov 1 (str "[" ++ pr_iorpat iorpat ++ str "]") + | IpatRw (occ, dir) -> pr_occ occ ++ pr_dir dir + | IpatAll -> str "*" + | IpatWild -> str "_" + | IpatAnon -> str "?" + | IpatView v -> pr_view v + | IpatNoop -> str "-" + | IpatNewHidden l -> str "[:" ++ pr_list spc pr_id l ++ str "]" +and pr_iorpat iorpat = pr_list pr_bar pr_ipats iorpat +and pr_ipats ipats = pr_list spc pr_ipat ipats + +let wit_ssripatrep = add_genarg "ssripatrep" pr_ipat + +let pr_ssripat _ _ _ = pr_ipat +let pr_ssripats _ _ _ = pr_ipats +let pr_ssriorpat _ _ _ = pr_iorpat + +let intern_ipat ist ipat = + let rec check_pat = function + | IpatSimpl (clr, _) -> ignore (List.map (intern_hyp ist) clr) + | IpatCase iorpat -> List.iter (List.iter check_pat) iorpat + | _ -> () in + check_pat ipat; ipat + +let intern_ipats ist = List.map (intern_ipat ist) + +let interp_introid ist gl id = + try IntroNaming (IntroIdentifier (hyp_id (snd (interp_hyp ist gl (SsrHyp (dummy_loc, id)))))) + with _ -> snd(snd (interp_intro_pattern ist gl (dummy_loc,IntroNaming (IntroIdentifier id)))) + +let rec add_intro_pattern_hyps (loc, ipat) hyps = match ipat with + | IntroNaming (IntroIdentifier id) -> + if not_section_id id then SsrHyp (loc, id) :: hyps else + hyp_err loc "Can't delete section hypothesis " id + | IntroAction IntroWildcard -> hyps + | IntroAction (IntroOrAndPattern iorpat) -> + List.fold_right (List.fold_right add_intro_pattern_hyps) iorpat hyps + | IntroNaming IntroAnonymous -> [] + | IntroNaming (IntroFresh _) -> [] + | IntroAction (IntroRewrite _) -> hyps + | IntroAction (IntroInjection ips) -> List.fold_right add_intro_pattern_hyps ips hyps + | IntroAction (IntroApplyOn (c,pat)) -> add_intro_pattern_hyps pat hyps + | IntroForthcoming _ -> + (* As in ipat_of_intro_pattern, was unable to determine which kind + of ipat interp_introid could return [HH] *) assert false + +let rec interp_ipat ist gl = + let ltacvar id = Id.Map.mem id ist.lfun in + let rec interp = function + | IpatId id when ltacvar id -> + ipat_of_intro_pattern (interp_introid ist gl id) + | IpatSimpl (clr, sim) -> + let add_hyps (SsrHyp (loc, id) as hyp) hyps = + if not (ltacvar id) then hyp :: hyps else + add_intro_pattern_hyps (loc, (interp_introid ist gl id)) hyps in + let clr' = List.fold_right add_hyps clr [] in + check_hyps_uniq [] clr'; IpatSimpl (clr', sim) + | IpatCase iorpat -> IpatCase (List.map (List.map interp) iorpat) + | IpatNewHidden l -> + IpatNewHidden + (List.map (function + | IntroNaming (IntroIdentifier id) -> id + | _ -> assert false) + (List.map (interp_introid ist gl) l)) + | ipat -> ipat in + interp + +let interp_ipats ist gl l = project gl, List.map (interp_ipat ist gl) l + +let pushIpatRw = function + | pats :: orpat -> (IpatRw (allocc, L2R) :: pats) :: orpat + | [] -> [] + +let pushIpatNoop = function + | pats :: orpat -> (IpatNoop :: pats) :: orpat + | [] -> [] + +ARGUMENT EXTEND ssripat TYPED AS ssripatrep list PRINTED BY pr_ssripats + INTERPRETED BY interp_ipats + GLOBALIZED BY intern_ipats + | [ "_" ] -> [ [IpatWild] ] + | [ "*" ] -> [ [IpatAll] ] + | [ ident(id) ] -> [ [IpatId id] ] + | [ "?" ] -> [ [IpatAnon] ] + | [ ssrsimpl_ne(sim) ] -> [ [IpatSimpl ([], sim)] ] + | [ ssrdocc(occ) "->" ] -> [ match occ with + | None, occ -> [IpatRw (occ, L2R)] + | Some clr, _ -> [IpatSimpl (clr, Nop); IpatRw (allocc, L2R)]] + | [ ssrdocc(occ) "<-" ] -> [ match occ with + | None, occ -> [IpatRw (occ, R2L)] + | Some clr, _ -> [IpatSimpl (clr, Nop); IpatRw (allocc, R2L)]] + | [ ssrdocc(occ) ] -> [ match occ with + | Some cl, _ -> check_hyps_uniq [] cl; [IpatSimpl (cl, Nop)] + | _ -> loc_error loc "Only identifiers are allowed here"] + | [ "->" ] -> [ [IpatRw (allocc, L2R)] ] + | [ "<-" ] -> [ [IpatRw (allocc, R2L)] ] + | [ "-" ] -> [ [IpatNoop] ] + | [ "-/" "=" ] -> [ [IpatNoop;IpatSimpl([],Simpl)] ] + | [ "-/=" ] -> [ [IpatNoop;IpatSimpl([],Simpl)] ] + | [ "-/" "/" ] -> [ [IpatNoop;IpatSimpl([],Cut)] ] + | [ "-//" ] -> [ [IpatNoop;IpatSimpl([],Cut)] ] + | [ "-/" "/=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ "-//" "=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ "-//=" ] -> [ [IpatNoop;IpatSimpl([],SimplCut)] ] + | [ ssrview(v) ] -> [ [IpatView v] ] + | [ "[" ":" ident_list(idl) "]" ] -> [ [IpatNewHidden idl] ] +END + +ARGUMENT EXTEND ssripats TYPED AS ssripat PRINTED BY pr_ssripats + | [ ssripat(i) ssripats(tl) ] -> [ i @ tl ] + | [ ] -> [ [] ] +END + +ARGUMENT EXTEND ssriorpat TYPED AS ssripat list PRINTED BY pr_ssriorpat +| [ ssripats(pats) "|" ssriorpat(orpat) ] -> [ pats :: orpat ] +| [ ssripats(pats) "|-" ">" ssriorpat(orpat) ] -> [ pats :: pushIpatRw orpat ] +| [ ssripats(pats) "|-" ssriorpat(orpat) ] -> [ pats :: pushIpatNoop orpat ] +| [ ssripats(pats) "|->" ssriorpat(orpat) ] -> [ pats :: pushIpatRw orpat ] +| [ ssripats(pats) "||" ssriorpat(orpat) ] -> [ pats :: [] :: orpat ] +| [ ssripats(pats) "|||" ssriorpat(orpat) ] -> [ pats :: [] :: [] :: orpat ] +| [ ssripats(pats) "||||" ssriorpat(orpat) ] -> [ [pats; []; []; []] @ orpat ] +| [ ssripats(pats) ] -> [ [pats] ] +END + +let reject_ssrhid strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "[" -> + (match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD ":" -> raise Stream.Failure + | _ -> ()) + | _ -> () + +let test_nohidden = Gram.Entry.of_parser "test_ssrhid" reject_ssrhid + +ARGUMENT EXTEND ssrcpat TYPED AS ssripatrep PRINTED BY pr_ssripat + | [ "YouShouldNotTypeThis" ssriorpat(x) ] -> [ IpatCase x ] +END + +GEXTEND Gram + GLOBAL: ssrcpat; + ssrcpat: [[ test_nohidden; "["; iorpat = ssriorpat; "]" -> IpatCase iorpat ]]; +END + +GEXTEND Gram + GLOBAL: ssripat; + ssripat: [[ pat = ssrcpat -> [pat] ]]; +END + +ARGUMENT EXTEND ssripats_ne TYPED AS ssripat PRINTED BY pr_ssripats + | [ ssripat(i) ssripats(tl) ] -> [ i @ tl ] +END + +(* subsets of patterns *) +let check_ssrhpats loc w_binders ipats = + let err_loc s = Errors.user_err_loc (loc, "ssreflect", s) in + let clr, ipats = + let rec aux clr = function + | IpatSimpl (cl, Nop) :: tl -> aux (clr @ cl) tl + | IpatSimpl (cl, sim) :: tl -> clr @ cl, IpatSimpl ([], sim) :: tl + | tl -> clr, tl + in aux [] ipats in + let simpl, ipats = + match List.rev ipats with + | IpatSimpl ([],_) as s :: tl -> [s], List.rev tl + | _ -> [], ipats in + if simpl <> [] && not w_binders then + err_loc (str "No s-item allowed here: " ++ pr_ipats simpl); + let ipat, binders = + let rec loop ipat = function + | [] -> ipat, [] + | ( IpatId _| IpatAnon| IpatCase _| IpatRw _ as i) :: tl -> + if w_binders then + if simpl <> [] && tl <> [] then + err_loc(str"binders XOR s-item allowed here: "++pr_ipats(tl@simpl)) + else if not (List.for_all (function IpatId _ -> true | _ -> false) tl) + then err_loc (str "Only binders allowed here: " ++ pr_ipats tl) + else ipat @ [i], tl + else + if tl = [] then ipat @ [i], [] + else err_loc (str "No binder or s-item allowed here: " ++ pr_ipats tl) + | hd :: tl -> loop (ipat @ [hd]) tl + in loop [] ipats in + ((clr, ipat), binders), simpl + +let single loc = + function [x] -> x | _ -> loc_error loc "Only one intro pattern is allowed" + +let pr_hpats (((clr, ipat), binders), simpl) = + pr_clear mt clr ++ pr_ipats ipat ++ pr_ipats binders ++ pr_ipats simpl +let pr_ssrhpats _ _ _ = pr_hpats +let pr_ssrhpats_wtransp _ _ _ (_, x) = pr_hpats x + +ARGUMENT EXTEND ssrhpats TYPED AS ((ssrclear * ssripat) * ssripat) * ssripat +PRINTED BY pr_ssrhpats + | [ ssripats(i) ] -> [ check_ssrhpats loc true i ] +END + +ARGUMENT EXTEND ssrhpats_wtransp + TYPED AS bool * (((ssrclear * ssripat) * ssripat) * ssripat) + PRINTED BY pr_ssrhpats_wtransp + | [ ssripats(i) ] -> [ false,check_ssrhpats loc true i ] + | [ ssripats(i) "@" ssripats(j) ] -> [ true,check_ssrhpats loc true (i @ j) ] +END + +ARGUMENT EXTEND ssrhpats_nobs +TYPED AS ((ssrclear * ssripat) * ssripat) * ssripat PRINTED BY pr_ssrhpats + | [ ssripats(i) ] -> [ check_ssrhpats loc false i ] +END + +ARGUMENT EXTEND ssrrpat TYPED AS ssripatrep PRINTED BY pr_ssripat + | [ "->" ] -> [ IpatRw (allocc, L2R) ] + | [ "<-" ] -> [ IpatRw (allocc, R2L) ] +END + +type ssrintros = ssripats + +let pr_intros sep intrs = + if intrs = [] then mt() else sep () ++ str "=> " ++ pr_ipats intrs +let pr_ssrintros _ _ _ = pr_intros mt + +ARGUMENT EXTEND ssrintros_ne TYPED AS ssripat + PRINTED BY pr_ssrintros + | [ "=>" ssripats_ne(pats) ] -> [ pats ] +END + +ARGUMENT EXTEND ssrintros TYPED AS ssrintros_ne PRINTED BY pr_ssrintros + | [ ssrintros_ne(intrs) ] -> [ intrs ] + | [ ] -> [ [] ] +END + +let injecteq_id = mk_internal_id "injection equation" + +let pf_nb_prod gl = nb_prod (pf_concl gl) + +let rev_id = mk_internal_id "rev concl" + +let revtoptac n0 gl = + let n = pf_nb_prod gl - n0 in + let dc, cl = decompose_prod_n n (pf_concl gl) in + let dc' = dc @ [Name rev_id, compose_prod (List.rev dc) cl] in + let f = compose_lam dc' (mkEtaApp (mkRel (n + 1)) (-n) 1) in + refine (mkApp (f, [|Evarutil.mk_new_meta ()|])) gl + +let equality_inj l b id c gl = + let msg = ref "" in + try Proofview.V82.of_tactic (Equality.inj l b None c) gl + with + | Compat.Exc_located(_,Errors.UserError (_,s)) + | Errors.UserError (_,s) + when msg := Pp.string_of_ppcmds s; + !msg = "Not a projectable equality but a discriminable one." || + !msg = "Nothing to inject." -> + msg_warning (str !msg); + discharge_hyp (id, (id, "")) gl + +let injectidl2rtac id c gl = + tclTHEN (equality_inj None true id c) (revtoptac (pf_nb_prod gl)) gl + +let injectl2rtac c = match kind_of_term c with +| Var id -> injectidl2rtac id (mkVar id, NoBindings) +| _ -> + let id = injecteq_id in + tclTHENLIST [havetac id c; injectidl2rtac id (mkVar id, NoBindings); clear [id]] + +let is_injection_case c gl = + let (mind,_), _ = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in + eq_gr (IndRef mind) (build_coq_eq ()) + +let perform_injection c gl = + let mind, t = pf_reduce_to_quantified_ind gl (pf_type_of gl c) in + let dc, eqt = decompose_prod t in + if dc = [] then injectl2rtac c gl else + if not (closed0 eqt) then + Errors.error "can't decompose a quantified equality" else + let cl = pf_concl gl in let n = List.length dc in + let c_eq = mkEtaApp c n 2 in + let cl1 = mkLambda (Anonymous, mkArrow eqt cl, mkApp (mkRel 1, [|c_eq|])) in + let id = injecteq_id in + let id_with_ebind = (mkVar id, NoBindings) in + let injtac = tclTHEN (introid id) (injectidl2rtac id id_with_ebind) in + tclTHENLAST (Proofview.V82.of_tactic (apply (compose_lam dc cl1))) injtac gl + +let simplest_newcase_ref = ref (fun t gl -> assert false) +let simplest_newcase x gl = !simplest_newcase_ref x gl + +let ssrscasetac c gl = + if is_injection_case c gl then perform_injection c gl + else simplest_newcase c gl + +let intro_all gl = + let dc, _ = Term.decompose_prod_assum (pf_concl gl) in + tclTHENLIST (List.map anontac (List.rev dc)) gl + +let rec intro_anon gl = + try anontac (List.hd (fst (Term.decompose_prod_n_assum 1 (pf_concl gl)))) gl + with err0 -> try tclTHEN red_in_concl intro_anon gl with _ -> raise err0 + (* with _ -> Errors.error "No product even after reduction" *) + +let with_top tac = + tclTHENLIST [introid top_id; tac (mkVar top_id); clear [top_id]] + +let rec mapLR f = function [] -> [] | x :: s -> let y = f x in y :: mapLR f s + +let wild_ids = ref [] + +let new_wild_id () = + let i = 1 + List.length !wild_ids in + let id = mk_wildcard_id i in + wild_ids := id :: !wild_ids; + id + +let clear_wilds wilds gl = + clear (List.filter (fun id -> List.mem id wilds) (pf_ids_of_hyps gl)) gl + +let clear_with_wilds wilds clr0 gl = + let extend_clr clr (id, _, _ as nd) = + if List.mem id clr || not (List.mem id wilds) then clr else + let vars = global_vars_set_of_decl (pf_env gl) nd in + let occurs id' = Idset.mem id' vars in + if List.exists occurs clr then id :: clr else clr in + clear (Context.fold_named_context_reverse extend_clr ~init:clr0 (pf_hyps gl)) gl + +let tclTHENS_nonstrict tac tacl taclname gl = + let tacres = tac gl in + let n_gls = List.length (sig_it tacres) in + let n_tac = List.length tacl in + if n_gls = n_tac then tclTHENS (fun _ -> tacres) tacl gl else + if n_gls = 0 then tacres else + let pr_only n1 n2 = if n1 < n2 then str "only " else mt () in + let pr_nb n1 n2 name = + pr_only n1 n2 ++ int n1 ++ str (" " ^ String.plural n1 name) in + errorstrm (pr_nb n_tac n_gls taclname ++ spc () + ++ str "for " ++ pr_nb n_gls n_tac "subgoal") + +(* Forward reference to extended rewrite *) +let ipat_rewritetac = ref (fun _ -> rewritetac) + +let rec is_name_in_ipats name = function + | IpatSimpl(clr,_) :: tl -> + List.exists (function SsrHyp(_,id) -> id = name) clr + || is_name_in_ipats name tl + | IpatId id :: tl -> id = name || is_name_in_ipats name tl + | IpatCase l :: tl -> is_name_in_ipats name (List.flatten l @ tl) + | _ :: tl -> is_name_in_ipats name tl + | [] -> false + +let move_top_with_view = ref (fun _ -> assert false) + +let rec nat_of_n n = + if n = 0 then mkConstruct path_of_O + else mkApp (mkConstruct path_of_S, [|nat_of_n (n-1)|]) + +let ssr_abstract_id = Summary.ref "~name:SSR:abstractid" 0 + +let mk_abstract_id () = incr ssr_abstract_id; nat_of_n !ssr_abstract_id + +let ssrmkabs id gl = + let env, concl = pf_env gl, pf_concl gl in + let step sigma = + let sigma, abstract_proof, abstract_ty = + let sigma, (ty, _) = + Evarutil.new_type_evar env sigma Evd.univ_flexible_alg in + let sigma, ablock = mkSsrConst "abstract_lock" env sigma in + let sigma, lock = Evarutil.new_evar env sigma ablock in + let sigma, abstract = mkSsrConst "abstract" env sigma in + let abstract_ty = mkApp(abstract, [|ty;mk_abstract_id ();lock|]) in + let sigma, m = Evarutil.new_evar env sigma abstract_ty in + sigma, m, abstract_ty in + let sigma, kont = + let rd = Name id, None, abstract_ty in + Evarutil.new_evar (Environ.push_rel rd env) sigma concl in + pp(lazy(pr_constr concl)); + let term = mkApp (mkLambda(Name id,abstract_ty,kont) ,[|abstract_proof|]) in + let sigma, _ = Typing.e_type_of env sigma term in + sigma, term in + Proofview.V82.of_tactic + (Proofview.tclTHEN + (Tactics.New.refine step) + (Proofview.tclFOCUS 1 3 Proofview.shelve)) gl + +let ssrmkabstac ids = + List.fold_right (fun id tac -> tclTHENFIRST (ssrmkabs id) tac) ids tclIDTAC + +(* introstac: for "move" and "clear", tclEQINTROS: for "case" and "elim" *) +(* This block hides the spaghetti-code needed to implement the only two *) +(* tactics that should be used to process intro patters. *) +(* The difficulty is that we don't want to always rename, but we can *) +(* compute needeed renamings only at runtime, so we theread a tree like *) +(* imperativestructure so that outer renamigs are inherited by inner *) +(* ipts and that the cler performed at the end of ipatstac clears hyps *) +(* eventually renamed at runtime. *) +(* TODO: hide wild_ids in this block too *) +let introstac, tclEQINTROS = + let rec map_acc_k f k = function + | [] -> (* tricky: we save wilds now, we get to_cler (aka k) later *) + let clear_ww = clear_with_wilds !wild_ids in + [fun gl -> clear_ww (hyps_ids (List.flatten (List.map (!) k))) gl] + | x :: xs -> let k, x = f k xs x in x :: map_acc_k f k xs in + let rename force to_clr rest clr gl = + let hyps = pf_hyps gl in + pp(lazy(str"rename " ++ pr_clear spc clr)); + let () = if not force then List.iter (check_hyp_exists hyps) clr in + if List.exists (fun x -> force || is_name_in_ipats (hyp_id x) rest) clr then + let ren_clr, ren = + List.split (List.map (fun x -> let x = hyp_id x in + let x' = mk_anon_id (string_of_id x) gl in + SsrHyp (dummy_loc, x'), (x, x')) clr) in + let () = to_clr := ren_clr in + Proofview.V82.of_tactic (rename_hyp ren) gl + else + let () = to_clr := clr in + tclIDTAC gl in + let rec ipattac ?ist k rest = function + | IpatWild -> k, introid (new_wild_id ()) + | IpatCase iorpat -> k, tclIORPAT ?ist k (with_top ssrscasetac) iorpat + | IpatRw (occ, dir) -> k, with_top (!ipat_rewritetac occ dir) + | IpatId id -> k, introid id + | IpatNewHidden idl -> k, ssrmkabstac idl + | IpatSimpl (clr, sim) -> + let to_clr = ref [] in + to_clr :: k, tclTHEN (rename false to_clr rest clr) (simpltac sim) + | IpatAll -> k, intro_all + | IpatAnon -> k, intro_anon + | IpatNoop -> k, tclIDTAC + | IpatView v -> match ist with + | None -> anomaly "ipattac with no ist but view" + | Some ist -> match rest with + | (IpatCase _ | IpatRw _)::_ -> + let to_clr = ref [] in let top_id = ref top_id in + to_clr :: k, + tclTHEN + (!move_top_with_view false top_id (false,v) ist) + (fun gl -> + rename true to_clr rest [SsrHyp (dummy_loc, !top_id)]gl) + | _ -> k, !move_top_with_view true (ref top_id) (true,v) ist + and tclIORPAT ?ist k tac = function + | [[]] -> tac + | orp -> + tclTHENS_nonstrict tac (mapLR (ipatstac ?ist k) orp) "intro pattern" + and ipatstac ?ist k ipats = + tclTHENLIST (map_acc_k (ipattac ?ist) k ipats) in + let introstac ?ist ipats = + wild_ids := []; + let tac = ipatstac ?ist [] ipats in + tclTHENLIST [tac; clear_wilds !wild_ids] in + let tclEQINTROS ?ist tac eqtac ipats = + wild_ids := []; + let rec split_itacs to_clr tac' = function + | (IpatSimpl _ as spat) :: ipats' -> + let to_clr, tac = ipattac ?ist to_clr ipats' spat in + split_itacs to_clr (tclTHEN tac' tac) ipats' + | IpatCase iorpat :: ipats' -> + to_clr, tclIORPAT ?ist to_clr tac' iorpat, ipats' + | ipats' -> to_clr, tac', ipats' in + let to_clr, tac1, ipats' = split_itacs [] tac ipats in + let tac2 = ipatstac ?ist to_clr ipats' in + tclTHENLIST [tac1; eqtac; tac2; clear_wilds !wild_ids] in + introstac, tclEQINTROS +;; + +let rec eqmoveipats eqpat = function + | (IpatSimpl _ as ipat) :: ipats -> ipat :: eqmoveipats eqpat ipats + | (IpatAll :: _ | []) as ipats -> IpatAnon :: eqpat :: ipats + | ipat :: ipats -> ipat :: eqpat :: ipats + +(* General case *) +let tclINTROS ist tac ipats = + tclEQINTROS ~ist (tac ist) tclIDTAC ipats + +(** The "=>" tactical *) + +let ssrintros_sep = + let atom_sep = function + | TacSplit (_, [NoBindings]) -> mt + (* | TacExtend (_, "ssrapply", []) -> mt *) + | _ -> spc in + function + | TacId [] -> mt + | TacArg (_, Tacexp _) -> mt + | TacArg (_, Reference _) -> mt + | TacAtom (_, atom) -> atom_sep atom + | _ -> spc + +let pr_ssrintrosarg _ _ prt (tac, ipats) = + prt tacltop tac ++ pr_intros (ssrintros_sep tac) ipats + +ARGUMENT EXTEND ssrintrosarg TYPED AS tactic * ssrintros + PRINTED BY pr_ssrintrosarg +| [ "YouShouldNotTypeThis" ssrtacarg(arg) ssrintros_ne(ipats) ] -> [ arg, ipats ] +END + +TACTIC EXTEND ssrtclintros +| [ "YouShouldNotTypeThis" ssrintrosarg(arg) ] -> + [ let tac, intros = arg in + Proofview.V82.tactic (tclINTROS ist (fun ist -> ssrevaltac ist tac) intros) ] +END +set_pr_ssrtac "tclintros" 0 [ArgSsr "introsarg"] + +let tclintros_expr loc tac ipats = + let args = [in_gen (rawwit wit_ssrintrosarg) (tac, ipats)] in + ssrtac_expr loc "tclintros" args + +GEXTEND Gram + GLOBAL: tactic_expr; + tactic_expr: LEVEL "1" [ RIGHTA + [ tac = tactic_expr; intros = ssrintros_ne -> tclintros_expr !@loc tac intros + ] ]; +END +(* }}} *) + +(** Multipliers {{{ ***********************************************************) + +(* modality *) + +type ssrmmod = May | Must | Once + +let pr_mmod = function May -> str "?" | Must -> str "!" | Once -> mt () + +let wit_ssrmmod = add_genarg "ssrmmod" pr_mmod +let ssrmmod = Pcoq.create_generic_entry "ssrmmod" (Genarg.rawwit wit_ssrmmod) +GEXTEND Gram + GLOBAL: ssrmmod; + ssrmmod: [[ "!" -> Must | LEFTQMARK -> May | "?" -> May]]; +END + +(* tactical *) + +let tclID tac = tac + +let tclDOTRY n tac = + if n <= 0 then tclIDTAC else + let rec loop i gl = + if i = n then tclTRY tac gl else + tclTRY (tclTHEN tac (loop (i + 1))) gl in + loop 1 + +let tclDO n tac = + let prefix i = str"At iteration " ++ int i ++ str": " in + let tac_err_at i gl = + try tac gl + with + | Errors.UserError (l, s) as e -> + let _, info = Errors.push e in + let e' = Errors.UserError (l, prefix i ++ s) in + Util.iraise (e', info) + | Compat.Exc_located(loc, Errors.UserError (l, s)) -> + raise (Compat.Exc_located(loc, Errors.UserError (l, prefix i ++ s))) in + let rec loop i gl = + if i = n then tac_err_at i gl else + (tclTHEN (tac_err_at i) (loop (i + 1))) gl in + loop 1 + +let tclMULT = function + | 0, May -> tclREPEAT + | 1, May -> tclTRY + | n, May -> tclDOTRY n + | 0, Must -> tclAT_LEAST_ONCE + | n, Must when n > 1 -> tclDO n + | _ -> tclID + +(** The "do" tactical. ********************************************************) + +(* +type ssrdoarg = ((ssrindex * ssrmmod) * ssrhint) * ssrclauses +*) + +let pr_ssrdoarg prc _ prt (((n, m), tac), clauses) = + pr_index n ++ pr_mmod m ++ pr_hintarg prt tac ++ pr_clauses clauses + +ARGUMENT EXTEND ssrdoarg + TYPED AS ((ssrindex * ssrmmod) * ssrhintarg) * ssrclauses + PRINTED BY pr_ssrdoarg +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let ssrdotac ist (((n, m), tac), clauses) = + let mul = get_index n, m in + tclCLAUSES ist (tclMULT mul (hinttac ist false tac)) clauses + +TACTIC EXTEND ssrtcldo +| [ "YouShouldNotTypeThis" "do" ssrdoarg(arg) ] -> [ Proofview.V82.tactic (ssrdotac ist arg) ] +END +set_pr_ssrtac "tcldo" 3 [ArgSep "do "; ArgSsr "doarg"] + +let ssrdotac_expr loc n m tac clauses = + let arg = ((n, m), tac), clauses in + ssrtac_expr loc "tcldo" [in_gen (rawwit wit_ssrdoarg) arg] + +GEXTEND Gram + GLOBAL: tactic_expr; + ssrdotac: [ + [ tac = tactic_expr LEVEL "3" -> mk_hint tac + | tacs = ssrortacarg -> tacs + ] ]; + tactic_expr: LEVEL "3" [ RIGHTA + [ IDENT "do"; m = ssrmmod; tac = ssrdotac; clauses = ssrclauses -> + ssrdotac_expr !@loc noindex m tac clauses + | IDENT "do"; tac = ssrortacarg; clauses = ssrclauses -> + ssrdotac_expr !@loc noindex Once tac clauses + | IDENT "do"; n = int_or_var; m = ssrmmod; + tac = ssrdotac; clauses = ssrclauses -> + ssrdotac_expr !@loc (mk_index !@loc n) m tac clauses + ] ]; +END +(* }}} *) + +(** The "first" and "last" tacticals. {{{ *************************************) + +(* type ssrseqarg = ssrindex * (ssrtacarg * ssrtac option) *) + +let pr_seqtacarg prt = function + | (is_first, []), _ -> str (if is_first then "first" else "last") + | tac, Some dtac -> + hv 0 (pr_hintarg prt tac ++ spc() ++ str "|| " ++ prt tacltop dtac) + | tac, _ -> pr_hintarg prt tac + +let pr_ssrseqarg _ _ prt = function + | ArgArg 0, tac -> pr_seqtacarg prt tac + | i, tac -> pr_index i ++ str " " ++ pr_seqtacarg prt tac + +(* We must parse the index separately to resolve the conflict with *) +(* an unindexed tactic. *) +ARGUMENT EXTEND ssrseqarg TYPED AS ssrindex * (ssrhintarg * tactic option) + PRINTED BY pr_ssrseqarg +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let sq_brace_tacnames = + ["first"; "solve"; "do"; "rewrite"; "have"; "suffices"; "wlog"] + (* "by" is a keyword *) +let accept_ssrseqvar strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT id when not (List.mem id sq_brace_tacnames) -> + accept_before_syms_or_ids ["["] ["first";"last"] strm + | _ -> raise Stream.Failure + +let test_ssrseqvar = Gram.Entry.of_parser "test_ssrseqvar" accept_ssrseqvar + +let swaptacarg (loc, b) = (b, []), Some (TacId []) + +let check_seqtacarg dir arg = match snd arg, dir with + | ((true, []), Some (TacAtom (loc, _))), L2R -> + loc_error loc "expected \"last\"" + | ((false, []), Some (TacAtom (loc, _))), R2L -> + loc_error loc "expected \"first\"" + | _, _ -> arg + +let ssrorelse = Gram.Entry.create "ssrorelse" +GEXTEND Gram + GLOBAL: ssrorelse ssrseqarg; + ssrseqidx: [ + [ test_ssrseqvar; id = Prim.ident -> ArgVar (!@loc, id) + | n = Prim.natural -> ArgArg (check_index !@loc n) + ] ]; + ssrswap: [[ IDENT "first" -> !@loc, true | IDENT "last" -> !@loc, false ]]; + ssrorelse: [[ "||"; tac = tactic_expr LEVEL "2" -> tac ]]; + ssrseqarg: [ + [ arg = ssrswap -> noindex, swaptacarg arg + | i = ssrseqidx; tac = ssrortacarg; def = OPT ssrorelse -> i, (tac, def) + | i = ssrseqidx; arg = ssrswap -> i, swaptacarg arg + | tac = tactic_expr LEVEL "3" -> noindex, (mk_hint tac, None) + ] ]; +END + +let tclPERM perm tac gls = + let subgls = tac gls in + let sigma, subgll = Refiner.unpackage subgls in + let subgll' = perm subgll in + Refiner.repackage sigma subgll' +(* +let tclPERM perm tac gls = + let mkpft n g r = + {Proof_type.open_subgoals = n; Proof_type.goal = g; Proof_type.ref = r} in + let mkleaf g = mkpft 0 g None in + let mkprpft n g pr a = mkpft n g (Some (Proof_type.Prim pr, a)) in + let mkrpft n g c = mkprpft n g (Proof_type.Refine c) in + let mkipft n g = + let mki pft (id, _, _ as d) = + let g' = {g with evar_concl = mkNamedProd_or_LetIn d g.evar_concl} in + mkprpft n g' (Proof_type.Intro id) [pft] in + List.fold_left mki in + let gl = Refiner.sig_it gls in + let mkhyp subgl = + let rec chop_section = function + | (x, _, _ as d) :: e when not_section_id x -> d :: chop_section e + | _ -> [] in + let lhyps = Environ.named_context_of_val subgl.evar_hyps in + mk_perm_id (), subgl, chop_section lhyps in + let mkpfvar (hyp, subgl, lhyps) = + let mkarg args (lhyp, body, _) = + if body = None then mkVar lhyp :: args else args in + mkrpft 0 subgl (applist (mkVar hyp, List.fold_left mkarg [] lhyps)) [] in + let mkpfleaf (_, subgl, lhyps) = mkipft 1 gl (mkleaf subgl) lhyps in + let mkmeta _ = Evarutil.mk_new_meta () in + let mkhypdecl (hyp, subgl, lhyps) = + hyp, None, it_mkNamedProd_or_LetIn subgl.evar_concl lhyps in + let subgls, v as res0 = tac gls in + let sigma, subgll = Refiner.unpackage subgls in + let n = List.length subgll in if n = 0 then res0 else + let hyps = List.map mkhyp subgll in + let hyp_decls = List.map mkhypdecl (List.rev (perm hyps)) in + let c = applist (mkmeta (), List.map mkmeta subgll) in + let pft0 = mkipft 0 gl (v (List.map mkpfvar hyps)) hyp_decls in + let pft1 = mkrpft n gl c (pft0 :: List.map mkpfleaf (perm hyps)) in + let subgll', v' = Refiner.frontier pft1 in + Refiner.repackage sigma subgll', v' +*) + +let tclREV tac gl = tclPERM List.rev tac gl + +let rot_hyps dir i hyps = + let n = List.length hyps in + if i = 0 then List.rev hyps else + if i > n then Errors.error "Not enough subgoals" else + let rec rot i l_hyps = function + | hyp :: hyps' when i > 0 -> rot (i - 1) (hyp :: l_hyps) hyps' + | hyps' -> hyps' @ (List.rev l_hyps) in + rot (match dir with L2R -> i | R2L -> n - i) [] hyps + +let tclSEQAT ist atac1 dir (ivar, ((_, atacs2), atac3)) = + let i = get_index ivar in + let evtac = ssrevaltac ist in + let tac1 = evtac atac1 in + if atacs2 = [] && atac3 <> None then tclPERM (rot_hyps dir i) tac1 else + let evotac = function Some atac -> evtac atac | _ -> tclIDTAC in + let tac3 = evotac atac3 in + let rec mk_pad n = if n > 0 then tac3 :: mk_pad (n - 1) else [] in + match dir, mk_pad (i - 1), List.map evotac atacs2 with + | L2R, [], [tac2] when atac3 = None -> tclTHENFIRST tac1 tac2 + | L2R, [], [tac2] when atac3 = None -> tclTHENLAST tac1 tac2 + | L2R, pad, tacs2 -> tclTHENSFIRSTn tac1 (Array.of_list (pad @ tacs2)) tac3 + | R2L, pad, tacs2 -> tclTHENSLASTn tac1 tac3 (Array.of_list (tacs2 @ pad)) + +(* We can't actually parse the direction separately because this *) +(* would introduce conflicts with the basic ltac syntax. *) +let pr_ssrseqdir _ _ _ = function + | L2R -> str ";" ++ spc () ++ str "first " + | R2L -> str ";" ++ spc () ++ str "last " + +ARGUMENT EXTEND ssrseqdir TYPED AS ssrdir PRINTED BY pr_ssrseqdir +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +TACTIC EXTEND ssrtclseq +| [ "YouShouldNotTypeThis" ssrtclarg(tac) ssrseqdir(dir) ssrseqarg(arg) ] -> + [ Proofview.V82.tactic (tclSEQAT ist tac dir arg) ] +END +set_pr_ssrtac "tclseq" 5 [ArgSsr "tclarg"; ArgSsr "seqdir"; ArgSsr "seqarg"] + +let tclseq_expr loc tac dir arg = + let arg1 = in_gen (rawwit wit_ssrtclarg) tac in + let arg2 = in_gen (rawwit wit_ssrseqdir) dir in + let arg3 = in_gen (rawwit wit_ssrseqarg) (check_seqtacarg dir arg) in + ssrtac_expr loc "tclseq" [arg1; arg2; arg3] + +GEXTEND Gram + GLOBAL: tactic_expr; + ssr_first: [ + [ tac = ssr_first; ipats = ssrintros_ne -> tclintros_expr !@loc tac ipats + | "["; tacl = LIST0 tactic_expr SEP "|"; "]" -> TacFirst tacl + ] ]; + ssr_first_else: [ + [ tac1 = ssr_first; tac2 = ssrorelse -> TacOrelse (tac1, tac2) + | tac = ssr_first -> tac ]]; + tactic_expr: LEVEL "4" [ LEFTA + [ tac1 = tactic_expr; ";"; IDENT "first"; tac2 = ssr_first_else -> + TacThen (tac1, tac2) + | tac = tactic_expr; ";"; IDENT "first"; arg = ssrseqarg -> + tclseq_expr !@loc tac L2R arg + | tac = tactic_expr; ";"; IDENT "last"; arg = ssrseqarg -> + tclseq_expr !@loc tac R2L arg + ] ]; +END +(* }}} *) + +(** 5. Bookkeeping tactics (clear, move, case, elim) *) + +(* post-interpretation of terms *) +let all_ok _ _ = true + +let pf_abs_ssrterm ?(resolve_typeclasses=false) ist gl t = + let sigma, ct as t = interp_term ist gl t in + let sigma, _ as t = + let env = pf_env gl in + if not resolve_typeclasses then t + else + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + sigma, Evarutil.nf_evar sigma ct in + let n, c, abstracted_away, ucst = pf_abs_evars gl t in + List.fold_left Evd.remove sigma abstracted_away, pf_abs_cterm gl n c, ucst + +let pf_interp_ty ?(resolve_typeclasses=false) ist gl ty = + let n_binders = ref 0 in + let ty = match ty with + | a, (t, None) -> + let rec force_type = function + | GProd (l, x, k, s, t) -> incr n_binders; GProd (l, x, k, s, force_type t) + | GLetIn (l, x, v, t) -> incr n_binders; GLetIn (l, x, v, force_type t) + | ty -> mkRCast ty mkRType in + a, (force_type t, None) + | _, (_, Some ty) -> + let rec force_type = function + | CProdN (l, abs, t) -> + n_binders := !n_binders + List.length (List.flatten (List.map pi1 abs)); + CProdN (l, abs, force_type t) + | CLetIn (l, n, v, t) -> incr n_binders; CLetIn (l, n, v, force_type t) + | ty -> mkCCast dummy_loc ty (mkCType dummy_loc) in + mk_term ' ' (force_type ty) in + let strip_cast (sigma, t) = + let rec aux t = match kind_of_type t with + | CastType (t, ty) when !n_binders = 0 && isSort ty -> t + | ProdType(n,s,t) -> decr n_binders; mkProd (n, s, aux t) + | LetInType(n,v,ty,t) -> decr n_binders; mkLetIn (n, v, ty, aux t) + | _ -> anomaly "pf_interp_ty: ssr Type cast deleted by typecheck" in + sigma, aux t in + let sigma, cty as ty = strip_cast (interp_term ist gl ty) in + let ty = + let env = pf_env gl in + if not resolve_typeclasses then ty + else + let sigma = Typeclasses.resolve_typeclasses ~fail:false env sigma in + sigma, Evarutil.nf_evar sigma cty in + let n, c, _, ucst = pf_abs_evars gl ty in + let lam_c = pf_abs_cterm gl n c in + let ctx, c = decompose_lam_n n lam_c in + n, compose_prod ctx c, lam_c, ucst +;; + +let whd_app f args = Reductionops.whd_betaiota Evd.empty (mkApp (f, args)) + +let pr_cargs a = + str "[" ++ pr_list pr_spc pr_constr (Array.to_list a) ++ str "]" + +let pp_term gl t = + let t = Reductionops.nf_evar (project gl) t in pr_constr t +let pp_concat hd ?(sep=str", ") = function [] -> hd | x :: xs -> + hd ++ List.fold_left (fun acc x -> acc ++ sep ++ x) x xs + +let fake_pmatcher_end () = + mkProp, L2R, (Evd.empty, Evd.empty_evar_universe_context, mkProp) + +(* TASSI: given (c : ty), generates (c ??? : ty[???/...]) with m evars *) +exception NotEnoughProducts +let prof_saturate_whd = mk_profiler "saturate.whd";; +let saturate ?(beta=false) ?(bi_types=false) env sigma c ?(ty=Retyping.get_type_of env sigma c) m += + let rec loop ty args sigma n = + if n = 0 then + let args = List.rev args in + (if beta then Reductionops.whd_beta sigma else fun x -> x) + (mkApp (c, Array.of_list (List.map snd args))), ty, args, sigma + else match kind_of_type ty with + | ProdType (_, src, tgt) -> + let sigma, x = + Evarutil.new_evar env (create_evar_defs sigma) + (if bi_types then Reductionops.nf_betaiota sigma src else src) in + loop (subst1 x tgt) ((m - n,x) :: args) sigma (n-1) + | CastType (t, _) -> loop t args sigma n + | LetInType (_, v, _, t) -> loop (subst1 v t) args sigma n + | SortType _ -> assert false + | AtomicType _ -> + let ty = + prof_saturate_whd.profile + (Reductionops.whd_betadeltaiota env sigma) ty in + match kind_of_type ty with + | ProdType _ -> loop ty args sigma n + | _ -> anomaly "saturate did not find enough products" + in + loop ty [] sigma m + +let pf_saturate ?beta ?bi_types gl c ?ty m = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + let t, ty, args, sigma = saturate ?beta ?bi_types env sigma c ?ty m in + t, ty, args, re_sig si sigma + +(** Rewrite redex switch *) + +(** Generalization (discharge) item *) + +(* An item is a switch + term pair. *) + +(* type ssrgen = ssrdocc * ssrterm *) + +let pr_gen (docc, dt) = pr_docc docc ++ pr_cpattern dt + +let pr_ssrgen _ _ _ = pr_gen + +ARGUMENT EXTEND ssrgen TYPED AS ssrdocc * cpattern PRINTED BY pr_ssrgen +| [ ssrdocc(docc) cpattern(dt) ] -> [ docc, dt ] +| [ cpattern(dt) ] -> [ nodocc, dt ] +END + +let has_occ ((_, occ), _) = occ <> None +let hyp_of_var v = SsrHyp (dummy_loc, destVar v) + +let interp_clr = function +| Some clr, (k, c) + when (k = ' ' || k = '@') && is_pf_var c -> hyp_of_var c :: clr +| Some clr, _ -> clr +| None, _ -> [] + +(* XXX the k of the redex should percolate out *) +let pf_interp_gen_aux ist gl to_ind ((oclr, occ), t) = + let pat = interp_cpattern ist gl t None in (* UGLY API *) + let cl, env, sigma = pf_concl gl, pf_env gl, project gl in + let (c, ucst), cl = + try fill_occ_pattern ~raise_NoMatch:true env sigma cl pat occ 1 + with NoMatch -> redex_of_pattern env pat, cl in + let clr = interp_clr (oclr, (tag_of_cpattern t, c)) in + if not(occur_existential c) then + if tag_of_cpattern t = '@' then + if not (isVar c) then + errorstrm (str "@ can be used with variables only") + else match pf_get_hyp gl (destVar c) with + | _, None, _ -> errorstrm (str "@ can be used with let-ins only") + | name, Some bo, ty -> true, pat, mkLetIn (Name name,bo,ty,cl),c,clr,ucst + else false, pat, pf_mkprod gl c cl, c, clr,ucst + else if to_ind && occ = None then + let nv, p, _, ucst' = pf_abs_evars gl (fst pat, c) in + let ucst = Evd.union_evar_universe_context ucst ucst' in + if nv = 0 then anomaly "occur_existential but no evars" else + false, pat, mkProd (constr_name c, pf_type_of gl p, pf_concl gl), p, clr,ucst + else loc_error (loc_of_cpattern t) "generalized term didn't match" + +let genclrtac cl cs clr = + let tclmyORELSE tac1 tac2 gl = + try tac1 gl + with e when Errors.noncritical e -> tac2 e gl in + (* apply_type may give a type error, but the useful message is + * the one of clear. You type "move: x" and you get + * "x is used in hyp H" instead of + * "The term H has type T x but is expected to have type T x0". *) + tclTHEN + (tclmyORELSE + (apply_type cl cs) + (fun type_err gl -> + tclTHEN + (tclTHEN (Proofview.V82.of_tactic (elim_type (build_coq_False ()))) (cleartac clr)) + (fun gl -> raise type_err) + gl)) + (cleartac clr) + +let gentac ist gen gl = +(* pp(lazy(str"sigma@gentac=" ++ pr_evar_map None (project gl))); *) + let conv, _, cl, c, clr, ucst = pf_interp_gen_aux ist gl false gen in + pp(lazy(str"c@gentac=" ++ pr_constr c)); + let gl = pf_merge_uc ucst gl in + if conv + then tclTHEN (Proofview.V82.of_tactic (convert_concl cl)) (cleartac clr) gl + else genclrtac cl [c] clr gl + +let pf_interp_gen ist gl to_ind gen = + let _, _, a, b, c, ucst = pf_interp_gen_aux ist gl to_ind gen in + a, b ,c, pf_merge_uc ucst gl + +(** Generalization (discharge) sequence *) + +(* A discharge sequence is represented as a list of up to two *) +(* lists of d-items, plus an ident list set (the possibly empty *) +(* final clear switch). The main list is empty iff the command *) +(* is defective, and has length two if there is a sequence of *) +(* dependent terms (and in that case it is the first of the two *) +(* lists). Thus, the first of the two lists is never empty. *) + +(* type ssrgens = ssrgen list *) +(* type ssrdgens = ssrgens list * ssrclear *) + +let gens_sep = function [], [] -> mt | _ -> spc + +let pr_dgens pr_gen (gensl, clr) = + let prgens s gens = str s ++ pr_list spc pr_gen gens in + let prdeps deps = prgens ": " deps ++ spc () ++ str "/" in + match gensl with + | [deps; []] -> prdeps deps ++ pr_clear pr_spc clr + | [deps; gens] -> prdeps deps ++ prgens " " gens ++ pr_clear spc clr + | [gens] -> prgens ": " gens ++ pr_clear spc clr + | _ -> pr_clear pr_spc clr + +let pr_ssrdgens _ _ _ = pr_dgens pr_gen + +let cons_gen gen = function + | gens :: gensl, clr -> (gen :: gens) :: gensl, clr + | _ -> anomaly "missing gen list" + +let cons_dep (gensl, clr) = + if List.length gensl = 1 then ([] :: gensl, clr) else + Errors.error "multiple dependents switches '/'" + +ARGUMENT EXTEND ssrdgens_tl TYPED AS ssrgen list list * ssrclear + PRINTED BY pr_ssrdgens +| [ "{" ne_ssrhyp_list(clr) "}" cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (mkclr clr, dt) dgens ] +| [ "{" ne_ssrhyp_list(clr) "}" ] -> + [ [[]], clr ] +| [ "{" ssrocc(occ) "}" cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (mkocc occ, dt) dgens ] +| [ "/" ssrdgens_tl(dgens) ] -> + [ cons_dep dgens ] +| [ cpattern(dt) ssrdgens_tl(dgens) ] -> + [ cons_gen (nodocc, dt) dgens ] +| [ ] -> + [ [[]], [] ] +END + +ARGUMENT EXTEND ssrdgens TYPED AS ssrdgens_tl PRINTED BY pr_ssrdgens +| [ ":" ssrgen(gen) ssrdgens_tl(dgens) ] -> [ cons_gen gen dgens ] +END + +let genstac (gens, clr) ist = + tclTHENLIST (cleartac clr :: List.rev_map (gentac ist) gens) + +(* Common code to handle generalization lists along with the defective case *) + +let with_defective maintac deps clr ist gl = + let top_id = + match kind_of_type (pf_concl gl) with + | ProdType (Name id, _, _) + when has_discharged_tag (string_of_id id) -> id + | _ -> top_id in + let top_gen = mkclr clr, cpattern_of_id top_id in + tclTHEN (introid top_id) (maintac deps top_gen ist) gl + +let with_dgens (gensl, clr) maintac ist = match gensl with + | [deps; []] -> with_defective maintac deps clr ist + | [deps; gen :: gens] -> + tclTHEN (genstac (gens, clr) ist) (maintac deps gen ist) + | [gen :: gens] -> tclTHEN (genstac (gens, clr) ist) (maintac [] gen ist) + | _ -> with_defective maintac [] clr ist + +let first_goal gls = + let gl = gls.Evd.it and sig_0 = gls.Evd.sigma in + if List.is_empty gl then Errors.error "first_goal"; + { Evd.it = List.hd gl; Evd.sigma = sig_0; } + +let with_deps deps0 maintac cl0 cs0 clr0 ist gl0 = + let rec loop gl cl cs clr args clrs = function + | [] -> + let n = List.length args in + maintac (if n > 0 then applist (to_lambda n cl, args) else cl) clrs ist gl0 + | dep :: deps -> + let gl' = first_goal (genclrtac cl cs clr gl) in + let cl', c', clr',gl' = pf_interp_gen ist gl' false dep in + loop gl' cl' [c'] clr' (c' :: args) (clr' :: clrs) deps in + loop gl0 cl0 cs0 clr0 cs0 [clr0] (List.rev deps0) + +(** Equations *) + +(* argument *) + +type ssreqid = ssripat option + +let pr_eqid = function Some pat -> str " " ++ pr_ipat pat | None -> mt () +let pr_ssreqid _ _ _ = pr_eqid + +(* We must use primitive parsing here to avoid conflicts with the *) +(* basic move, case, and elim tactics. *) +ARGUMENT EXTEND ssreqid TYPED AS ssripatrep option PRINTED BY pr_ssreqid +| [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let accept_ssreqid strm = + match Compat.get_tok (Util.stream_nth 0 strm) with + | Tok.IDENT _ -> accept_before_syms [":"] strm + | Tok.KEYWORD ":" -> () + | Tok.KEYWORD pat when List.mem pat ["_"; "?"; "->"; "<-"] -> + accept_before_syms [":"] strm + | _ -> raise Stream.Failure + +let test_ssreqid = Gram.Entry.of_parser "test_ssreqid" accept_ssreqid + +GEXTEND Gram + GLOBAL: ssreqid; + ssreqpat: [ + [ id = Prim.ident -> IpatId id + | "_" -> IpatWild + | "?" -> IpatAnon + | occ = ssrdocc; "->" -> (match occ with + | None, occ -> IpatRw (occ, L2R) + | _ -> loc_error !@loc "Only occurrences are allowed here") + | occ = ssrdocc; "<-" -> (match occ with + | None, occ -> IpatRw (occ, R2L) + | _ -> loc_error !@loc "Only occurrences are allowed here") + | "->" -> IpatRw (allocc, L2R) + | "<-" -> IpatRw (allocc, R2L) + ]]; + ssreqid: [ + [ test_ssreqid; pat = ssreqpat -> Some pat + | test_ssreqid -> None + ]]; +END + +(* creation *) + +let mkEq dir cl c t n gl = + let eqargs = [|t; c; c|] in eqargs.(dir_org dir) <- mkRel n; + let eq, gl = pf_fresh_global (build_coq_eq()) gl in + let refl, gl = mkRefl t c gl in + mkArrow (mkApp (eq, eqargs)) (lift 1 cl), refl, gl + +let pushmoveeqtac cl c gl = + let x, t, cl1 = destProd cl in + let cl2, eqc, gl = mkEq R2L cl1 c t 1 gl in + apply_type (mkProd (x, t, cl2)) [c; eqc] gl + +let pushcaseeqtac cl gl = + let cl1, args = destApplication cl in + let n = Array.length args in + let dc, cl2 = decompose_lam_n n cl1 in + let _, t = List.nth dc (n - 1) in + let cl3, eqc, gl = mkEq R2L cl2 args.(0) t n gl in + let prot, gl = mkProt (pf_type_of gl cl) cl3 gl in + let cl4 = mkApp (compose_lam dc prot, args) in + let gl, _ = pf_e_type_of gl cl4 in + tclTHEN (apply_type cl4 [eqc]) + (Proofview.V82.of_tactic (convert_concl cl4)) gl + +let pushelimeqtac gl = + let _, args = destApplication (pf_concl gl) in + let x, t, _ = destLambda args.(1) in + let cl1 = mkApp (args.(1), Array.sub args 2 (Array.length args - 2)) in + let cl2, eqc, gl = mkEq L2R cl1 args.(2) t 1 gl in + tclTHEN (apply_type (mkProd (x, t, cl2)) [args.(2); eqc]) + (Proofview.V82.of_tactic intro) gl + +(** Bookkeeping (discharge-intro) argument *) + +(* Since all bookkeeping ssr commands have the same discharge-intro *) +(* argument format we use a single grammar entry point to parse them. *) +(* the entry point parses only non-empty arguments to avoid conflicts *) +(* with the basic Coq tactics. *) + +(* type ssrarg = ssrview * (ssreqid * (ssrdgens * ssripats)) *) + +let pr_ssrarg _ _ _ (view, (eqid, (dgens, ipats))) = + let pri = pr_intros (gens_sep dgens) in + pr_view view ++ pr_eqid eqid ++ pr_dgens pr_gen dgens ++ pri ipats + +ARGUMENT EXTEND ssrarg TYPED AS ssrview * (ssreqid * (ssrdgens * ssrintros)) + PRINTED BY pr_ssrarg +| [ ssrview(view) ssreqid(eqid) ssrdgens(dgens) ssrintros(ipats) ] -> + [ view, (eqid, (dgens, ipats)) ] +| [ ssrview(view) ssrclear(clr) ssrintros(ipats) ] -> + [ view, (None, (([], clr), ipats)) ] +| [ ssreqid(eqid) ssrdgens(dgens) ssrintros(ipats) ] -> + [ [], (eqid, (dgens, ipats)) ] +| [ ssrclear_ne(clr) ssrintros(ipats) ] -> + [ [], (None, (([], clr), ipats)) ] +| [ ssrintros_ne(ipats) ] -> + [ [], (None, (([], []), ipats)) ] +END + +(** The "clear" tactic *) + +(* We just add a numeric version that clears the n top assumptions. *) + +let poptac ist n = introstac ~ist (List.init n (fun _ -> IpatWild)) + +TACTIC EXTEND ssrclear + | [ "clear" natural(n) ] -> [ Proofview.V82.tactic (poptac ist n) ] +END + +(** The "move" tactic *) + +let rec improper_intros = function + | IpatSimpl _ :: ipats -> improper_intros ipats + | (IpatId _ | IpatAnon | IpatCase _ | IpatAll) :: _ -> false + | _ -> true + +let check_movearg = function + | view, (eqid, _) when view <> [] && eqid <> None -> + Errors.error "incompatible view and equation in move tactic" + | view, (_, (([gen :: _], _), _)) when view <> [] && has_occ gen -> + Errors.error "incompatible view and occurrence switch in move tactic" + | _, (_, ((dgens, _), _)) when List.length dgens > 1 -> + Errors.error "dependents switch `/' in move tactic" + | _, (eqid, (_, ipats)) when eqid <> None && improper_intros ipats -> + Errors.error "no proper intro pattern for equation in move tactic" + | arg -> arg + +ARGUMENT EXTEND ssrmovearg TYPED AS ssrarg PRINTED BY pr_ssrarg +| [ ssrarg(arg) ] -> [ check_movearg arg ] +END + +let viewmovetac_aux clear name_ref (_, vl as v) _ gen ist gl = + let cl, c, clr, gl = pf_interp_gen ist gl false gen in + let cl, c, gl = if vl = [] then cl, c, gl else pf_with_view ist gl v cl c in + let clr = if clear then clr else [] in + name_ref := (match id_of_cpattern (snd gen) with Some id -> id | _ -> top_id); + genclrtac cl [c] clr gl + +let () = move_top_with_view := + (fun c r v -> with_defective (viewmovetac_aux c r v) [] []) + +let viewmovetac v deps gen ist gl = + viewmovetac_aux true (ref top_id) v deps gen ist gl + +let eqmovetac _ gen ist gl = + let cl, c, _, gl = pf_interp_gen ist gl false gen in pushmoveeqtac cl c gl + +let movehnftac gl = match kind_of_term (pf_concl gl) with + | Prod _ | LetIn _ -> tclIDTAC gl + | _ -> hnf_in_concl gl + +let ssrmovetac ist = function + | _::_ as view, (_, (dgens, ipats)) -> + let dgentac = with_dgens dgens (viewmovetac (true, view)) ist in + tclTHEN dgentac (introstac ~ist ipats) + | _, (Some pat, (dgens, ipats)) -> + let dgentac = with_dgens dgens eqmovetac ist in + tclTHEN dgentac (introstac ~ist (eqmoveipats pat ipats)) + | _, (_, (([gens], clr), ipats)) -> + let gentac = genstac (gens, clr) ist in + tclTHEN gentac (introstac ~ist ipats) + | _, (_, ((_, clr), ipats)) -> + tclTHENLIST [movehnftac; cleartac clr; introstac ~ist ipats] + +TACTIC EXTEND ssrmove +| [ "move" ssrmovearg(arg) ssrrpat(pat) ] -> + [ Proofview.V82.tactic (tclTHEN (ssrmovetac ist arg) (introstac ~ist [pat])) ] +| [ "move" ssrmovearg(arg) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrmovetac ist arg) clauses) ] +| [ "move" ssrrpat(pat) ] -> [ Proofview.V82.tactic (introstac ~ist [pat]) ] +| [ "move" ] -> [ Proofview.V82.tactic (movehnftac) ] +END + +(* TASSI: given the type of an elimination principle, it finds the higher order + * argument (index), it computes it's arity and the arity of the eliminator and + * checks if the eliminator is recursive or not *) +let analyze_eliminator elimty env sigma = + let rec loop ctx t = match kind_of_type t with + | AtomicType (hd, args) when isRel hd -> + ctx, destRel hd, not (noccurn 1 t), Array.length args + | CastType (t, _) -> loop ctx t + | ProdType (x, ty, t) -> loop ((x,None,ty) :: ctx) t + | LetInType (x,b,ty,t) -> loop ((x,Some b,ty) :: ctx) (subst1 b t) + | _ -> + let env' = Environ.push_rel_context ctx env in + let t' = Reductionops.whd_betadeltaiota env' sigma t in + if not (Term.eq_constr t t') then loop ctx t' else + errorstrm (str"The eliminator has the wrong shape."++spc()++ + str"A (applied) bound variable was expected as the conclusion of "++ + str"the eliminator's"++Pp.cut()++str"type:"++spc()++pr_constr elimty) in + let ctx, pred_id, elim_is_dep, n_pred_args = loop [] elimty in + let n_elim_args = rel_context_nhyps ctx in + let is_rec_elim = + let count_occurn n term = + let count = ref 0 in + let rec occur_rec n c = match kind_of_term c with + | Rel m -> if m = n then incr count + | _ -> iter_constr_with_binders succ occur_rec n c + in + occur_rec n term; !count in + let occurr2 n t = count_occurn n t > 1 in + not (List.for_all_i + (fun i (_,rd) -> pred_id <= i || not (occurr2 (pred_id - i) rd)) + 1 (assums_of_rel_context ctx)) + in + n_elim_args - pred_id, n_elim_args, is_rec_elim, elim_is_dep, n_pred_args + +(* TASSI: This version of unprotects inlines the unfold tactic definition, + * since we don't want to wipe out let-ins, and it seems there is no flag + * to change that behaviour in the standard unfold code *) +let unprotecttac gl = + let c, gl = pf_mkSsrConst "protect_term" gl in + let prot, _ = destConst c in + onClause (fun idopt -> + let hyploc = Option.map (fun id -> id, InHyp) idopt in + reduct_option + (Reductionops.clos_norm_flags + (Closure.RedFlags.mkflags + [Closure.RedFlags.fBETA; + Closure.RedFlags.fCONST prot; + Closure.RedFlags.fIOTA]), DEFAULTcast) hyploc) + allHypsAndConcl gl + +let dependent_apply_error = + try Errors.error "Could not fill dependent hole in \"apply\"" with err -> err + +(* TASSI: Sometimes Coq's apply fails. According to my experience it may be + * related to goals that are products and with beta redexes. In that case it + * guesses the wrong number of implicit arguments for your lemma. What follows + * is just like apply, but with a user-provided number n of implicits. + * + * Refine.refine function that handles type classes and evars but fails to + * handle "dependently typed higher order evars". + * + * Refiner.refiner that does not handle metas with a non ground type but works + * with dependently typed higher order metas. *) +let applyn ~with_evars ?beta ?(with_shelve=false) n t gl = + if with_evars then + let refine gl = + let t, ty, args, gl = pf_saturate ?beta ~bi_types:true gl t n in +(* pp(lazy(str"sigma@saturate=" ++ pr_evar_map None (project gl))); *) + let gl = pf_unify_HO gl ty (pf_concl gl) in + let gs = CList.map_filter (fun (_, e) -> + if isEvar (pf_nf_evar gl e) then Some e else None) + args in + pf_partial_solution gl t gs + in + Proofview.(V82.of_tactic + (tclTHEN (V82.tactic refine) + (if with_shelve then shelve_unifiable else tclUNIT ()))) gl + else + let t, gl = if n = 0 then t, gl else + let sigma, si = project gl, sig_it gl in + let rec loop sigma bo args = function (* saturate with metas *) + | 0 -> mkApp (t, Array.of_list (List.rev args)), re_sig si sigma + | n -> match kind_of_term bo with + | Lambda (_, ty, bo) -> + if not (closed0 ty) then raise dependent_apply_error; + let m = Evarutil.new_meta () in + loop (meta_declare m ty sigma) bo ((mkMeta m)::args) (n-1) + | _ -> assert false + in loop sigma t [] n in + pp(lazy(str"Refiner.refiner " ++ pr_constr t)); + Refiner.refiner (Proof_type.Refine t) gl + +let refine_with ?(first_goes_last=false) ?beta ?(with_evars=true) oc gl = + let rec mkRels = function 1 -> [] | n -> mkRel n :: mkRels (n-1) in + let uct = Evd.evar_universe_context (fst oc) in + let n, oc = pf_abs_evars_pirrel gl oc in + let gl = pf_merge_uc uct gl in + let oc = if not first_goes_last || n <= 1 then oc else + let l, c = decompose_lam oc in + if not (List.for_all_i (fun i (_,t) -> closedn ~-i t) (1-n) l) then oc else + compose_lam (let xs,y = List.chop (n-1) l in y @ xs) + (mkApp (compose_lam l c, Array.of_list (mkRel 1 :: mkRels n))) + in + pp(lazy(str"after: " ++ pr_constr oc)); + try applyn ~with_evars ~with_shelve:true ?beta n oc gl with _ -> raise dependent_apply_error + +let pf_fresh_inductive_instance ind gl = + let sigma, env, it = project gl, pf_env gl, sig_it gl in + let sigma, indu = Evd.fresh_inductive_instance env sigma ind in + indu, re_sig it sigma + +(** The "case" and "elim" tactic *) + +(* A case without explicit dependent terms but with both a view and an *) +(* occurrence switch and/or an equation is treated as dependent, with the *) +(* viewed term as the dependent term (the occurrence switch would be *) +(* meaningless otherwise). When both a view and explicit dependents are *) +(* present, it is forbidden to put a (meaningless) occurrence switch on *) +(* the viewed term. *) + +(* This is both elim and case (defaulting to the former). If ~elim is omitted + * the standard eliminator is chosen. The code is made of 4 parts: + * 1. find the eliminator if not given as ~elim and analyze it + * 2. build the patterns to be matched against the conclusion, looking at + * (occ, c), deps and the pattern inferred from the type of the eliminator + * 3. build the new predicate matching the patterns, and the tactic to + * generalize the equality in case eqid is not None + * 4. build the tactic handle intructions and clears as required in ipats and + * by eqid *) +let ssrelim ?(is_case=false) ?ist deps what ?elim eqid ipats gl = + (* some sanity checks *) + let oc, orig_clr, occ, c_gen, gl = match what with + | `EConstr(_,_,t) when isEvar t -> + anomaly "elim called on a constr evar" + | `EGen _ when ist = None -> + anomaly "no ist and non simple elimination" + | `EGen (_, g) when elim = None && is_wildcard g -> + errorstrm(str"Indeterminate pattern and no eliminator") + | `EGen ((Some clr,occ), g) when is_wildcard g -> + None, clr, occ, None, gl + | `EGen ((None, occ), g) when is_wildcard g -> None,[],occ,None,gl + | `EGen ((_, occ), p as gen) -> + let _, c, clr,gl = pf_interp_gen (Option.get ist) gl true gen in + Some c, clr, occ, Some p,gl + | `EConstr (clr, occ, c) -> Some c, clr, occ, None,gl in + let orig_gl, concl, env = gl, pf_concl gl, pf_env gl in + pp(lazy(str(if is_case then "==CASE==" else "==ELIM=="))); + (* Utils of local interest only *) + let iD s ?t gl = let t = match t with None -> pf_concl gl | Some x -> x in + pp(lazy(str s ++ pr_constr t)); tclIDTAC gl in + let eq, gl = pf_fresh_global (build_coq_eq ()) gl in + let protectC, gl = pf_mkSsrConst "protect_term" gl in + let fire_subst gl t = Reductionops.nf_evar (project gl) t in + let fire_sigma sigma t = Reductionops.nf_evar sigma t in + let is_undef_pat = function + | sigma, T t -> + (match kind_of_term (fire_sigma sigma t) with Evar _ -> true | _ -> false) + | _ -> false in + let match_pat env p occ h cl = + let sigma0 = project orig_gl in + pp(lazy(str"matching: " ++ pp_pattern p)); + let (c,ucst), cl = + fill_occ_pattern ~raise_NoMatch:true env sigma0 cl p occ h in + pp(lazy(str" got: " ++ pr_constr c)); + c, cl, ucst in + let mkTpat gl t = (* takes a term, refreshes it and makes a T pattern *) + let n, t, _, ucst = pf_abs_evars orig_gl (project gl, fire_subst gl t) in + let t, _, _, sigma = saturate ~beta:true env (project gl) t n in + Evd.merge_universe_context sigma ucst, T t in + let unif_redex gl (sigma, r as p) t = (* t is a hint for the redex of p *) + let n, t, _, ucst = pf_abs_evars orig_gl (project gl, fire_subst gl t) in + let t, _, _, sigma = saturate ~beta:true env sigma t n in + let sigma = Evd.merge_universe_context sigma ucst in + match r with + | X_In_T (e, p) -> sigma, E_As_X_In_T (t, e, p) + | _ -> + try unify_HO env sigma t (fst (redex_of_pattern env p)), r + with e when Errors.noncritical e -> p in + (* finds the eliminator applies it to evars and c saturated as needed *) + (* obtaining "elim ??? (c ???)". pred is the higher order evar *) + let cty, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl = + match elim with + | Some elim -> + let gl, elimty = pf_e_type_of gl elim in + let pred_id, n_elim_args, is_rec, elim_is_dep, n_pred_args = + analyze_eliminator elimty env (project gl) in + let elim, elimty, elim_args, gl = + pf_saturate ~beta:is_case gl elim ~ty:elimty n_elim_args in + let pred = List.assoc pred_id elim_args in + let elimty = Reductionops.whd_betadeltaiota env (project gl) elimty in + None, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl + | None -> + let c = Option.get oc in let c_ty = pf_type_of gl c in + let ((kn, i) as ind, _ as indu), unfolded_c_ty = + pf_reduce_to_quantified_ind gl c_ty in + let sort = elimination_sort_of_goal gl in + let gl, elim = + if not is_case then + let t, gl = pf_fresh_global (Indrec.lookup_eliminator ind sort) gl in + gl, t + else + pf_eapply (fun env sigma -> + Indrec.build_case_analysis_scheme env sigma indu true) gl sort in + let elimty = pf_type_of gl elim in + let pred_id,n_elim_args,is_rec,elim_is_dep,n_pred_args = + analyze_eliminator elimty env (project gl) in + let rctx = fst (decompose_prod_assum unfolded_c_ty) in + let n_c_args = rel_context_length rctx in + let c, c_ty, t_args, gl = pf_saturate gl c ~ty:c_ty n_c_args in + let elim, elimty, elim_args, gl = + pf_saturate ~beta:is_case gl elim ~ty:elimty n_elim_args in + let pred = List.assoc pred_id elim_args in + let pc = match n_c_args, c_gen with + | 0, Some p -> interp_cpattern (Option.get ist) orig_gl p None + | _ -> mkTpat gl c in + let cty = Some (c, c_ty, pc) in + let elimty = Reductionops.whd_betadeltaiota env (project gl) elimty in + cty, elim, elimty, elim_args, n_elim_args, elim_is_dep, is_rec, pred, gl + in + pp(lazy(str"elim= "++ pr_constr_pat elim)); + pp(lazy(str"elimty= "++ pr_constr elimty)); + let inf_deps_r = match kind_of_type elimty with + | AtomicType (_, args) -> List.rev (Array.to_list args) + | _ -> assert false in + let saturate_until gl c c_ty f = + let rec loop n = try + let c, c_ty, _, gl = pf_saturate gl c ~ty:c_ty n in + let gl' = f c c_ty gl in + Some (c, c_ty, gl, gl') + with NotEnoughProducts -> None | _ -> loop (n+1) in loop 0 in + let elim_is_dep, gl = match cty with + | None -> true, gl + | Some (c, c_ty, _) -> + let res = + if elim_is_dep then None else + let arg = List.assoc (n_elim_args - 1) elim_args in + let arg_ty = pf_type_of gl arg in + match saturate_until gl c c_ty (fun c c_ty gl -> + pf_unify_HO (pf_unify_HO gl c_ty arg_ty) arg c) with + | Some (c, _, _, gl) -> Some (false, gl) + | None -> None in + match res with + | Some x -> x + | None -> + let inf_arg = List.hd inf_deps_r in + let inf_arg_ty = pf_type_of gl inf_arg in + match saturate_until gl c c_ty (fun _ c_ty gl -> + pf_unify_HO gl c_ty inf_arg_ty) with + | Some (c, _, _,gl) -> true, gl + | None -> + errorstrm (str"Unable to apply the eliminator to the term"++ + spc()++pr_constr c++spc()++str"or to unify it's type with"++ + pr_constr inf_arg_ty) in + pp(lazy(str"elim_is_dep= " ++ bool elim_is_dep)); + let predty = pf_type_of gl pred in + (* Patterns for the inductive types indexes to be bound in pred are computed + * looking at the ones provided by the user and the inferred ones looking at + * the type of the elimination principle *) + let pp_pat (_,p,_,_) = pp_pattern p in + let pp_inf_pat gl (_,_,t,_) = pr_constr_pat (fire_subst gl t) in + let patterns, clr, gl = + let rec loop patterns clr i = function + | [],[] -> patterns, clr, gl + | ((oclr, occ), t):: deps, inf_t :: inf_deps -> + let ist = match ist with Some x -> x | None -> assert false in + let p = interp_cpattern ist orig_gl t None in + let clr_t = + interp_clr (oclr,(tag_of_cpattern t,fst (redex_of_pattern env p)))in + (* if we are the index for the equation we do not clear *) + let clr_t = if deps = [] && eqid <> None then [] else clr_t in + let p = if is_undef_pat p then mkTpat gl inf_t else p in + loop (patterns @ [i, p, inf_t, occ]) + (clr_t @ clr) (i+1) (deps, inf_deps) + | [], c :: inf_deps -> + pp(lazy(str"adding inferred pattern " ++ pr_constr_pat c)); + loop (patterns @ [i, mkTpat gl c, c, allocc]) + clr (i+1) ([], inf_deps) + | _::_, [] -> errorstrm (str "Too many dependent abstractions") in + let deps, head_p, inf_deps_r = match what, elim_is_dep, cty with + | `EConstr _, _, None -> anomaly "Simple welim with no term" + | _, false, _ -> deps, [], inf_deps_r + | `EGen gen, true, None -> deps @ [gen], [], inf_deps_r + | _, true, Some (c, _, pc) -> + let occ = if occ = None then allocc else occ in + let inf_p, inf_deps_r = List.hd inf_deps_r, List.tl inf_deps_r in + deps, [1, pc, inf_p, occ], inf_deps_r in + let patterns, clr, gl = + loop [] orig_clr (List.length head_p+1) (List.rev deps, inf_deps_r) in + head_p @ patterns, Util.List.uniquize clr, gl + in + pp(lazy(pp_concat (str"patterns=") (List.map pp_pat patterns))); + pp(lazy(pp_concat (str"inf. patterns=") (List.map (pp_inf_pat gl) patterns))); + (* Predicate generation, and (if necessary) tactic to generalize the + * equation asked by the user *) + let elim_pred, gen_eq_tac, clr, gl = + let error gl t inf_t = errorstrm (str"The given pattern matches the term"++ + spc()++pp_term gl t++spc()++str"while the inferred pattern"++ + spc()++pr_constr_pat (fire_subst gl inf_t)++spc()++ str"doesn't") in + let match_or_postpone (cl, gl, post) (h, p, inf_t, occ) = + let p = unif_redex gl p inf_t in + if is_undef_pat p then + let () = pp(lazy(str"postponing " ++ pp_pattern p)) in + cl, gl, post @ [h, p, inf_t, occ] + else try + let c, cl, ucst = match_pat env p occ h cl in + let gl = pf_merge_uc ucst gl in + let gl = try pf_unify_HO gl inf_t c with _ -> error gl c inf_t in + cl, gl, post + with + | NoMatch | NoProgress -> + let e, ucst = redex_of_pattern env p in + let gl = pf_merge_uc ucst gl in + let n, e, _, _ucst = pf_abs_evars gl (fst p, e) in + let e, _, _, gl = pf_saturate ~beta:true gl e n in + let gl = try pf_unify_HO gl inf_t e with _ -> error gl e inf_t in + cl, gl, post + in + let rec match_all concl gl patterns = + let concl, gl, postponed = + List.fold_left match_or_postpone (concl, gl, []) patterns in + if postponed = [] then concl, gl + else if List.length postponed = List.length patterns then + errorstrm (str "Some patterns are undefined even after all"++spc()++ + str"the defined ones matched") + else match_all concl gl postponed in + let concl, gl = match_all concl gl patterns in + let pred_rctx, _ = decompose_prod_assum (fire_subst gl predty) in + let concl, gen_eq_tac, clr, gl = match eqid with + | Some (IpatId _) when not is_rec -> + let k = List.length deps in + let c = fire_subst gl (List.assoc (n_elim_args - k - 1) elim_args) in + let t = pf_type_of gl c in + let gen_eq_tac, gl = + let refl = mkApp (eq, [|t; c; c|]) in + let new_concl = mkArrow refl (lift 1 (pf_concl orig_gl)) in + let new_concl = fire_subst gl new_concl in + let erefl, gl = mkRefl t c gl in + let erefl = fire_subst gl erefl in + apply_type new_concl [erefl], gl in + let rel = k + if elim_is_dep then 1 else 0 in + let src, gl = mkProt mkProp (mkApp (eq,[|t; c; mkRel rel|])) gl in + let concl = mkArrow src (lift 1 concl) in + let clr = if deps <> [] then clr else [] in + concl, gen_eq_tac, clr, gl + | _ -> concl, tclIDTAC, clr, gl in + let mk_lam t r = mkLambda_or_LetIn r t in + let concl = List.fold_left mk_lam concl pred_rctx in + let gl, concl = + if eqid <> None && is_rec then + let concl, gl = mkProt (pf_type_of gl concl) concl gl in + let gl, _ = pf_e_type_of gl concl in + gl, concl + else gl, concl in + concl, gen_eq_tac, clr, gl in + let gl, pty = pf_e_type_of gl elim_pred in + pp(lazy(str"elim_pred=" ++ pp_term gl elim_pred)); + pp(lazy(str"elim_pred_ty=" ++ pp_term gl pty)); + let gl = pf_unify_HO gl pred elim_pred in + let elim = fire_subst gl elim in + let gl, _ = pf_e_type_of gl elim in + (* check that the patterns do not contain non instantiated dependent metas *) + let () = + let evars_of_term = Evarutil.undefined_evars_of_term (project gl) in + let patterns = List.map (fun (_,_,t,_) -> fire_subst gl t) patterns in + let patterns_ev = List.map evars_of_term patterns in + let ev = List.fold_left Intset.union Intset.empty patterns_ev in + let ty_ev = Intset.fold (fun i e -> + let ex = i in + let i_ty = Evd.evar_concl (Evd.find (project gl) ex) in + Intset.union e (evars_of_term i_ty)) + ev Intset.empty in + let inter = Intset.inter ev ty_ev in + if not (Intset.is_empty inter) then begin + let i = Intset.choose inter in + let pat = List.find (fun t -> Intset.mem i (evars_of_term t)) patterns in + errorstrm(str"Pattern"++spc()++pr_constr_pat pat++spc()++ + str"was not completely instantiated and one of its variables"++spc()++ + str"occurs in the type of another non instantieted pattern variable"); + end + in + (* the elim tactic, with the eliminator and the predicated we computed *) + let elim = project gl, elim in + let elim_tac gl = + tclTHENLIST [refine_with ~with_evars:false elim; cleartac clr] gl in + (* handling of following intro patterns and equation introduction if any *) + let elim_intro_tac gl = + let intro_eq = + match eqid with + | Some (IpatId ipat) when not is_rec -> + let rec intro_eq gl = match kind_of_type (pf_concl gl) with + | ProdType (_, src, tgt) -> + (match kind_of_type src with + | AtomicType (hd, _) when Term.eq_constr hd protectC -> + tclTHENLIST [unprotecttac; introid ipat] gl + | _ -> tclTHENLIST [ iD "IA"; introstac [IpatAnon]; intro_eq] gl) + |_ -> errorstrm (str "Too many names in intro pattern") in + intro_eq + | Some (IpatId ipat) -> + let name gl = mk_anon_id "K" gl in + let intro_lhs gl = + let elim_name = match orig_clr, what with + | [SsrHyp(_, x)], _ -> x + | _, `EConstr(_,_,t) when isVar t -> destVar t + | _ -> name gl in + if is_name_in_ipats elim_name ipats then introid (name gl) gl + else introid elim_name gl + in + let rec gen_eq_tac gl = + let concl = pf_concl gl in + let ctx, last = decompose_prod_assum concl in + let args = match kind_of_type last with + | AtomicType (hd, args) -> assert(Term.eq_constr hd protectC); args + | _ -> assert false in + let case = args.(Array.length args-1) in + if not(closed0 case) then tclTHEN (introstac [IpatAnon]) gen_eq_tac gl + else + let case_ty = pf_type_of gl case in + let refl = mkApp (eq, [|lift 1 case_ty; mkRel 1; lift 1 case|]) in + let new_concl = fire_subst gl + (mkProd (Name (name gl), case_ty, mkArrow refl (lift 2 concl))) in + let erefl, gl = mkRefl case_ty case gl in + let erefl = fire_subst gl erefl in + apply_type new_concl [case;erefl] gl in + tclTHENLIST [gen_eq_tac; intro_lhs; introid ipat] + | _ -> tclIDTAC in + let unprot = if eqid <> None && is_rec then unprotecttac else tclIDTAC in + tclEQINTROS ?ist elim_tac (tclTHENLIST [intro_eq; unprot]) ipats gl + in + tclTHENLIST [gen_eq_tac; elim_intro_tac] orig_gl +;; + +let simplest_newelim x= ssrelim ~is_case:false [] (`EConstr ([],None,x)) None [] +let simplest_newcase x= ssrelim ~is_case:true [] (`EConstr ([],None,x)) None [] +let _ = simplest_newcase_ref := simplest_newcase + +let check_casearg = function +| view, (_, (([_; gen :: _], _), _)) when view <> [] && has_occ gen -> + Errors.error "incompatible view and occurrence switch in dependent case tactic" +| arg -> arg + +ARGUMENT EXTEND ssrcasearg TYPED AS ssrarg PRINTED BY pr_ssrarg +| [ ssrarg(arg) ] -> [ check_casearg arg ] +END + +let ssrcasetac ist (view, (eqid, (dgens, ipats))) = + let ndefectcasetac view eqid ipats deps ((_, occ), _ as gen) ist gl = + let simple = (eqid = None && deps = [] && occ = None) in + let cl, c, clr, gl = pf_interp_gen ist gl true gen in + let _, vc, gl = + if view = [] then c, c, gl else pf_with_view ist gl (false, view) cl c in + if simple && is_injection_case vc gl then + tclTHENLIST [perform_injection vc; cleartac clr; introstac ~ist ipats] gl + else + (* macro for "case/v E: x" ---> "case E: x / (v x)" *) + let deps, clr, occ = + if view <> [] && eqid <> None && deps = [] then [gen], [], None + else deps, clr, occ in + ssrelim ~is_case:true ~ist deps (`EConstr (clr,occ, vc)) eqid ipats gl + in + with_dgens dgens (ndefectcasetac view eqid ipats) ist + +TACTIC EXTEND ssrcase +| [ "case" ssrcasearg(arg) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrcasetac ist arg) clauses) ] +| [ "case" ] -> [ Proofview.V82.tactic (with_top ssrscasetac) ] +END + +(** The "elim" tactic *) + +(* Elim views are elimination lemmas, so the eliminated term is not addded *) +(* to the dependent terms as for "case", unless it actually occurs in the *) +(* goal, the "all occurrences" {+} switch is used, or the equation switch *) +(* is used and there are no dependents. *) + +let ssrelimtac ist (view, (eqid, (dgens, ipats))) = + let ndefectelimtac view eqid ipats deps gen ist gl = + let elim = match view with [v] -> Some (snd(force_term ist gl v)) | _ -> None in + ssrelim ~ist deps (`EGen gen) ?elim eqid ipats gl + in + with_dgens dgens (ndefectelimtac view eqid ipats) ist + +TACTIC EXTEND ssrelim +| [ "elim" ssrarg(arg) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrelimtac ist arg) clauses) ] +| [ "elim" ] -> [ Proofview.V82.tactic (with_top simplest_newelim) ] +END + +(** 6. Backward chaining tactics: apply, exact, congr. *) + +(** The "apply" tactic *) + +let pr_agen (docc, dt) = pr_docc docc ++ pr_term dt +let pr_ssragen _ _ _ = pr_agen +let pr_ssragens _ _ _ = pr_dgens pr_agen + +ARGUMENT EXTEND ssragen TYPED AS ssrdocc * ssrterm PRINTED BY pr_ssragen +| [ "{" ne_ssrhyp_list(clr) "}" ssrterm(dt) ] -> [ mkclr clr, dt ] +| [ ssrterm(dt) ] -> [ nodocc, dt ] +END + +ARGUMENT EXTEND ssragens TYPED AS ssragen list list * ssrclear +PRINTED BY pr_ssragens +| [ "{" ne_ssrhyp_list(clr) "}" ssrterm(dt) ssragens(agens) ] -> + [ cons_gen (mkclr clr, dt) agens ] +| [ "{" ne_ssrhyp_list(clr) "}" ] -> [ [[]], clr] +| [ ssrterm(dt) ssragens(agens) ] -> + [ cons_gen (nodocc, dt) agens ] +| [ ] -> [ [[]], [] ] +END + +let mk_applyarg views agens intros = views, (None, (agens, intros)) + +let pr_ssraarg _ _ _ (view, (eqid, (dgens, ipats))) = + let pri = pr_intros (gens_sep dgens) in + pr_view view ++ pr_eqid eqid ++ pr_dgens pr_agen dgens ++ pri ipats + +ARGUMENT EXTEND ssrapplyarg +TYPED AS ssrview * (ssreqid * (ssragens * ssrintros)) +PRINTED BY pr_ssraarg +| [ ":" ssragen(gen) ssragens(dgens) ssrintros(intros) ] -> + [ mk_applyarg [] (cons_gen gen dgens) intros ] +| [ ssrclear_ne(clr) ssrintros(intros) ] -> + [ mk_applyarg [] ([], clr) intros ] +| [ ssrintros_ne(intros) ] -> + [ mk_applyarg [] ([], []) intros ] +| [ ssrview(view) ":" ssragen(gen) ssragens(dgens) ssrintros(intros) ] -> + [ mk_applyarg view (cons_gen gen dgens) intros ] +| [ ssrview(view) ssrclear(clr) ssrintros(intros) ] -> + [ mk_applyarg view ([], clr) intros ] +END + +let interp_agen ist gl ((goclr, _), (k, gc)) (clr, rcs) = +(* pp(lazy(str"sigma@interp_agen=" ++ pr_evar_map None (project gl))); *) + let rc = glob_constr ist (pf_env gl) gc in + let rcs' = rc :: rcs in + match goclr with + | None -> clr, rcs' + | Some ghyps -> + let clr' = snd (interp_hyps ist gl ghyps) @ clr in + if k <> ' ' then clr', rcs' else + match rc with + | GVar (loc, id) when not_section_id id -> SsrHyp (loc, id) :: clr', rcs' + | GRef (loc, VarRef id, _) when not_section_id id -> + SsrHyp (loc, id) :: clr', rcs' + | _ -> clr', rcs' + +let interp_agens ist gl gagens = + match List.fold_right (interp_agen ist gl) gagens ([], []) with + | clr, rlemma :: args -> + let n = interp_nbargs ist gl rlemma - List.length args in + let rec loop i = + if i > n then + errorstrm (str "Cannot apply lemma " ++ pf_pr_glob_constr gl rlemma) + else + try interp_refine ist gl (mkRApp rlemma (mkRHoles i @ args)) + with _ -> loop (i + 1) in + clr, loop 0 + | _ -> assert false + +let apply_rconstr ?ist t gl = +(* pp(lazy(str"sigma@apply_rconstr=" ++ pr_evar_map None (project gl))); *) + let n = match ist, t with + | None, (GVar (_, id) | GRef (_, VarRef id,_)) -> pf_nbargs gl (mkVar id) + | Some ist, _ -> interp_nbargs ist gl t + | _ -> anomaly "apply_rconstr without ist and not RVar" in + let mkRlemma i = mkRApp t (mkRHoles i) in + let cl = pf_concl gl in + let rec loop i = + if i > n then + errorstrm (str"Cannot apply lemma "++pf_pr_glob_constr gl t) + else try pf_match gl (mkRlemma i) (OfType cl) with _ -> loop (i + 1) in + refine_with (loop 0) gl + +let mkRAppView ist gl rv gv = + let nb_view_imps = interp_view_nbimps ist gl rv in + mkRApp rv (mkRHoles (abs nb_view_imps)) + +let prof_apply_interp_with = mk_profiler "ssrapplytac.interp_with";; + +let refine_interp_apply_view i ist gl gv = + let pair i = List.map (fun x -> i, x) in + let rv = pf_intern_term ist gl gv in + let v = mkRAppView ist gl rv gv in + let interp_with (i, hint) = + interp_refine ist gl (mkRApp hint (v :: mkRHoles i)) in + let interp_with x = prof_apply_interp_with.profile interp_with x in + let rec loop = function + | [] -> (try apply_rconstr ~ist rv gl with _ -> view_error "apply" gv) + | h :: hs -> (try refine_with (snd (interp_with h)) gl with _ -> loop hs) in + loop (pair i viewtab.(i) @ if i = 2 then pair 1 viewtab.(1) else []) + +let apply_top_tac gl = + tclTHENLIST [introid top_id; apply_rconstr (mkRVar top_id); clear [top_id]] gl + +let inner_ssrapplytac gviews ggenl gclr ist gl = + let _, clr = interp_hyps ist gl gclr in + let vtac gv i gl' = refine_interp_apply_view i ist gl' gv in + let ggenl, tclGENTAC = + if gviews <> [] && ggenl <> [] then + let ggenl= List.map (fun (x,g) -> x, cpattern_of_term g) (List.hd ggenl) in + [], tclTHEN (genstac (ggenl,[]) ist) + else ggenl, tclTHEN tclIDTAC in + tclGENTAC (fun gl -> + match gviews, ggenl with + | v :: tl, [] -> + let dbl = if List.length tl = 1 then 2 else 1 in + tclTHEN + (List.fold_left (fun acc v -> tclTHENLAST acc (vtac v dbl)) (vtac v 1) tl) + (cleartac clr) gl + | [], [agens] -> + let clr', (sigma, lemma) = interp_agens ist gl agens in + let gl = pf_merge_uc_of sigma gl in + tclTHENLIST [cleartac clr; refine_with ~beta:true lemma; cleartac clr'] gl + | _, _ -> tclTHEN apply_top_tac (cleartac clr) gl) gl + +let ssrapplytac ist (views, (_, ((gens, clr), intros))) = + tclINTROS ist (inner_ssrapplytac views gens clr) intros + +TACTIC EXTEND ssrapply +| [ "apply" ssrapplyarg(arg) ] -> [ Proofview.V82.tactic (ssrapplytac ist arg) ] +| [ "apply" ] -> [ Proofview.V82.tactic apply_top_tac ] +END + +(** The "exact" tactic *) + +let mk_exactarg views dgens = mk_applyarg views dgens [] + +ARGUMENT EXTEND ssrexactarg TYPED AS ssrapplyarg PRINTED BY pr_ssraarg +| [ ":" ssragen(gen) ssragens(dgens) ] -> + [ mk_exactarg [] (cons_gen gen dgens) ] +| [ ssrview(view) ssrclear(clr) ] -> + [ mk_exactarg view ([], clr) ] +| [ ssrclear_ne(clr) ] -> + [ mk_exactarg [] ([], clr) ] +END + +let vmexacttac pf gl = exact_no_check (mkCast (pf, VMcast, pf_concl gl)) gl + +TACTIC EXTEND ssrexact +| [ "exact" ssrexactarg(arg) ] -> [ Proofview.V82.tactic (tclBY (ssrapplytac ist arg)) ] +| [ "exact" ] -> [ Proofview.V82.tactic (tclORELSE donetac (tclBY apply_top_tac)) ] +| [ "exact" "<:" lconstr(pf) ] -> [ Proofview.V82.tactic (vmexacttac pf) ] +END + +(** The "congr" tactic *) + +(* type ssrcongrarg = open_constr * (int * constr) *) + +let pr_ssrcongrarg _ _ _ ((n, f), dgens) = + (if n <= 0 then mt () else str " " ++ int n) ++ + str " " ++ pr_term f ++ pr_dgens pr_gen dgens + +ARGUMENT EXTEND ssrcongrarg TYPED AS (int * ssrterm) * ssrdgens + PRINTED BY pr_ssrcongrarg +| [ natural(n) constr(c) ssrdgens(dgens) ] -> [ (n, mk_term ' ' c), dgens ] +| [ natural(n) constr(c) ] -> [ (n, mk_term ' ' c),([[]],[]) ] +| [ constr(c) ssrdgens(dgens) ] -> [ (0, mk_term ' ' c), dgens ] +| [ constr(c) ] -> [ (0, mk_term ' ' c), ([[]],[]) ] +END + +let rec mkRnat n = + if n <= 0 then GRef (dummy_loc, glob_O, None) else + mkRApp (GRef (dummy_loc, glob_S, None)) [mkRnat (n - 1)] + +let interp_congrarg_at ist gl n rf ty m = + pp(lazy(str"===interp_congrarg_at===")); + let congrn, _ = mkSsrRRef "nary_congruence" in + let args1 = mkRnat n :: mkRHoles n @ [ty] in + let args2 = mkRHoles (3 * n) in + let rec loop i = + if i + n > m then None else + try + let rt = mkRApp congrn (args1 @ mkRApp rf (mkRHoles i) :: args2) in + pp(lazy(str"rt=" ++ pr_glob_constr rt)); + Some (interp_refine ist gl rt) + with _ -> loop (i + 1) in + loop 0 + +let pattern_id = mk_internal_id "pattern value" + +let congrtac ((n, t), ty) ist gl = + pp(lazy(str"===congr===")); + pp(lazy(str"concl=" ++ pr_constr (pf_concl gl))); + let sigma, _ as it = interp_term ist gl t in + let gl = pf_merge_uc_of sigma gl in + let _, f, _, _ucst = pf_abs_evars gl it in + let ist' = {ist with lfun = + Id.Map.add pattern_id (Value.of_constr f) Id.Map.empty } in + let rf = mkRltacVar pattern_id in + let m = pf_nbargs gl f in + let _, cf = if n > 0 then + match interp_congrarg_at ist' gl n rf ty m with + | Some cf -> cf + | None -> errorstrm (str "No " ++ int n ++ str "-congruence with " + ++ pr_term t) + else let rec loop i = + if i > m then errorstrm (str "No congruence with " ++ pr_term t) + else match interp_congrarg_at ist' gl i rf ty m with + | Some cf -> cf + | None -> loop (i + 1) in + loop 1 in + tclTHEN (refine_with cf) (tclTRY (Proofview.V82.of_tactic reflexivity)) gl + +let newssrcongrtac arg ist gl = + pp(lazy(str"===newcongr===")); + pp(lazy(str"concl=" ++ pr_constr (pf_concl gl))); + (* utils *) + let fs gl t = Reductionops.nf_evar (project gl) t in + let tclMATCH_GOAL (c, gl_c) proj t_ok t_fail gl = + match try Some (pf_unify_HO gl_c (pf_concl gl) c) with _ -> None with + | Some gl_c -> + tclTHEN (Proofview.V82.of_tactic (convert_concl (fs gl_c c))) + (t_ok (proj gl_c)) gl + | None -> t_fail () gl in + let mk_evar gl ty = + let env, sigma, si = pf_env gl, project gl, sig_it gl in + let sigma, x = Evarutil.new_evar env (create_evar_defs sigma) ty in + x, re_sig si sigma in + let arr, gl = pf_mkSsrConst "ssr_congr_arrow" gl in + let ssr_congr lr = mkApp (arr, lr) in + (* here thw two cases: simple equality or arrow *) + let equality, _, eq_args, gl' = + let eq, gl = pf_fresh_global (build_coq_eq ()) gl in + pf_saturate gl eq 3 in + tclMATCH_GOAL (equality, gl') (fun gl' -> fs gl' (List.assoc 0 eq_args)) + (fun ty -> congrtac (arg, Detyping.detype false [] (pf_env gl) (project gl) ty) ist) + (fun () -> + let lhs, gl' = mk_evar gl mkProp in let rhs, gl' = mk_evar gl' mkProp in + let arrow = mkArrow lhs (lift 1 rhs) in + tclMATCH_GOAL (arrow, gl') (fun gl' -> [|fs gl' lhs;fs gl' rhs|]) + (fun lr -> tclTHEN (Proofview.V82.of_tactic (apply (ssr_congr lr))) (congrtac (arg, mkRType) ist)) + (fun _ _ -> errorstrm (str"Conclusion is not an equality nor an arrow"))) + gl +;; + +TACTIC EXTEND ssrcongr +| [ "congr" ssrcongrarg(arg) ] -> +[ let arg, dgens = arg in + Proofview.V82.tactic begin + match dgens with + | [gens], clr -> tclTHEN (genstac (gens,clr) ist) (newssrcongrtac arg ist) + | _ -> errorstrm (str"Dependent family abstractions not allowed in congr") + end] +END + +(** 7. Rewriting tactics (rewrite, unlock) *) + +(** Coq rewrite compatibility flag *) + +let ssr_strict_match = ref false + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "strict redex matching"; + Goptions.optkey = ["Match"; "Strict"]; + Goptions.optread = (fun () -> !ssr_strict_match); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssr_strict_match := b) } + +(** Rewrite multiplier *) + +type ssrmult = int * ssrmmod + +let notimes = 0 +let nomult = 1, Once + +let pr_mult (n, m) = + if n > 0 && m <> Once then int n ++ pr_mmod m else pr_mmod m + +let pr_ssrmult _ _ _ = pr_mult + +ARGUMENT EXTEND ssrmult_ne TYPED AS int * ssrmmod PRINTED BY pr_ssrmult + | [ natural(n) ssrmmod(m) ] -> [ check_index loc n, m ] + | [ ssrmmod(m) ] -> [ notimes, m ] +END + +ARGUMENT EXTEND ssrmult TYPED AS ssrmult_ne PRINTED BY pr_ssrmult + | [ ssrmult_ne(m) ] -> [ m ] + | [ ] -> [ nomult ] +END + +(** Rewrite clear/occ switches *) + +let pr_rwocc = function + | None, None -> mt () + | None, occ -> pr_occ occ + | Some clr, _ -> pr_clear_ne clr + +let pr_ssrrwocc _ _ _ = pr_rwocc + +ARGUMENT EXTEND ssrrwocc TYPED AS ssrdocc PRINTED BY pr_ssrrwocc +| [ "{" ssrhyp_list(clr) "}" ] -> [ mkclr clr ] +| [ "{" ssrocc(occ) "}" ] -> [ mkocc occ ] +| [ ] -> [ noclr ] +END + +(** Rewrite rules *) + +type ssrwkind = RWred of ssrsimpl | RWdef | RWeq +(* type ssrrule = ssrwkind * ssrterm *) + +let pr_rwkind = function + | RWred s -> pr_simpl s + | RWdef -> str "/" + | RWeq -> mt () + +let wit_ssrrwkind = add_genarg "ssrrwkind" pr_rwkind + +let pr_rule = function + | RWred s, _ -> pr_simpl s + | RWdef, r-> str "/" ++ pr_term r + | RWeq, r -> pr_term r + +let pr_ssrrule _ _ _ = pr_rule + +let noruleterm loc = mk_term ' ' (mkCProp loc) + +ARGUMENT EXTEND ssrrule_ne TYPED AS ssrrwkind * ssrterm PRINTED BY pr_ssrrule + | [ ssrsimpl_ne(s) ] -> [ RWred s, noruleterm loc ] + | [ "/" ssrterm(t) ] -> [ RWdef, t ] + | [ ssrterm(t) ] -> [ RWeq, t ] +END + +ARGUMENT EXTEND ssrrule TYPED AS ssrrule_ne PRINTED BY pr_ssrrule + | [ ssrrule_ne(r) ] -> [ r ] + | [ ] -> [ RWred Nop, noruleterm loc ] +END + +(** Rewrite arguments *) + +(* type ssrrwarg = (ssrdir * ssrmult) * ((ssrdocc * ssrpattern) * ssrrule) *) + +let pr_option f = function None -> mt() | Some x -> f x +let pr_pattern_squarep= pr_option (fun r -> str "[" ++ pr_rpattern r ++ str "]") +let pr_ssrpattern_squarep _ _ _ = pr_pattern_squarep +let pr_rwarg ((d, m), ((docc, rx), r)) = + pr_rwdir d ++ pr_mult m ++ pr_rwocc docc ++ pr_pattern_squarep rx ++ pr_rule r + +let pr_ssrrwarg _ _ _ = pr_rwarg + +let mk_rwarg (d, (n, _ as m)) ((clr, occ as docc), rx) (rt, _ as r) = + if rt <> RWeq then begin + if rt = RWred Nop && not (m = nomult && occ = None && rx = None) + && (clr = None || clr = Some []) then + anomaly "Improper rewrite clear switch"; + if d = R2L && rt <> RWdef then + Errors.error "Right-to-left switch on simplification"; + if n <> 1 && rt = RWred Cut then + Errors.error "Bad or useless multiplier"; + if occ <> None && rx = None && rt <> RWdef then + Errors.error "Missing redex for simplification occurrence" + end; (d, m), ((docc, rx), r) + +let norwmult = L2R, nomult +let norwocc = noclr, None + +(* +let pattern_ident = Prim.pattern_ident in +GEXTEND Gram +GLOBAL: pattern_ident; +pattern_ident: +[[c = pattern_ident -> (CRef (Ident (loc,c)), NoBindings)]]; +END +*) + +ARGUMENT EXTEND ssrpattern_squarep +TYPED AS rpattern option PRINTED BY pr_ssrpattern_squarep + | [ "[" rpattern(rdx) "]" ] -> [ Some rdx ] + | [ ] -> [ None ] +END + +ARGUMENT EXTEND ssrpattern_ne_squarep +TYPED AS rpattern option PRINTED BY pr_ssrpattern_squarep + | [ "[" rpattern(rdx) "]" ] -> [ Some rdx ] +END + + +ARGUMENT EXTEND ssrrwarg + TYPED AS (ssrdir * ssrmult) * ((ssrdocc * rpattern option) * ssrrule) + PRINTED BY pr_ssrrwarg + | [ "-" ssrmult(m) ssrrwocc(docc) ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg (R2L, m) (docc, rx) r ] + | [ "-/" ssrterm(t) ] -> (* just in case '-/' should become a token *) + [ mk_rwarg (R2L, nomult) norwocc (RWdef, t) ] + | [ ssrmult_ne(m) ssrrwocc(docc) ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg (L2R, m) (docc, rx) r ] + | [ "{" ne_ssrhyp_list(clr) "}" ssrpattern_ne_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (mkclr clr, rx) r ] + | [ "{" ne_ssrhyp_list(clr) "}" ssrrule(r) ] -> + [ mk_rwarg norwmult (mkclr clr, None) r ] + | [ "{" ssrocc(occ) "}" ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (mkocc occ, rx) r ] + | [ "{" "}" ssrpattern_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (nodocc, rx) r ] + | [ ssrpattern_ne_squarep(rx) ssrrule_ne(r) ] -> + [ mk_rwarg norwmult (noclr, rx) r ] + | [ ssrrule_ne(r) ] -> + [ mk_rwarg norwmult norwocc r ] +END + +let simplintac occ rdx sim gl = + let simptac gl = + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let simp env c _ _ = red_safe Tacred.simpl env sigma0 c in + Proofview.V82.of_tactic + (convert_concl_no_check (eval_pattern env0 sigma0 concl0 rdx occ simp)) + gl in + match sim with + | Simpl -> simptac gl + | SimplCut -> tclTHEN simptac (tclTRY donetac) gl + | _ -> simpltac sim gl + +let rec get_evalref c = match kind_of_term c with + | Var id -> EvalVarRef id + | Const (k,_) -> EvalConstRef k + | App (c', _) -> get_evalref c' + | Cast (c', _, _) -> get_evalref c' + | _ -> errorstrm (str "The term " ++ pr_constr c ++ str " is not unfoldable") + +(* Strip a pattern generated by a prenex implicit to its constant. *) +let strip_unfold_term ((sigma, t) as p) kt = match kind_of_term t with + | App (f, a) when kt = ' ' && Array.for_all isEvar a && isConst f -> + (sigma, f), true + | Const _ | Var _ -> p, true + | _ -> p, false + +let unfoldintac occ rdx t (kt,_) gl = + let fs sigma x = Reductionops.nf_evar sigma x in + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let (sigma, t), const = strip_unfold_term t kt in + let body env t c = + Tacred.unfoldn [OnlyOccurrences [1], get_evalref t] env sigma0 c in + let easy = occ = None && rdx = None in + let red_flags = if easy then Closure.betaiotazeta else Closure.betaiota in + let beta env = Reductionops.clos_norm_flags red_flags env sigma0 in + let unfold, conclude = match rdx with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let ise = create_evar_defs sigma in + let ise, u = mk_tpattern env0 sigma0 (ise,t) all_ok L2R t in + let find_T, end_T = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[u]) in + (fun env c _ h -> + try find_T env c h (fun env c _ _ -> body env t c) + with NoMatch when easy -> c + | NoMatch | NoProgress -> errorstrm (str"No occurrence of " + ++ pr_constr_pat t ++ spc() ++ str "in " ++ pr_constr c)), + (fun () -> try end_T () with + | NoMatch when easy -> fake_pmatcher_end () + | NoMatch -> anomaly "unfoldintac") + | _ -> + (fun env (c as orig_c) _ h -> + if const then + let rec aux c = + match kind_of_term c with + | Const _ when Term.eq_constr c t -> body env t t + | App (f,a) when Term.eq_constr f t -> mkApp (body env f f,a) + | _ -> let c = Reductionops.whd_betaiotazeta sigma0 c in + match kind_of_term c with + | Const _ when Term.eq_constr c t -> body env t t + | App (f,a) when Term.eq_constr f t -> mkApp (body env f f,a) + | Const f -> aux (body env c c) + | App (f, a) -> aux (mkApp (body env f f, a)) + | _ -> errorstrm (str "The term "++pr_constr orig_c++ + str" contains no " ++ pr_constr t ++ str" even after unfolding") + in aux c + else + try body env t (fs (unify_HO env sigma c t) t) + with _ -> errorstrm (str "The term " ++ + pr_constr c ++spc()++ str "does not unify with " ++ pr_constr_pat t)), + fake_pmatcher_end in + let concl = + try beta env0 (eval_pattern env0 sigma0 concl0 rdx occ unfold) + with Option.IsNone -> errorstrm (str"Failed to unfold " ++ pr_constr_pat t) in + let _ = conclude () in + Proofview.V82.of_tactic (convert_concl concl) gl +;; + +let foldtac occ rdx ft gl = + let fs sigma x = Reductionops.nf_evar sigma x in + let sigma0, concl0, env0 = project gl, pf_concl gl, pf_env gl in + let sigma, t = ft in + let fold, conclude = match rdx with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let ise = create_evar_defs sigma in + let ut = red_product_skip_id env0 sigma t in + let ise, ut = mk_tpattern env0 sigma0 (ise,t) all_ok L2R ut in + let find_T, end_T = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[ut]) in + (fun env c _ h -> try find_T env c h (fun env t _ _ -> t) with NoMatch ->c), + (fun () -> try end_T () with NoMatch -> fake_pmatcher_end ()) + | _ -> + (fun env c _ h -> try let sigma = unify_HO env sigma c t in fs sigma t + with _ -> errorstrm (str "fold pattern " ++ pr_constr_pat t ++ spc () + ++ str "does not match redex " ++ pr_constr_pat c)), + fake_pmatcher_end in + let concl = eval_pattern env0 sigma0 concl0 rdx occ fold in + let _ = conclude () in + Proofview.V82.of_tactic (convert_concl concl) gl +;; + +let converse_dir = function L2R -> R2L | R2L -> L2R + +let rw_progress rhs lhs ise = not (Term.eq_constr lhs (Evarutil.nf_evar ise rhs)) + +(* Coq has a more general form of "equation" (any type with a single *) +(* constructor with no arguments with_rect_r elimination lemmas). *) +(* However there is no clear way of determining the LHS and RHS of *) +(* such a generic Leibnitz equation -- short of inspecting the type *) +(* of the elimination lemmas. *) + +let rec strip_prod_assum c = match kind_of_term c with + | Prod (_, _, c') -> strip_prod_assum c' + | LetIn (_, v, _, c') -> strip_prod_assum (subst1 v c) + | Cast (c', _, _) -> strip_prod_assum c' + | _ -> c + +let rule_id = mk_internal_id "rewrite rule" + +exception PRtype_error +exception PRindetermined_rhs of constr + +let pirrel_rewrite pred rdx rdx_ty new_rdx dir (sigma, c) c_ty gl = +(* pp(lazy(str"sigma@pirrel_rewrite=" ++ pr_evar_map None sigma)); *) + let env = pf_env gl in + let beta = Reductionops.clos_norm_flags Closure.beta env sigma in + let sigma, p = + let sigma = create_evar_defs sigma in + Evarutil.new_evar env sigma (beta (subst1 new_rdx pred)) in + let pred = mkNamedLambda pattern_id rdx_ty pred in + let elim, gl = + let ((kn, i) as ind, _), unfolded_c_ty = pf_reduce_to_quantified_ind gl c_ty in + let sort = elimination_sort_of_goal gl in + let elim, gl = pf_fresh_global (Indrec.lookup_eliminator ind sort) gl in + if dir = R2L then elim, gl else (* taken from Coq's rewrite *) + let elim, _ = destConst elim in + let mp,dp,l = repr_con (constant_of_kn (canonical_con elim)) in + let l' = label_of_id (Nameops.add_suffix (id_of_label l) "_r") in + let c1' = Global.constant_of_delta_kn (canonical_con (make_con mp dp l')) in + mkConst c1', gl in + let proof = mkApp (elim, [| rdx_ty; new_rdx; pred; p; rdx; c |]) in + (* We check the proof is well typed *) + let sigma, proof_ty = + try Typing.e_type_of env sigma proof with _ -> raise PRtype_error in + pp(lazy(str"pirrel_rewrite proof term of type: " ++ pr_constr proof_ty)); + try refine_with + ~first_goes_last:(not !ssroldreworder) ~with_evars:false (sigma, proof) gl + with _ -> + (* we generate a msg like: "Unable to find an instance for the variable" *) + let c = Reductionops.nf_evar sigma c in + let hd_ty, miss = match kind_of_term c with + | App (hd, args) -> + let hd_ty = Retyping.get_type_of env sigma hd in + let names = let rec aux t = function 0 -> [] | n -> + let t = Reductionops.whd_betadeltaiota env sigma t in + match kind_of_type t with + | ProdType (name, _, t) -> name :: aux t (n-1) + | _ -> assert false in aux hd_ty (Array.length args) in + hd_ty, Util.List.map_filter (fun (t, name) -> + let evs = Intset.elements (Evarutil.undefined_evars_of_term sigma t) in + let open_evs = List.filter (fun k -> + InProp <> Retyping.get_sort_family_of + env sigma (Evd.evar_concl (Evd.find sigma k))) + evs in + if open_evs <> [] then Some name else None) + (List.combine (Array.to_list args) names) + | _ -> anomaly "rewrite rule not an application" in + errorstrm (Himsg.explain_refiner_error (Logic.UnresolvedBindings miss)++ + (Pp.fnl()++str"Rule's type:" ++ spc() ++ pr_constr hd_ty)) +;; + +let is_const_ref c r = isConst c && eq_gr (ConstRef (fst(destConst c))) r +let is_construct_ref c r = + isConstruct c && eq_gr (ConstructRef (fst(destConstruct c))) r +let is_ind_ref c r = isInd c && eq_gr (IndRef (fst(destInd c))) r + +let rwcltac cl rdx dir sr gl = + let n, r_n,_, ucst = pf_abs_evars gl sr in + let r_n' = pf_abs_cterm gl n r_n in + let r' = subst_var pattern_id r_n' in + let gl = pf_merge_uc ucst gl in + let rdxt = Retyping.get_type_of (pf_env gl) (fst sr) rdx in +(* pp(lazy(str"sigma@rwcltac=" ++ pr_evar_map None (fst sr))); *) + pp(lazy(str"r@rwcltac=" ++ pr_constr (snd sr))); + let cvtac, rwtac, gl = + if closed0 r' then + let env, sigma, c, c_eq = pf_env gl, fst sr, snd sr, build_coq_eq () in + let sigma, c_ty = Typing.e_type_of env sigma c in + pp(lazy(str"c_ty@rwcltac=" ++ pr_constr c_ty)); + match kind_of_type (Reductionops.whd_betadeltaiota env sigma c_ty) with + | AtomicType(e, a) when is_ind_ref e c_eq -> + let new_rdx = if dir = L2R then a.(2) else a.(1) in + pirrel_rewrite cl rdx rdxt new_rdx dir (sigma,c) c_ty, tclIDTAC, gl + | _ -> + let cl' = mkApp (mkNamedLambda pattern_id rdxt cl, [|rdx|]) in + let sigma, _ = Typing.e_type_of env sigma cl' in + let gl = pf_merge_uc_of sigma gl in + Proofview.V82.of_tactic (convert_concl cl'), rewritetac dir r', gl + else + let dc, r2 = decompose_lam_n n r' in + let r3, _, r3t = + try destCast r2 with _ -> + errorstrm (str "no cast from " ++ pr_constr_pat (snd sr) + ++ str " to " ++ pr_constr r2) in + let cl' = mkNamedProd rule_id (compose_prod dc r3t) (lift 1 cl) in + let cl'' = mkNamedProd pattern_id rdxt cl' in + let itacs = [introid pattern_id; introid rule_id] in + let cltac = clear [pattern_id; rule_id] in + let rwtacs = [rewritetac dir (mkVar rule_id); cltac] in + apply_type cl'' [rdx; compose_lam dc r3], tclTHENLIST (itacs @ rwtacs), gl + in + let cvtac' _ = + try cvtac gl with + | PRtype_error -> + if occur_existential (pf_concl gl) + then errorstrm (str "Rewriting impacts evars") + else errorstrm (str "Dependent type error in rewrite of " + ++ pf_pr_constr gl (project gl) (mkNamedLambda pattern_id rdxt cl)) + | Errors.UserError _ as e -> raise e + | e -> anomaly ("cvtac's exception: " ^ Printexc.to_string e); + in + tclTHEN cvtac' rwtac gl + +let prof_rwcltac = mk_profiler "rwrxtac.rwcltac";; +let rwcltac cl rdx dir sr gl = + prof_rwcltac.profile (rwcltac cl rdx dir sr) gl +;; + + +let lz_coq_prod = + let prod = lazy (build_prod ()) in fun () -> Lazy.force prod + +let lz_setoid_relation = + let sdir = ["Classes"; "RelationClasses"] in + let last_srel = ref (Environ.empty_env, None) in + fun env -> match !last_srel with + | env', srel when env' == env -> srel + | _ -> + let srel = + try Some (coq_constant "Class_setoid" sdir "RewriteRelation") + with _ -> None in + last_srel := (env, srel); srel + +let ssr_is_setoid env = + match lz_setoid_relation env with + | None -> fun _ _ _ -> false + | Some srel -> + fun sigma r args -> + Rewrite.is_applied_rewrite_relation env + sigma [] (mkApp (r, args)) <> None + +let prof_rwxrtac_find_rule = mk_profiler "rwrxtac.find_rule";; + +let closed0_check cl p gl = + if closed0 cl then + errorstrm (str"No occurrence of redex "++pf_pr_constr gl (project gl) p) + +let rwprocess_rule dir rule gl = + let env = pf_env gl in + let coq_prod = lz_coq_prod () in + let is_setoid = ssr_is_setoid env in + let r_sigma, rules = + let rec loop d sigma r t0 rs red = + let t = + if red = 1 then Tacred.hnf_constr env sigma t0 + else Reductionops.whd_betaiotazeta sigma t0 in + pp(lazy(str"rewrule="++pr_constr t)); + match kind_of_term t with + | Prod (_, xt, at) -> + let ise, x = Evarutil.new_evar env (create_evar_defs sigma) xt in + loop d ise (mkApp (r, [|x|])) (subst1 x at) rs 0 + | App (pr, a) when is_ind_ref pr coq_prod.Coqlib.typ -> + let sr sigma = match kind_of_term (Tacred.hnf_constr env sigma r) with + | App (c, ra) when is_construct_ref c coq_prod.Coqlib.intro -> + fun i -> ra.(i + 1), sigma + | _ -> let ra = Array.append a [|r|] in + function 1 -> + let sigma, pi1 = Evd.fresh_global env sigma coq_prod.Coqlib.proj1 in + mkApp (pi1, ra), sigma + | _ -> + let sigma, pi2 = Evd.fresh_global env sigma coq_prod.Coqlib.proj2 in + mkApp (pi2, ra), sigma in + if Term.eq_constr a.(0) (build_coq_True ()) then + let s, sigma = sr sigma 2 in + loop (converse_dir d) sigma s a.(1) rs 0 + else + let s, sigma = sr sigma 2 in + let sigma, rs2 = loop d sigma s a.(1) rs 0 in + let s, sigma = sr sigma 1 in + loop d sigma s a.(0) rs2 0 + | App (r_eq, a) when Hipattern.match_with_equality_type t != None -> + let indu = destInd r_eq and rhs = Array.last a in + let np = Inductiveops.inductive_nparamdecls (fst indu) in + let ind_ct = Inductiveops.type_of_constructors env indu in + let lhs0 = last_arg (strip_prod_assum ind_ct.(0)) in + let rdesc = match kind_of_term lhs0 with + | Rel i -> + let lhs = a.(np - i) in + let lhs, rhs = if d = L2R then lhs, rhs else rhs, lhs in +(* msgnl (str "RW: " ++ pr_rwdir d ++ str " " ++ pr_constr_pat r ++ str " : " + ++ pr_constr_pat lhs ++ str " ~> " ++ pr_constr_pat rhs); *) + d, r, lhs, rhs +(* + let l_i, r_i = if d = L2R then i, 1 - ndep else 1 - ndep, i in + let lhs = a.(np - l_i) and rhs = a.(np - r_i) in + let a' = Array.copy a in let _ = a'.(np - l_i) <- mkVar pattern_id in + let r' = mkCast (r, DEFAULTcast, mkApp (r_eq, a')) in + (d, r', lhs, rhs) +*) + | _ -> + let lhs = substl (array_list_of_tl (Array.sub a 0 np)) lhs0 in + let lhs, rhs = if d = R2L then lhs, rhs else rhs, lhs in + let d' = if Array.length a = 1 then d else converse_dir d in + d', r, lhs, rhs in + sigma, rdesc :: rs + | App (s_eq, a) when is_setoid sigma s_eq a -> + let np = Array.length a and i = 3 - dir_org d in + let lhs = a.(np - i) and rhs = a.(np + i - 3) in + let a' = Array.copy a in let _ = a'.(np - i) <- mkVar pattern_id in + let r' = mkCast (r, DEFAULTcast, mkApp (s_eq, a')) in + sigma, (d, r', lhs, rhs) :: rs + | _ -> + if red = 0 then loop d sigma r t rs 1 + else errorstrm (str "not a rewritable relation: " ++ pr_constr_pat t + ++ spc() ++ str "in rule " ++ pr_constr_pat (snd rule)) + in + let sigma, r = rule in + let t = Retyping.get_type_of env sigma r in + loop dir sigma r t [] 0 + in + r_sigma, rules + +let rwrxtac occ rdx_pat dir rule gl = + let env = pf_env gl in + let r_sigma, rules = rwprocess_rule dir rule gl in + let find_rule rdx = + let rec rwtac = function + | [] -> + errorstrm (str "pattern " ++ pr_constr_pat rdx ++ + str " does not match " ++ pr_dir_side dir ++ + str " of " ++ pr_constr_pat (snd rule)) + | (d, r, lhs, rhs) :: rs -> + try + let ise = unify_HO env (create_evar_defs r_sigma) lhs rdx in + if not (rw_progress rhs rdx ise) then raise NoMatch else + d, (ise, Evd.evar_universe_context ise, Reductionops.nf_evar ise r) + with _ -> rwtac rs in + rwtac rules in + let find_rule rdx = prof_rwxrtac_find_rule.profile find_rule rdx in + let sigma0, env0, concl0 = project gl, pf_env gl, pf_concl gl in + let find_R, conclude = match rdx_pat with + | Some (_, (In_T _ | In_X_In_T _)) | None -> + let upats_origin = dir, snd rule in + let rpat env sigma0 (sigma, pats) (d, r, lhs, rhs) = + let sigma, pat = + mk_tpattern env sigma0 (sigma,r) (rw_progress rhs) d lhs in + sigma, pats @ [pat] in + let rpats = List.fold_left (rpat env0 sigma0) (r_sigma,[]) rules in + let find_R, end_R = mk_tpattern_matcher sigma0 occ ~upats_origin rpats in + (fun e c _ i -> find_R ~k:(fun _ _ _ h -> mkRel h) e c i), + fun cl -> let rdx,d,r = end_R () in closed0_check cl rdx gl; (d,r),rdx + | Some(_, (T e | X_In_T (_,e) | E_As_X_In_T (e,_,_) | E_In_X_In_T (e,_,_))) -> + let r = ref None in + (fun env c _ h -> do_once r (fun () -> find_rule c, c); mkRel h), + (fun concl -> closed0_check concl e gl; assert_done r) in + let concl = eval_pattern env0 sigma0 concl0 rdx_pat occ find_R in + let (d, r), rdx = conclude concl in + let r = Evd.merge_universe_context (pi1 r) (pi2 r), pi3 r in + rwcltac concl rdx d r gl +;; + +let prof_rwxrtac = mk_profiler "rwrxtac";; +let rwrxtac occ rdx_pat dir rule gl = + prof_rwxrtac.profile (rwrxtac occ rdx_pat dir rule) gl +;; + +let ssrinstancesofrule ist dir arg gl = + let sigma0, env0, concl0 = project gl, pf_env gl, pf_concl gl in + let rule = interp_term ist gl arg in + let r_sigma, rules = rwprocess_rule dir rule gl in + let find, conclude = + let upats_origin = dir, snd rule in + let rpat env sigma0 (sigma, pats) (d, r, lhs, rhs) = + let sigma, pat = + mk_tpattern env sigma0 (sigma,r) (rw_progress rhs) d lhs in + sigma, pats @ [pat] in + let rpats = List.fold_left (rpat env0 sigma0) (r_sigma,[]) rules in + mk_tpattern_matcher ~all_instances:true ~raise_NoMatch:true sigma0 None ~upats_origin rpats in + let print env p c _ = ppnl (hov 1 (str"instance:" ++ spc() ++ pr_constr p ++ spc() ++ str "matches:" ++ spc() ++ pr_constr c)); c in + ppnl (str"BEGIN INSTANCES"); + try + while true do + ignore(find env0 concl0 1 ~k:print) + done; raise NoMatch + with NoMatch -> ppnl (str"END INSTANCES"); tclIDTAC gl + +TACTIC EXTEND ssrinstofruleL2R +| [ "ssrinstancesofruleL2R" ssrterm(arg) ] -> [ Proofview.V82.tactic (ssrinstancesofrule ist L2R arg) ] +END +TACTIC EXTEND ssrinstofruleR2L +| [ "ssrinstancesofruleR2L" ssrterm(arg) ] -> [ Proofview.V82.tactic (ssrinstancesofrule ist R2L arg) ] +END + +(* Resolve forward reference *) +let _ = + ipat_rewritetac := fun occ dir c gl -> rwrxtac occ None dir (project gl, c) gl + +let rwargtac ist ((dir, mult), (((oclr, occ), grx), (kind, gt))) gl = + let fail = ref false in + let interp_rpattern ist gl gc = + try interp_rpattern ist gl gc + with _ when snd mult = May -> fail := true; project gl, T mkProp in + let interp gc gl = + try interp_term ist gl gc + with _ when snd mult = May -> fail := true; (project gl, mkProp) in + let rwtac gl = + let rx = Option.map (interp_rpattern ist gl) grx in + let t = interp gt gl in + (match kind with + | RWred sim -> simplintac occ rx sim + | RWdef -> if dir = R2L then foldtac occ rx t else unfoldintac occ rx t gt + | RWeq -> rwrxtac occ rx dir t) gl in + let ctac = cleartac (interp_clr (oclr, (fst gt, snd (interp gt gl)))) in + if !fail then ctac gl else tclTHEN (tclMULT mult rwtac) ctac gl + +(** Rewrite argument sequence *) + +(* type ssrrwargs = ssrrwarg list *) + +let pr_ssrrwargs _ _ _ rwargs = pr_list spc pr_rwarg rwargs + +ARGUMENT EXTEND ssrrwargs TYPED AS ssrrwarg list PRINTED BY pr_ssrrwargs + | [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let ssr_rw_syntax = Summary.ref ~name:"SSR:rewrite" true + +let _ = + Goptions.declare_bool_option + { Goptions.optsync = true; + Goptions.optname = "ssreflect rewrite"; + Goptions.optkey = ["SsrRewrite"]; + Goptions.optread = (fun _ -> !ssr_rw_syntax); + Goptions.optdepr = false; + Goptions.optwrite = (fun b -> ssr_rw_syntax := b) } + +let test_ssr_rw_syntax = + let test strm = + if not !ssr_rw_syntax then raise Stream.Failure else + if is_ssr_loaded () then () else + match Compat.get_tok (Util.stream_nth 0 strm) with + | Tok.KEYWORD key when List.mem key.[0] ['{'; '['; '/'] -> () + | _ -> raise Stream.Failure in + Gram.Entry.of_parser "test_ssr_rw_syntax" test + +GEXTEND Gram + GLOBAL: ssrrwargs; + ssrrwargs: [[ test_ssr_rw_syntax; a = LIST1 ssrrwarg -> a ]]; +END + +(** The "rewrite" tactic *) + +let ssrrewritetac ist rwargs = + tclTHENLIST (List.map (rwargtac ist) rwargs) + +TACTIC EXTEND ssrrewrite + | [ "rewrite" ssrrwargs(args) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrrewritetac ist args) clauses) ] +END + +(** The "unlock" tactic *) + +let pr_unlockarg (occ, t) = pr_occ occ ++ pr_term t +let pr_ssrunlockarg _ _ _ = pr_unlockarg + +ARGUMENT EXTEND ssrunlockarg TYPED AS ssrocc * ssrterm + PRINTED BY pr_ssrunlockarg + | [ "{" ssrocc(occ) "}" ssrterm(t) ] -> [ occ, t ] + | [ ssrterm(t) ] -> [ None, t ] +END + +let pr_ssrunlockargs _ _ _ args = pr_list spc pr_unlockarg args + +ARGUMENT EXTEND ssrunlockargs TYPED AS ssrunlockarg list + PRINTED BY pr_ssrunlockargs + | [ ssrunlockarg_list(args) ] -> [ args ] +END + +let unfoldtac occ ko t kt gl = + let cl, c = pf_fill_occ_term gl occ (fst (strip_unfold_term t kt)) in + let cl' = subst1 (pf_unfoldn [OnlyOccurrences [1], get_evalref c] gl c) cl in + let f = if ko = None then Closure.betaiotazeta else Closure.betaiota in + Proofview.V82.of_tactic + (convert_concl (pf_reduce (Reductionops.clos_norm_flags f) gl cl')) gl + +let unlocktac ist args gl = + let utac (occ, gt) gl = + unfoldtac occ occ (interp_term ist gl gt) (fst gt) gl in + let locked, gl = pf_mkSsrConst "locked" gl in + let key, gl = pf_mkSsrConst "master_key" gl in + let ktacs = [ + (fun gl -> unfoldtac None None (project gl,locked) '(' gl); + simplest_newcase key ] in + tclTHENLIST (List.map utac args @ ktacs) gl + +TACTIC EXTEND ssrunlock + | [ "unlock" ssrunlockargs(args) ssrclauses(clauses) ] -> +[ Proofview.V82.tactic (tclCLAUSES ist (unlocktac ist args) clauses) ] +END + +(** 8. Forward chaining tactics (pose, set, have, suffice, wlog) *) + +(** Defined identifier *) + +type ssrfwdid = identifier + +let pr_ssrfwdid _ _ _ id = pr_spc () ++ pr_id id + +(* We use a primitive parser for the head identifier of forward *) +(* tactis to avoid syntactic conflicts with basic Coq tactics. *) +ARGUMENT EXTEND ssrfwdid TYPED AS ident PRINTED BY pr_ssrfwdid + | [ "YouShouldNotTypeThis" ] -> [ anomaly "Grammar placeholder match" ] +END + +let accept_ssrfwdid strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT id -> accept_before_syms_or_any_id [":"; ":="; "("] strm + | _ -> raise Stream.Failure + + +let test_ssrfwdid = Gram.Entry.of_parser "test_ssrfwdid" accept_ssrfwdid + +GEXTEND Gram + GLOBAL: ssrfwdid; + ssrfwdid: [[ test_ssrfwdid; id = Prim.ident -> id ]]; + END + + + +(** Definition value formatting *) + +(* We use an intermediate structure to correctly render the binder list *) +(* abbreviations. We use a list of hints to extract the binders and *) +(* base term from a term, for the two first levels of representation of *) +(* of constr terms. *) + +type 'term ssrbind = + | Bvar of name + | Bdecl of name list * 'term + | Bdef of name * 'term option * 'term + | Bstruct of name + | Bcast of 'term + +let pr_binder prl = function + | Bvar x -> + pr_name x + | Bdecl (xs, t) -> + str "(" ++ pr_list pr_spc pr_name xs ++ str " : " ++ prl t ++ str ")" + | Bdef (x, None, v) -> + str "(" ++ pr_name x ++ str " := " ++ prl v ++ str ")" + | Bdef (x, Some t, v) -> + str "(" ++ pr_name x ++ str " : " ++ prl t ++ + str " := " ++ prl v ++ str ")" + | Bstruct x -> + str "{struct " ++ pr_name x ++ str "}" + | Bcast t -> + str ": " ++ prl t + +type 'term ssrbindval = 'term ssrbind list * 'term + +type ssrbindfmt = + | BFvar + | BFdecl of int (* #xs *) + | BFcast (* final cast *) + | BFdef of bool (* has cast? *) + | BFrec of bool * bool (* has struct? * has cast? *) + +let rec mkBstruct i = function + | Bvar x :: b -> + if i = 0 then [Bstruct x] else mkBstruct (i - 1) b + | Bdecl (xs, _) :: b -> + let i' = i - List.length xs in + if i' < 0 then [Bstruct (List.nth xs i)] else mkBstruct i' b + | _ :: b -> mkBstruct i b + | [] -> [] + +let rec format_local_binders h0 bl0 = match h0, bl0 with + | BFvar :: h, LocalRawAssum ([_, x], _, _) :: bl -> + Bvar x :: format_local_binders h bl + | BFdecl _ :: h, LocalRawAssum (lxs, _, t) :: bl -> + Bdecl (List.map snd lxs, t) :: format_local_binders h bl + | BFdef false :: h, LocalRawDef ((_, x), v) :: bl -> + Bdef (x, None, v) :: format_local_binders h bl + | BFdef true :: h, + LocalRawDef ((_, x), CCast (_, v, CastConv t)) :: bl -> + Bdef (x, Some t, v) :: format_local_binders h bl + | _ -> [] + +let rec format_constr_expr h0 c0 = match h0, c0 with + | BFvar :: h, CLambdaN (_, [[_, x], _, _], c) -> + let bs, c' = format_constr_expr h c in + Bvar x :: bs, c' + | BFdecl _:: h, CLambdaN (_, [lxs, _, t], c) -> + let bs, c' = format_constr_expr h c in + Bdecl (List.map snd lxs, t) :: bs, c' + | BFdef false :: h, CLetIn(_, (_, x), v, c) -> + let bs, c' = format_constr_expr h c in + Bdef (x, None, v) :: bs, c' + | BFdef true :: h, CLetIn(_, (_, x), CCast (_, v, CastConv t), c) -> + let bs, c' = format_constr_expr h c in + Bdef (x, Some t, v) :: bs, c' + | [BFcast], CCast (_, c, CastConv t) -> + [Bcast t], c + | BFrec (has_str, has_cast) :: h, + CFix (_, _, [_, (Some locn, CStructRec), bl, t, c]) -> + let bs = format_local_binders h bl in + let bstr = if has_str then [Bstruct (Name (snd locn))] else [] in + bs @ bstr @ (if has_cast then [Bcast t] else []), c + | BFrec (_, has_cast) :: h, CCoFix (_, _, [_, bl, t, c]) -> + format_local_binders h bl @ (if has_cast then [Bcast t] else []), c + | _, c -> + [], c + +let rec format_glob_decl h0 d0 = match h0, d0 with + | BFvar :: h, (x, _, None, _) :: d -> + Bvar x :: format_glob_decl h d + | BFdecl 1 :: h, (x, _, None, t) :: d -> + Bdecl ([x], t) :: format_glob_decl h d + | BFdecl n :: h, (x, _, None, t) :: d when n > 1 -> + begin match format_glob_decl (BFdecl (n - 1) :: h) d with + | Bdecl (xs, _) :: bs -> Bdecl (x :: xs, t) :: bs + | bs -> Bdecl ([x], t) :: bs + end + | BFdef false :: h, (x, _, Some v, _) :: d -> + Bdef (x, None, v) :: format_glob_decl h d + | BFdef true:: h, (x, _, Some (GCast (_, v, CastConv t)), _) :: d -> + Bdef (x, Some t, v) :: format_glob_decl h d + | _, (x, _, None, t) :: d -> + Bdecl ([x], t) :: format_glob_decl [] d + | _, (x, _, Some v, _) :: d -> + Bdef (x, None, v) :: format_glob_decl [] d + | _, [] -> [] + +let rec format_glob_constr h0 c0 = match h0, c0 with + | BFvar :: h, GLambda (_, x, _, _, c) -> + let bs, c' = format_glob_constr h c in + Bvar x :: bs, c' + | BFdecl 1 :: h, GLambda (_, x, _, t, c) -> + let bs, c' = format_glob_constr h c in + Bdecl ([x], t) :: bs, c' + | BFdecl n :: h, GLambda (_, x, _, t, c) when n > 1 -> + begin match format_glob_constr (BFdecl (n - 1) :: h) c with + | Bdecl (xs, _) :: bs, c' -> Bdecl (x :: xs, t) :: bs, c' + | _ -> [Bdecl ([x], t)], c + end + | BFdef false :: h, GLetIn(_, x, v, c) -> + let bs, c' = format_glob_constr h c in + Bdef (x, None, v) :: bs, c' + | BFdef true :: h, GLetIn(_, x, GCast (_, v, CastConv t), c) -> + let bs, c' = format_glob_constr h c in + Bdef (x, Some t, v) :: bs, c' + | [BFcast], GCast (_, c, CastConv t) -> + [Bcast t], c + | BFrec (has_str, has_cast) :: h, GRec (_, f, _, bl, t, c) + when Array.length c = 1 -> + let bs = format_glob_decl h bl.(0) in + let bstr = match has_str, f with + | true, GFix ([|Some i, GStructRec|], _) -> mkBstruct i bs + | _ -> [] in + bs @ bstr @ (if has_cast then [Bcast t.(0)] else []), c.(0) + | _, c -> + [], c + +(** Forward chaining argument *) + +(* There are three kinds of forward definitions: *) +(* - Hint: type only, cast to Type, may have proof hint. *) +(* - Have: type option + value, no space before type *) +(* - Pose: binders + value, space before binders. *) + +type ssrfwdkind = FwdHint of string * bool | FwdHave | FwdPose + +type ssrfwdfmt = ssrfwdkind * ssrbindfmt list + +let pr_fwdkind = function + | FwdHint (s,_) -> str (s ^ " ") | _ -> str " :=" ++ spc () +let pr_fwdfmt (fk, _ : ssrfwdfmt) = pr_fwdkind fk + +let wit_ssrfwdfmt = add_genarg "ssrfwdfmt" pr_fwdfmt + +(* type ssrfwd = ssrfwdfmt * ssrterm *) + +let mkFwdVal fk c = ((fk, []), mk_term ' ' c) +let mkssrFwdVal fk c = ((fk, []), (c,None)) + +let mkFwdCast fk loc t c = ((fk, [BFcast]), mk_term ' ' (CCast (loc, c, dC t))) +let mkssrFwdCast fk loc t c = ((fk, [BFcast]), (c, Some t)) + +let mkFwdHint s t = + let loc = constr_loc t in + mkFwdCast (FwdHint (s,false)) loc t (mkCHole loc) +let mkFwdHintNoTC s t = + let loc = constr_loc t in + mkFwdCast (FwdHint (s,true)) loc t (mkCHole loc) + +let pr_gen_fwd prval prc prlc fk (bs, c) = + let prc s = str s ++ spc () ++ prval prc prlc c in + match fk, bs with + | FwdHint (s,_), [Bcast t] -> str s ++ spc () ++ prlc t + | FwdHint (s,_), _ -> prc (s ^ "(* typeof *)") + | FwdHave, [Bcast t] -> str ":" ++ spc () ++ prlc t ++ prc " :=" + | _, [] -> prc " :=" + | _, _ -> spc () ++ pr_list spc (pr_binder prlc) bs ++ prc " :=" + +let pr_fwd_guarded prval prval' = function +| (fk, h), (_, (_, Some c)) -> + pr_gen_fwd prval pr_constr_expr prl_constr_expr fk (format_constr_expr h c) +| (fk, h), (_, (c, None)) -> + pr_gen_fwd prval' pr_glob_constr prl_glob_constr fk (format_glob_constr h c) + +let pr_unguarded prc prlc = prlc + +let pr_fwd = pr_fwd_guarded pr_unguarded pr_unguarded +let pr_ssrfwd _ _ _ = pr_fwd + +ARGUMENT EXTEND ssrfwd TYPED AS (ssrfwdfmt * ssrterm) PRINTED BY pr_ssrfwd + | [ ":=" lconstr(c) ] -> [ mkFwdVal FwdPose c ] + | [ ":" lconstr (t) ":=" lconstr(c) ] -> [ mkFwdCast FwdPose loc t c ] +END + +(** Independent parsing for binders *) + +(* The pose, pose fix, and pose cofix tactics use these internally to *) +(* parse argument fragments. *) + +let pr_ssrbvar prc _ _ v = prc v + +ARGUMENT EXTEND ssrbvar TYPED AS constr PRINTED BY pr_ssrbvar +| [ ident(id) ] -> [ mkCVar loc id ] +| [ "_" ] -> [ mkCHole loc ] +END + +let bvar_lname = function + | CRef (Ident (loc, id), _) -> loc, Name id + | c -> constr_loc c, Anonymous + +let pr_ssrbinder prc _ _ (_, c) = prc c + +ARGUMENT EXTEND ssrbinder TYPED AS ssrfwdfmt * constr PRINTED BY pr_ssrbinder + | [ ssrbvar(bv) ] -> + [ let xloc, _ as x = bvar_lname bv in + (FwdPose, [BFvar]), + CLambdaN (loc,[[x],Default Explicit,mkCHole xloc],mkCHole loc) ] + | [ "(" ssrbvar(bv) ")" ] -> + [ let xloc, _ as x = bvar_lname bv in + (FwdPose, [BFvar]), + CLambdaN (loc,[[x],Default Explicit,mkCHole xloc],mkCHole loc) ] + | [ "(" ssrbvar(bv) ":" lconstr(t) ")" ] -> + [ let x = bvar_lname bv in + (FwdPose, [BFdecl 1]), + CLambdaN (loc, [[x], Default Explicit, t], mkCHole loc) ] + | [ "(" ssrbvar(bv) ne_ssrbvar_list(bvs) ":" lconstr(t) ")" ] -> + [ let xs = List.map bvar_lname (bv :: bvs) in + let n = List.length xs in + (FwdPose, [BFdecl n]), + CLambdaN (loc, [xs, Default Explicit, t], mkCHole loc) ] + | [ "(" ssrbvar(id) ":" lconstr(t) ":=" lconstr(v) ")" ] -> + [ let loc' = Loc.join_loc (constr_loc t) (constr_loc v) in + let v' = CCast (loc', v, dC t) in + (FwdPose,[BFdef true]), CLetIn (loc,bvar_lname id, v',mkCHole loc) ] + | [ "(" ssrbvar(id) ":=" lconstr(v) ")" ] -> + [ (FwdPose,[BFdef false]), CLetIn (loc,bvar_lname id, v,mkCHole loc) ] +END + +GEXTEND Gram + GLOBAL: ssrbinder; + ssrbinder: [ + [ ["of" | "&"]; c = operconstr LEVEL "99" -> + let loc = !@loc in + (FwdPose, [BFvar]), + CLambdaN (loc,[[loc,Anonymous],Default Explicit,c],mkCHole loc) ] + ]; +END + +let rec binders_fmts = function + | ((_, h), _) :: bs -> h @ binders_fmts bs + | _ -> [] + +let push_binders c2 bs = + let loc2 = constr_loc c2 in let mkloc loc1 = Loc.join_loc loc1 loc2 in + let rec loop ty c = function + | (_, CLambdaN (loc1, b, _)) :: bs when ty -> + CProdN (mkloc loc1, b, loop ty c bs) + | (_, CLambdaN (loc1, b, _)) :: bs -> + CLambdaN (mkloc loc1, b, loop ty c bs) + | (_, CLetIn (loc1, x, v, _)) :: bs -> + CLetIn (mkloc loc1, x, v, loop ty c bs) + | [] -> c + | _ -> anomaly "binder not a lambda nor a let in" in + match c2 with + | CCast (x, ct, CastConv cty) -> + (CCast (x, loop false ct bs, CastConv (loop true cty bs))) + | ct -> loop false ct bs + +let rec fix_binders = function + | (_, CLambdaN (_, [xs, _, t], _)) :: bs -> + LocalRawAssum (xs, Default Explicit, t) :: fix_binders bs + | (_, CLetIn (_, x, v, _)) :: bs -> + LocalRawDef (x, v) :: fix_binders bs + | _ -> [] + +let pr_ssrstruct _ _ _ = function + | Some id -> str "{struct " ++ pr_id id ++ str "}" + | None -> mt () + +ARGUMENT EXTEND ssrstruct TYPED AS ident option PRINTED BY pr_ssrstruct +| [ "{" "struct" ident(id) "}" ] -> [ Some id ] +| [ ] -> [ None ] +END + +(** The "pose" tactic *) + +(* The plain pose form. *) + +let bind_fwd bs = function + | (fk, h), (ck, (rc, Some c)) -> + (fk,binders_fmts bs @ h), (ck,(rc,Some (push_binders c bs))) + | fwd -> fwd + +ARGUMENT EXTEND ssrposefwd TYPED AS ssrfwd PRINTED BY pr_ssrfwd + | [ ssrbinder_list(bs) ssrfwd(fwd) ] -> [ bind_fwd bs fwd ] +END + +(* The pose fix form. *) + +let pr_ssrfixfwd _ _ _ (id, fwd) = str " fix " ++ pr_id id ++ pr_fwd fwd + +let bvar_locid = function + | CRef (Ident (loc, id), _) -> loc, id + | _ -> Errors.error "Missing identifier after \"(co)fix\"" + + +ARGUMENT EXTEND ssrfixfwd TYPED AS ident * ssrfwd PRINTED BY pr_ssrfixfwd + | [ "fix" ssrbvar(bv) ssrbinder_list(bs) ssrstruct(sid) ssrfwd(fwd) ] -> + [ let (_, id) as lid = bvar_locid bv in + let (fk, h), (ck, (rc, oc)) = fwd in + let c = Option.get oc in + let has_cast, t', c' = match format_constr_expr h c with + | [Bcast t'], c' -> true, t', c' + | _ -> false, mkCHole (constr_loc c), c in + let lb = fix_binders bs in + let has_struct, i = + let rec loop = function + (l', Name id') :: _ when Option.equal Id.equal sid (Some id') -> true, (l', id') + | [l', Name id'] when sid = None -> false, (l', id') + | _ :: bn -> loop bn + | [] -> Errors.error "Bad structural argument" in + loop (names_of_local_assums lb) in + let h' = BFrec (has_struct, has_cast) :: binders_fmts bs in + let fix = CFix (loc, lid, [lid, (Some i, CStructRec), lb, t', c']) in + id, ((fk, h'), (ck, (rc, Some fix))) ] +END + + +(* The pose cofix form. *) + +let pr_ssrcofixfwd _ _ _ (id, fwd) = str " cofix " ++ pr_id id ++ pr_fwd fwd + +ARGUMENT EXTEND ssrcofixfwd TYPED AS ssrfixfwd PRINTED BY pr_ssrcofixfwd + | [ "cofix" ssrbvar(bv) ssrbinder_list(bs) ssrfwd(fwd) ] -> + [ let _, id as lid = bvar_locid bv in + let (fk, h), (ck, (rc, oc)) = fwd in + let c = Option.get oc in + let has_cast, t', c' = match format_constr_expr h c with + | [Bcast t'], c' -> true, t', c' + | _ -> false, mkCHole (constr_loc c), c in + let h' = BFrec (false, has_cast) :: binders_fmts bs in + let cofix = CCoFix (loc, lid, [lid, fix_binders bs, t', c']) in + id, ((fk, h'), (ck, (rc, Some cofix))) + ] +END + +let ssrposetac ist (id, (_, t)) gl = + let sigma, t, ucst = pf_abs_ssrterm ist gl t in + posetac id t (pf_merge_uc ucst gl) + + +let prof_ssrposetac = mk_profiler "ssrposetac";; +let ssrposetac arg gl = prof_ssrposetac.profile (ssrposetac arg) gl;; + +TACTIC EXTEND ssrpose +| [ "pose" ssrfixfwd(ffwd) ] -> [ Proofview.V82.tactic (ssrposetac ist ffwd) ] +| [ "pose" ssrcofixfwd(ffwd) ] -> [ Proofview.V82.tactic (ssrposetac ist ffwd) ] +| [ "pose" ssrfwdid(id) ssrposefwd(fwd) ] -> [ Proofview.V82.tactic (ssrposetac ist (id, fwd)) ] +END + +(** The "set" tactic *) + +(* type ssrsetfwd = ssrfwd * ssrdocc *) + +let guard_setrhs s i = s.[i] = '{' + +let pr_setrhs occ prc prlc c = + if occ = nodocc then pr_guarded guard_setrhs prlc c else pr_docc occ ++ prc c + +let pr_fwd_guarded prval prval' = function +| (fk, h), (_, (_, Some c)) -> + pr_gen_fwd prval pr_constr_expr prl_constr_expr fk (format_constr_expr h c) +| (fk, h), (_, (c, None)) -> + pr_gen_fwd prval' pr_glob_constr prl_glob_constr fk (format_glob_constr h c) + +(* This does not print the type, it should be fixed... *) +let pr_ssrsetfwd _ _ _ (((fk,_),(t,_)), docc) = + pr_gen_fwd (fun _ _ -> pr_cpattern) + (fun _ -> mt()) (fun _ -> mt()) fk ([Bcast ()],t) + +ARGUMENT EXTEND ssrsetfwd +TYPED AS (ssrfwdfmt * (lcpattern * ssrterm option)) * ssrdocc +PRINTED BY pr_ssrsetfwd +| [ ":" lconstr(t) ":=" "{" ssrocc(occ) "}" cpattern(c) ] -> + [ mkssrFwdCast FwdPose loc (mk_lterm t) c, mkocc occ ] +| [ ":" lconstr(t) ":=" lcpattern(c) ] -> + [ mkssrFwdCast FwdPose loc (mk_lterm t) c, nodocc ] +| [ ":=" "{" ssrocc(occ) "}" cpattern(c) ] -> + [ mkssrFwdVal FwdPose c, mkocc occ ] +| [ ":=" lcpattern(c) ] -> [ mkssrFwdVal FwdPose c, nodocc ] +END + +let ssrsettac ist id ((_, (pat, pty)), (_, occ)) gl = + let pat = interp_cpattern ist gl pat (Option.map snd pty) in + let cl, sigma, env = pf_concl gl, project gl, pf_env gl in + let (c, ucst), cl = + try fill_occ_pattern ~raise_NoMatch:true env sigma cl pat occ 1 + with NoMatch -> redex_of_pattern ~resolve_typeclasses:true env pat, cl in + if occur_existential c then errorstrm(str"The pattern"++spc()++ + pr_constr_pat c++spc()++str"did not match and has holes."++spc()++ + str"Did you mean pose?") else + let c, cty = match kind_of_term c with + | Cast(t, DEFAULTcast, ty) -> t, ty + | _ -> c, pf_type_of gl c in + let cl' = mkLetIn (Name id, c, cty, cl) in + let gl = pf_merge_uc ucst gl in + tclTHEN (Proofview.V82.of_tactic (convert_concl cl')) (introid id) gl + +TACTIC EXTEND ssrset +| [ "set" ssrfwdid(id) ssrsetfwd(fwd) ssrclauses(clauses) ] -> + [ Proofview.V82.tactic (tclCLAUSES ist (ssrsettac ist id fwd) clauses) ] +END + +(** The "have" tactic *) + +(* type ssrhavefwd = ssrfwd * ssrhint *) + +let pr_ssrhavefwd _ _ prt (fwd, hint) = pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrhavefwd TYPED AS ssrfwd * ssrhint PRINTED BY pr_ssrhavefwd +| [ ":" lconstr(t) ssrhint(hint) ] -> [ mkFwdHint ":" t, hint ] +| [ ":" lconstr(t) ":=" lconstr(c) ] -> [ mkFwdCast FwdHave loc t c, nohint ] +| [ ":" lconstr(t) ":=" ] -> [ mkFwdHintNoTC ":" t, nohint ] +| [ ":=" lconstr(c) ] -> [ mkFwdVal FwdHave c, nohint ] +END + +let intro_id_to_binder = List.map (function + | IpatId id -> + let xloc, _ as x = bvar_lname (mkCVar dummy_loc id) in + (FwdPose, [BFvar]), + CLambdaN (dummy_loc, [[x], Default Explicit, mkCHole xloc], + mkCHole dummy_loc) + | _ -> anomaly "non-id accepted as binder") + +let binder_to_intro_id = List.map (function + | (FwdPose, [BFvar]), CLambdaN (_,[ids,_,_],_) + | (FwdPose, [BFdecl _]), CLambdaN (_,[ids,_,_],_) -> + List.map (function (_, Name id) -> IpatId id | _ -> IpatAnon) ids + | (FwdPose, [BFdef _]), CLetIn (_,(_,Name id),_,_) -> [IpatId id] + | (FwdPose, [BFdef _]), CLetIn (_,(_,Anonymous),_,_) -> [IpatAnon] + | _ -> anomaly "ssrbinder is not a binder") + +let pr_ssrhavefwdwbinders _ _ prt (tr,((hpats, (fwd, hint)))) = + pr_hpats hpats ++ pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrhavefwdwbinders + TYPED AS bool * (ssrhpats * (ssrfwd * ssrhint)) + PRINTED BY pr_ssrhavefwdwbinders +| [ ssrhpats_wtransp(trpats) ssrbinder_list(bs) ssrhavefwd(fwd) ] -> + [ let tr, pats = trpats in + let ((clr, pats), binders), simpl = pats in + let allbs = intro_id_to_binder binders @ bs in + let allbinders = binders @ List.flatten (binder_to_intro_id bs) in + let hint = bind_fwd allbs (fst fwd), snd fwd in + tr, ((((clr, pats), allbinders), simpl), hint) ] +END + +(* Tactic. *) + +let is_Evar_or_CastedMeta x = + isEvar_or_Meta x || + (isCast x && isEvar_or_Meta (pi1 (destCast x))) + +let occur_existential_or_casted_meta c = + let rec occrec c = match kind_of_term c with + | Evar _ -> raise Not_found + | Cast (m,_,_) when isMeta m -> raise Not_found + | _ -> iter_constr occrec c + in try occrec c; false with Not_found -> true + +let examine_abstract id gl = + let tid = pf_type_of gl id in + let abstract, gl = pf_mkSsrConst "abstract" gl in + if not (isApp tid) || not (Term.eq_constr (fst(destApp tid)) abstract) then + errorstrm(strbrk"not an abstract constant: "++pr_constr id); + let _, args_id = destApp tid in + if Array.length args_id <> 3 then + errorstrm(strbrk"not a proper abstract constant: "++pr_constr id); + if not (is_Evar_or_CastedMeta args_id.(2)) then + errorstrm(strbrk"abstract constant "++pr_constr id++str" already used"); + tid, args_id + +let pf_find_abstract_proof check_lock gl abstract_n = + let fire gl t = Reductionops.nf_evar (project gl) t in + let abstract, gl = pf_mkSsrConst "abstract" gl in + let l = Evd.fold_undefined (fun e ei l -> + match kind_of_term ei.Evd.evar_concl with + | App(hd, [|ty; n; lock|]) + when (not check_lock || + (occur_existential_or_casted_meta (fire gl ty) && + is_Evar_or_CastedMeta (fire gl lock))) && + Term.eq_constr hd abstract && Term.eq_constr n abstract_n -> e::l + | _ -> l) (project gl) [] in + match l with + | [e] -> e + | _ -> errorstrm(strbrk"abstract constant "++pr_constr abstract_n++ + strbrk" not found in the evar map exactly once. "++ + strbrk"Did you tamper with it?") + +let unfold cl = + let module R = Reductionops in let module F = Closure.RedFlags in + reduct_in_concl (R.clos_norm_flags (F.mkflags + (List.map (fun c -> F.fCONST (fst (destConst c))) cl @ [F.fBETA; F.fIOTA]))) + +let havegentac ist t gl = + let sigma, c, ucst = pf_abs_ssrterm ist gl t in + let gl = pf_merge_uc ucst gl in + apply_type (mkArrow (pf_type_of gl c) (pf_concl gl)) [c] gl + +let havetac ist + (transp,((((clr, pats), binders), simpl), (((fk, _), t), hint))) + suff namefst gl += + let concl = pf_concl gl in + let skols, pats = + List.partition (function IpatNewHidden _ -> true | _ -> false) pats in + let itac_mkabs = introstac ~ist skols in + let itac_c = introstac ~ist (IpatSimpl(clr,Nop) :: pats) in + let itac, id, clr = introstac ~ist pats, tclIDTAC, cleartac clr in + let binderstac n = + let rec aux = function 0 -> [] | n -> IpatAnon :: aux (n-1) in + tclTHEN (if binders <> [] then introstac ~ist (aux n) else tclIDTAC) + (introstac ~ist binders) in + let simpltac = introstac ~ist simpl in + let fixtc = + not !ssrhaveNOtcresolution && + match fk with FwdHint(_,true) -> false | _ -> true in + let hint = hinttac ist true hint in + let cuttac t gl = + if transp then + let have_let, gl = pf_mkSsrConst "ssr_have_let" gl in + let step = mkApp (have_let, [|concl;t|]) in + let gl, _ = pf_e_type_of gl step in + applyn ~with_evars:true ~with_shelve:false 2 step gl + else basecuttac "ssr_have" t gl in + (* Introduce now abstract constants, so that everything sees them *) + let abstract_key, gl = pf_mkSsrConst "abstract_key" gl in + let unlock_abs (idty,args_id) gl = + let gl, _ = pf_e_type_of gl idty in + pf_unify_HO gl args_id.(2) abstract_key in + tclTHENFIRST itac_mkabs (fun gl -> + let mkt t = mk_term ' ' t in + let mkl t = (' ', (t, None)) in + let interp gl rtc t = pf_abs_ssrterm ~resolve_typeclasses:rtc ist gl t in + let interp_ty gl rtc t = + let a,b,_,u = pf_interp_ty ~resolve_typeclasses:rtc ist gl t in a,b,u in + let ct, cty, hole, loc = match t with + | _, (_, Some (CCast (loc, ct, CastConv cty))) -> + mkt ct, mkt cty, mkt (mkCHole dummy_loc), loc + | _, (_, Some ct) -> + mkt ct, mkt (mkCHole dummy_loc), mkt (mkCHole dummy_loc), dummy_loc + | _, (GCast (loc, ct, CastConv cty), None) -> + mkl ct, mkl cty, mkl mkRHole, loc + | _, (t, None) -> mkl t, mkl mkRHole, mkl mkRHole, dummy_loc in + let gl, cut, sol, itac1, itac2 = + match fk, namefst, suff with + | FwdHave, true, true -> + errorstrm (str"Suff have does not accept a proof term") + | FwdHave, false, true -> + let cty = combineCG cty hole (mkCArrow loc) mkRArrow in + let _, t, uc = interp gl false (combineCG ct cty (mkCCast loc) mkRCast) in + let gl = pf_merge_uc uc gl in + let ty = pf_type_of gl t in + let ctx, _ = decompose_prod_n 1 ty in + let assert_is_conv gl = + try Proofview.V82.of_tactic (convert_concl (compose_prod ctx concl)) gl + with _ -> errorstrm (str "Given proof term is not of type " ++ + pr_constr (mkArrow (mkVar (id_of_string "_")) concl)) in + gl, ty, tclTHEN assert_is_conv (Proofview.V82.of_tactic (apply t)), id, itac_c + | FwdHave, false, false -> + let skols = List.flatten (List.map (function + | IpatNewHidden ids -> ids + | _ -> assert false) skols) in + let skols_args = + List.map (fun id -> examine_abstract (mkVar id) gl) skols in + let gl = List.fold_right unlock_abs skols_args gl in + let sigma, t, uc = + interp gl false (combineCG ct cty (mkCCast loc) mkRCast) in + let gl = re_sig (sig_it gl) (Evd.merge_universe_context sigma uc) in + let gs = + List.map (fun (_,a) -> + pf_find_abstract_proof false gl a.(1)) skols_args in + let tacopen_skols gl = + let stuff, g = Refiner.unpackage gl in + Refiner.repackage stuff (gs @ [g]) in + let gl, ty = pf_e_type_of gl t in + gl, ty, Proofview.V82.of_tactic (apply t), id, + tclTHEN (tclTHEN itac_c simpltac) + (tclTHEN tacopen_skols (fun gl -> + let abstract, gl = pf_mkSsrConst "abstract" gl in + unfold [abstract; abstract_key] gl)) + | _,true,true -> + let _, ty, uc = interp_ty gl fixtc cty in let gl = pf_merge_uc uc gl in + gl, mkArrow ty concl, hint, itac, clr + | _,false,true -> + let _, ty, uc = interp_ty gl fixtc cty in let gl = pf_merge_uc uc gl in + gl, mkArrow ty concl, hint, id, itac_c + | _, false, false -> + let n, cty, uc = interp_ty gl fixtc cty in let gl = pf_merge_uc uc gl in + gl, cty, tclTHEN (binderstac n) hint, id, tclTHEN itac_c simpltac + | _, true, false -> assert false in + tclTHENS (cuttac cut) [ tclTHEN sol itac1; itac2 ] gl) + gl +;; + +(* to extend the abstract value one needs: + Utility lemma to partially instantiate an abstract constant type. + Lemma use_abstract T n l (x : abstract T n l) : T. + Proof. by case: l x. Qed. +*) +let ssrabstract ist gens (*last*) gl = + let main _ (_,cid) ist gl = +(* + let proj1, proj2, prod = + let pdata = build_prod () in + pdata.Coqlib.proj1, pdata.Coqlib.proj2, pdata.Coqlib.typ in +*) + let concl, env = pf_concl gl, pf_env gl in + let fire gl t = Reductionops.nf_evar (project gl) t in + let abstract, gl = pf_mkSsrConst "abstract" gl in + let abstract_key, gl = pf_mkSsrConst "abstract_key" gl in + let id = mkVar (Option.get (id_of_cpattern cid)) in + let idty, args_id = examine_abstract id gl in + let abstract_n = args_id.(1) in + let abstract_proof = pf_find_abstract_proof true gl abstract_n in + let gl, proof = + let pf_unify_HO gl a b = + try pf_unify_HO gl a b + with _ -> errorstrm(strbrk"The abstract variable "++pr_constr id++ + strbrk" cannot abstract this goal. Did you generalize it?") in + let rec find_hole p t = + match kind_of_term t with + | Evar _ (*when last*) -> pf_unify_HO gl concl t, p + | Meta _ (*when last*) -> pf_unify_HO gl concl t, p + | Cast(m,_,_) when isEvar_or_Meta m (*when last*) -> pf_unify_HO gl concl t, p +(* + | Evar _ -> + let sigma, it = project gl, sig_it gl in + let sigma, ty = Evarutil.new_type_evar sigma env in + let gl = re_sig it sigma in + let p = mkApp (proj2,[|ty;concl;p|]) in + let concl = mkApp(prod,[|ty; concl|]) in + pf_unify_HO gl concl t, p + | App(hd, [|left; right|]) when Term.eq_constr hd prod -> + find_hole (mkApp (proj1,[|left;right;p|])) left +*) + | _ -> errorstrm(strbrk"abstract constant "++pr_constr abstract_n++ + strbrk" has an unexpected shape. Did you tamper with it?") + in + find_hole + ((*if last then*) id + (*else mkApp(mkSsrConst "use_abstract",Array.append args_id [|id|])*)) + (fire gl args_id.(0)) in + let gl = (*if last then*) pf_unify_HO gl abstract_key args_id.(2) (*else gl*) in + let gl, _ = pf_e_type_of gl idty in + let proof = fire gl proof in +(* if last then *) + let tacopen gl = + let stuff, g = Refiner.unpackage gl in + Refiner.repackage stuff [ g; abstract_proof ] in + tclTHENS tacopen [tclSOLVE [Proofview.V82.of_tactic (apply proof)];unfold[abstract;abstract_key]] gl +(* else apply proof gl *) + in + let introback ist (gens, _) = + introstac ~ist + (List.map (fun (_,cp) -> match id_of_cpattern cp with + | None -> IpatAnon + | Some id -> IpatId id) + (List.tl (List.hd gens))) in + tclTHEN (with_dgens gens main ist) (introback ist gens) gl + +(* The standard TACTIC EXTEND does not work for abstract *) +GEXTEND Gram + GLOBAL: tactic_expr; + tactic_expr: LEVEL "3" + [ RIGHTA [ IDENT "abstract"; gens = ssrdgens -> + Tacexpr.TacML (!@loc, ssrtac_name "abstract", + [Genarg.in_gen (Genarg.rawwit wit_ssrdgens) gens]) ]]; +END +TACTIC EXTEND ssrabstract +| [ "abstract" ssrdgens(gens) ] -> [ + if List.length (fst gens) <> 1 then + errorstrm (str"dependents switches '/' not allowed here"); + Proofview.V82.tactic (ssrabstract ist gens) ] +END + +let prof_havetac = mk_profiler "havetac";; +let havetac arg a b gl = prof_havetac.profile (havetac arg a b) gl;; + +TACTIC EXTEND ssrhave +| [ "have" ssrhavefwdwbinders(fwd) ] -> + [ Proofview.V82.tactic (havetac ist fwd false false) ] +END + +TACTIC EXTEND ssrhavesuff +| [ "have" "suff" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ Proofview.V82.tactic (havetac ist (false,(pats,fwd)) true false) ] +END + +TACTIC EXTEND ssrhavesuffices +| [ "have" "suffices" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ Proofview.V82.tactic (havetac ist (false,(pats,fwd)) true false) ] +END + +TACTIC EXTEND ssrsuffhave +| [ "suff" "have" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ Proofview.V82.tactic (havetac ist (false,(pats,fwd)) true true) ] +END + +TACTIC EXTEND ssrsufficeshave +| [ "suffices" "have" ssrhpats_nobs(pats) ssrhavefwd(fwd) ] -> + [ Proofview.V82.tactic (havetac ist (false,(pats,fwd)) true true) ] +END + +(** The "suffice" tactic *) + +let pr_ssrsufffwdwbinders _ _ prt (hpats, (fwd, hint)) = + pr_hpats hpats ++ pr_fwd fwd ++ pr_hint prt hint + +ARGUMENT EXTEND ssrsufffwd + TYPED AS ssrhpats * (ssrfwd * ssrhint) PRINTED BY pr_ssrsufffwdwbinders +| [ ssrhpats(pats) ssrbinder_list(bs) ":" lconstr(t) ssrhint(hint) ] -> + [ let ((clr, pats), binders), simpl = pats in + let allbs = intro_id_to_binder binders @ bs in + let allbinders = binders @ List.flatten (binder_to_intro_id bs) in + let fwd = mkFwdHint ":" t in + (((clr, pats), allbinders), simpl), (bind_fwd allbs fwd, hint) ] +END + +let sufftac ist ((((clr, pats),binders),simpl), ((_, c), hint)) = + let htac = tclTHEN (introstac ~ist pats) (hinttac ist true hint) in + let c = match c with + | (a, (b, Some (CCast (_, _, CastConv cty)))) -> a, (b, Some cty) + | (a, (GCast (_, _, CastConv cty), None)) -> a, (cty, None) + | _ -> anomaly "suff: ssr cast hole deleted by typecheck" in + let ctac gl = + let _,ty,_,uc = pf_interp_ty ist gl c in let gl = pf_merge_uc uc gl in + basecuttac "ssr_suff" ty gl in + tclTHENS ctac [htac; tclTHEN (cleartac clr) (introstac ~ist (binders@simpl))] + +TACTIC EXTEND ssrsuff +| [ "suff" ssrsufffwd(fwd) ] -> [ Proofview.V82.tactic (sufftac ist fwd) ] +END + +TACTIC EXTEND ssrsuffices +| [ "suffices" ssrsufffwd(fwd) ] -> [ Proofview.V82.tactic (sufftac ist fwd) ] +END + +(** The "wlog" (Without Loss Of Generality) tactic *) + +(* type ssrwlogfwd = ssrwgen list * ssrfwd *) + +let pr_ssrwlogfwd _ _ _ (gens, t) = + str ":" ++ pr_list mt pr_wgen gens ++ spc() ++ pr_fwd t + +ARGUMENT EXTEND ssrwlogfwd TYPED AS ssrwgen list * ssrfwd + PRINTED BY pr_ssrwlogfwd +| [ ":" ssrwgen_list(gens) "/" lconstr(t) ] -> [ gens, mkFwdHint "/" t] +END + +let destProd_or_LetIn c = + match kind_of_term c with + | Prod (n,ty,c) -> (n,None,ty), c + | LetIn (n,bo,ty,c) -> (n,Some bo,ty), c + | _ -> raise DestKO + +let wlogtac ist (((clr0, pats),_),_) (gens, ((_, ct))) hint suff ghave gl = + let mkabs gen = abs_wgen false ist (fun x -> x) gen in + let mkclr gen clrs = clr_of_wgen gen clrs in + let mkpats = function + | _, Some ((x, _), _) -> fun pats -> IpatId (hoi_id x) :: pats + | _ -> fun x -> x in + let ct = match ct with + | (a, (b, Some (CCast (_, _, CastConv cty)))) -> a, (b, Some cty) + | (a, (GCast (_, _, CastConv cty), None)) -> a, (cty, None) + | _ -> anomaly "wlog: ssr cast hole deleted by typecheck" in + let cut_implies_goal = not (suff || ghave <> `NoGen) in + let c, args, ct, gl = + let gens = List.filter (function _, Some _ -> true | _ -> false) gens in + let concl = pf_concl gl in + let c = mkProp in + let c = if cut_implies_goal then mkArrow c concl else c in + let gl, args, c = List.fold_right mkabs gens (gl,[],c) in + let env, _ = + List.fold_left (fun (env, c) _ -> + let rd, c = destProd_or_LetIn c in + Environ.push_rel rd env, c) (pf_env gl, c) gens in + let sigma, ev = Evarutil.new_evar env (project gl) Term.mkProp in + let k, _ = Term.destEvar ev in + let fake_gl = {Evd.it = k; Evd.sigma = sigma} in + let _, ct, _, uc = pf_interp_ty ist fake_gl ct in + let rec var2rel c g s = match kind_of_term c, g with + | Prod(Anonymous,_,c), [] -> mkProd(Anonymous, Vars.subst_vars s ct, c) + | Sort _, [] -> Vars.subst_vars s ct + | LetIn(Name id as n,b,ty,c), _::g -> mkLetIn (n,b,ty,var2rel c g (id::s)) + | Prod(Name id as n,ty,c), _::g -> mkProd (n,ty,var2rel c g (id::s)) + | _ -> Errors.anomaly(str"SSR: wlog: var2rel: " ++ pr_constr c) in + let c = var2rel c gens [] in + let rec pired c = function + | [] -> c + | t::ts as args -> match kind_of_term c with + | Prod(_,_,c) -> pired (subst1 t c) ts + | LetIn(id,b,ty,c) -> mkLetIn (id,b,ty,pired c args) + | _ -> Errors.anomaly(str"SSR: wlog: pired: " ++ pr_constr c) in + c, args, pired c args, pf_merge_uc uc gl in + let tacipat pats = introstac ~ist pats in + let tacigens = + tclTHEN + (tclTHENLIST(List.rev(List.fold_right mkclr gens [cleartac clr0]))) + (introstac ~ist (List.fold_right mkpats gens [])) in + let hinttac = hinttac ist true hint in + let cut_kind, fst_goal_tac, snd_goal_tac = + match suff, ghave with + | true, `NoGen -> "ssr_wlog", tclTHEN hinttac (tacipat pats), tacigens + | false, `NoGen -> "ssr_wlog", hinttac, tclTHEN tacigens (tacipat pats) + | true, `Gen _ -> assert false + | false, `Gen id -> + if gens = [] then errorstrm(str"gen have requires some generalizations"); + let clear0 = cleartac clr0 in + let id, name_general_hyp, cleanup, pats = match id, pats with + | None, (IpatId id as ip)::pats -> Some id, tacipat [ip], clear0, pats + | None, _ -> None, tclIDTAC, clear0, pats + | Some (Some id),_ -> Some id, introid id, clear0, pats + | Some _,_ -> + let id = mk_anon_id "tmp" gl in + Some id, introid id, tclTHEN clear0 (clear [id]), pats in + let tac_specialize = match id with + | None -> tclIDTAC + | Some id -> + if pats = [] then tclIDTAC else + let args = Array.of_list args in + pp(lazy(str"specialized="++pr_constr (mkApp (mkVar id,args)))); + pp(lazy(str"specialized_ty="++pr_constr ct)); + tclTHENS (basecuttac "ssr_have" ct) + [Proofview.V82.of_tactic (apply (mkApp (mkVar id,args))); tclIDTAC] in + "ssr_have", + (if hint = nohint then tacigens else hinttac), + tclTHENLIST [name_general_hyp; tac_specialize; tacipat pats; cleanup] + in + tclTHENS (basecuttac cut_kind c) [fst_goal_tac; snd_goal_tac] gl + +TACTIC EXTEND ssrwlog +| [ "wlog" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint false `NoGen) ] +END + +TACTIC EXTEND ssrwlogs +| [ "wlog" "suff" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint true `NoGen) ] +END + +TACTIC EXTEND ssrwlogss +| [ "wlog" "suffices" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ]-> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint true `NoGen) ] +END + +TACTIC EXTEND ssrwithoutloss +| [ "without" "loss" ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint false `NoGen) ] +END + +TACTIC EXTEND ssrwithoutlosss +| [ "without" "loss" "suff" + ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint true `NoGen) ] +END + +TACTIC EXTEND ssrwithoutlossss +| [ "without" "loss" "suffices" + ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ]-> + [ Proofview.V82.tactic (wlogtac ist pats fwd hint true `NoGen) ] +END + +(* Generally have *) +let pr_idcomma _ _ _ = function + | None -> mt() + | Some None -> str"_, " + | Some (Some id) -> pr_id id ++ str", " + +ARGUMENT EXTEND ssr_idcomma TYPED AS ident option option PRINTED BY pr_idcomma + | [ ] -> [ None ] +END + +let accept_idcomma strm = + match Compat.get_tok (stream_nth 0 strm) with + | Tok.IDENT _ | Tok.KEYWORD "_" -> accept_before_syms [","] strm + | _ -> raise Stream.Failure + +let test_idcomma = Gram.Entry.of_parser "test_idcomma" accept_idcomma + +GEXTEND Gram + GLOBAL: ssr_idcomma; + ssr_idcomma: [ [ test_idcomma; + ip = [ id = IDENT -> Some (id_of_string id) | "_" -> None ]; "," -> + Some ip + ] ]; +END + +let augment_preclr clr1 (((clr0, x),y),z) = (((clr1 @ clr0, x),y),z) + +TACTIC EXTEND ssrgenhave +| [ "gen" "have" ssrclear(clr) + ssr_idcomma(id) ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ let pats = augment_preclr clr pats in + Proofview.V82.tactic (wlogtac ist pats fwd hint false (`Gen id)) ] +END + +TACTIC EXTEND ssrgenhave2 +| [ "generally" "have" ssrclear(clr) + ssr_idcomma(id) ssrhpats_nobs(pats) ssrwlogfwd(fwd) ssrhint(hint) ] -> + [ let pats = augment_preclr clr pats in + Proofview.V82.tactic (wlogtac ist pats fwd hint false (`Gen id)) ] +END + +(** Canonical Structure alias *) + +let def_body : Vernacexpr.definition_expr Gram.Entry.e = Obj.magic + (Grammar.Entry.find (Obj.magic gallina_ext) "vernac:def_body") in + +GEXTEND Gram + GLOBAL: gallina_ext; + + gallina_ext: + (* Canonical structure *) + [[ IDENT "Canonical"; qid = Constr.global -> + Vernacexpr.VernacCanonical (AN qid) + | IDENT "Canonical"; ntn = Prim.by_notation -> + Vernacexpr.VernacCanonical (ByNotation ntn) + | IDENT "Canonical"; qid = Constr.global; + d = def_body -> + let s = coerce_reference_to_id qid in + Vernacexpr.VernacDefinition + ((Some Decl_kinds.Global,Decl_kinds.CanonicalStructure), + (dummy_loc,s),(d )) + ]]; +END + +(** 9. Keyword compatibility fixes. *) + +(* Coq v8.1 notation uses "by" and "of" quasi-keywords, i.e., reserved *) +(* identifiers used as keywords. This is incompatible with ssreflect.v *) +(* which makes "by" and "of" true keywords, because of technicalities *) +(* in the internal lexer-parser API of Coq. We patch this here by *) +(* adding new parsing rules that recognize the new keywords. *) +(* To make matters worse, the Coq grammar for tactics fails to *) +(* export the non-terminals we need to patch. Fortunately, the CamlP5 *) +(* API provides a backdoor access (with loads of Obj.magic trickery). *) + +(* Coq v8.3 defines "by" as a keyword, some hacks are not needed any *) +(* longer and thus comment out. Such comments are marked with v8.3 *) + +let tac_ent = List.fold_left Grammar.Entry.find (Obj.magic simple_tactic) in +let hypident_ent = + tac_ent ["clause_dft_all"; "in_clause"; "hypident_occ"; "hypident"] in +let id_or_meta : Obj.t Gram.Entry.e = Obj.magic + (Grammar.Entry.find hypident_ent "id_or_meta") in +let hypident : (Obj.t * hyp_location_flag) Gram.Entry.e = + Obj.magic hypident_ent in +GEXTEND Gram + GLOBAL: hypident; +hypident: [ + [ "("; IDENT "type"; "of"; id = id_or_meta; ")" -> id, InHypTypeOnly + | "("; IDENT "value"; "of"; id = id_or_meta; ")" -> id, InHypValueOnly + ] ]; +END + +GEXTEND Gram + GLOBAL: hloc; +hloc: [ + [ "in"; "("; "Type"; "of"; id = ident; ")" -> + HypLocation ((dummy_loc,id), InHypTypeOnly) + | "in"; "("; IDENT "Value"; "of"; id = ident; ")" -> + HypLocation ((dummy_loc,id), InHypValueOnly) + ] ]; +END + +let constr_eval + : (Constrexpr.constr_expr,Obj.t,Obj.t) Genredexpr.may_eval Gram.Entry.e + = Obj.magic (Grammar.Entry.find (Obj.magic constr_may_eval) "constr_eval") + +GEXTEND Gram + GLOBAL: constr_eval; + constr_eval: [ + [ IDENT "type"; "of"; c = Constr.constr -> Genredexpr.ConstrTypeOf c ] + ]; +END + +(* We wipe out all the keywords generated by the grammar rules we defined. *) +(* The user is supposed to Require Import ssreflect or Require ssreflect *) +(* and Import ssreflect.SsrSyntax to obtain these keywords and as a *) +(* consequence the extended ssreflect grammar. *) +let () = Lexer.unfreeze frozen_lexer ;; + +(* vim: set filetype=ocaml foldmethod=marker: *) diff --git a/mathcomp/ssreflect/plugin/v8.5/ssreflect.mllib b/mathcomp/ssreflect/plugin/v8.5/ssreflect.mllib new file mode 100644 index 0000000..006b70f --- /dev/null +++ b/mathcomp/ssreflect/plugin/v8.5/ssreflect.mllib @@ -0,0 +1,2 @@ +Ssrmatching +Ssreflect diff --git a/mathcomp/ssreflect/plugin/v8.5/ssrmatching.ml4 b/mathcomp/ssreflect/plugin/v8.5/ssrmatching.ml4 new file mode 100644 index 0000000..2fd0fe6 --- /dev/null +++ b/mathcomp/ssreflect/plugin/v8.5/ssrmatching.ml4 @@ -0,0 +1,1290 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +(* Defining grammar rules with "xx" in it automatically declares keywords too, + * we thus save the lexer to restore it at the end of the file *) +let frozen_lexer = Lexer.freeze () ;; + +(*i camlp4use: "pa_extend.cmo" i*) +(*i camlp4deps: "grammar/grammar.cma" i*) + +open Names +open Pp +open Pcoq +open Genarg +open Constrarg +open Term +open Vars +open Topconstr +open Libnames +open Tactics +open Tacticals +open Termops +open Namegen +open Recordops +open Tacmach +open Coqlib +open Glob_term +open Util +open Evd +open Extend +open Goptions +open Tacexpr +open Tacinterp +open Pretyping +open Constr +open Tactic +open Extraargs +open Ppconstr +open Printer + +open Globnames +open Misctypes +open Decl_kinds +open Evar_kinds +open Constrexpr +open Constrexpr_ops +open Notation_term +open Notation_ops +open Locus +open Locusops + +DECLARE PLUGIN "ssreflect" + +type loc = Loc.t +let dummy_loc = Loc.ghost +let errorstrm = Errors.errorlabstrm "ssreflect" +let loc_error loc msg = Errors.user_err_loc (loc, msg, str msg) + +(* 0 cost pp function. Active only if env variable SSRDEBUG is set *) +(* or if SsrDebug is Set *) +let pp_ref = ref (fun _ -> ()) +let ssr_pp s = pperrnl (str"SSR: "++Lazy.force s) +let _ = try ignore(Sys.getenv "SSRDEBUG"); pp_ref := ssr_pp with Not_found -> () +let debug b = + if b then pp_ref := ssr_pp else pp_ref := fun _ -> () +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssrmatching debugging"; + Goptions.optkey = ["SsrMatchingDebug"]; + Goptions.optdepr = false; + Goptions.optread = (fun _ -> !pp_ref == ssr_pp); + Goptions.optwrite = debug } +let pp s = !pp_ref s + +(** Utils {{{ *****************************************************************) +let env_size env = List.length (Environ.named_context env) +let safeDestApp c = + match kind_of_term c with App (f, a) -> f, a | _ -> c, [| |] +let get_index = function ArgArg i -> i | _ -> + Errors.anomaly (str"Uninterpreted index") +(* Toplevel constr must be globalized twice ! *) +let glob_constr ist genv = function + | _, Some ce -> + let vars = Id.Map.fold (fun x _ accu -> Id.Set.add x accu) ist.lfun Id.Set.empty in + let ltacvars = { Constrintern.empty_ltac_sign with Constrintern.ltac_vars = vars } in + Constrintern.intern_gen WithoutTypeConstraint ~ltacvars:ltacvars genv ce + | rc, None -> rc + +(* Term printing utilities functions for deciding bracketing. *) +let pr_paren prx x = hov 1 (str "(" ++ prx x ++ str ")") +(* String lexing utilities *) +let skip_wschars s = + let rec loop i = match s.[i] with '\n'..' ' -> loop (i + 1) | _ -> i in loop +(* We also guard characters that might interfere with the ssreflect *) +(* tactic syntax. *) +let guard_term ch1 s i = match s.[i] with + | '(' -> false + | '{' | '/' | '=' -> true + | _ -> ch1 = '(' +(* The call 'guard s i' should return true if the contents of s *) +(* starting at i need bracketing to avoid ambiguities. *) +let pr_guarded guard prc c = + msg_with Format.str_formatter (prc c); + let s = Format.flush_str_formatter () ^ "$" in + if guard s (skip_wschars s 0) then pr_paren prc c else prc c +(* More sensible names for constr printers *) +let pr_constr = pr_constr +let prl_glob_constr c = pr_lglob_constr_env (Global.env ()) c +let pr_glob_constr c = pr_glob_constr_env (Global.env ()) c +let prl_constr_expr = pr_lconstr_expr +let pr_constr_expr = pr_constr_expr +let prl_glob_constr_and_expr = function + | _, Some c -> prl_constr_expr c + | c, None -> prl_glob_constr c +let pr_glob_constr_and_expr = function + | _, Some c -> pr_constr_expr c + | c, None -> pr_glob_constr c +let pr_term (k, c) = pr_guarded (guard_term k) pr_glob_constr_and_expr c +let prl_term (k, c) = pr_guarded (guard_term k) prl_glob_constr_and_expr c + +(** Adding a new uninterpreted generic argument type *) +let add_genarg tag pr = + let wit = Genarg.make0 None tag in + let glob ist x = (ist, x) in + let subst _ x = x in + let interp ist gl x = (gl.Evd.sigma, x) in + let gen_pr _ _ _ = pr in + let () = Genintern.register_intern0 wit glob in + let () = Genintern.register_subst0 wit subst in + let () = Geninterp.register_interp0 wit interp in + Pptactic.declare_extra_genarg_pprule wit gen_pr gen_pr gen_pr; + wit + +(** Constructors for cast type *) +let dC t = CastConv t +(** Constructors for constr_expr *) +let isCVar = function CRef (Ident _, _) -> true | _ -> false +let destCVar = function CRef (Ident (_, id), _) -> id | _ -> + Errors.anomaly (str"not a CRef") +let mkCHole loc = CHole (loc, None, IntroAnonymous, None) +let mkCLambda loc name ty t = + CLambdaN (loc, [[loc, name], Default Explicit, ty], t) +let mkCLetIn loc name bo t = + CLetIn (loc, (loc, name), bo, t) +let mkCCast loc t ty = CCast (loc,t, dC ty) +(** Constructors for rawconstr *) +let mkRHole = GHole (dummy_loc, InternalHole, IntroAnonymous, None) +let mkRApp f args = if args = [] then f else GApp (dummy_loc, f, args) +let mkRCast rc rt = GCast (dummy_loc, rc, dC rt) +let mkRLambda n s t = GLambda (dummy_loc, n, Explicit, s, t) + +(* ssrterm conbinators *) +let combineCG t1 t2 f g = match t1, t2 with + | (x, (t1, None)), (_, (t2, None)) -> x, (g t1 t2, None) + | (x, (_, Some t1)), (_, (_, Some t2)) -> x, (mkRHole, Some (f t1 t2)) + | _, (_, (_, None)) -> Errors.anomaly (str"have: mixed C-G constr") + | _ -> Errors.anomaly (str"have: mixed G-C constr") +let loc_ofCG = function + | (_, (s, None)) -> Glob_ops.loc_of_glob_constr s + | (_, (_, Some s)) -> Constrexpr_ops.constr_loc s + +let mk_term k c = k, (mkRHole, Some c) +let mk_lterm = mk_term ' ' + +(* }}} *) + +(** Profiling {{{ *************************************************************) +type profiler = { + profile : 'a 'b. ('a -> 'b) -> 'a -> 'b; + reset : unit -> unit; + print : unit -> unit } +let profile_now = ref false +let something_profiled = ref false +let profilers = ref [] +let add_profiler f = profilers := f :: !profilers;; +let profile b = + profile_now := b; + if b then List.iter (fun f -> f.reset ()) !profilers; + if not b then List.iter (fun f -> f.print ()) !profilers +;; +let _ = + Goptions.declare_bool_option + { Goptions.optsync = false; + Goptions.optname = "ssrmatching profiling"; + Goptions.optkey = ["SsrMatchingProfiling"]; + Goptions.optread = (fun _ -> !profile_now); + Goptions.optdepr = false; + Goptions.optwrite = profile } +let () = + let prof_total = + let init = ref 0.0 in { + profile = (fun f x -> assert false); + reset = (fun () -> init := Unix.gettimeofday ()); + print = (fun () -> if !something_profiled then + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + "total" 0 (Unix.gettimeofday() -. !init) 0.0 0.0)) } in + let prof_legenda = { + profile = (fun f x -> assert false); + reset = (fun () -> ()); + print = (fun () -> if !something_profiled then begin + prerr_endline + (Printf.sprintf "!! %39s ---------- --------- --------- ---------" + (String.make 39 '-')); + prerr_endline + (Printf.sprintf "!! %-39s %10s %9s %9s %9s" + "function" "#calls" "total" "max" "average") end) } in + add_profiler prof_legenda; + add_profiler prof_total +;; + +let mk_profiler s = + let total, calls, max = ref 0.0, ref 0, ref 0.0 in + let reset () = total := 0.0; calls := 0; max := 0.0 in + let profile f x = + if not !profile_now then f x else + let before = Unix.gettimeofday () in + try + incr calls; + let res = f x in + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + res + with exc -> + let after = Unix.gettimeofday () in + let delta = after -. before in + total := !total +. delta; + if delta > !max then max := delta; + raise exc in + let print () = + if !calls <> 0 then begin + something_profiled := true; + prerr_endline + (Printf.sprintf "!! %-39s %10d %9.4f %9.4f %9.4f" + s !calls !total !max (!total /. (float_of_int !calls))) end in + let prof = { profile = profile; reset = reset; print = print } in + add_profiler prof; + prof +;; +(* }}} *) + +exception NoProgress + +(** Unification procedures. *) + +(* To enforce the rigidity of the rooted match we always split *) +(* top applications, so the unification procedures operate on *) +(* arrays of patterns and terms. *) +(* We perform three kinds of unification: *) +(* EQ: exact conversion check *) +(* FO: first-order unification of evars, without conversion *) +(* HO: higher-order unification with conversion *) +(* The subterm unification strategy is to find the first FO *) +(* match, if possible, and the first HO match otherwise, then *) +(* compute all the occurrences that are EQ matches for the *) +(* relevant subterm. *) +(* Additional twists: *) +(* - If FO/HO fails then we attempt to fill evars using *) +(* typeclasses before raising an outright error. We also *) +(* fill typeclasses even after a successful match, since *) +(* beta-reduction and canonical instances may leave *) +(* undefined evars. *) +(* - We do postchecks to rule out matches that are not *) +(* closed or that assign to a global evar; these can be *) +(* disabled for rewrite or dependent family matches. *) +(* - We do a full FO scan before turning to HO, as the FO *) +(* comparison can be much faster than the HO one. *) + +let unif_EQ env sigma p c = + let evars = existential_opt_value sigma in + try let _ = Reduction.conv env p ~evars c in true with _ -> false + +let unif_EQ_args env sigma pa a = + let n = Array.length pa in + let rec loop i = (i = n) || unif_EQ env sigma pa.(i) a.(i) && loop (i + 1) in + loop 0 + +let prof_unif_eq_args = mk_profiler "unif_EQ_args";; +let unif_EQ_args env sigma pa a = + prof_unif_eq_args.profile (unif_EQ_args env sigma pa) a +;; + +let unif_HO env ise p c = Evarconv.the_conv_x env p c ise + +let unif_HOtype env ise p c = Evarconv.the_conv_x_leq env p c ise + +let unif_HO_args env ise0 pa i ca = + let n = Array.length pa in + let rec loop ise j = + if j = n then ise else loop (unif_HO env ise pa.(j) ca.(i + j)) (j + 1) in + loop ise0 0 + +(* FO unification should boil down to calling w_unify with no_delta, but *) +(* alas things are not so simple: w_unify does partial type-checking, *) +(* which breaks down when the no-delta flag is on (as the Coq type system *) +(* requires full convertibility. The workaround here is to convert all *) +(* evars into metas, since 8.2 does not TC metas. This means some lossage *) +(* for HO evars, though hopefully Miller patterns can pick up some of *) +(* those cases, and HO matching will mop up the rest. *) +let flags_FO = + let flags = + { (Unification.default_no_delta_unify_flags ()).Unification.core_unify_flags + with + Unification.modulo_conv_on_closed_terms = None; + Unification.modulo_eta = true; + Unification.modulo_betaiota = true; + Unification.modulo_delta_types = full_transparent_state} + in + { Unification.core_unify_flags = flags; + Unification.merge_unify_flags = flags; + Unification.subterm_unify_flags = flags; + Unification.allow_K_in_toplevel_higher_order_unification = false; + Unification.resolve_evars = + (Unification.default_no_delta_unify_flags ()).Unification.resolve_evars + } +let unif_FO env ise p c = + Unification.w_unify env ise Reduction.CONV ~flags:flags_FO p c + +(* Perform evar substitution in main term and prune substitution. *) +let nf_open_term sigma0 ise c = + let s = ise and s' = ref sigma0 in + let rec nf c' = match kind_of_term c' with + | Evar ex -> + begin try nf (existential_value s ex) with _ -> + let k, a = ex in let a' = Array.map nf a in + if not (Evd.mem !s' k) then + s' := Evd.add !s' k (Evarutil.nf_evar_info s (Evd.find s k)); + mkEvar (k, a') + end + | _ -> map_constr nf c' in + let copy_def k evi () = + if evar_body evi != Evd.Evar_empty then () else + match Evd.evar_body (Evd.find s k) with + | Evar_defined c' -> s' := Evd.define k (nf c') !s' + | _ -> () in + let c' = nf c in let _ = Evd.fold copy_def sigma0 () in + !s', Evd.evar_universe_context s, c' + +let unif_end env sigma0 ise0 pt ok = + let ise = Evarconv.consider_remaining_unif_problems env ise0 in + let s, uc, t = nf_open_term sigma0 ise pt in + let ise1 = create_evar_defs s in + let ise2 = Typeclasses.resolve_typeclasses ~fail:true env ise1 in + if not (ok ise) then raise NoProgress else + if ise2 == ise1 then (s, uc, t) + else + let s, uc', t = nf_open_term sigma0 ise2 t in + s, Evd.union_evar_universe_context uc uc', t + +let pf_unif_HO gl sigma pt p c = + let env = pf_env gl in + let ise = unif_HO env (create_evar_defs sigma) p c in + unif_end env (project gl) ise pt (fun _ -> true) + +let unify_HO env sigma0 t1 t2 = + let sigma = unif_HO env sigma0 t1 t2 in + let sigma, uc, _ = unif_end env sigma0 sigma t2 (fun _ -> true) in + Evd.merge_universe_context sigma uc + +let pf_unify_HO gl t1 t2 = + let env, sigma0, si = pf_env gl, project gl, sig_it gl in + let sigma = unify_HO env sigma0 t1 t2 in + re_sig si sigma + +(* This is what the definition of iter_constr should be... *) +let iter_constr_LR f c = match kind_of_term c with + | Evar (k, a) -> Array.iter f a + | Cast (cc, _, t) -> f cc; f t + | Prod (_, t, b) | Lambda (_, t, b) -> f t; f b + | LetIn (_, v, t, b) -> f v; f t; f b + | App (cf, a) -> f cf; Array.iter f a + | Case (_, p, v, b) -> f v; f p; Array.iter f b + | Fix (_, (_, t, b)) | CoFix (_, (_, t, b)) -> + for i = 0 to Array.length t - 1 do f t.(i); f b.(i) done + | _ -> () + +(* The comparison used to determine which subterms matches is KEYED *) +(* CONVERSION. This looks for convertible terms that either have the same *) +(* same head constant as pat if pat is an application (after beta-iota), *) +(* or start with the same constr constructor (esp. for LetIn); this is *) +(* disregarded if the head term is let x := ... in x, and casts are always *) +(* ignored and removed). *) +(* Record projections get special treatment: in addition to the projection *) +(* constant itself, ssreflect also recognizes head constants of canonical *) +(* projections. *) + +exception NoMatch +type ssrdir = L2R | R2L +let pr_dir_side = function L2R -> str "LHS" | R2L -> str "RHS" +let inv_dir = function L2R -> R2L | R2L -> L2R + + +type pattern_class = + | KpatFixed + | KpatConst + | KpatEvar of existential_key + | KpatLet + | KpatLam + | KpatRigid + | KpatFlex + | KpatProj of constant + +type tpattern = { + up_k : pattern_class; + up_FO : constr; + up_f : constr; + up_a : constr array; + up_t : constr; (* equation proof term or matched term *) + up_dir : ssrdir; (* direction of the rule *) + up_ok : constr -> evar_map -> bool; (* progess test for rewrite *) + } + +let all_ok _ _ = true + +let proj_nparams c = + try 1 + Recordops.find_projection_nparams (ConstRef c) with _ -> 0 + +let isFixed c = match kind_of_term c with + | Var _ | Ind _ | Construct _ | Const _ -> true + | _ -> false + +let isRigid c = match kind_of_term c with + | Prod _ | Sort _ | Lambda _ | Case _ | Fix _ | CoFix _ -> true + | _ -> false + +exception UndefPat + +let hole_var = mkVar (id_of_string "_") +let pr_constr_pat c0 = + let rec wipe_evar c = + if isEvar c then hole_var else map_constr wipe_evar c in + pr_constr (wipe_evar c0) + +(* Turn (new) evars into metas *) +let evars_for_FO ~hack env sigma0 (ise0:evar_map) c0 = + let ise = ref ise0 in + let sigma = ref ise0 in + let nenv = env_size env + if hack then 1 else 0 in + let rec put c = match kind_of_term c with + | Evar (k, a as ex) -> + begin try put (existential_value !sigma ex) + with NotInstantiatedEvar -> + if Evd.mem sigma0 k then map_constr put c else + let evi = Evd.find !sigma k in + let dc = List.firstn (max 0 (Array.length a - nenv)) (evar_filtered_context evi) in + let abs_dc (d, c) = function + | x, Some b, t -> d, mkNamedLetIn x (put b) (put t) c + | x, None, t -> mkVar x :: d, mkNamedProd x (put t) c in + let a, t = + Context.fold_named_context_reverse abs_dc ~init:([], (put evi.evar_concl)) dc in + let m = Evarutil.new_meta () in + ise := meta_declare m t !ise; + sigma := Evd.define k (applist (mkMeta m, a)) !sigma; + put (existential_value !sigma ex) + end + | _ -> map_constr put c in + let c1 = put c0 in !ise, c1 + +(* Compile a match pattern from a term; t is the term to fill. *) +(* p_origin can be passed to obtain a better error message *) +let mk_tpattern ?p_origin ?(hack=false) env sigma0 (ise, t) ok dir p = + let k, f, a = + let f, a = Reductionops.whd_betaiota_stack ise p in + match kind_of_term f with + | Const (p,_) -> + let np = proj_nparams p in + if np = 0 || np > List.length a then KpatConst, f, a else + let a1, a2 = List.chop np a in KpatProj p, applist(f, a1), a2 + | Var _ | Ind _ | Construct _ -> KpatFixed, f, a + | Evar (k, _) -> + if Evd.mem sigma0 k then KpatEvar k, f, a else + if a <> [] then KpatFlex, f, a else + (match p_origin with None -> Errors.error "indeterminate pattern" + | Some (dir, rule) -> + errorstrm (str "indeterminate " ++ pr_dir_side dir + ++ str " in " ++ pr_constr_pat rule)) + | LetIn (_, v, _, b) -> + if b <> mkRel 1 then KpatLet, f, a else KpatFlex, v, a + | Lambda _ -> KpatLam, f, a + | _ -> KpatRigid, f, a in + let aa = Array.of_list a in + let ise', p' = evars_for_FO ~hack env sigma0 ise (mkApp (f, aa)) in + ise', + { up_k = k; up_FO = p'; up_f = f; + up_a = aa; up_ok = ok; up_dir = dir; up_t = t} + +(* Specialize a pattern after a successful match: assign a precise head *) +(* kind and arity for Proj and Flex patterns. *) +let ungen_upat lhs (sigma, uc, t) u = + let f, a = safeDestApp lhs in + let k = match kind_of_term f with + | Var _ | Ind _ | Construct _ -> KpatFixed + | Const _ -> KpatConst + | Evar (k, _) -> if is_defined sigma k then raise NoMatch else KpatEvar k + | LetIn _ -> KpatLet + | Lambda _ -> KpatLam + | _ -> KpatRigid in + sigma, uc, {u with up_k = k; up_FO = lhs; up_f = f; up_a = a; up_t = t} + +let nb_cs_proj_args pc f u = + let na k = + List.length (snd (lookup_canonical_conversion (ConstRef pc, k))).o_TCOMPS in + try match kind_of_term f with + | Prod _ -> na Prod_cs + | Sort s -> na (Sort_cs (family_of_sort s)) + | Const (c',_) when Constant.equal c' pc -> Array.length (snd (destApp u.up_f)) + | Var _ | Ind _ | Construct _ | Const _ -> na (Const_cs (global_of_constr f)) + | _ -> -1 + with Not_found -> -1 + +let isEvar_k k f = + match kind_of_term f with Evar (k', _) -> k = k' | _ -> false + +let nb_args c = + match kind_of_term c with App (_, a) -> Array.length a | _ -> 0 + +let mkSubArg i a = if i = Array.length a then a else Array.sub a 0 i +let mkSubApp f i a = if i = 0 then f else mkApp (f, mkSubArg i a) + +let splay_app ise = + let rec loop c a = match kind_of_term c with + | App (f, a') -> loop f (Array.append a' a) + | Cast (c', _, _) -> loop c' a + | Evar ex -> + (try loop (existential_value ise ex) a with _ -> c, a) + | _ -> c, a in + fun c -> match kind_of_term c with + | App (f, a) -> loop f a + | Cast _ | Evar _ -> loop c [| |] + | _ -> c, [| |] + +let filter_upat i0 f n u fpats = + let na = Array.length u.up_a in + if n < na then fpats else + let np = match u.up_k with + | KpatConst when Term.eq_constr u.up_f f -> na + | KpatFixed when Term.eq_constr u.up_f f -> na + | KpatEvar k when isEvar_k k f -> na + | KpatLet when isLetIn f -> na + | KpatLam when isLambda f -> na + | KpatRigid when isRigid f -> na + | KpatFlex -> na + | KpatProj pc -> + let np = na + nb_cs_proj_args pc f u in if n < np then -1 else np + | _ -> -1 in + if np < na then fpats else + let () = if !i0 < np then i0 := n in (u, np) :: fpats + +let filter_upat_FO i0 f n u fpats = + let np = nb_args u.up_FO in + if n < np then fpats else + let ok = match u.up_k with + | KpatConst -> Term.eq_constr u.up_f f + | KpatFixed -> Term.eq_constr u.up_f f + | KpatEvar k -> isEvar_k k f + | KpatLet -> isLetIn f + | KpatLam -> isLambda f + | KpatRigid -> isRigid f + | KpatProj pc -> Term.eq_constr f (mkConst pc) + | KpatFlex -> i0 := n; true in + if ok then begin if !i0 < np then i0 := np; (u, np) :: fpats end else fpats + +exception FoundUnif of (evar_map * evar_universe_context * tpattern) +(* Note: we don't update env as we descend into the term, as the primitive *) +(* unification procedure always rejects subterms with bound variables. *) + +(* We are forced to duplicate code between the FO/HO matching because we *) +(* have to work around several kludges in unify.ml: *) +(* - w_unify drops into second-order unification when the pattern is an *) +(* application whose head is a meta. *) +(* - w_unify tries to unify types without subsumption when the pattern *) +(* head is an evar or meta (e.g., it fails on ?1 = nat when ?1 : Type). *) +(* - w_unify expands let-in (zeta conversion) eagerly, whereas we want to *) +(* match a head let rigidly. *) +let match_upats_FO upats env sigma0 ise = + let rec loop c = + let f, a = splay_app ise c in let i0 = ref (-1) in + let fpats = + List.fold_right (filter_upat_FO i0 f (Array.length a)) upats [] in + while !i0 >= 0 do + let i = !i0 in i0 := -1; + let c' = mkSubApp f i a in + let one_match (u, np) = + let skip = + if i <= np then i < np else + if u.up_k == KpatFlex then begin i0 := i - 1; false end else + begin if !i0 < np then i0 := np; true end in + if skip || not (closed0 c') then () else try + let _ = match u.up_k with + | KpatFlex -> + let kludge v = mkLambda (Anonymous, mkProp, v) in + unif_FO env ise (kludge u.up_FO) (kludge c') + | KpatLet -> + let kludge vla = + let vl, a = safeDestApp vla in + let x, v, t, b = destLetIn vl in + mkApp (mkLambda (x, t, b), Array.cons v a) in + unif_FO env ise (kludge u.up_FO) (kludge c') + | _ -> unif_FO env ise u.up_FO c' in + let ise' = (* Unify again using HO to assign evars *) + let p = mkApp (u.up_f, u.up_a) in + try unif_HO env ise p c' with _ -> raise NoMatch in + let lhs = mkSubApp f i a in + let pt' = unif_end env sigma0 ise' u.up_t (u.up_ok lhs) in + raise (FoundUnif (ungen_upat lhs pt' u)) + with FoundUnif _ as sigma_u -> raise sigma_u + | Not_found -> Errors.anomaly (str"incomplete ise in match_upats_FO") + | _ -> () in + List.iter one_match fpats + done; + iter_constr_LR loop f; Array.iter loop a in + fun c -> try loop c with Invalid_argument _ -> Errors.anomaly (str"IN FO") + +let prof_FO = mk_profiler "match_upats_FO";; +let match_upats_FO upats env sigma0 ise c = + prof_FO.profile (match_upats_FO upats env sigma0) ise c +;; + + +let match_upats_HO ~on_instance upats env sigma0 ise c = + let it_did_match = ref false in + let rec aux upats env sigma0 ise c = + let f, a = splay_app ise c in let i0 = ref (-1) in + let fpats = List.fold_right (filter_upat i0 f (Array.length a)) upats [] in + while !i0 >= 0 do + let i = !i0 in i0 := -1; + let one_match (u, np) = + let skip = + if i <= np then i < np else + if u.up_k == KpatFlex then begin i0 := i - 1; false end else + begin if !i0 < np then i0 := np; true end in + if skip then () else try + let ise' = match u.up_k with + | KpatFixed | KpatConst -> ise + | KpatEvar _ -> + let _, pka = destEvar u.up_f and _, ka = destEvar f in + unif_HO_args env ise pka 0 ka + | KpatLet -> + let x, v, t, b = destLetIn f in + let _, pv, _, pb = destLetIn u.up_f in + let ise' = unif_HO env ise pv v in + unif_HO (Environ.push_rel (x, None, t) env) ise' pb b + | KpatFlex | KpatProj _ -> + unif_HO env ise u.up_f (mkSubApp f (i - Array.length u.up_a) a) + | _ -> unif_HO env ise u.up_f f in + let ise'' = unif_HO_args env ise' u.up_a (i - Array.length u.up_a) a in + let lhs = mkSubApp f i a in + let pt' = unif_end env sigma0 ise'' u.up_t (u.up_ok lhs) in + on_instance (ungen_upat lhs pt' u) + with FoundUnif _ as sigma_u -> raise sigma_u + | NoProgress -> it_did_match := true + | _ -> () in + List.iter one_match fpats + done; + iter_constr_LR (aux upats env sigma0 ise) f; + Array.iter (aux upats env sigma0 ise) a + in + aux upats env sigma0 ise c; + if !it_did_match then raise NoProgress + +let prof_HO = mk_profiler "match_upats_HO";; +let match_upats_HO ~on_instance upats env sigma0 ise c = + prof_HO.profile (match_upats_HO ~on_instance upats env sigma0) ise c +;; + + +let fixed_upat = function +| {up_k = KpatFlex | KpatEvar _ | KpatProj _} -> false +| {up_t = t} -> not (occur_existential t) + +let do_once r f = match !r with Some _ -> () | None -> r := Some (f ()) + +let assert_done r = + match !r with Some x -> x | None -> Errors.anomaly (str"do_once never called") + +let assert_done_multires r = + match !r with + | None -> Errors.anomaly (str"do_once never called") + | Some (n, xs) -> + r := Some (n+1,xs); + try List.nth xs n with Failure _ -> raise NoMatch + +type subst = Environ.env -> Term.constr -> Term.constr -> int -> Term.constr +type find_P = + Environ.env -> Term.constr -> int -> + k:subst -> + Term.constr +type conclude = unit -> + Term.constr * ssrdir * (Evd.evar_map * Evd.evar_universe_context * Term.constr) + +(* upats_origin makes a better error message only *) +let mk_tpattern_matcher ?(all_instances=false) + ?(raise_NoMatch=false) ?upats_origin sigma0 occ (ise, upats) += + let nocc = ref 0 and skip_occ = ref false in + let use_occ, occ_list = match occ with + | Some (true, ol) -> ol = [], ol + | Some (false, ol) -> ol <> [], ol + | None -> false, [] in + let max_occ = List.fold_right max occ_list 0 in + let subst_occ = + let occ_set = Array.make max_occ (not use_occ) in + let _ = List.iter (fun i -> occ_set.(i - 1) <- use_occ) occ_list in + let _ = if max_occ = 0 then skip_occ := use_occ in + fun () -> incr nocc; + if !nocc = max_occ then skip_occ := use_occ; + if !nocc <= max_occ then occ_set.(!nocc - 1) else not use_occ in + let upat_that_matched = ref None in + let match_EQ env sigma u = + match u.up_k with + | KpatLet -> + let x, pv, t, pb = destLetIn u.up_f in + let env' = Environ.push_rel (x, None, t) env in + let match_let f = match kind_of_term f with + | LetIn (_, v, _, b) -> unif_EQ env sigma pv v && unif_EQ env' sigma pb b + | _ -> false in match_let + | KpatFixed -> Term.eq_constr u.up_f + | KpatConst -> Term.eq_constr u.up_f + | KpatLam -> fun c -> + (match kind_of_term c with + | Lambda _ -> unif_EQ env sigma u.up_f c + | _ -> false) + | _ -> unif_EQ env sigma u.up_f in +let p2t p = mkApp(p.up_f,p.up_a) in +let source () = match upats_origin, upats with + | None, [p] -> + (if fixed_upat p then str"term " else str"partial term ") ++ + pr_constr_pat (p2t p) ++ spc() + | Some (dir,rule), [p] -> str"The " ++ pr_dir_side dir ++ str" of " ++ + pr_constr_pat rule ++ fnl() ++ ws 4 ++ pr_constr_pat (p2t p) ++ fnl() + | Some (dir,rule), _ -> str"The " ++ pr_dir_side dir ++ str" of " ++ + pr_constr_pat rule ++ spc() + | _, [] | None, _::_::_ -> + Errors.anomaly (str"mk_tpattern_matcher with no upats_origin") in +let on_instance, instances = + let instances = ref [] in + (fun x -> + if all_instances then instances := !instances @ [x] + else raise (FoundUnif x)), + (fun () -> !instances) in +let rec uniquize = function + | [] -> [] + | (sigma,_,{ up_f = f; up_a = a; up_t = t } as x) :: xs -> + let t = Reductionops.nf_evar sigma t in + let f = Reductionops.nf_evar sigma f in + let a = Array.map (Reductionops.nf_evar sigma) a in + let neq (sigma1,_,{ up_f = f1; up_a = a1; up_t = t1 }) = + let t1 = Reductionops.nf_evar sigma1 t1 in + let f1 = Reductionops.nf_evar sigma1 f1 in + let a1 = Array.map (Reductionops.nf_evar sigma1) a1 in + not (Term.eq_constr t t1 && + Term.eq_constr f f1 && CArray.for_all2 Term.eq_constr a a1) in + x :: uniquize (List.filter neq xs) in + +((fun env c h ~k -> + do_once upat_that_matched (fun () -> + try + if not all_instances then match_upats_FO upats env sigma0 ise c; + match_upats_HO ~on_instance upats env sigma0 ise c; + raise NoMatch + with FoundUnif sigma_u -> 0,[sigma_u] + | (NoMatch|NoProgress) when all_instances && instances () <> [] -> + 0, uniquize (instances ()) + | NoMatch when (not raise_NoMatch) -> + errorstrm (source () ++ str "does not match any subterm of the goal") + | NoProgress when (not raise_NoMatch) -> + let dir = match upats_origin with Some (d,_) -> d | _ -> + Errors.anomaly (str"mk_tpattern_matcher with no upats_origin") in + errorstrm (str"all matches of "++source()++ + str"are equal to the " ++ pr_dir_side (inv_dir dir)) + | NoProgress -> raise NoMatch); + let sigma, _, ({up_f = pf; up_a = pa} as u) = + if all_instances then assert_done_multires upat_that_matched + else List.hd (snd(assert_done upat_that_matched)) in + pp(lazy(str"sigma@tmatch=" ++ pr_evar_map None sigma)); + if !skip_occ then ((*ignore(k env u.up_t 0);*) c) else + let match_EQ = match_EQ env sigma u in + let pn = Array.length pa in + let rec subst_loop (env,h as acc) c' = + if !skip_occ then c' else + let f, a = splay_app sigma c' in + if Array.length a >= pn && match_EQ f && unif_EQ_args env sigma pa a then + let a1, a2 = Array.chop (Array.length pa) a in + let fa1 = mkApp (f, a1) in + let f' = if subst_occ () then k env u.up_t fa1 h else fa1 in + mkApp (f', Array.map_left (subst_loop acc) a2) + else + (* TASSI: clear letin values to avoid unfolding *) + let inc_h (n,_,ty) (env,h') = Environ.push_rel (n,None,ty) env, h' + 1 in + let f' = map_constr_with_binders_left_to_right inc_h subst_loop acc f in + mkApp (f', Array.map_left (subst_loop acc) a) in + subst_loop (env,h) c) : find_P), +((fun () -> + let sigma, uc, ({up_f = pf; up_a = pa} as u) = + match !upat_that_matched with + | Some (_,x) -> List.hd x | None when raise_NoMatch -> raise NoMatch + | None -> Errors.anomaly (str"companion function never called") in + let p' = mkApp (pf, pa) in + if max_occ <= !nocc then p', u.up_dir, (sigma, uc, u.up_t) + else errorstrm (str"Only " ++ int !nocc ++ str" < " ++ int max_occ ++ + str(String.plural !nocc " occurence") ++ match upats_origin with + | None -> str" of" ++ spc() ++ pr_constr_pat p' + | Some (dir,rule) -> str" of the " ++ pr_dir_side dir ++ fnl() ++ + ws 4 ++ pr_constr_pat p' ++ fnl () ++ + str"of " ++ pr_constr_pat rule)) : conclude) + +type ('ident, 'term) ssrpattern = + | T of 'term + | In_T of 'term + | X_In_T of 'ident * 'term + | In_X_In_T of 'ident * 'term + | E_In_X_In_T of 'term * 'ident * 'term + | E_As_X_In_T of 'term * 'ident * 'term + +let pr_pattern = function + | T t -> prl_term t + | In_T t -> str "in " ++ prl_term t + | X_In_T (x,t) -> prl_term x ++ str " in " ++ prl_term t + | In_X_In_T (x,t) -> str "in " ++ prl_term x ++ str " in " ++ prl_term t + | E_In_X_In_T (e,x,t) -> + prl_term e ++ str " in " ++ prl_term x ++ str " in " ++ prl_term t + | E_As_X_In_T (e,x,t) -> + prl_term e ++ str " as " ++ prl_term x ++ str " in " ++ prl_term t + +let pr_pattern_w_ids = function + | T t -> prl_term t + | In_T t -> str "in " ++ prl_term t + | X_In_T (x,t) -> pr_id x ++ str " in " ++ prl_term t + | In_X_In_T (x,t) -> str "in " ++ pr_id x ++ str " in " ++ prl_term t + | E_In_X_In_T (e,x,t) -> + prl_term e ++ str " in " ++ pr_id x ++ str " in " ++ prl_term t + | E_As_X_In_T (e,x,t) -> + prl_term e ++ str " as " ++ pr_id x ++ str " in " ++ prl_term t + +let pr_pattern_aux pr_constr = function + | T t -> pr_constr t + | In_T t -> str "in " ++ pr_constr t + | X_In_T (x,t) -> pr_constr x ++ str " in " ++ pr_constr t + | In_X_In_T (x,t) -> str "in " ++ pr_constr x ++ str " in " ++ pr_constr t + | E_In_X_In_T (e,x,t) -> + pr_constr e ++ str " in " ++ pr_constr x ++ str " in " ++ pr_constr t + | E_As_X_In_T (e,x,t) -> + pr_constr e ++ str " as " ++ pr_constr x ++ str " in " ++ pr_constr t +let pp_pattern (sigma, p) = + pr_pattern_aux (fun t -> pr_constr (pi3 (nf_open_term sigma sigma t))) p +let pr_cpattern = pr_term +let pr_rpattern _ _ _ = pr_pattern + +let pr_option f = function None -> mt() | Some x -> f x +let pr_ssrpattern _ _ _ = pr_option pr_pattern +let pr_pattern_squarep = pr_option (fun r -> str "[" ++ pr_pattern r ++ str "]") +let pr_ssrpattern_squarep _ _ _ = pr_pattern_squarep +let pr_pattern_roundp = pr_option (fun r -> str "(" ++ pr_pattern r ++ str ")") +let pr_ssrpattern_roundp _ _ _ = pr_pattern_roundp + +let wit_rpatternty = add_genarg "rpatternty" pr_pattern + +ARGUMENT EXTEND rpattern TYPED AS rpatternty PRINTED BY pr_rpattern + | [ lconstr(c) ] -> [ T (mk_lterm c) ] + | [ "in" lconstr(c) ] -> [ In_T (mk_lterm c) ] + | [ lconstr(x) "in" lconstr(c) ] -> + [ X_In_T (mk_lterm x, mk_lterm c) ] + | [ "in" lconstr(x) "in" lconstr(c) ] -> + [ In_X_In_T (mk_lterm x, mk_lterm c) ] + | [ lconstr(e) "in" lconstr(x) "in" lconstr(c) ] -> + [ E_In_X_In_T (mk_lterm e, mk_lterm x, mk_lterm c) ] + | [ lconstr(e) "as" lconstr(x) "in" lconstr(c) ] -> + [ E_As_X_In_T (mk_lterm e, mk_lterm x, mk_lterm c) ] +END + +type cpattern = char * glob_constr_and_expr +let tag_of_cpattern = fst +let loc_of_cpattern = loc_ofCG +let cpattern_of_term t = t +type occ = (bool * int list) option + +type rpattern = (cpattern, cpattern) ssrpattern +let pr_rpattern = pr_pattern + +type pattern = Evd.evar_map * (Term.constr, Term.constr) ssrpattern + + +let id_of_cpattern = function + | _,(_,Some (CRef (Ident (_, x), _))) -> Some x + | _,(_,Some (CAppExpl (_, (_, Ident (_, x), _), []))) -> Some x + | _,(GRef (_, VarRef x, _) ,None) -> Some x + | _ -> None +let id_of_Cterm t = match id_of_cpattern t with + | Some x -> x + | None -> loc_error (loc_of_cpattern t) "Only identifiers are allowed here" + +let interp_wit wit ist gl x = + let globarg = in_gen (glbwit wit) x in + let sigma, arg = interp_genarg ist (pf_env gl) (project gl) (pf_concl gl) gl.Evd.it globarg in + sigma, out_gen (topwit wit) arg +let interp_constr = interp_wit wit_constr +let interp_open_constr ist gl gc = + interp_wit wit_open_constr ist gl ((), gc) +let pf_intern_term ist gl (_, c) = glob_constr ist (pf_env gl) c +let interp_term ist gl (_, c) = snd (interp_open_constr ist gl c) +let glob_ssrterm gs = function + | k, (_, Some c) -> k, Tacintern.intern_constr gs c + | ct -> ct +let subst_ssrterm s (k, c) = k, Tacsubst.subst_glob_constr_and_expr s c +let pr_ssrterm _ _ _ = pr_term +let input_ssrtermkind strm = match Compat.get_tok (stream_nth 0 strm) with + | Tok.KEYWORD "(" -> '(' + | Tok.KEYWORD "@" -> '@' + | _ -> ' ' +let ssrtermkind = Gram.Entry.of_parser "ssrtermkind" input_ssrtermkind + +(* This piece of code asserts the following notations are reserved *) +(* Reserved Notation "( a 'in' b )" (at level 0). *) +(* Reserved Notation "( a 'as' b )" (at level 0). *) +(* Reserved Notation "( a 'in' b 'in' c )" (at level 0). *) +(* Reserved Notation "( a 'as' b 'in' c )" (at level 0). *) +let glob_cpattern gs p = + pp(lazy(str"globbing pattern: " ++ pr_term p)); + let glob x = snd (glob_ssrterm gs (mk_lterm x)) in + let encode k s l = + let name = Name (id_of_string ("_ssrpat_" ^ s)) in + k, (mkRCast mkRHole (mkRLambda name mkRHole (mkRApp mkRHole l)), None) in + let bind_in t1 t2 = + let d = dummy_loc in let n = Name (destCVar t1) in + fst (glob (mkCCast d (mkCHole d) (mkCLambda d n (mkCHole d) t2))) in + let check_var t2 = if not (isCVar t2) then + loc_error (constr_loc t2) "Only identifiers are allowed here" in + match p with + | _, (_, None) as x -> x + | k, (v, Some t) as orig -> + if k = 'x' then glob_ssrterm gs ('(', (v, Some t)) else + match t with + | CNotation(_, "( _ in _ )", ([t1; t2], [], [])) -> + (try match glob t1, glob t2 with + | (r1, None), (r2, None) -> encode k "In" [r1;r2] + | (r1, Some _), (r2, Some _) when isCVar t1 -> + encode k "In" [r1; r2; bind_in t1 t2] + | (r1, Some _), (r2, Some _) -> encode k "In" [r1; r2] + | _ -> Errors.anomaly (str"where are we?") + with _ when isCVar t1 -> encode k "In" [bind_in t1 t2]) + | CNotation(_, "( _ in _ in _ )", ([t1; t2; t3], [], [])) -> + check_var t2; encode k "In" [fst (glob t1); bind_in t2 t3] + | CNotation(_, "( _ as _ )", ([t1; t2], [], [])) -> + encode k "As" [fst (glob t1); fst (glob t2)] + | CNotation(_, "( _ as _ in _ )", ([t1; t2; t3], [], [])) -> + check_var t2; encode k "As" [fst (glob t1); bind_in t2 t3] + | _ -> glob_ssrterm gs orig +;; + +let interp_ssrterm _ gl t = Tacmach.project gl, t + +ARGUMENT EXTEND cpattern + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_cpattern SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "Qed" constr(c) ] -> [ mk_lterm c ] +END + +let (!@) = Compat.to_coqloc + +GEXTEND Gram + GLOBAL: cpattern; + cpattern: [[ k = ssrtermkind; c = constr -> + let pattern = mk_term k c in + if loc_ofCG pattern <> !@loc && k = '(' then mk_term 'x' c else pattern ]]; +END + +ARGUMENT EXTEND lcpattern + PRINTED BY pr_ssrterm + INTERPRETED BY interp_ssrterm + GLOBALIZED BY glob_cpattern SUBSTITUTED BY subst_ssrterm + RAW_TYPED AS cpattern RAW_PRINTED BY pr_ssrterm + GLOB_TYPED AS cpattern GLOB_PRINTED BY pr_ssrterm +| [ "Qed" lconstr(c) ] -> [ mk_lterm c ] +END + +GEXTEND Gram + GLOBAL: lcpattern; + lcpattern: [[ k = ssrtermkind; c = lconstr -> + let pattern = mk_term k c in + if loc_ofCG pattern <> !@loc && k = '(' then mk_term 'x' c else pattern ]]; +END + +let interp_pattern ist gl red redty = + pp(lazy(str"interpreting: " ++ pr_pattern red)); + let xInT x y = X_In_T(x,y) and inXInT x y = In_X_In_T(x,y) in + let inT x = In_T x and eInXInT e x t = E_In_X_In_T(e,x,t) in + let eAsXInT e x t = E_As_X_In_T(e,x,t) in + let mkG ?(k=' ') x = k,(x,None) in + let decode t f g = + try match (pf_intern_term ist gl t) with + | GCast(_,GHole _,CastConv(GLambda(_,Name x,_,_,c))) -> f x (' ',(c,None)) + | it -> g t with _ -> g t in + let decodeG t f g = decode (mkG t) f g in + let bad_enc id _ = Errors.anomaly (str"bad encoding for pattern "++str id) in + let cleanup_XinE h x rp sigma = + let h_k = match kind_of_term h with Evar (k,_) -> k | _ -> assert false in + let to_clean, update = (* handle rename if x is already used *) + let ctx = pf_hyps gl in + let len = Context.named_context_length ctx in + let name = ref None in + try ignore(Context.lookup_named x ctx); (name, fun k -> + if !name = None then + let nctx = Evd.evar_context (Evd.find sigma k) in + let nlen = Context.named_context_length nctx in + if nlen > len then begin + name := Some (pi1 (List.nth nctx (nlen - len - 1))) + end) + with Not_found -> ref (Some x), fun _ -> () in + let sigma0 = project gl in + let new_evars = + let rec aux acc t = match kind_of_term t with + | Evar (k,_) -> + if k = h_k || List.mem k acc || Evd.mem sigma0 k then acc else + (update k; k::acc) + | _ -> fold_constr aux acc t in + aux [] (Evarutil.nf_evar sigma rp) in + let sigma = + List.fold_left (fun sigma e -> + if Evd.is_defined sigma e then sigma else (* clear may be recursiv *) + let name = Option.get !to_clean in + pp(lazy(pr_id name)); + try snd(Logic.prim_refiner (Proof_type.Thin [name]) sigma e) + with Evarutil.ClearDependencyError _ -> sigma) + sigma new_evars in + sigma in + let red = match red with + | T(k,(GCast (_,GHole _,(CastConv(GLambda (_,Name id,_,_,t)))),None)) + when let id = string_of_id id in let len = String.length id in + (len > 8 && String.sub id 0 8 = "_ssrpat_") -> + let id = string_of_id id in let len = String.length id in + (match String.sub id 8 (len - 8), t with + | "In", GApp(_, _, [t]) -> decodeG t xInT (fun x -> T x) + | "In", GApp(_, _, [e; t]) -> decodeG t (eInXInT (mkG e)) (bad_enc id) + | "In", GApp(_, _, [e; t; e_in_t]) -> + decodeG t (eInXInT (mkG e)) + (fun _ -> decodeG e_in_t xInT (fun _ -> assert false)) + | "As", GApp(_, _, [e; t]) -> decodeG t (eAsXInT (mkG e)) (bad_enc id) + | _ -> bad_enc id ()) + | T t -> decode t xInT (fun x -> T x) + | In_T t -> decode t inXInT inT + | X_In_T (e,t) -> decode t (eInXInT e) (fun x -> xInT (id_of_Cterm e) x) + | In_X_In_T (e,t) -> inXInT (id_of_Cterm e) t + | E_In_X_In_T (e,x,rp) -> eInXInT e (id_of_Cterm x) rp + | E_As_X_In_T (e,x,rp) -> eAsXInT e (id_of_Cterm x) rp in + pp(lazy(str"decoded as: " ++ pr_pattern_w_ids red)); + let red = match redty with None -> red | Some ty -> let ty = ' ', ty in + match red with + | T t -> T (combineCG t ty (mkCCast (loc_ofCG t)) mkRCast) + | X_In_T (x,t) -> + let ty = pf_intern_term ist gl ty in + E_As_X_In_T (mkG (mkRCast mkRHole ty), x, t) + | E_In_X_In_T (e,x,t) -> + let ty = mkG (pf_intern_term ist gl ty) in + E_In_X_In_T (combineCG e ty (mkCCast (loc_ofCG t)) mkRCast, x, t) + | E_As_X_In_T (e,x,t) -> + let ty = mkG (pf_intern_term ist gl ty) in + E_As_X_In_T (combineCG e ty (mkCCast (loc_ofCG t)) mkRCast, x, t) + | red -> red in + pp(lazy(str"typed as: " ++ pr_pattern_w_ids red)); + let mkXLetIn loc x (a,(g,c)) = match c with + | Some b -> a,(g,Some (mkCLetIn loc x (mkCHole loc) b)) + | None -> a,(GLetIn (loc,x,(GHole (loc, BinderType x, IntroAnonymous, None)), g), None) in + match red with + | T t -> let sigma, t = interp_term ist gl t in sigma, T t + | In_T t -> let sigma, t = interp_term ist gl t in sigma, In_T t + | X_In_T (x, rp) | In_X_In_T (x, rp) -> + let mk x p = match red with X_In_T _ -> X_In_T(x,p) | _ -> In_X_In_T(x,p) in + let rp = mkXLetIn dummy_loc (Name x) rp in + let sigma, rp = interp_term ist gl rp in + let _, h, _, rp = destLetIn rp in + let sigma = cleanup_XinE h x rp sigma in + let rp = subst1 h (Evarutil.nf_evar sigma rp) in + sigma, mk h rp + | E_In_X_In_T(e, x, rp) | E_As_X_In_T (e, x, rp) -> + let mk e x p = + match red with E_In_X_In_T _ ->E_In_X_In_T(e,x,p)|_->E_As_X_In_T(e,x,p) in + let rp = mkXLetIn dummy_loc (Name x) rp in + let sigma, rp = interp_term ist gl rp in + let _, h, _, rp = destLetIn rp in + let sigma = cleanup_XinE h x rp sigma in + let rp = subst1 h (Evarutil.nf_evar sigma rp) in + let sigma, e = interp_term ist (re_sig (sig_it gl) sigma) e in + sigma, mk e h rp +;; +let interp_cpattern ist gl red redty = interp_pattern ist gl (T red) redty;; +let interp_rpattern ist gl red = interp_pattern ist gl red None;; + +(* The full occurrence set *) +let noindex = Some(false,[]) + +(* calls do_subst on every sub-term identified by (pattern,occ) *) +let eval_pattern ?raise_NoMatch env0 sigma0 concl0 pattern occ do_subst = + let fs sigma x = Reductionops.nf_evar sigma x in + let pop_evar sigma e p = + let { Evd.evar_body = e_body } as e_def = Evd.find sigma e in + let e_body = match e_body with Evar_defined c -> c + | _ -> errorstrm (str "Matching the pattern " ++ pr_constr p ++ + str " did not instantiate ?" ++ int (Evar.repr e) ++ spc () ++ + str "Does the variable bound by the \"in\" construct occur "++ + str "in the pattern?") in + let sigma = + Evd.add (Evd.remove sigma e) e {e_def with Evd.evar_body = Evar_empty} in + sigma, e_body in + let ex_value hole = + match kind_of_term hole with Evar (e,_) -> e | _ -> assert false in + let mk_upat_for ?hack env sigma0 (sigma, t) ?(p=t) ok = + let sigma,pat= mk_tpattern ?hack env sigma0 (sigma,p) ok L2R (fs sigma t) in + sigma, [pat] in + match pattern with + | None -> do_subst env0 concl0 concl0 1 + | Some (sigma, (T rp | In_T rp)) -> + let rp = fs sigma rp in + let ise = create_evar_defs sigma in + let occ = match pattern with Some (_, T _) -> occ | _ -> noindex in + let rp = mk_upat_for env0 sigma0 (ise, rp) all_ok in + let find_T, end_T = mk_tpattern_matcher ?raise_NoMatch sigma0 occ rp in + let concl = find_T env0 concl0 1 do_subst in + let _ = end_T () in + concl + | Some (sigma, (X_In_T (hole, p) | In_X_In_T (hole, p))) -> + let p = fs sigma p in + let occ = match pattern with Some (_, X_In_T _) -> occ | _ -> noindex in + let ex = ex_value hole in + let rp = mk_upat_for ~hack:true env0 sigma0 (sigma, p) all_ok in + let find_T, end_T = mk_tpattern_matcher sigma0 noindex rp in + (* we start from sigma, so hole is considered a rigid head *) + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher ?raise_NoMatch sigma occ holep in + let concl = find_T env0 concl0 1 (fun env c _ h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h + (fun env _ -> do_subst env e_body))) in + let _ = end_X () in let _ = end_T () in + concl + | Some (sigma, E_In_X_In_T (e, hole, p)) -> + let p, e = fs sigma p, fs sigma e in + let ex = ex_value hole in + let rp = mk_upat_for ~hack:true env0 sigma0 (sigma, p) all_ok in + let find_T, end_T = mk_tpattern_matcher sigma0 noindex rp in + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher sigma noindex holep in + let re = mk_upat_for env0 sigma0 (sigma, e) all_ok in + let find_E, end_E = mk_tpattern_matcher ?raise_NoMatch sigma0 occ re in + let concl = find_T env0 concl0 1 (fun env c _ h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h (fun env c _ h -> + find_E env e_body h do_subst))) in + let _ = end_E () in let _ = end_X () in let _ = end_T () in + concl + | Some (sigma, E_As_X_In_T (e, hole, p)) -> + let p, e = fs sigma p, fs sigma e in + let ex = ex_value hole in + let rp = + let e_sigma = unify_HO env0 sigma hole e in + e_sigma, fs e_sigma p in + let rp = mk_upat_for ~hack:true env0 sigma0 rp all_ok in + let find_TE, end_TE = mk_tpattern_matcher sigma0 noindex rp in + let holep = mk_upat_for env0 sigma (sigma, hole) all_ok in + let find_X, end_X = mk_tpattern_matcher sigma occ holep in + let concl = find_TE env0 concl0 1 (fun env c _ h -> + let p_sigma = unify_HO env (create_evar_defs sigma) c p in + let sigma, e_body = pop_evar p_sigma ex p in + fs p_sigma (find_X env (fs sigma p) h (fun env c _ h -> + let e_sigma = unify_HO env sigma e_body e in + let e_body = fs e_sigma e in + do_subst env e_body e_body h))) in + let _ = end_X () in let _ = end_TE () in + concl +;; + +let redex_of_pattern ?(resolve_typeclasses=false) env (sigma, p) = + let e = match p with + | In_T _ | In_X_In_T _ -> Errors.anomaly (str"pattern without redex") + | T e | X_In_T (e, _) | E_As_X_In_T (e, _, _) | E_In_X_In_T (e, _, _) -> e in + let sigma = + if not resolve_typeclasses then sigma + else Typeclasses.resolve_typeclasses ~fail:false env sigma in + Reductionops.nf_evar sigma e, Evd.evar_universe_context sigma + +let fill_occ_pattern ?raise_NoMatch env sigma cl pat occ h = + let find_R, conclude = let r = ref None in + (fun env c _ h' -> do_once r (fun () -> c, Evd.empty_evar_universe_context); + mkRel (h'+h-1)), + (fun _ -> if !r = None then redex_of_pattern env pat else assert_done r) in + let cl = eval_pattern ?raise_NoMatch env sigma cl (Some pat) occ find_R in + let e = conclude cl in + e, cl +;; + +(* clenup interface for external use *) +let mk_tpattern ?p_origin env sigma0 sigma_t f dir c = + mk_tpattern ?p_origin env sigma0 sigma_t f dir c +;; + +let pf_fill_occ env concl occ sigma0 p (sigma, t) ok h = + let ise = create_evar_defs sigma in + let ise, u = mk_tpattern env sigma0 (ise,t) ok L2R p in + let find_U, end_U = + mk_tpattern_matcher ~raise_NoMatch:true sigma0 occ (ise,[u]) in + let concl = find_U env concl h (fun _ _ _ -> mkRel) in + let rdx, _, (sigma, uc, p) = end_U () in + sigma, uc, p, concl, rdx + +let fill_occ_term env cl occ sigma0 (sigma, t) = + try + let sigma',uc,t',cl,_= pf_fill_occ env cl occ sigma0 t (sigma, t) all_ok 1 in + if sigma' != sigma0 then Errors.error "matching impacts evars" + else cl, (Evd.merge_universe_context sigma' uc, t') + with NoMatch -> try + let sigma', uc, t' = + unif_end env sigma0 (create_evar_defs sigma) t (fun _ -> true) in + if sigma' != sigma0 then raise NoMatch + else cl, (Evd.merge_universe_context sigma' uc, t') + with _ -> + errorstrm (str "partial term " ++ pr_constr_pat t + ++ str " does not match any subterm of the goal") + +let pf_fill_occ_term gl occ t = + let sigma0 = project gl and env = pf_env gl and concl = pf_concl gl in + let cl,(_,t) = fill_occ_term env concl occ sigma0 t in + cl, t + +let cpattern_of_id id = ' ', (GRef (dummy_loc, VarRef id, None), None) + +let is_wildcard = function + | _,(_,Some (CHole _)|GHole _,None) -> true + | _ -> false + +(* "ssrpattern" *) +let pr_ssrpatternarg _ _ _ cpat = pr_rpattern cpat + +ARGUMENT EXTEND ssrpatternarg + TYPED AS rpattern + PRINTED BY pr_ssrpatternarg +| [ "[" rpattern(pat) "]" ] -> [ pat ] +END + +let pf_merge_uc uc gl = + re_sig (sig_it gl) (Evd.merge_universe_context (project gl) uc) + +let ssrpatterntac ist arg gl = + let pat = interp_rpattern ist gl arg in + let sigma0 = project gl in + let concl0 = pf_concl gl in + let (t, uc), concl_x = + fill_occ_pattern (Global.env()) sigma0 concl0 pat noindex 1 in + let tty = pf_type_of gl t in + let concl = mkLetIn (Name (id_of_string "toto"), t, tty, concl_x) in + Proofview.V82.of_tactic (convert_concl concl DEFAULTcast) gl + +TACTIC EXTEND ssrat +| [ "ssrpattern" ssrpatternarg(arg) ] -> [ Proofview.V82.tactic (ssrpatterntac ist arg) ] +END + +let ssrinstancesof ist arg gl = + let ok rhs lhs ise = true in +(* not (Term.eq_constr lhs (Evarutil.nf_evar ise rhs)) in *) + let env, sigma, concl = pf_env gl, project gl, pf_concl gl in + let sigma0, cpat = interp_cpattern ist gl arg None in + let pat = match cpat with T x -> x | _ -> errorstrm (str"Not supported") in + let etpat, tpat = mk_tpattern env sigma (sigma0,pat) (ok pat) L2R pat in + let find, conclude = + mk_tpattern_matcher ~all_instances:true ~raise_NoMatch:true + sigma None (etpat,[tpat]) in + let print env p c _ = ppnl (hov 1 (str"instance:" ++ spc() ++ pr_constr p ++ spc() ++ str "matches:" ++ spc() ++ pr_constr c)); c in + ppnl (str"BEGIN INSTANCES"); + try + while true do + ignore(find env concl 1 ~k:print) + done; raise NoMatch + with NoMatch -> ppnl (str"END INSTANCES"); tclIDTAC gl + +TACTIC EXTEND ssrinstoftpat +| [ "ssrinstancesoftpat" cpattern(arg) ] -> [ Proofview.V82.tactic (ssrinstancesof ist arg) ] +END + +(* We wipe out all the keywords generated by the grammar rules we defined. *) +(* The user is supposed to Require Import ssreflect or Require ssreflect *) +(* and Import ssreflect.SsrSyntax to obtain these keywords and as a *) +(* consequence the extended ssreflect grammar. *) +let () = Lexer.unfreeze frozen_lexer ;; + +(* vim: set filetype=ocaml foldmethod=marker: *) diff --git a/mathcomp/ssreflect/plugin/v8.5/ssrmatching.mli b/mathcomp/ssreflect/plugin/v8.5/ssrmatching.mli new file mode 100644 index 0000000..e8b4d81 --- /dev/null +++ b/mathcomp/ssreflect/plugin/v8.5/ssrmatching.mli @@ -0,0 +1,239 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) + +open Genarg +open Tacexpr +open Environ +open Evd +open Proof_type +open Term + +(** ******** Small Scale Reflection pattern matching facilities ************* *) + +(** Pattern parsing *) + +(** The type of context patterns, the patterns of the [set] tactic and + [:] tactical. These are patterns that identify a precise subterm. *) +type cpattern +val pr_cpattern : cpattern -> Pp.std_ppcmds + +(** CS cpattern: (f _), (X in t), (t in X in t), (t as X in t) *) +val cpattern : cpattern Pcoq.Gram.entry +val wit_cpattern : cpattern uniform_genarg_type + +(** OS cpattern: f _, (X in t), (t in X in t), (t as X in t) *) +val lcpattern : cpattern Pcoq.Gram.entry +val wit_lcpattern : cpattern uniform_genarg_type + +(** The type of rewrite patterns, the patterns of the [rewrite] tactic. + These patterns also include patterns that identify all the subterms + of a context (i.e. "in" prefix) *) +type rpattern +val pr_rpattern : rpattern -> Pp.std_ppcmds + +(** OS rpattern: f _, in t, X in t, in X in t, t in X in t, t as X in t *) +val rpattern : rpattern Pcoq.Gram.entry +val wit_rpattern : rpattern uniform_genarg_type + +(** Pattern interpretation and matching *) + +exception NoMatch +exception NoProgress + +(** AST for [rpattern] (and consequently [cpattern]) *) +type ('ident, 'term) ssrpattern = + | T of 'term + | In_T of 'term + | X_In_T of 'ident * 'term + | In_X_In_T of 'ident * 'term + | E_In_X_In_T of 'term * 'ident * 'term + | E_As_X_In_T of 'term * 'ident * 'term + +type pattern = evar_map * (constr, constr) ssrpattern +val pp_pattern : pattern -> Pp.std_ppcmds + +(** Extracts the redex and applies to it the substitution part of the pattern. + @raise Anomaly if called on [In_T] or [In_X_In_T] *) +val redex_of_pattern : + ?resolve_typeclasses:bool -> env -> pattern -> + constr Evd.in_evar_universe_context + +(** [interp_rpattern ise gl rpat] "internalizes" and "interprets" [rpat] + in the current [Ltac] interpretation signature [ise] and tactic input [gl]*) +val interp_rpattern : + Tacinterp.interp_sign -> goal Tacmach.sigma -> + rpattern -> + pattern + +(** [interp_cpattern ise gl cpat ty] "internalizes" and "interprets" [cpat] + in the current [Ltac] interpretation signature [ise] and tactic input [gl]. + [ty] is an optional type for the redex of [cpat] *) +val interp_cpattern : + Tacinterp.interp_sign -> goal Tacmach.sigma -> + cpattern -> glob_constr_and_expr option -> + pattern + +(** The set of occurrences to be matched. The boolean is set to true + * to signal the complement of this set (i.e. {-1 3}) *) +type occ = (bool * int list) option + +(** [subst e p t i]. [i] is the number of binders + traversed so far, [p] the term from the pattern, [t] the matched one *) +type subst = env -> constr -> constr -> int -> constr + +(** [eval_pattern b env sigma t pat occ subst] maps [t] calling [subst] on every + [occ] occurrence of [pat]. The [int] argument is the number of + binders traversed. If [pat] is [None] then then subst is called on [t]. + [t] must live in [env] and [sigma], [pat] must have been interpreted in + (an extension of) [sigma]. + @raise NoMatch if [pat] has no occurrence and [b] is [true] (default [false]) + @return [t] where all [occ] occurrences of [pat] have been mapped using + [subst] *) +val eval_pattern : + ?raise_NoMatch:bool -> + env -> evar_map -> constr -> + pattern option -> occ -> subst -> + constr + +(** [fill_occ_pattern b env sigma t pat occ h] is a simplified version of + [eval_pattern]. + It replaces all [occ] occurrences of [pat] in [t] with Rel [h]. + [t] must live in [env] and [sigma], [pat] must have been interpreted in + (an extension of) [sigma]. + @raise NoMatch if [pat] has no occurrence and [b] is [true] (default [false]) + @return the instance of the redex of [pat] that was matched and [t] + transformed as described above. *) +val fill_occ_pattern : + ?raise_NoMatch:bool -> + env -> evar_map -> constr -> + pattern -> occ -> int -> + constr Evd.in_evar_universe_context * constr + +(** *************************** Low level APIs ****************************** *) + +(* The primitive matching facility. It matches of a term with holes, like + the T pattern above, and calls a continuation on its occurrences. *) + +type ssrdir = L2R | R2L +val pr_dir_side : ssrdir -> Pp.std_ppcmds + +(** a pattern for a term with wildcards *) +type tpattern + +(** [mk_tpattern env sigma0 sigma_p ok p_origin dir t] compiles a term [t] + living in [env] [sigma] (an extension of [sigma0]) intro a [tpattern]. + The [tpattern] can hold a (proof) term [p] and a diction [dir]. The [ok] + callback is used to filter occurrences. + @return the compiled [tpattern] and its [evar_map] + @raise UserEerror is the pattern is a wildcard *) +val mk_tpattern : + ?p_origin:ssrdir * constr -> + env -> evar_map -> + evar_map * constr -> + (constr -> evar_map -> bool) -> + ssrdir -> constr -> + evar_map * tpattern + +(** [findP env t i k] is a stateful function that finds the next occurrence + of a tpattern and calls the callback [k] to map the subterm matched. + The [int] argument passed to [k] is the number of binders traversed so far + plus the initial value [i]. + @return [t] where the subterms identified by the selected occurrences of + the patter have been mapped using [k] + @raise NoMatch if the raise_NoMatch flag given to [mk_tpattern_matcher] is + [true] and if the pattern did not match + @raise UserEerror if the raise_NoMatch flag given to [mk_tpattern_matcher] is + [false] and if the pattern did not match *) +type find_P = + env -> constr -> int -> k:subst -> constr + +(** [conclude ()] asserts that all mentioned ocurrences have been visited. + @return the instance of the pattern, the evarmap after the pattern + instantiation, the proof term and the ssrdit stored in the tpattern + @raise UserEerror if too many occurrences were specified *) +type conclude = + unit -> constr * ssrdir * (evar_map * Evd.evar_universe_context * constr) + +(** [mk_tpattern_matcher b o sigma0 occ sigma_tplist] creates a pair + a function [find_P] and [conclude] with the behaviour explained above. + The flag [b] (default [false]) changes the error reporting behaviour + of [find_P] if none of the [tpattern] matches. The argument [o] can + be passed to tune the [UserError] eventually raised (useful if the + pattern is coming from the LHS/RHS of an equation) *) +val mk_tpattern_matcher : + ?all_instances:bool -> + ?raise_NoMatch:bool -> + ?upats_origin:ssrdir * constr -> + evar_map -> occ -> evar_map * tpattern list -> + find_P * conclude + +(** Example of [mk_tpattern_matcher] to implement + [rewrite \{occ\}\[in t\]rules]. + It first matches "in t" (called [pat]), then in all matched subterms + it matches the LHS of the rules using [find_R]. + [concl0] is the initial goal, [concl] will be the goal where some terms + are replaced by a De Bruijn index. The [rw_progress] extra check + selects only occurrences that are not rewritten to themselves (e.g. + an occurrence "x + x" rewritten with the commutativity law of addition + is skipped) {[ + let find_R, conclude = match pat with + | Some (_, In_T _) -> + let aux (sigma, pats) (d, r, lhs, rhs) = + let sigma, pat = + mk_tpattern env0 sigma0 (sigma, r) (rw_progress rhs) d lhs in + sigma, pats @ [pat] in + let rpats = List.fold_left aux (r_sigma, []) rules in + let find_R, end_R = mk_tpattern_matcher sigma0 occ rpats in + find_R ~k:(fun _ _ h -> mkRel h), + fun cl -> let rdx, d, r = end_R () in (d,r),rdx + | _ -> ... in + let concl = eval_pattern env0 sigma0 concl0 pat occ find_R in + let (d, r), rdx = conclude concl in ]} *) + +(* convenience shortcut: [pf_fill_occ_term gl occ (sigma,t)] returns + * the conclusion of [gl] where [occ] occurrences of [t] have been replaced + * by [Rel 1] and the instance of [t] *) +val pf_fill_occ_term : + goal Tacmach.sigma -> occ -> evar_map * constr -> constr * constr + +(* It may be handy to inject a simple term into the first form of cpattern *) +val cpattern_of_term : char * glob_constr_and_expr -> cpattern + +(** Helpers to make stateful closures. Example: a [find_P] function may be + called many times, but the pattern instantiation phase is performed only the + first time. The corresponding [conclude] has to return the instantiated + pattern redex. Since it is up to [find_P] to raise [NoMatch] if the pattern + has no instance, [conclude] considers it an anomaly if the pattern did + not match *) + +(** [do_once r f] calls [f] and updates the ref only once *) +val do_once : 'a option ref -> (unit -> 'a) -> unit +(** [assert_done r] return the content of r. @raise Anomaly is r is [None] *) +val assert_done : 'a option ref -> 'a + +(** Very low level APIs. + these are calls to evarconv's [the_conv_x] followed by + [consider_remaining_unif_problems] and [resolve_typeclasses]. + In case of failure they raise [NoMatch] *) + +val unify_HO : env -> evar_map -> constr -> constr -> evar_map +val pf_unify_HO : goal Tacmach.sigma -> constr -> constr -> goal Tacmach.sigma + +(** Some more low level functions needed to implement the full SSR language + on top of the former APIs *) +val tag_of_cpattern : cpattern -> char +val loc_of_cpattern : cpattern -> Loc.t +val id_of_cpattern : cpattern -> Names.variable option +val is_wildcard : cpattern -> bool +val cpattern_of_id : Names.variable -> cpattern +val cpattern_of_id : Names.variable -> cpattern +val pr_constr_pat : constr -> Pp.std_ppcmds +val pf_merge_uc : Evd.evar_universe_context -> goal Evd.sigma -> goal Evd.sigma + +(* One can also "Set SsrMatchingDebug" from a .v *) +val debug : bool -> unit + +(* One should delimit a snippet with "Set SsrMatchingProfiling" and + * "Unset SsrMatchingProfiling" to get timings *) +val profile : bool -> unit + +(* eof *) diff --git a/mathcomp/ssreflect/seq.v b/mathcomp/ssreflect/seq.v new file mode 100644 index 0000000..8522017 --- /dev/null +++ b/mathcomp/ssreflect/seq.v @@ -0,0 +1,2552 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Require Import ssreflect ssrfun ssrbool eqtype ssrnat. + +(******************************************************************************) +(* The seq type is the ssreflect type for sequences; it is an alias for the *) +(* standard Coq list type. The ssreflect library equips it with many *) +(* operations, as well as eqType and predType (and, later, choiceType) *) +(* structures. The operations are geared towards reflection: they generally *) +(* expect and provide boolean predicates, e.g., the membership predicate *) +(* expects an eqType. To avoid any confusion we do not Import the Coq List *) +(* module. *) +(* As there is no true subtyping in Coq, we don't use a type for non-empty *) +(* sequences; rather, we pass explicitly the head and tail of the sequence. *) +(* The empty sequence is especially bothersome for subscripting, since it *) +(* forces us to pass a default value. This default value can often be hidden *) +(* by a notation. *) +(* Here is the list of seq operations: *) +(* ** Constructors: *) +(* seq T == the type of sequences with items of type T *) +(* bitseq == seq bool *) +(* [::], nil, Nil T == the empty sequence (of type T) *) +(* x :: s, cons x s, Cons T x s == the sequence x followed by s (of type T) *) +(* [:: x] == the singleton sequence *) +(* [:: x_0; ...; x_n] == the explicit sequence of the x_i *) +(* [:: x_0, ..., x_n & s] == the sequence of the x_i, followed by s *) +(* rcons s x == the sequence s, followed by x *) +(* All of the above, except rcons, can be used in patterns. We define a view *) +(* lastP and an induction principle last_ind that can be used to decompose *) +(* or traverse a sequence in a right to left order. The view lemma lastP has *) +(* a dependent family type, so the ssreflect tactic case/lastP: p => [|p' x] *) +(* will generate two subgoals in which p has been replaced by [::] and by *) +(* rcons p' x, respectively. *) +(* ** Factories: *) +(* nseq n x == a sequence of n x's *) +(* ncons n x s == a sequence of n x's, followed by s *) +(* seqn n x_0 ... x_n-1 == the sequence of the x_i (can be partially applied) *) +(* iota m n == the sequence m, m + 1, ..., m + n - 1 *) +(* mkseq f n == the sequence f 0, f 1, ..., f (n - 1) *) +(* ** Sequential access: *) +(* head x0 s == the head (zero'th item) of s if s is non-empty, else x0 *) +(* ohead s == None if s is empty, else Some x where x is the head of s *) +(* behead s == s minus its head, i.e., s' if s = x :: s', else [::] *) +(* last x s == the last element of x :: s (which is non-empty) *) +(* belast x s == x :: s minus its last item *) +(* ** Dimensions: *) +(* size s == the number of items (length) in s *) +(* shape ss == the sequence of sizes of the items of the sequence of *) +(* sequences ss *) +(* ** Random access: *) +(* nth x0 s i == the item i of s (numbered from 0), or x0 if s does *) +(* not have at least i+1 items (i.e., size x <= i) *) +(* s`_i == standard notation for nth x0 s i for a default x0, *) +(* e.g., 0 for rings. *) +(* set_nth x0 s i y == s where item i has been changed to y; if s does not *) +(* have an item i, it is first padded with copies of x0 *) +(* to size i+1. *) +(* incr_nth s i == the nat sequence s with item i incremented (s is *) +(* first padded with 0's to size i+1, if needed). *) +(* ** Predicates: *) +(* nilp s == s is [::] *) +(* := (size s == 0) *) +(* x \in s == x appears in s (this requires an eqType for T) *) +(* index x s == the first index at which x appears in s, or size s if *) +(* x \notin s *) +(* has p s == the (applicative, boolean) predicate p holds for some *) +(* item in s *) +(* all p s == p holds for all items in s *) +(* find p s == the index of the first item in s for which p holds, *) +(* or size s if no such item is found *) +(* count p s == the number of items of s for which p holds *) +(* count_mem x s == the number of times x occurs in s := count (pred1 x) s *) +(* constant s == all items in s are identical (trivial if s = [::]) *) +(* uniq s == all the items in s are pairwise different *) +(* subseq s1 s2 == s1 is a subsequence of s2, i.e., s1 = mask m s2 for *) +(* some m : bitseq (see below). *) +(* perm_eq s1 s2 == s2 is a permutation of s1, i.e., s1 and s2 have the *) +(* items (with the same repetitions), but possibly in a *) +(* different order. *) +(* perm_eql s1 s2 <-> s1 and s2 behave identically on the left of perm_eq *) +(* perm_eqr s1 s2 <-> s1 and s2 behave identically on the rightt of perm_eq *) +(* These left/right transitive versions of perm_eq make it easier to chain *) +(* a sequence of equivalences. *) +(* ** Filtering: *) +(* filter p s == the subsequence of s consisting of all the items *) +(* for which the (boolean) predicate p holds *) +(* subfilter s : seq sT == when sT has a subType p structure, the sequence *) +(* of items of type sT corresponding to items of s *) +(* for which p holds. *) +(* rem x s == the subsequence of s, where the first occurrence *) +(* of x has been removed (compare filter (predC1 x) s *) +(* where ALL occurrences of x are removed). *) +(* undup s == the subsequence of s containing only the first *) +(* occurrence of each item in s, i.e., s with all *) +(* duplicates removed. *) +(* mask m s == the subsequence of s selected by m : bitseq, with *) +(* item i of s selected by bit i in m (extra items or *) +(* bits are ignored. *) +(* ** Surgery: *) +(* s1 ++ s2, cat s1 s2 == the concatenation of s1 and s2. *) +(* take n s == the sequence containing only the first n items of s *) +(* (or all of s if size s <= n). *) +(* drop n s == s minus its first n items ([::] if size s <= n) *) +(* rot n s == s rotated left n times (or s if size s <= n). *) +(* := drop n s ++ take n s *) +(* rotr n s == s rotated right n times (or s if size s <= n). *) +(* rev s == the (linear time) reversal of s. *) +(* catrev s1 s2 == the reversal of s1 followed by s2 (this is the *) +(* recursive form of rev). *) +(* ** Iterators: for s == [:: x_1, ..., x_n], t == [:: y_1, ..., y_m], *) +(* map f s == the sequence [:: f x_1, ..., f x_n]. *) +(* allpairs f s t == the sequence of all the f x y, with x and y drawn from *) +(* s and t, respectively, in row-major order. *) +(* := [:: f x_1 y_1; ...; f x_1 y_m; f x_2 y_1; ...; f x_n y_m] *) +(* pmap pf s == the sequence [:: y_i1, ..., y_ik] where i1 < ... < ik, *) +(* pf x_i = Some y_i, and pf x_j = None iff j is not in *) +(* {i1, ..., ik}. *) +(* foldr f a s == the right fold of s by f (i.e., the natural iterator). *) +(* := f x_1 (f x_2 ... (f x_n a)) *) +(* sumn s == x_1 + (x_2 + ... + (x_n + 0)) (when s : seq nat). *) +(* foldl f a s == the left fold of s by f. *) +(* := f (f ... (f a x_1) ... x_n-1) x_n *) +(* scanl f a s == the sequence of partial accumulators of foldl f a s. *) +(* := [:: f a x_1; ...; foldl f a s] *) +(* pairmap f a s == the sequence of f applied to consecutive items in a :: s. *) +(* := [:: f a x_1; f x_1 x_2; ...; f x_n-1 x_n] *) +(* zip s t == itemwise pairing of s and t (dropping any extra items). *) +(* := [:: (x_1, y_1); ...; (x_mn, y_mn)] with mn = minn n m. *) +(* unzip1 s == [:: (x_1).1; ...; (x_n).1] when s : seq (S * T). *) +(* unzip2 s == [:: (x_1).2; ...; (x_n).2] when s : seq (S * T). *) +(* flatten s == x_1 ++ ... ++ x_n ++ [::] when s : seq (seq T). *) +(* reshape r s == s reshaped into a sequence of sequences whose sizes are *) +(* given by r (truncating if s is too long or too short). *) +(* := [:: [:: x_1; ...; x_r1]; *) +(* [:: x_(r1 + 1); ...; x_(r0 + r1)]; *) +(* ...; *) +(* [:: x_(r1 + ... + r(k-1) + 1); ...; x_(r0 + ... rk)]] *) +(* ** Notation for manifest comprehensions: *) +(* [seq x <- s | C] := filter (fun x => C) s. *) +(* [seq E | x <- s] := map (fun x => E) s. *) +(* [seq E | x <- s, y <- t] := allpairs (fun x y => E) s t. *) +(* [seq x <- s | C1 & C2] := [seq x <- s | C1 && C2]. *) +(* [seq E | x <- s & C] := [seq E | x <- [seq x | C]]. *) +(* --> The above allow optional type casts on the eigenvariables, as in *) +(* [seq x : T <- s | C] or [seq E | x : T <- s, y : U <- t]. The cast may be *) +(* needed as type inference considers E or C before s. *) +(* We are quite systematic in providing lemmas to rewrite any composition *) +(* of two operations. "rev", whose simplifications are not natural, is *) +(* protected with nosimpl. *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Delimit Scope seq_scope with SEQ. +Open Scope seq_scope. + +(* Inductive seq (T : Type) : Type := Nil | Cons of T & seq T. *) +Notation seq := list. +Prenex Implicits cons. +Notation Cons T := (@cons T) (only parsing). +Notation Nil T := (@nil T) (only parsing). + +Bind Scope seq_scope with list. +Arguments Scope cons [type_scope _ seq_scope]. + +(* As :: and ++ are (improperly) declared in Init.datatypes, we only rebind *) +(* them here. *) +Infix "::" := cons : seq_scope. + +(* GG - this triggers a camlp4 warning, as if this Notation had been Reserved *) +Notation "[ :: ]" := nil (at level 0, format "[ :: ]") : seq_scope. + +Notation "[ :: x1 ]" := (x1 :: [::]) + (at level 0, format "[ :: x1 ]") : seq_scope. + +Notation "[ :: x & s ]" := (x :: s) (at level 0, only parsing) : seq_scope. + +Notation "[ :: x1 , x2 , .. , xn & s ]" := (x1 :: x2 :: .. (xn :: s) ..) + (at level 0, format + "'[hv' [ :: '[' x1 , '/' x2 , '/' .. , '/' xn ']' '/ ' & s ] ']'" + ) : seq_scope. + +Notation "[ :: x1 ; x2 ; .. ; xn ]" := (x1 :: x2 :: .. [:: xn] ..) + (at level 0, format "[ :: '[' x1 ; '/' x2 ; '/' .. ; '/' xn ']' ]" + ) : seq_scope. + +Section Sequences. + +Variable n0 : nat. (* numerical parameter for take, drop et al *) +Variable T : Type. (* must come before the implicit Type *) +Variable x0 : T. (* default for head/nth *) + +Implicit Types x y z : T. +Implicit Types m n : nat. +Implicit Type s : seq T. + +Fixpoint size s := if s is _ :: s' then (size s').+1 else 0. + +Lemma size0nil s : size s = 0 -> s = [::]. Proof. by case: s. Qed. + +Definition nilp s := size s == 0. + +Lemma nilP s : reflect (s = [::]) (nilp s). +Proof. by case: s => [|x s]; constructor. Qed. + +Definition ohead s := if s is x :: _ then Some x else None. +Definition head s := if s is x :: _ then x else x0. + +Definition behead s := if s is _ :: s' then s' else [::]. + +Lemma size_behead s : size (behead s) = (size s).-1. +Proof. by case: s. Qed. + +(* Factories *) + +Definition ncons n x := iter n (cons x). +Definition nseq n x := ncons n x [::]. + +Lemma size_ncons n x s : size (ncons n x s) = n + size s. +Proof. by elim: n => //= n ->. Qed. + +Lemma size_nseq n x : size (nseq n x) = n. +Proof. by rewrite size_ncons addn0. Qed. + +(* n-ary, dependently typed constructor. *) + +Fixpoint seqn_type n := if n is n'.+1 then T -> seqn_type n' else seq T. + +Fixpoint seqn_rec f n : seqn_type n := + if n is n'.+1 return seqn_type n then + fun x => seqn_rec (fun s => f (x :: s)) n' + else f [::]. +Definition seqn := seqn_rec id. + +(* Sequence catenation "cat". *) + +Fixpoint cat s1 s2 := if s1 is x :: s1' then x :: s1' ++ s2 else s2 +where "s1 ++ s2" := (cat s1 s2) : seq_scope. + +Lemma cat0s s : [::] ++ s = s. Proof. by []. Qed. +Lemma cat1s x s : [:: x] ++ s = x :: s. Proof. by []. Qed. +Lemma cat_cons x s1 s2 : (x :: s1) ++ s2 = x :: s1 ++ s2. Proof. by []. Qed. + +Lemma cat_nseq n x s : nseq n x ++ s = ncons n x s. +Proof. by elim: n => //= n ->. Qed. + +Lemma cats0 s : s ++ [::] = s. +Proof. by elim: s => //= x s ->. Qed. + +Lemma catA s1 s2 s3 : s1 ++ s2 ++ s3 = (s1 ++ s2) ++ s3. +Proof. by elim: s1 => //= x s1 ->. Qed. + +Lemma size_cat s1 s2 : size (s1 ++ s2) = size s1 + size s2. +Proof. by elim: s1 => //= x s1 ->. Qed. + +(* last, belast, rcons, and last induction. *) + +Fixpoint rcons s z := if s is x :: s' then x :: rcons s' z else [:: z]. + +Lemma rcons_cons x s z : rcons (x :: s) z = x :: rcons s z. +Proof. by []. Qed. + +Lemma cats1 s z : s ++ [:: z] = rcons s z. +Proof. by elim: s => //= x s ->. Qed. + +Fixpoint last x s := if s is x' :: s' then last x' s' else x. +Fixpoint belast x s := if s is x' :: s' then x :: (belast x' s') else [::]. + +Lemma lastI x s : x :: s = rcons (belast x s) (last x s). +Proof. by elim: s x => [|y s IHs] x //=; rewrite IHs. Qed. + +Lemma last_cons x y s : last x (y :: s) = last y s. +Proof. by []. Qed. + +Lemma size_rcons s x : size (rcons s x) = (size s).+1. +Proof. by rewrite -cats1 size_cat addnC. Qed. + +Lemma size_belast x s : size (belast x s) = size s. +Proof. by elim: s x => [|y s IHs] x //=; rewrite IHs. Qed. + +Lemma last_cat x s1 s2 : last x (s1 ++ s2) = last (last x s1) s2. +Proof. by elim: s1 x => [|y s1 IHs] x //=; rewrite IHs. Qed. + +Lemma last_rcons x s z : last x (rcons s z) = z. +Proof. by rewrite -cats1 last_cat. Qed. + +Lemma belast_cat x s1 s2 : + belast x (s1 ++ s2) = belast x s1 ++ belast (last x s1) s2. +Proof. by elim: s1 x => [|y s1 IHs] x //=; rewrite IHs. Qed. + +Lemma belast_rcons x s z : belast x (rcons s z) = x :: s. +Proof. by rewrite lastI -!cats1 belast_cat. Qed. + +Lemma cat_rcons x s1 s2 : rcons s1 x ++ s2 = s1 ++ x :: s2. +Proof. by rewrite -cats1 -catA. Qed. + +Lemma rcons_cat x s1 s2 : rcons (s1 ++ s2) x = s1 ++ rcons s2 x. +Proof. by rewrite -!cats1 catA. Qed. + +CoInductive last_spec : seq T -> Type := + | LastNil : last_spec [::] + | LastRcons s x : last_spec (rcons s x). + +Lemma lastP s : last_spec s. +Proof. case: s => [|x s]; [left | rewrite lastI; right]. Qed. + +Lemma last_ind P : + P [::] -> (forall s x, P s -> P (rcons s x)) -> forall s, P s. +Proof. +move=> Hnil Hlast s; rewrite -(cat0s s). +elim: s [::] Hnil => [|x s2 IHs] s1 Hs1; first by rewrite cats0. +by rewrite -cat_rcons; auto. +Qed. + +(* Sequence indexing. *) + +Fixpoint nth s n {struct n} := + if s is x :: s' then if n is n'.+1 then @nth s' n' else x else x0. + +Fixpoint set_nth s n y {struct n} := + if s is x :: s' then if n is n'.+1 then x :: @set_nth s' n' y else y :: s' + else ncons n x0 [:: y]. + +Lemma nth0 s : nth s 0 = head s. Proof. by []. Qed. + +Lemma nth_default s n : size s <= n -> nth s n = x0. +Proof. by elim: s n => [|x s IHs] [|n]; try exact (IHs n). Qed. + +Lemma nth_nil n : nth [::] n = x0. +Proof. by case: n. Qed. + +Lemma last_nth x s : last x s = nth (x :: s) (size s). +Proof. by elim: s x => [|y s IHs] x /=. Qed. + +Lemma nth_last s : nth s (size s).-1 = last x0 s. +Proof. by case: s => //= x s; rewrite last_nth. Qed. + +Lemma nth_behead s n : nth (behead s) n = nth s n.+1. +Proof. by case: s n => [|x s] [|n]. Qed. + +Lemma nth_cat s1 s2 n : + nth (s1 ++ s2) n = if n < size s1 then nth s1 n else nth s2 (n - size s1). +Proof. by elim: s1 n => [|x s1 IHs] [|n]; try exact (IHs n). Qed. + +Lemma nth_rcons s x n : + nth (rcons s x) n = + if n < size s then nth s n else if n == size s then x else x0. +Proof. by elim: s n => [|y s IHs] [|n] //=; rewrite ?nth_nil ?IHs. Qed. + +Lemma nth_ncons m x s n : + nth (ncons m x s) n = if n < m then x else nth s (n - m). +Proof. by elim: m n => [|m IHm] [|n] //=; exact: IHm. Qed. + +Lemma nth_nseq m x n : nth (nseq m x) n = (if n < m then x else x0). +Proof. by elim: m n => [|m IHm] [|n] //=; exact: IHm. Qed. + +Lemma eq_from_nth s1 s2 : + size s1 = size s2 -> (forall i, i < size s1 -> nth s1 i = nth s2 i) -> + s1 = s2. +Proof. +elim: s1 s2 => [|x1 s1 IHs1] [|x2 s2] //= [eq_sz] eq_s12. +rewrite [x1](eq_s12 0) // (IHs1 s2) // => i; exact: (eq_s12 i.+1). +Qed. + +Lemma size_set_nth s n y : size (set_nth s n y) = maxn n.+1 (size s). +Proof. +elim: s n => [|x s IHs] [|n] //=. +- by rewrite size_ncons addn1 maxn0. +- by rewrite maxnE subn1. +by rewrite IHs -add1n addn_maxr. +Qed. + +Lemma set_nth_nil n y : set_nth [::] n y = ncons n x0 [:: y]. +Proof. by case: n. Qed. + +Lemma nth_set_nth s n y : nth (set_nth s n y) =1 [eta nth s with n |-> y]. +Proof. +elim: s n => [|x s IHs] [|n] [|m] //=; rewrite ?nth_nil ?IHs // nth_ncons eqSS. +case: ltngtP => // [lt_nm | ->]; last by rewrite subnn. +by rewrite nth_default // subn_gt0. +Qed. + +Lemma set_set_nth s n1 y1 n2 y2 (s2 := set_nth s n2 y2) : + set_nth (set_nth s n1 y1) n2 y2 = if n1 == n2 then s2 else set_nth s2 n1 y1. +Proof. +have [-> | ne_n12] := altP eqP. + apply: eq_from_nth => [|i _]; first by rewrite !size_set_nth maxnA maxnn. + by do 2!rewrite !nth_set_nth /=; case: eqP. +apply: eq_from_nth => [|i _]; first by rewrite !size_set_nth maxnCA. +do 2!rewrite !nth_set_nth /=; case: eqP => // ->. +by rewrite eq_sym -if_neg ne_n12. +Qed. + +(* find, count, has, all. *) + +Section SeqFind. + +Variable a : pred T. + +Fixpoint find s := if s is x :: s' then if a x then 0 else (find s').+1 else 0. + +Fixpoint filter s := + if s is x :: s' then if a x then x :: filter s' else filter s' else [::]. + +Fixpoint count s := if s is x :: s' then a x + count s' else 0. + +Fixpoint has s := if s is x :: s' then a x || has s' else false. + +Fixpoint all s := if s is x :: s' then a x && all s' else true. + +Lemma size_filter s : size (filter s) = count s. +Proof. by elim: s => //= x s <-; case (a x). Qed. + +Lemma has_count s : has s = (0 < count s). +Proof. by elim: s => //= x s ->; case (a x). Qed. + +Lemma count_size s : count s <= size s. +Proof. by elim: s => //= x s; case: (a x); last exact: leqW. Qed. + +Lemma all_count s : all s = (count s == size s). +Proof. +elim: s => //= x s; case: (a x) => _ //=. +by rewrite add0n eqn_leq andbC ltnNge count_size. +Qed. + +Lemma filter_all s : all (filter s). +Proof. by elim: s => //= x s IHs; case: ifP => //= ->. Qed. + +Lemma all_filterP s : reflect (filter s = s) (all s). +Proof. +apply: (iffP idP) => [| <-]; last exact: filter_all. +by elim: s => //= x s IHs /andP[-> Hs]; rewrite IHs. +Qed. + +Lemma filter_id s : filter (filter s) = filter s. +Proof. by apply/all_filterP; exact: filter_all. Qed. + +Lemma has_find s : has s = (find s < size s). +Proof. by elim: s => //= x s IHs; case (a x); rewrite ?leqnn. Qed. + +Lemma find_size s : find s <= size s. +Proof. by elim: s => //= x s IHs; case (a x). Qed. + +Lemma find_cat s1 s2 : + find (s1 ++ s2) = if has s1 then find s1 else size s1 + find s2. +Proof. +by elim: s1 => //= x s1 IHs; case: (a x) => //; rewrite IHs (fun_if succn). +Qed. + +Lemma has_nil : has [::] = false. Proof. by []. Qed. + +Lemma has_seq1 x : has [:: x] = a x. +Proof. exact: orbF. Qed. + +Lemma has_nseq n x : has (nseq n x) = (0 < n) && a x. +Proof. by elim: n => //= n ->; apply: andKb. Qed. + +Lemma has_seqb (b : bool) x : has (nseq b x) = b && a x. +Proof. by rewrite has_nseq lt0b. Qed. + +Lemma all_nil : all [::] = true. Proof. by []. Qed. + +Lemma all_seq1 x : all [:: x] = a x. +Proof. exact: andbT. Qed. + +Lemma all_nseq n x : all (nseq n x) = (n == 0) || a x. +Proof. by elim: n => //= n ->; apply: orKb. Qed. + +Lemma all_nseqb (b : bool) x : all (nseq b x) = b ==> a x. +Proof. by rewrite all_nseq eqb0 implybE. Qed. + +Lemma find_nseq n x : find (nseq n x) = ~~ a x * n. +Proof. by elim: n => //= n ->; case: (a x). Qed. + +Lemma nth_find s : has s -> a (nth s (find s)). +Proof. by elim: s => //= x s IHs; case Hx: (a x). Qed. + +Lemma before_find s i : i < find s -> a (nth s i) = false. +Proof. +by elim: s i => //= x s IHs; case Hx: (a x) => [|] // [|i] //; apply: (IHs i). +Qed. + +Lemma filter_cat s1 s2 : filter (s1 ++ s2) = filter s1 ++ filter s2. +Proof. by elim: s1 => //= x s1 ->; case (a x). Qed. + +Lemma filter_rcons s x : + filter (rcons s x) = if a x then rcons (filter s) x else filter s. +Proof. by rewrite -!cats1 filter_cat /=; case (a x); rewrite /= ?cats0. Qed. + +Lemma count_cat s1 s2 : count (s1 ++ s2) = count s1 + count s2. +Proof. by rewrite -!size_filter filter_cat size_cat. Qed. + +Lemma has_cat s1 s2 : has (s1 ++ s2) = has s1 || has s2. +Proof. by elim: s1 => [|x s1 IHs] //=; rewrite IHs orbA. Qed. + +Lemma has_rcons s x : has (rcons s x) = a x || has s. +Proof. by rewrite -cats1 has_cat has_seq1 orbC. Qed. + +Lemma all_cat s1 s2 : all (s1 ++ s2) = all s1 && all s2. +Proof. by elim: s1 => [|x s1 IHs] //=; rewrite IHs andbA. Qed. + +Lemma all_rcons s x : all (rcons s x) = a x && all s. +Proof. by rewrite -cats1 all_cat all_seq1 andbC. Qed. + +End SeqFind. + +Lemma eq_find a1 a2 : a1 =1 a2 -> find a1 =1 find a2. +Proof. by move=> Ea; elim=> //= x s IHs; rewrite Ea IHs. Qed. + +Lemma eq_filter a1 a2 : a1 =1 a2 -> filter a1 =1 filter a2. +Proof. by move=> Ea; elim=> //= x s IHs; rewrite Ea IHs. Qed. + +Lemma eq_count a1 a2 : a1 =1 a2 -> count a1 =1 count a2. +Proof. by move=> Ea s; rewrite -!size_filter (eq_filter Ea). Qed. + +Lemma eq_has a1 a2 : a1 =1 a2 -> has a1 =1 has a2. +Proof. by move=> Ea s; rewrite !has_count (eq_count Ea). Qed. + +Lemma eq_all a1 a2 : a1 =1 a2 -> all a1 =1 all a2. +Proof. by move=> Ea s; rewrite !all_count (eq_count Ea). Qed. + +Section SubPred. + +Variable (a1 a2 : pred T). +Hypothesis s12 : subpred a1 a2. + +Lemma sub_find s : find a2 s <= find a1 s. +Proof. by elim: s => //= x s IHs; case: ifP => // /(contraFF (@s12 x))->. Qed. + +Lemma sub_has s : has a1 s -> has a2 s. +Proof. by rewrite !has_find; exact: leq_ltn_trans (sub_find s). Qed. + +Lemma sub_count s : count a1 s <= count a2 s. +Proof. +by elim: s => //= x s; apply: leq_add; case a1x: (a1 x); rewrite // s12. +Qed. + +Lemma sub_all s : all a1 s -> all a2 s. +Proof. +by rewrite !all_count !eqn_leq !count_size => /leq_trans-> //; apply: sub_count. +Qed. + +End SubPred. + +Lemma filter_pred0 s : filter pred0 s = [::]. Proof. by elim: s. Qed. + +Lemma filter_predT s : filter predT s = s. +Proof. by elim: s => //= x s ->. Qed. + +Lemma filter_predI a1 a2 s : filter (predI a1 a2) s = filter a1 (filter a2 s). +Proof. +elim: s => //= x s IHs; rewrite andbC IHs. +by case: (a2 x) => //=; case (a1 x). +Qed. + +Lemma count_pred0 s : count pred0 s = 0. +Proof. by rewrite -size_filter filter_pred0. Qed. + +Lemma count_predT s : count predT s = size s. +Proof. by rewrite -size_filter filter_predT. Qed. + +Lemma count_predUI a1 a2 s : + count (predU a1 a2) s + count (predI a1 a2) s = count a1 s + count a2 s. +Proof. +elim: s => //= x s IHs; rewrite /= addnCA -addnA IHs addnA addnC. +by rewrite -!addnA; do 2 nat_congr; case (a1 x); case (a2 x). +Qed. + +Lemma count_predC a s : count a s + count (predC a) s = size s. +Proof. +by elim: s => //= x s IHs; rewrite addnCA -addnA IHs addnA addn_negb. +Qed. + +Lemma count_filter a1 a2 s : count a1 (filter a2 s) = count (predI a1 a2) s. +Proof. by rewrite -!size_filter filter_predI. Qed. + +Lemma has_pred0 s : has pred0 s = false. +Proof. by rewrite has_count count_pred0. Qed. + +Lemma has_predT s : has predT s = (0 < size s). +Proof. by rewrite has_count count_predT. Qed. + +Lemma has_predC a s : has (predC a) s = ~~ all a s. +Proof. by elim: s => //= x s ->; case (a x). Qed. + +Lemma has_predU a1 a2 s : has (predU a1 a2) s = has a1 s || has a2 s. +Proof. by elim: s => //= x s ->; rewrite -!orbA; do !bool_congr. Qed. + +Lemma all_pred0 s : all pred0 s = (size s == 0). +Proof. by rewrite all_count count_pred0 eq_sym. Qed. + +Lemma all_predT s : all predT s. +Proof. by rewrite all_count count_predT. Qed. + +Lemma all_predC a s : all (predC a) s = ~~ has a s. +Proof. by elim: s => //= x s ->; case (a x). Qed. + +Lemma all_predI a1 a2 s : all (predI a1 a2) s = all a1 s && all a2 s. +Proof. +apply: (can_inj negbK); rewrite negb_and -!has_predC -has_predU. +by apply: eq_has => x; rewrite /= negb_and. +Qed. + +(* Surgery: drop, take, rot, rotr. *) + +Fixpoint drop n s {struct s} := + match s, n with + | _ :: s', n'.+1 => drop n' s' + | _, _ => s + end. + +Lemma drop_behead : drop n0 =1 iter n0 behead. +Proof. by elim: n0 => [|n IHn] [|x s] //; rewrite iterSr -IHn. Qed. + +Lemma drop0 s : drop 0 s = s. Proof. by case: s. Qed. + +Lemma drop1 : drop 1 =1 behead. Proof. by case=> [|x [|y s]]. Qed. + +Lemma drop_oversize n s : size s <= n -> drop n s = [::]. +Proof. by elim: n s => [|n IHn] [|x s]; try exact (IHn s). Qed. + +Lemma drop_size s : drop (size s) s = [::]. +Proof. by rewrite drop_oversize // leqnn. Qed. + +Lemma drop_cons x s : + drop n0 (x :: s) = if n0 is n.+1 then drop n s else x :: s. +Proof. by []. Qed. + +Lemma size_drop s : size (drop n0 s) = size s - n0. +Proof. by elim: s n0 => [|x s IHs] [|n]; try exact (IHs n). Qed. + +Lemma drop_cat s1 s2 : + drop n0 (s1 ++ s2) = + if n0 < size s1 then drop n0 s1 ++ s2 else drop (n0 - size s1) s2. +Proof. by elim: s1 n0 => [|x s1 IHs] [|n]; try exact (IHs n). Qed. + +Lemma drop_size_cat n s1 s2 : size s1 = n -> drop n (s1 ++ s2) = s2. +Proof. by move <-; elim: s1 => //=; rewrite drop0. Qed. + +Lemma nconsK n x : cancel (ncons n x) (drop n). +Proof. by elim: n => //; case. Qed. + +Fixpoint take n s {struct s} := + match s, n with + | x :: s', n'.+1 => x :: take n' s' + | _, _ => [::] + end. + +Lemma take0 s : take 0 s = [::]. Proof. by case: s. Qed. + +Lemma take_oversize n s : size s <= n -> take n s = s. +Proof. by elim: n s => [|n IHn] [|x s] Hsn; try (congr cons; apply: IHn). Qed. + +Lemma take_size s : take (size s) s = s. +Proof. by rewrite take_oversize // leqnn. Qed. + +Lemma take_cons x s : + take n0 (x :: s) = if n0 is n.+1 then x :: (take n s) else [::]. +Proof. by []. Qed. + +Lemma drop_rcons s : n0 <= size s -> + forall x, drop n0 (rcons s x) = rcons (drop n0 s) x. +Proof. by elim: s n0 => [|y s IHs] [|n]; try exact (IHs n). Qed. + +Lemma cat_take_drop s : take n0 s ++ drop n0 s = s. +Proof. by elim: s n0 => [|x s IHs] [|n]; try exact (congr1 _ (IHs n)). Qed. + +Lemma size_takel s : n0 <= size s -> size (take n0 s) = n0. +Proof. +by move/subnKC; rewrite -{2}(cat_take_drop s) size_cat size_drop => /addIn. +Qed. + +Lemma size_take s : size (take n0 s) = if n0 < size s then n0 else size s. +Proof. +have [le_sn | lt_ns] := leqP (size s) n0; first by rewrite take_oversize. +by rewrite size_takel // ltnW. +Qed. + +Lemma take_cat s1 s2 : + take n0 (s1 ++ s2) = + if n0 < size s1 then take n0 s1 else s1 ++ take (n0 - size s1) s2. +Proof. +elim: s1 n0 => [|x s1 IHs] [|n] //=. +by rewrite ltnS subSS -(fun_if (cons x)) -IHs. +Qed. + +Lemma take_size_cat n s1 s2 : size s1 = n -> take n (s1 ++ s2) = s1. +Proof. by move <-; elim: s1 => [|x s1 IHs]; rewrite ?take0 //= IHs. Qed. + +Lemma takel_cat s1 : + n0 <= size s1 -> + forall s2, take n0 (s1 ++ s2) = take n0 s1. +Proof. +move=> Hn0 s2; rewrite take_cat ltn_neqAle Hn0 andbT. +by case: (n0 =P size s1) => //= ->; rewrite subnn take0 cats0 take_size. +Qed. + +Lemma nth_drop s i : nth (drop n0 s) i = nth s (n0 + i). +Proof. +have [lt_n0_s | le_s_n0] := ltnP n0 (size s). + rewrite -{2}[s]cat_take_drop nth_cat size_take lt_n0_s /= addKn. + by rewrite ltnNge leq_addr. +rewrite !nth_default //; first exact: leq_trans (leq_addr _ _). +by rewrite size_drop (eqnP le_s_n0). +Qed. + +Lemma nth_take i : i < n0 -> forall s, nth (take n0 s) i = nth s i. +Proof. +move=> lt_i_n0 s; case lt_n0_s: (n0 < size s). + by rewrite -{2}[s]cat_take_drop nth_cat size_take lt_n0_s /= lt_i_n0. +by rewrite -{1}[s]cats0 take_cat lt_n0_s /= cats0. +Qed. + +(* drop_nth and take_nth below do NOT use the default n0, because the "n" *) +(* can be inferred from the condition, whereas the nth default value x0 *) +(* will have to be given explicitly (and this will provide "d" as well). *) + +Lemma drop_nth n s : n < size s -> drop n s = nth s n :: drop n.+1 s. +Proof. by elim: s n => [|x s IHs] [|n] Hn //=; rewrite ?drop0 1?IHs. Qed. + +Lemma take_nth n s : n < size s -> take n.+1 s = rcons (take n s) (nth s n). +Proof. by elim: s n => [|x s IHs] //= [|n] Hn /=; rewrite ?take0 -?IHs. Qed. + +(* Rotation *) + +Definition rot n s := drop n s ++ take n s. + +Lemma rot0 s : rot 0 s = s. +Proof. by rewrite /rot drop0 take0 cats0. Qed. + +Lemma size_rot s : size (rot n0 s) = size s. +Proof. by rewrite -{2}[s]cat_take_drop /rot !size_cat addnC. Qed. + +Lemma rot_oversize n s : size s <= n -> rot n s = s. +Proof. by move=> le_s_n; rewrite /rot take_oversize ?drop_oversize. Qed. + +Lemma rot_size s : rot (size s) s = s. +Proof. exact: rot_oversize. Qed. + +Lemma has_rot s a : has a (rot n0 s) = has a s. +Proof. by rewrite has_cat orbC -has_cat cat_take_drop. Qed. + +Lemma rot_size_cat s1 s2 : rot (size s1) (s1 ++ s2) = s2 ++ s1. +Proof. by rewrite /rot take_size_cat ?drop_size_cat. Qed. + +Definition rotr n s := rot (size s - n) s. + +Lemma rotK : cancel (rot n0) (rotr n0). +Proof. +move=> s; rewrite /rotr size_rot -size_drop {2}/rot. +by rewrite rot_size_cat cat_take_drop. +Qed. + +Lemma rot_inj : injective (rot n0). Proof. exact (can_inj rotK). Qed. + +Lemma rot1_cons x s : rot 1 (x :: s) = rcons s x. +Proof. by rewrite /rot /= take0 drop0 -cats1. Qed. + +(* (efficient) reversal *) + +Fixpoint catrev s1 s2 := if s1 is x :: s1' then catrev s1' (x :: s2) else s2. + +End Sequences. + +(* rev must be defined outside a Section because Coq's end of section *) +(* "cooking" removes the nosimpl guard. *) + +Definition rev T (s : seq T) := nosimpl (catrev s [::]). + +Implicit Arguments nilP [T s]. +Implicit Arguments all_filterP [T a s]. + +Prenex Implicits size nilP head ohead behead last rcons belast. +Prenex Implicits cat take drop rev rot rotr. +Prenex Implicits find count nth all has filter all_filterP. + +Notation count_mem x := (count (pred_of_simpl (pred1 x))). + +Infix "++" := cat : seq_scope. + +Notation "[ 'seq' x <- s | C ]" := (filter (fun x => C%B) s) + (at level 0, x at level 99, + format "[ '[hv' 'seq' x <- s '/ ' | C ] ']'") : seq_scope. +Notation "[ 'seq' x <- s | C1 & C2 ]" := [seq x <- s | C1 && C2] + (at level 0, x at level 99, + format "[ '[hv' 'seq' x <- s '/ ' | C1 '/ ' & C2 ] ']'") : seq_scope. +Notation "[ 'seq' x : T <- s | C ]" := (filter (fun x : T => C%B) s) + (at level 0, x at level 99, only parsing). +Notation "[ 'seq' x : T <- s | C1 & C2 ]" := [seq x : T <- s | C1 && C2] + (at level 0, x at level 99, only parsing). + + +(* Double induction/recursion. *) +Lemma seq2_ind T1 T2 (P : seq T1 -> seq T2 -> Type) : + P [::] [::] -> (forall x1 x2 s1 s2, P s1 s2 -> P (x1 :: s1) (x2 :: s2)) -> + forall s1 s2, size s1 = size s2 -> P s1 s2. +Proof. by move=> Pnil Pcons; elim=> [|x s IHs] [] //= x2 s2 [] /IHs/Pcons. Qed. + +Section Rev. + +Variable T : Type. +Implicit Types s t : seq T. + +Lemma catrev_catl s t u : catrev (s ++ t) u = catrev t (catrev s u). +Proof. by elim: s u => /=. Qed. + +Lemma catrev_catr s t u : catrev s (t ++ u) = catrev s t ++ u. +Proof. by elim: s t => //= x s IHs t; rewrite -IHs. Qed. + +Lemma catrevE s t : catrev s t = rev s ++ t. +Proof. by rewrite -catrev_catr. Qed. + +Lemma rev_cons x s : rev (x :: s) = rcons (rev s) x. +Proof. by rewrite -cats1 -catrevE. Qed. + +Lemma size_rev s : size (rev s) = size s. +Proof. by elim: s => // x s IHs; rewrite rev_cons size_rcons IHs. Qed. + +Lemma rev_cat s t : rev (s ++ t) = rev t ++ rev s. +Proof. by rewrite -catrev_catr -catrev_catl. Qed. + +Lemma rev_rcons s x : rev (rcons s x) = x :: rev s. +Proof. by rewrite -cats1 rev_cat. Qed. + +Lemma revK : involutive (@rev T). +Proof. by elim=> //= x s IHs; rewrite rev_cons rev_rcons IHs. Qed. + +Lemma nth_rev x0 n s : + n < size s -> nth x0 (rev s) n = nth x0 s (size s - n.+1). +Proof. +elim/last_ind: s => // s x IHs in n *. +rewrite rev_rcons size_rcons ltnS subSS -cats1 nth_cat /=. +case: n => [|n] lt_n_s; first by rewrite subn0 ltnn subnn. +by rewrite -{2}(subnK lt_n_s) -addSnnS leq_addr /= -IHs. +Qed. + +Lemma filter_rev a s : filter a (rev s) = rev (filter a s). +Proof. by elim: s => //= x s IH; rewrite fun_if !rev_cons filter_rcons IH. Qed. + +Lemma count_rev a s : count a (rev s) = count a s. +Proof. by rewrite -!size_filter filter_rev size_rev. Qed. + +Lemma has_rev a s : has a (rev s) = has a s. +Proof. by rewrite !has_count count_rev. Qed. + +Lemma all_rev a s : all a (rev s) = all a s. +Proof. by rewrite !all_count count_rev size_rev. Qed. + +End Rev. + +Implicit Arguments revK [[T]]. + +(* Equality and eqType for seq. *) + +Section EqSeq. + +Variables (n0 : nat) (T : eqType) (x0 : T). +Notation Local nth := (nth x0). +Implicit Type s : seq T. +Implicit Types x y z : T. + +Fixpoint eqseq s1 s2 {struct s2} := + match s1, s2 with + | [::], [::] => true + | x1 :: s1', x2 :: s2' => (x1 == x2) && eqseq s1' s2' + | _, _ => false + end. + +Lemma eqseqP : Equality.axiom eqseq. +Proof. +move; elim=> [|x1 s1 IHs] [|x2 s2]; do [by constructor | simpl]. +case: (x1 =P x2) => [<-|neqx]; last by right; case. +by apply: (iffP (IHs s2)) => [<-|[]]. +Qed. + +Canonical seq_eqMixin := EqMixin eqseqP. +Canonical seq_eqType := Eval hnf in EqType (seq T) seq_eqMixin. + +Lemma eqseqE : eqseq = eq_op. Proof. by []. Qed. + +Lemma eqseq_cons x1 x2 s1 s2 : + (x1 :: s1 == x2 :: s2) = (x1 == x2) && (s1 == s2). +Proof. by []. Qed. + +Lemma eqseq_cat s1 s2 s3 s4 : + size s1 = size s2 -> (s1 ++ s3 == s2 ++ s4) = (s1 == s2) && (s3 == s4). +Proof. +elim: s1 s2 => [|x1 s1 IHs] [|x2 s2] //= [sz12]. +by rewrite !eqseq_cons -andbA IHs. +Qed. + +Lemma eqseq_rcons s1 s2 x1 x2 : + (rcons s1 x1 == rcons s2 x2) = (s1 == s2) && (x1 == x2). +Proof. by rewrite -(can_eq revK) !rev_rcons eqseq_cons andbC (can_eq revK). Qed. + +Lemma size_eq0 s : (size s == 0) = (s == [::]). +Proof. exact: (sameP nilP eqP). Qed. + +Lemma has_filter a s : has a s = (filter a s != [::]). +Proof. by rewrite -size_eq0 size_filter has_count lt0n. Qed. + +(* mem_seq and index. *) +(* mem_seq defines a predType for seq. *) + +Fixpoint mem_seq (s : seq T) := + if s is y :: s' then xpredU1 y (mem_seq s') else xpred0. + +Definition eqseq_class := seq T. +Identity Coercion seq_of_eqseq : eqseq_class >-> seq. + +Coercion pred_of_eq_seq (s : eqseq_class) : pred_class := [eta mem_seq s]. + +Canonical seq_predType := @mkPredType T (seq T) pred_of_eq_seq. +(* The line below makes mem_seq a canonical instance of topred. *) +Canonical mem_seq_predType := mkPredType mem_seq. + +Lemma in_cons y s x : (x \in y :: s) = (x == y) || (x \in s). +Proof. by []. Qed. + +Lemma in_nil x : (x \in [::]) = false. +Proof. by []. Qed. + +Lemma mem_seq1 x y : (x \in [:: y]) = (x == y). +Proof. by rewrite in_cons orbF. Qed. + + (* to be repeated after the Section discharge. *) +Let inE := (mem_seq1, in_cons, inE). + +Lemma mem_seq2 x y1 y2 : (x \in [:: y1; y2]) = xpred2 y1 y2 x. +Proof. by rewrite !inE. Qed. + +Lemma mem_seq3 x y1 y2 y3 : (x \in [:: y1; y2; y3]) = xpred3 y1 y2 y3 x. +Proof. by rewrite !inE. Qed. + +Lemma mem_seq4 x y1 y2 y3 y4 : + (x \in [:: y1; y2; y3; y4]) = xpred4 y1 y2 y3 y4 x. +Proof. by rewrite !inE. Qed. + +Lemma mem_cat x s1 s2 : (x \in s1 ++ s2) = (x \in s1) || (x \in s2). +Proof. by elim: s1 => //= y s1 IHs; rewrite !inE /= -orbA -IHs. Qed. + +Lemma mem_rcons s y : rcons s y =i y :: s. +Proof. by move=> x; rewrite -cats1 /= mem_cat mem_seq1 orbC in_cons. Qed. + +Lemma mem_head x s : x \in x :: s. +Proof. exact: predU1l. Qed. + +Lemma mem_last x s : last x s \in x :: s. +Proof. by rewrite lastI mem_rcons mem_head. Qed. + +Lemma mem_behead s : {subset behead s <= s}. +Proof. by case: s => // y s x; exact: predU1r. Qed. + +Lemma mem_belast s y : {subset belast y s <= y :: s}. +Proof. by move=> x ys'x; rewrite lastI mem_rcons mem_behead. Qed. + +Lemma mem_nth s n : n < size s -> nth s n \in s. +Proof. +by elim: s n => [|x s IHs] // [_|n sz_s]; rewrite ?mem_head // mem_behead ?IHs. +Qed. + +Lemma mem_take s x : x \in take n0 s -> x \in s. +Proof. by move=> s0x; rewrite -(cat_take_drop n0 s) mem_cat /= s0x. Qed. + +Lemma mem_drop s x : x \in drop n0 s -> x \in s. +Proof. by move=> s0'x; rewrite -(cat_take_drop n0 s) mem_cat /= s0'x orbT. Qed. + +Section Filters. + +Variable a : pred T. + +Lemma hasP s : reflect (exists2 x, x \in s & a x) (has a s). +Proof. +elim: s => [|y s IHs] /=; first by right; case. +case ay: (a y); first by left; exists y; rewrite ?mem_head. +apply: (iffP IHs) => [] [x ysx ax]; exists x => //; first exact: mem_behead. +by case: (predU1P ysx) ax => [->|//]; rewrite ay. +Qed. + +Lemma hasPn s : reflect (forall x, x \in s -> ~~ a x) (~~ has a s). +Proof. +apply: (iffP idP) => not_a_s => [x s_x|]. + by apply: contra not_a_s => a_x; apply/hasP; exists x. +by apply/hasP=> [[x s_x]]; apply/negP; exact: not_a_s. +Qed. + +Lemma allP s : reflect (forall x, x \in s -> a x) (all a s). +Proof. +elim: s => [|x s IHs]; first by left. +rewrite /= andbC; case: IHs => IHs /=. + apply: (iffP idP) => [Hx y|]; last by apply; exact: mem_head. + by case/predU1P=> [->|Hy]; auto. +by right=> H; case IHs => y Hy; apply H; exact: mem_behead. +Qed. + +Lemma allPn s : reflect (exists2 x, x \in s & ~~ a x) (~~ all a s). +Proof. +elim: s => [|x s IHs]; first by right=> [[x Hx _]]. +rewrite /= andbC negb_and; case: IHs => IHs /=. + by left; case: IHs => y Hy Hay; exists y; first exact: mem_behead. +apply: (iffP idP) => [|[y]]; first by exists x; rewrite ?mem_head. +by case/predU1P=> [-> // | s_y not_a_y]; case: IHs; exists y. +Qed. + +Lemma mem_filter x s : (x \in filter a s) = a x && (x \in s). +Proof. +rewrite andbC; elim: s => //= y s IHs. +rewrite (fun_if (fun s' : seq T => x \in s')) !in_cons {}IHs. +by case: eqP => [->|_]; case (a y); rewrite /= ?andbF. +Qed. + +End Filters. + +Section EqIn. + +Variables a1 a2 : pred T. + +Lemma eq_in_filter s : {in s, a1 =1 a2} -> filter a1 s = filter a2 s. +Proof. +elim: s => //= x s IHs eq_a. +rewrite eq_a ?mem_head ?IHs // => y s_y; apply: eq_a; exact: mem_behead. +Qed. + +Lemma eq_in_find s : {in s, a1 =1 a2} -> find a1 s = find a2 s. +Proof. +elim: s => //= x s IHs eq_a12; rewrite eq_a12 ?mem_head // IHs // => y s'y. +by rewrite eq_a12 // mem_behead. +Qed. + +Lemma eq_in_count s : {in s, a1 =1 a2} -> count a1 s = count a2 s. +Proof. by move/eq_in_filter=> eq_a12; rewrite -!size_filter eq_a12. Qed. + +Lemma eq_in_all s : {in s, a1 =1 a2} -> all a1 s = all a2 s. +Proof. by move=> eq_a12; rewrite !all_count eq_in_count. Qed. + +Lemma eq_in_has s : {in s, a1 =1 a2} -> has a1 s = has a2 s. +Proof. by move/eq_in_filter=> eq_a12; rewrite !has_filter eq_a12. Qed. + +End EqIn. + +Lemma eq_has_r s1 s2 : s1 =i s2 -> has^~ s1 =1 has^~ s2. +Proof. +move=> Es12 a; apply/(hasP a s1)/(hasP a s2) => [] [x Hx Hax]; + by exists x; rewrite // ?Es12 // -Es12. +Qed. + +Lemma eq_all_r s1 s2 : s1 =i s2 -> all^~ s1 =1 all^~ s2. +Proof. +by move=> Es12 a; apply/(allP a s1)/(allP a s2) => Hs x Hx; + apply: Hs; rewrite Es12 in Hx *. +Qed. + +Lemma has_sym s1 s2 : has (mem s1) s2 = has (mem s2) s1. +Proof. by apply/(hasP _ s2)/(hasP _ s1) => [] [x]; exists x. Qed. + +Lemma has_pred1 x s : has (pred1 x) s = (x \in s). +Proof. by rewrite -(eq_has (mem_seq1^~ x)) (has_sym [:: x]) /= orbF. Qed. + +Lemma mem_rev s : rev s =i s. +Proof. by move=> a; rewrite -!has_pred1 has_rev. Qed. + +(* Constant sequences, i.e., the image of nseq. *) + +Definition constant s := if s is x :: s' then all (pred1 x) s' else true. + +Lemma all_pred1P x s : reflect (s = nseq (size s) x) (all (pred1 x) s). +Proof. +elim: s => [|y s IHs] /=; first by left. +case: eqP => [->{y} | ne_xy]; last by right=> [] [? _]; case ne_xy. +by apply: (iffP IHs) => [<- //| []]. +Qed. + +Lemma all_pred1_constant x s : all (pred1 x) s -> constant s. +Proof. by case: s => //= y s /andP[/eqP->]. Qed. + +Lemma all_pred1_nseq x n : all (pred1 x) (nseq n x). +Proof. by rewrite all_nseq /= eqxx orbT. Qed. + +Lemma nseqP n x y : reflect (y = x /\ n > 0) (y \in nseq n x). +Proof. +by rewrite -has_pred1 has_nseq /= eq_sym andbC; apply: (iffP andP) => -[/eqP]. +Qed. + +Lemma constant_nseq n x : constant (nseq n x). +Proof. exact: all_pred1_constant (all_pred1_nseq x n). Qed. + +(* Uses x0 *) +Lemma constantP s : reflect (exists x, s = nseq (size s) x) (constant s). +Proof. +apply: (iffP idP) => [| [x ->]]; last exact: constant_nseq. +case: s => [|x s] /=; first by exists x0. +by move/all_pred1P=> def_s; exists x; rewrite -def_s. +Qed. + +(* Duplicate-freenes. *) + +Fixpoint uniq s := if s is x :: s' then (x \notin s') && uniq s' else true. + +Lemma cons_uniq x s : uniq (x :: s) = (x \notin s) && uniq s. +Proof. by []. Qed. + +Lemma cat_uniq s1 s2 : + uniq (s1 ++ s2) = [&& uniq s1, ~~ has (mem s1) s2 & uniq s2]. +Proof. +elim: s1 => [|x s1 IHs]; first by rewrite /= has_pred0. +by rewrite has_sym /= mem_cat !negb_or has_sym IHs -!andbA; do !bool_congr. +Qed. + +Lemma uniq_catC s1 s2 : uniq (s1 ++ s2) = uniq (s2 ++ s1). +Proof. by rewrite !cat_uniq has_sym andbCA andbA andbC. Qed. + +Lemma uniq_catCA s1 s2 s3 : uniq (s1 ++ s2 ++ s3) = uniq (s2 ++ s1 ++ s3). +Proof. +by rewrite !catA -!(uniq_catC s3) !(cat_uniq s3) uniq_catC !has_cat orbC. +Qed. + +Lemma rcons_uniq s x : uniq (rcons s x) = (x \notin s) && uniq s. +Proof. by rewrite -cats1 uniq_catC. Qed. + +Lemma filter_uniq s a : uniq s -> uniq (filter a s). +Proof. +elim: s => [|x s IHs] //= /andP[Hx Hs]; case (a x); auto. +by rewrite /= mem_filter /= (negbTE Hx) andbF; auto. +Qed. + +Lemma rot_uniq s : uniq (rot n0 s) = uniq s. +Proof. by rewrite /rot uniq_catC cat_take_drop. Qed. + +Lemma rev_uniq s : uniq (rev s) = uniq s. +Proof. +elim: s => // x s IHs. +by rewrite rev_cons -cats1 cat_uniq /= andbT andbC mem_rev orbF IHs. +Qed. + +Lemma count_memPn x s : reflect (count_mem x s = 0) (x \notin s). +Proof. by rewrite -has_pred1 has_count -eqn0Ngt; apply: eqP. Qed. + +Lemma count_uniq_mem s x : uniq s -> count_mem x s = (x \in s). +Proof. +elim: s => //= y s IHs /andP[/negbTE s'y /IHs-> {IHs}]. +by rewrite in_cons eq_sym; case: eqP => // ->; rewrite s'y. +Qed. + +Lemma filter_pred1_uniq s x : uniq s -> x \in s -> filter (pred1 x) s = [:: x]. +Proof. +move=> uniq_s s_x; rewrite (all_pred1P _ _ (filter_all _ _)). +by rewrite size_filter count_uniq_mem ?s_x. +Qed. + +(* Removing duplicates *) + +Fixpoint undup s := + if s is x :: s' then if x \in s' then undup s' else x :: undup s' else [::]. + +Lemma size_undup s : size (undup s) <= size s. +Proof. by elim: s => //= x s IHs; case: (x \in s) => //=; exact: ltnW. Qed. + +Lemma mem_undup s : undup s =i s. +Proof. +move=> x; elim: s => //= y s IHs. +by case Hy: (y \in s); rewrite in_cons IHs //; case: eqP => // ->. +Qed. + +Lemma undup_uniq s : uniq (undup s). +Proof. +by elim: s => //= x s IHs; case s_x: (x \in s); rewrite //= mem_undup s_x. +Qed. + +Lemma undup_id s : uniq s -> undup s = s. +Proof. by elim: s => //= x s IHs /andP[/negbTE-> /IHs->]. Qed. + +Lemma ltn_size_undup s : (size (undup s) < size s) = ~~ uniq s. +Proof. +by elim: s => //= x s IHs; case Hx: (x \in s); rewrite //= ltnS size_undup. +Qed. + +Lemma filter_undup p s : filter p (undup s) = undup (filter p s). +Proof. +elim: s => //= x s IHs; rewrite (fun_if undup) fun_if /= mem_filter /=. +by rewrite (fun_if (filter p)) /= IHs; case: ifP => -> //=; exact: if_same. +Qed. + +Lemma undup_nil s : undup s = [::] -> s = [::]. +Proof. by case: s => //= x s; rewrite -mem_undup; case: ifP; case: undup. Qed. + +(* Lookup *) + +Definition index x := find (pred1 x). + +Lemma index_size x s : index x s <= size s. +Proof. by rewrite /index find_size. Qed. + +Lemma index_mem x s : (index x s < size s) = (x \in s). +Proof. by rewrite -has_pred1 has_find. Qed. + +Lemma nth_index x s : x \in s -> nth s (index x s) = x. +Proof. by rewrite -has_pred1 => /(nth_find x0)/eqP. Qed. + +Lemma index_cat x s1 s2 : + index x (s1 ++ s2) = if x \in s1 then index x s1 else size s1 + index x s2. +Proof. by rewrite /index find_cat has_pred1. Qed. + +Lemma index_uniq i s : i < size s -> uniq s -> index (nth s i) s = i. +Proof. +elim: s i => [|x s IHs] //= [|i]; rewrite /= ?eqxx // ltnS => lt_i_s. +case/andP=> not_s_x /(IHs i)-> {IHs}//. +by case: eqP not_s_x => // ->; rewrite mem_nth. +Qed. + +Lemma index_head x s : index x (x :: s) = 0. +Proof. by rewrite /= eqxx. Qed. + +Lemma index_last x s : uniq (x :: s) -> index (last x s) (x :: s) = size s. +Proof. +rewrite lastI rcons_uniq -cats1 index_cat size_belast. +by case: ifP => //=; rewrite eqxx addn0. +Qed. + +Lemma nth_uniq s i j : + i < size s -> j < size s -> uniq s -> (nth s i == nth s j) = (i == j). +Proof. +move=> lt_i_s lt_j_s Us; apply/eqP/eqP=> [eq_sij|-> //]. +by rewrite -(index_uniq lt_i_s Us) eq_sij index_uniq. +Qed. + +Lemma mem_rot s : rot n0 s =i s. +Proof. by move=> x; rewrite -{2}(cat_take_drop n0 s) !mem_cat /= orbC. Qed. + +Lemma eqseq_rot s1 s2 : (rot n0 s1 == rot n0 s2) = (s1 == s2). +Proof. by apply: inj_eq; exact: rot_inj. Qed. + +CoInductive rot_to_spec s x := RotToSpec i s' of rot i s = x :: s'. + +Lemma rot_to s x : x \in s -> rot_to_spec s x. +Proof. +move=> s_x; pose i := index x s; exists i (drop i.+1 s ++ take i s). +rewrite -cat_cons {}/i; congr cat; elim: s s_x => //= y s IHs. +by rewrite eq_sym in_cons; case: eqP => // -> _; rewrite drop0. +Qed. + +End EqSeq. + +Definition inE := (mem_seq1, in_cons, inE). + +Prenex Implicits mem_seq1 uniq undup index. + +Implicit Arguments eqseqP [T x y]. +Implicit Arguments hasP [T a s]. +Implicit Arguments hasPn [T a s]. +Implicit Arguments allP [T a s]. +Implicit Arguments allPn [T a s]. +Implicit Arguments nseqP [T n x y]. +Implicit Arguments count_memPn [T x s]. +Prenex Implicits eqseqP hasP hasPn allP allPn nseqP count_memPn. + +Section NthTheory. + +Lemma nthP (T : eqType) (s : seq T) x x0 : + reflect (exists2 i, i < size s & nth x0 s i = x) (x \in s). +Proof. +apply: (iffP idP) => [|[n Hn <-]]; last by apply mem_nth. +by exists (index x s); [rewrite index_mem | apply nth_index]. +Qed. + +Variable T : Type. + +Lemma has_nthP (a : pred T) s x0 : + reflect (exists2 i, i < size s & a (nth x0 s i)) (has a s). +Proof. +elim: s => [|x s IHs] /=; first by right; case. +case nax: (a x); first by left; exists 0. +by apply: (iffP IHs) => [[i]|[[|i]]]; [exists i.+1 | rewrite nax | exists i]. +Qed. + +Lemma all_nthP (a : pred T) s x0 : + reflect (forall i, i < size s -> a (nth x0 s i)) (all a s). +Proof. +rewrite -(eq_all (fun x => negbK (a x))) all_predC. +case: (has_nthP _ _ x0) => [na_s | a_s]; [right=> a_s | left=> i lti]. + by case: na_s => i lti; rewrite a_s. +by apply/idPn=> na_si; case: a_s; exists i. +Qed. + +End NthTheory. + +Lemma set_nth_default T s (y0 x0 : T) n : n < size s -> nth x0 s n = nth y0 s n. +Proof. by elim: s n => [|y s' IHs] [|n] /=; auto. Qed. + +Lemma headI T s (x : T) : rcons s x = head x s :: behead (rcons s x). +Proof. by case: s. Qed. + +Implicit Arguments nthP [T s x]. +Implicit Arguments has_nthP [T a s]. +Implicit Arguments all_nthP [T a s]. +Prenex Implicits nthP has_nthP all_nthP. + +Definition bitseq := seq bool. +Canonical bitseq_eqType := Eval hnf in [eqType of bitseq]. +Canonical bitseq_predType := Eval hnf in [predType of bitseq]. + +(* Incrementing the ith nat in a seq nat, padding with 0's if needed. This *) +(* allows us to use nat seqs as bags of nats. *) + +Fixpoint incr_nth v i {struct i} := + if v is n :: v' then if i is i'.+1 then n :: incr_nth v' i' else n.+1 :: v' + else ncons i 0 [:: 1]. + +Lemma nth_incr_nth v i j : nth 0 (incr_nth v i) j = (i == j) + nth 0 v j. +Proof. +elim: v i j => [|n v IHv] [|i] [|j] //=; rewrite ?eqSS ?addn0 //; try by case j. +elim: i j => [|i IHv] [|j] //=; rewrite ?eqSS //; by case j. +Qed. + +Lemma size_incr_nth v i : + size (incr_nth v i) = if i < size v then size v else i.+1. +Proof. +elim: v i => [|n v IHv] [|i] //=; first by rewrite size_ncons /= addn1. +rewrite IHv; exact: fun_if. +Qed. + +(* Equality up to permutation *) + +Section PermSeq. + +Variable T : eqType. +Implicit Type s : seq T. + +Definition perm_eq s1 s2 := + all [pred x | count_mem x s1 == count_mem x s2] (s1 ++ s2). + +Lemma perm_eqP s1 s2 : reflect (count^~ s1 =1 count^~ s2) (perm_eq s1 s2). +Proof. +apply: (iffP allP) => /= [eq_cnt1 a | eq_cnt x _]; last exact/eqP. +elim: {a}_.+1 {-2}a (ltnSn (count a (s1 ++ s2))) => // n IHn a le_an. +have [/eqP|] := posnP (count a (s1 ++ s2)). + by rewrite count_cat addn_eq0; do 2!case: eqP => // ->. +rewrite -has_count => /hasP[x s12x a_x]; pose a' := predD1 a x. +have cnt_a' s: count a s = count_mem x s + count a' s. + rewrite -count_predUI -[LHS]addn0 -(count_pred0 s). + by congr (_ + _); apply: eq_count => y /=; case: eqP => // ->. +rewrite !cnt_a' (eqnP (eq_cnt1 _ s12x)) (IHn a') // -ltnS. +apply: leq_trans le_an. +by rewrite ltnS cnt_a' -add1n leq_add2r -has_count has_pred1. +Qed. + +Lemma perm_eq_refl s : perm_eq s s. +Proof. exact/perm_eqP. Qed. +Hint Resolve perm_eq_refl. + +Lemma perm_eq_sym : symmetric perm_eq. +Proof. by move=> s1 s2; apply/perm_eqP/perm_eqP=> ? ?. Qed. + +Lemma perm_eq_trans : transitive perm_eq. +Proof. by move=> s2 s1 s3 /perm_eqP-eq12 /perm_eqP/(ftrans eq12)/perm_eqP. Qed. + +Notation perm_eql s1 s2 := (perm_eq s1 =1 perm_eq s2). +Notation perm_eqr s1 s2 := (perm_eq^~ s1 =1 perm_eq^~ s2). + +Lemma perm_eqlE s1 s2 : perm_eql s1 s2 -> perm_eq s1 s2. Proof. by move->. Qed. + +Lemma perm_eqlP s1 s2 : reflect (perm_eql s1 s2) (perm_eq s1 s2). +Proof. +apply: (iffP idP) => [eq12 s3 | -> //]. +apply/idP/idP; last exact: perm_eq_trans. +by rewrite -!(perm_eq_sym s3); move/perm_eq_trans; apply. +Qed. + +Lemma perm_eqrP s1 s2 : reflect (perm_eqr s1 s2) (perm_eq s1 s2). +Proof. +apply: (iffP idP) => [/perm_eqlP eq12 s3| <- //]. +by rewrite !(perm_eq_sym s3) eq12. +Qed. + +Lemma perm_catC s1 s2 : perm_eql (s1 ++ s2) (s2 ++ s1). +Proof. by apply/perm_eqlP; apply/perm_eqP=> a; rewrite !count_cat addnC. Qed. + +Lemma perm_cat2l s1 s2 s3 : perm_eq (s1 ++ s2) (s1 ++ s3) = perm_eq s2 s3. +Proof. +apply/perm_eqP/perm_eqP=> eq23 a; apply/eqP; + by move/(_ a)/eqP: eq23; rewrite !count_cat eqn_add2l. +Qed. + +Lemma perm_cons x s1 s2 : perm_eq (x :: s1) (x :: s2) = perm_eq s1 s2. +Proof. exact: (perm_cat2l [::x]). Qed. + +Lemma perm_cat2r s1 s2 s3 : perm_eq (s2 ++ s1) (s3 ++ s1) = perm_eq s2 s3. +Proof. by do 2!rewrite perm_eq_sym perm_catC; exact: perm_cat2l. Qed. + +Lemma perm_catAC s1 s2 s3 : perm_eql ((s1 ++ s2) ++ s3) ((s1 ++ s3) ++ s2). +Proof. by apply/perm_eqlP; rewrite -!catA perm_cat2l perm_catC. Qed. + +Lemma perm_catCA s1 s2 s3 : perm_eql (s1 ++ s2 ++ s3) (s2 ++ s1 ++ s3). +Proof. by apply/perm_eqlP; rewrite !catA perm_cat2r perm_catC. Qed. + +Lemma perm_rcons x s : perm_eql (rcons s x) (x :: s). +Proof. by move=> /= s2; rewrite -cats1 perm_catC. Qed. + +Lemma perm_rot n s : perm_eql (rot n s) s. +Proof. by move=> /= s2; rewrite perm_catC cat_take_drop. Qed. + +Lemma perm_rotr n s : perm_eql (rotr n s) s. +Proof. exact: perm_rot. Qed. + +Lemma perm_filterC a s : perm_eql (filter a s ++ filter (predC a) s) s. +Proof. +apply/perm_eqlP; elim: s => //= x s IHs. +by case: (a x); last rewrite /= -cat1s perm_catCA; rewrite perm_cons. +Qed. + +Lemma perm_eq_mem s1 s2 : perm_eq s1 s2 -> s1 =i s2. +Proof. by move/perm_eqP=> eq12 x; rewrite -!has_pred1 !has_count eq12. Qed. + +Lemma perm_eq_size s1 s2 : perm_eq s1 s2 -> size s1 = size s2. +Proof. by move/perm_eqP=> eq12; rewrite -!count_predT eq12. Qed. + +Lemma perm_eq_small s1 s2 : size s2 <= 1 -> perm_eq s1 s2 -> s1 = s2. +Proof. +move=> s2_le1 eqs12; move/perm_eq_size: eqs12 s2_le1 (perm_eq_mem eqs12). +by case: s2 s1 => [|x []] // [|y []] // _ _ /(_ x); rewrite !inE eqxx => /eqP->. +Qed. + +Lemma uniq_leq_size s1 s2 : uniq s1 -> {subset s1 <= s2} -> size s1 <= size s2. +Proof. +elim: s1 s2 => //= x s1 IHs s2 /andP[not_s1x Us1] /allP/=/andP[s2x /allP ss12]. +have [i s3 def_s2] := rot_to s2x; rewrite -(size_rot i s2) def_s2. +apply: IHs => // y s1y; have:= ss12 y s1y. +by rewrite -(mem_rot i) def_s2 inE (negPf (memPn _ y s1y)). +Qed. + +Lemma leq_size_uniq s1 s2 : + uniq s1 -> {subset s1 <= s2} -> size s2 <= size s1 -> uniq s2. +Proof. +elim: s1 s2 => [[] | x s1 IHs s2] // Us1x; have /andP[not_s1x Us1] := Us1x. +case/allP/andP=> /rot_to[i s3 def_s2] /allP ss12 le_s21. +rewrite -(rot_uniq i) -(size_rot i) def_s2 /= in le_s21 *. +have ss13 y (s1y : y \in s1): y \in s3. + by have:= ss12 y s1y; rewrite -(mem_rot i) def_s2 inE (negPf (memPn _ y s1y)). +rewrite IHs // andbT; apply: contraL _ le_s21 => s3x; rewrite -leqNgt. +by apply/(uniq_leq_size Us1x)/allP; rewrite /= s3x; exact/allP. +Qed. + +Lemma uniq_size_uniq s1 s2 : + uniq s1 -> s1 =i s2 -> uniq s2 = (size s2 == size s1). +Proof. +move=> Us1 eqs12; apply/idP/idP=> [Us2 | /eqP eq_sz12]. + by rewrite eqn_leq !uniq_leq_size // => y; rewrite eqs12. +by apply: (leq_size_uniq Us1) => [y|]; rewrite (eqs12, eq_sz12). +Qed. + +Lemma leq_size_perm s1 s2 : + uniq s1 -> {subset s1 <= s2} -> size s2 <= size s1 -> + s1 =i s2 /\ size s1 = size s2. +Proof. +move=> Us1 ss12 le_s21; have Us2: uniq s2 := leq_size_uniq Us1 ss12 le_s21. +suffices: s1 =i s2 by split; last by apply/eqP; rewrite -uniq_size_uniq. +move=> x; apply/idP/idP=> [/ss12// | s2x]; apply: contraLR le_s21 => not_s1x. +rewrite -ltnNge (@uniq_leq_size (x :: s1)) /= ?not_s1x //. +by apply/allP; rewrite /= s2x; apply/allP. +Qed. + +Lemma perm_uniq s1 s2 : s1 =i s2 -> size s1 = size s2 -> uniq s1 = uniq s2. +Proof. +move=> Es12 Esz12. +by apply/idP/idP=> Us; rewrite (uniq_size_uniq Us) ?Esz12 ?eqxx. +Qed. + +Lemma perm_eq_uniq s1 s2 : perm_eq s1 s2 -> uniq s1 = uniq s2. +Proof. +by move=> eq_s12; apply: perm_uniq; [apply: perm_eq_mem | apply: perm_eq_size]. +Qed. + +Lemma uniq_perm_eq s1 s2 : uniq s1 -> uniq s2 -> s1 =i s2 -> perm_eq s1 s2. +Proof. +move=> Us1 Us2 eq12; apply/allP=> x _; apply/eqP. +by rewrite !count_uniq_mem ?eq12. +Qed. + +Lemma count_mem_uniq s : (forall x, count_mem x s = (x \in s)) -> uniq s. +Proof. +move=> count1_s; have Uus := undup_uniq s. +suffices: perm_eq s (undup s) by move/perm_eq_uniq->. +by apply/allP=> x _; apply/eqP; rewrite (count_uniq_mem x Uus) mem_undup. +Qed. + +Lemma catCA_perm_ind P : + (forall s1 s2 s3, P (s1 ++ s2 ++ s3) -> P (s2 ++ s1 ++ s3)) -> + (forall s1 s2, perm_eq s1 s2 -> P s1 -> P s2). +Proof. +move=> PcatCA s1 s2 eq_s12; rewrite -[s1]cats0 -[s2]cats0. +elim: s2 nil => [| x s2 IHs] s3 in s1 eq_s12 *. + by case: s1 {eq_s12}(perm_eq_size eq_s12). +have /rot_to[i s' def_s1]: x \in s1 by rewrite (perm_eq_mem eq_s12) mem_head. +rewrite -(cat_take_drop i s1) -catA => /PcatCA. +rewrite catA -/(rot i s1) def_s1 /= -cat1s => /PcatCA/IHs/PcatCA; apply. +by rewrite -(perm_cons x) -def_s1 perm_rot. +Qed. + +Lemma catCA_perm_subst R F : + (forall s1 s2 s3, F (s1 ++ s2 ++ s3) = F (s2 ++ s1 ++ s3) :> R) -> + (forall s1 s2, perm_eq s1 s2 -> F s1 = F s2). +Proof. +move=> FcatCA s1 s2 /catCA_perm_ind => ind_s12. +by apply: (ind_s12 (eq _ \o F)) => //= *; rewrite FcatCA. +Qed. + +End PermSeq. + +Notation perm_eql s1 s2 := (perm_eq s1 =1 perm_eq s2). +Notation perm_eqr s1 s2 := (perm_eq^~ s1 =1 perm_eq^~ s2). + +Implicit Arguments perm_eqP [T s1 s2]. +Implicit Arguments perm_eqlP [T s1 s2]. +Implicit Arguments perm_eqrP [T s1 s2]. +Prenex Implicits perm_eq perm_eqP perm_eqlP perm_eqrP. +Hint Resolve perm_eq_refl. + +Section RotrLemmas. + +Variables (n0 : nat) (T : Type) (T' : eqType). +Implicit Type s : seq T. + +Lemma size_rotr s : size (rotr n0 s) = size s. +Proof. by rewrite size_rot. Qed. + +Lemma mem_rotr (s : seq T') : rotr n0 s =i s. +Proof. by move=> x; rewrite mem_rot. Qed. + +Lemma rotr_size_cat s1 s2 : rotr (size s2) (s1 ++ s2) = s2 ++ s1. +Proof. by rewrite /rotr size_cat addnK rot_size_cat. Qed. + +Lemma rotr1_rcons x s : rotr 1 (rcons s x) = x :: s. +Proof. by rewrite -rot1_cons rotK. Qed. + +Lemma has_rotr a s : has a (rotr n0 s) = has a s. +Proof. by rewrite has_rot. Qed. + +Lemma rotr_uniq (s : seq T') : uniq (rotr n0 s) = uniq s. +Proof. by rewrite rot_uniq. Qed. + +Lemma rotrK : cancel (@rotr T n0) (rot n0). +Proof. +move=> s; have [lt_n0s | ge_n0s] := ltnP n0 (size s). + by rewrite -{1}(subKn (ltnW lt_n0s)) -{1}[size s]size_rotr; exact: rotK. +by rewrite -{2}(rot_oversize ge_n0s) /rotr (eqnP ge_n0s) rot0. +Qed. + +Lemma rotr_inj : injective (@rotr T n0). +Proof. exact (can_inj rotrK). Qed. + +Lemma rev_rot s : rev (rot n0 s) = rotr n0 (rev s). +Proof. +rewrite /rotr size_rev -{3}(cat_take_drop n0 s) {1}/rot !rev_cat. +by rewrite -size_drop -size_rev rot_size_cat. +Qed. + +Lemma rev_rotr s : rev (rotr n0 s) = rot n0 (rev s). +Proof. +apply: canRL rotrK _; rewrite {1}/rotr size_rev size_rotr /rotr {2}/rot rev_cat. +set m := size s - n0; rewrite -{1}(@size_takel m _ _ (leq_subr _ _)). +by rewrite -size_rev rot_size_cat -rev_cat cat_take_drop. +Qed. + +End RotrLemmas. + +Section RotCompLemmas. + +Variable T : Type. +Implicit Type s : seq T. + +Lemma rot_addn m n s : m + n <= size s -> rot (m + n) s = rot m (rot n s). +Proof. +move=> sz_s; rewrite {1}/rot -[take _ s](cat_take_drop n). +rewrite 5!(catA, =^~ rot_size_cat) !cat_take_drop. +by rewrite size_drop !size_takel ?leq_addl ?addnK. +Qed. + +Lemma rotS n s : n < size s -> rot n.+1 s = rot 1 (rot n s). +Proof. exact: (@rot_addn 1). Qed. + +Lemma rot_add_mod m n s : n <= size s -> m <= size s -> + rot m (rot n s) = rot (if m + n <= size s then m + n else m + n - size s) s. +Proof. +move=> Hn Hm; case: leqP => [/rot_addn // | /ltnW Hmn]; symmetry. +by rewrite -{2}(rotK n s) /rotr -rot_addn size_rot addnBA ?subnK ?addnK. +Qed. + +Lemma rot_rot m n s : rot m (rot n s) = rot n (rot m s). +Proof. +case: (ltnP (size s) m) => Hm. + by rewrite !(@rot_oversize T m) ?size_rot 1?ltnW. +case: (ltnP (size s) n) => Hn. + by rewrite !(@rot_oversize T n) ?size_rot 1?ltnW. +by rewrite !rot_add_mod 1?addnC. +Qed. + +Lemma rot_rotr m n s : rot m (rotr n s) = rotr n (rot m s). +Proof. by rewrite {2}/rotr size_rot rot_rot. Qed. + +Lemma rotr_rotr m n s : rotr m (rotr n s) = rotr n (rotr m s). +Proof. by rewrite /rotr !size_rot rot_rot. Qed. + +End RotCompLemmas. + +Section Mask. + +Variables (n0 : nat) (T : Type). +Implicit Types (m : bitseq) (s : seq T). + +Fixpoint mask m s {struct m} := + match m, s with + | b :: m', x :: s' => if b then x :: mask m' s' else mask m' s' + | _, _ => [::] + end. + +Lemma mask_false s n : mask (nseq n false) s = [::]. +Proof. by elim: s n => [|x s IHs] [|n] /=. Qed. + +Lemma mask_true s n : size s <= n -> mask (nseq n true) s = s. +Proof. by elim: s n => [|x s IHs] [|n] //= Hn; congr (_ :: _); apply: IHs. Qed. + +Lemma mask0 m : mask m [::] = [::]. +Proof. by case: m. Qed. + +Lemma mask1 b x : mask [:: b] [:: x] = nseq b x. +Proof. by case: b. Qed. + +Lemma mask_cons b m x s : mask (b :: m) (x :: s) = nseq b x ++ mask m s. +Proof. by case: b. Qed. + +Lemma size_mask m s : size m = size s -> size (mask m s) = count id m. +Proof. by move: m s; apply: seq2_ind => // -[] x m s /= ->. Qed. + +Lemma mask_cat m1 m2 s1 s2 : + size m1 = size s1 -> mask (m1 ++ m2) (s1 ++ s2) = mask m1 s1 ++ mask m2 s2. +Proof. by move: m1 s1; apply: seq2_ind => // -[] m1 x1 s1 /= ->. Qed. + +Lemma has_mask_cons a b m x s : + has a (mask (b :: m) (x :: s)) = b && a x || has a (mask m s). +Proof. by case: b. Qed. + +Lemma has_mask a m s : has a (mask m s) -> has a s. +Proof. +elim: m s => [|b m IHm] [|x s] //; rewrite has_mask_cons /= andbC. +by case: (a x) => //= /IHm. +Qed. + +Lemma mask_rot m s : size m = size s -> + mask (rot n0 m) (rot n0 s) = rot (count id (take n0 m)) (mask m s). +Proof. +move=> Ems; rewrite mask_cat ?size_drop ?Ems // -rot_size_cat. +by rewrite size_mask -?mask_cat ?size_take ?Ems // !cat_take_drop. +Qed. + +Lemma resize_mask m s : {m1 | size m1 = size s & mask m s = mask m1 s}. +Proof. +by exists (take (size s) m ++ nseq (size s - size m) false); + elim: s m => [|x s IHs] [|b m] //=; rewrite (size_nseq, mask_false, IHs). +Qed. + +End Mask. + +Section EqMask. + +Variables (n0 : nat) (T : eqType). +Implicit Types (s : seq T) (m : bitseq). + +Lemma mem_mask_cons x b m y s : + (x \in mask (b :: m) (y :: s)) = b && (x == y) || (x \in mask m s). +Proof. by case: b. Qed. + +Lemma mem_mask x m s : x \in mask m s -> x \in s. +Proof. by rewrite -!has_pred1 => /has_mask. Qed. + +Lemma mask_uniq s : uniq s -> forall m, uniq (mask m s). +Proof. +elim: s => [|x s IHs] Uxs [|b m] //=. +case: b Uxs => //= /andP[s'x Us]; rewrite {}IHs // andbT. +by apply: contra s'x; exact: mem_mask. +Qed. + +Lemma mem_mask_rot m s : + size m = size s -> mask (rot n0 m) (rot n0 s) =i mask m s. +Proof. by move=> Ems x; rewrite mask_rot // mem_rot. Qed. + +End EqMask. + +Section Subseq. + +Variable T : eqType. +Implicit Type s : seq T. + +Fixpoint subseq s1 s2 := + if s2 is y :: s2' then + if s1 is x :: s1' then subseq (if x == y then s1' else s1) s2' else true + else s1 == [::]. + +Lemma sub0seq s : subseq [::] s. Proof. by case: s. Qed. + +Lemma subseq0 s : subseq s [::] = (s == [::]). Proof. by []. Qed. + +Lemma subseqP s1 s2 : + reflect (exists2 m, size m = size s2 & s1 = mask m s2) (subseq s1 s2). +Proof. +elim: s2 s1 => [|y s2 IHs2] [|x s1]. +- by left; exists [::]. +- by right; do 2!case. +- by left; exists (nseq (size s2).+1 false); rewrite ?size_nseq //= mask_false. +apply: {IHs2}(iffP (IHs2 _)) => [] [m sz_m def_s1]. + by exists ((x == y) :: m); rewrite /= ?sz_m // -def_s1; case: eqP => // ->. +case: eqP => [_ | ne_xy]; last first. + by case: m def_s1 sz_m => [//|[m []//|m]] -> [<-]; exists m. +pose i := index true m; have def_m_i: take i m = nseq (size (take i m)) false. + apply/all_pred1P; apply/(all_nthP true) => j. + rewrite size_take ltnNge geq_min negb_or -ltnNge; case/andP=> lt_j_i _. + rewrite nth_take //= -negb_add addbF -addbT -negb_eqb. + by rewrite [_ == _](before_find _ lt_j_i). +have lt_i_m: i < size m. + rewrite ltnNge; apply/negP=> le_m_i; rewrite take_oversize // in def_m_i. + by rewrite def_m_i mask_false in def_s1. +rewrite size_take lt_i_m in def_m_i. +exists (take i m ++ drop i.+1 m). + rewrite size_cat size_take size_drop lt_i_m. + by rewrite sz_m in lt_i_m *; rewrite subnKC. +rewrite {s1 def_s1}[s1](congr1 behead def_s1). +rewrite -[s2](cat_take_drop i) -{1}[m](cat_take_drop i) {}def_m_i -cat_cons. +have sz_i_s2: size (take i s2) = i by apply: size_takel; rewrite sz_m in lt_i_m. +rewrite lastI cat_rcons !mask_cat ?size_nseq ?size_belast ?mask_false //=. +by rewrite (drop_nth true) // nth_index -?index_mem. +Qed. + +Lemma mask_subseq m s : subseq (mask m s) s. +Proof. by apply/subseqP; have [m1] := resize_mask m s; exists m1. Qed. + +Lemma subseq_trans : transitive subseq. +Proof. +move=> _ _ s /subseqP[m2 _ ->] /subseqP[m1 _ ->]. +elim: s => [|x s IHs] in m2 m1 *; first by rewrite !mask0. +case: m1 => [|[] m1]; first by rewrite mask0. + case: m2 => [|[] m2] //; first by rewrite /= eqxx IHs. + case/subseqP: (IHs m2 m1) => m sz_m def_s; apply/subseqP. + by exists (false :: m); rewrite //= sz_m. +case/subseqP: (IHs m2 m1) => m sz_m def_s; apply/subseqP. +by exists (false :: m); rewrite //= sz_m. +Qed. + +Lemma subseq_refl s : subseq s s. +Proof. by elim: s => //= x s IHs; rewrite eqxx. Qed. +Hint Resolve subseq_refl. + +Lemma cat_subseq s1 s2 s3 s4 : + subseq s1 s3 -> subseq s2 s4 -> subseq (s1 ++ s2) (s3 ++ s4). +Proof. +case/subseqP=> m1 sz_m1 ->; case/subseqP=> m2 sz_m2 ->; apply/subseqP. +by exists (m1 ++ m2); rewrite ?size_cat ?mask_cat ?sz_m1 ?sz_m2. +Qed. + +Lemma prefix_subseq s1 s2 : subseq s1 (s1 ++ s2). +Proof. by rewrite -{1}[s1]cats0 cat_subseq ?sub0seq. Qed. + +Lemma suffix_subseq s1 s2 : subseq s2 (s1 ++ s2). +Proof. by rewrite -{1}[s2]cat0s cat_subseq ?sub0seq. Qed. + +Lemma mem_subseq s1 s2 : subseq s1 s2 -> {subset s1 <= s2}. +Proof. by case/subseqP=> m _ -> x; exact: mem_mask. Qed. + +Lemma sub1seq x s : subseq [:: x] s = (x \in s). +Proof. +by elim: s => //= y s; rewrite inE; case: (x == y); rewrite ?sub0seq. +Qed. + +Lemma size_subseq s1 s2 : subseq s1 s2 -> size s1 <= size s2. +Proof. by case/subseqP=> m sz_m ->; rewrite size_mask -sz_m ?count_size. Qed. + +Lemma size_subseq_leqif s1 s2 : + subseq s1 s2 -> size s1 <= size s2 ?= iff (s1 == s2). +Proof. +move=> sub12; split; first exact: size_subseq. +apply/idP/eqP=> [|-> //]; case/subseqP: sub12 => m sz_m ->{s1}. +rewrite size_mask -sz_m // -all_count -(eq_all eqb_id). +by move/(@all_pred1P _ true)->; rewrite sz_m mask_true. +Qed. + +Lemma subseq_cons s x : subseq s (x :: s). +Proof. by exact: (@cat_subseq [::] s [:: x]). Qed. + +Lemma subseq_rcons s x : subseq s (rcons s x). +Proof. by rewrite -{1}[s]cats0 -cats1 cat_subseq. Qed. + +Lemma subseq_uniq s1 s2 : subseq s1 s2 -> uniq s2 -> uniq s1. +Proof. by case/subseqP=> m _ -> Us2; exact: mask_uniq. Qed. + +End Subseq. + +Prenex Implicits subseq. +Implicit Arguments subseqP [T s1 s2]. + +Hint Resolve subseq_refl. + +Section Rem. + +Variables (T : eqType) (x : T). + +Fixpoint rem s := if s is y :: t then (if y == x then t else y :: rem t) else s. + +Lemma rem_id s : x \notin s -> rem s = s. +Proof. +by elim: s => //= y s IHs /norP[neq_yx /IHs->]; rewrite eq_sym (negbTE neq_yx). +Qed. + +Lemma perm_to_rem s : x \in s -> perm_eq s (x :: rem s). +Proof. +elim: s => // y s IHs; rewrite inE /= eq_sym perm_eq_sym. +case: eqP => [-> // | _ /IHs]. +by rewrite (perm_catCA [:: x] [:: y]) perm_cons perm_eq_sym. +Qed. + +Lemma size_rem s : x \in s -> size (rem s) = (size s).-1. +Proof. by move/perm_to_rem/perm_eq_size->. Qed. + +Lemma rem_subseq s : subseq (rem s) s. +Proof. +elim: s => //= y s IHs; rewrite eq_sym. +by case: ifP => _; [exact: subseq_cons | rewrite eqxx]. +Qed. + +Lemma rem_uniq s : uniq s -> uniq (rem s). +Proof. by apply: subseq_uniq; exact: rem_subseq. Qed. + +Lemma mem_rem s : {subset rem s <= s}. +Proof. exact: mem_subseq (rem_subseq s). Qed. + +Lemma rem_filter s : uniq s -> rem s = filter (predC1 x) s. +Proof. +elim: s => //= y s IHs /andP[not_s_y /IHs->]. +by case: eqP => //= <-; apply/esym/all_filterP; rewrite all_predC has_pred1. +Qed. + +Lemma mem_rem_uniq s : uniq s -> rem s =i [predD1 s & x]. +Proof. by move/rem_filter=> -> y; rewrite mem_filter. Qed. + +End Rem. + +Section Map. + +Variables (n0 : nat) (T1 : Type) (x1 : T1). +Variables (T2 : Type) (x2 : T2) (f : T1 -> T2). + +Fixpoint map s := if s is x :: s' then f x :: map s' else [::]. + +Lemma map_cons x s : map (x :: s) = f x :: map s. +Proof. by []. Qed. + +Lemma map_nseq x : map (nseq n0 x) = nseq n0 (f x). +Proof. by elim: n0 => // *; congr (_ :: _). Qed. + +Lemma map_cat s1 s2 : map (s1 ++ s2) = map s1 ++ map s2. +Proof. by elim: s1 => [|x s1 IHs] //=; rewrite IHs. Qed. + +Lemma size_map s : size (map s) = size s. +Proof. by elim: s => //= x s ->. Qed. + +Lemma behead_map s : behead (map s) = map (behead s). +Proof. by case: s. Qed. + +Lemma nth_map n s : n < size s -> nth x2 (map s) n = f (nth x1 s n). +Proof. by elim: s n => [|x s IHs] [|n]; try exact (IHs n). Qed. + +Lemma map_rcons s x : map (rcons s x) = rcons (map s) (f x). +Proof. by rewrite -!cats1 map_cat. Qed. + +Lemma last_map s x : last (f x) (map s) = f (last x s). +Proof. by elim: s x => /=. Qed. + +Lemma belast_map s x : belast (f x) (map s) = map (belast x s). +Proof. by elim: s x => //= y s IHs x; rewrite IHs. Qed. + +Lemma filter_map a s : filter a (map s) = map (filter (preim f a) s). +Proof. by elim: s => //= x s IHs; rewrite (fun_if map) /= IHs. Qed. + +Lemma find_map a s : find a (map s) = find (preim f a) s. +Proof. by elim: s => //= x s ->. Qed. + +Lemma has_map a s : has a (map s) = has (preim f a) s. +Proof. by elim: s => //= x s ->. Qed. + +Lemma all_map a s : all a (map s) = all (preim f a) s. +Proof. by elim: s => //= x s ->. Qed. + +Lemma count_map a s : count a (map s) = count (preim f a) s. +Proof. by elim: s => //= x s ->. Qed. + +Lemma map_take s : map (take n0 s) = take n0 (map s). +Proof. by elim: n0 s => [|n IHn] [|x s] //=; rewrite IHn. Qed. + +Lemma map_drop s : map (drop n0 s) = drop n0 (map s). +Proof. by elim: n0 s => [|n IHn] [|x s] //=; rewrite IHn. Qed. + +Lemma map_rot s : map (rot n0 s) = rot n0 (map s). +Proof. by rewrite /rot map_cat map_take map_drop. Qed. + +Lemma map_rotr s : map (rotr n0 s) = rotr n0 (map s). +Proof. by apply: canRL (@rotK n0 T2) _; rewrite -map_rot rotrK. Qed. + +Lemma map_rev s : map (rev s) = rev (map s). +Proof. by elim: s => //= x s IHs; rewrite !rev_cons -!cats1 map_cat IHs. Qed. + +Lemma map_mask m s : map (mask m s) = mask m (map s). +Proof. by elim: m s => [|[|] m IHm] [|x p] //=; rewrite IHm. Qed. + +Lemma inj_map : injective f -> injective map. +Proof. +by move=> injf; elim=> [|y1 s1 IHs] [|y2 s2] //= [/injf-> /IHs->]. +Qed. + +End Map. + +Notation "[ 'seq' E | i <- s ]" := (map (fun i => E) s) + (at level 0, E at level 99, i ident, + format "[ '[hv' 'seq' E '/ ' | i <- s ] ']'") : seq_scope. + +Notation "[ 'seq' E | i <- s & C ]" := [seq E | i <- [seq i <- s | C]] + (at level 0, E at level 99, i ident, + format "[ '[hv' 'seq' E '/ ' | i <- s '/ ' & C ] ']'") : seq_scope. + +Notation "[ 'seq' E | i : T <- s ]" := (map (fun i : T => E) s) + (at level 0, E at level 99, i ident, only parsing) : seq_scope. + +Notation "[ 'seq' E | i : T <- s & C ]" := + [seq E | i : T <- [seq i : T <- s | C]] + (at level 0, E at level 99, i ident, only parsing) : seq_scope. + +Lemma filter_mask T a (s : seq T) : filter a s = mask (map a s) s. +Proof. by elim: s => //= x s <-; case: (a x). Qed. + +Section FilterSubseq. + +Variable T : eqType. +Implicit Types (s : seq T) (a : pred T). + +Lemma filter_subseq a s : subseq (filter a s) s. +Proof. by apply/subseqP; exists (map a s); rewrite ?size_map ?filter_mask. Qed. + +Lemma subseq_filter s1 s2 a : + subseq s1 (filter a s2) = all a s1 && subseq s1 s2. +Proof. +elim: s2 s1 => [|x s2 IHs] [|y s1] //=; rewrite ?andbF ?sub0seq //. +by case a_x: (a x); rewrite /= !IHs /=; case: eqP => // ->; rewrite a_x. +Qed. + +Lemma subseq_uniqP s1 s2 : + uniq s2 -> reflect (s1 = filter (mem s1) s2) (subseq s1 s2). +Proof. +move=> uniq_s2; apply: (iffP idP) => [ss12 | ->]; last exact: filter_subseq. +apply/eqP; rewrite -size_subseq_leqif ?subseq_filter ?(introT allP) //. +apply/eqP/esym/perm_eq_size. +rewrite uniq_perm_eq ?filter_uniq ?(subseq_uniq ss12) // => x. +by rewrite mem_filter; apply: andb_idr; exact: (mem_subseq ss12). +Qed. + +Lemma perm_to_subseq s1 s2 : + subseq s1 s2 -> {s3 | perm_eq s2 (s1 ++ s3)}. +Proof. +elim Ds2: s2 s1 => [|y s2' IHs] [|x s1] //=; try by exists s2; rewrite Ds2. +case: eqP => [-> | _] /IHs[s3 perm_s2] {IHs}. + by exists s3; rewrite perm_cons. +by exists (rcons s3 y); rewrite -cat_cons -perm_rcons -!cats1 catA perm_cat2r. +Qed. + +End FilterSubseq. + +Implicit Arguments subseq_uniqP [T s1 s2]. + +Section EqMap. + +Variables (n0 : nat) (T1 : eqType) (x1 : T1). +Variables (T2 : eqType) (x2 : T2) (f : T1 -> T2). +Implicit Type s : seq T1. + +Lemma map_f s x : x \in s -> f x \in map f s. +Proof. +elim: s => [|y s IHs] //=. +by case/predU1P=> [->|Hx]; [exact: predU1l | apply: predU1r; auto]. +Qed. + +Lemma mapP s y : reflect (exists2 x, x \in s & y = f x) (y \in map f s). +Proof. +elim: s => [|x s IHs]; first by right; case. +rewrite /= in_cons eq_sym; case Hxy: (f x == y). + by left; exists x; [rewrite mem_head | rewrite (eqP Hxy)]. +apply: (iffP IHs) => [[x' Hx' ->]|[x' Hx' Dy]]. + by exists x'; first exact: predU1r. +by move: Dy Hxy => ->; case/predU1P: Hx' => [->|]; [rewrite eqxx | exists x']. +Qed. + +Lemma map_uniq s : uniq (map f s) -> uniq s. +Proof. +elim: s => //= x s IHs /andP[not_sfx /IHs->]; rewrite andbT. +by apply: contra not_sfx => sx; apply/mapP; exists x. +Qed. + +Lemma map_inj_in_uniq s : {in s &, injective f} -> uniq (map f s) = uniq s. +Proof. +elim: s => //= x s IHs //= injf; congr (~~ _ && _). + apply/mapP/idP=> [[y sy /injf] | ]; last by exists x. + by rewrite mem_head mem_behead // => ->. +apply: IHs => y z sy sz; apply: injf => //; exact: predU1r. +Qed. + +Lemma map_subseq s1 s2 : subseq s1 s2 -> subseq (map f s1) (map f s2). +Proof. +case/subseqP=> m sz_m ->; apply/subseqP. +by exists m; rewrite ?size_map ?map_mask. +Qed. + +Lemma nth_index_map s x0 x : + {in s &, injective f} -> x \in s -> nth x0 s (index (f x) (map f s)) = x. +Proof. +elim: s => //= y s IHs inj_f s_x; rewrite (inj_in_eq inj_f) ?mem_head //. +move: s_x; rewrite inE eq_sym; case: eqP => [-> | _] //=; apply: IHs. +by apply: sub_in2 inj_f => z; exact: predU1r. +Qed. + +Lemma perm_map s t : perm_eq s t -> perm_eq (map f s) (map f t). +Proof. by move/perm_eqP=> Est; apply/perm_eqP=> a; rewrite !count_map Est. Qed. + +Hypothesis Hf : injective f. + +Lemma mem_map s x : (f x \in map f s) = (x \in s). +Proof. by apply/mapP/idP=> [[y Hy /Hf->] //|]; exists x. Qed. + +Lemma index_map s x : index (f x) (map f s) = index x s. +Proof. by rewrite /index; elim: s => //= y s IHs; rewrite (inj_eq Hf) IHs. Qed. + +Lemma map_inj_uniq s : uniq (map f s) = uniq s. +Proof. apply: map_inj_in_uniq; exact: in2W. Qed. + +End EqMap. + +Implicit Arguments mapP [T1 T2 f s y]. +Prenex Implicits mapP. + +Lemma map_of_seq (T1 : eqType) T2 (s : seq T1) (fs : seq T2) (y0 : T2) : + {f | uniq s -> size fs = size s -> map f s = fs}. +Proof. +exists (fun x => nth y0 fs (index x s)) => uAs eq_sz. +apply/esym/(@eq_from_nth _ y0); rewrite ?size_map eq_sz // => i ltis. +have x0 : T1 by [case: (s) ltis]; by rewrite (nth_map x0) // index_uniq. +Qed. + +Section MapComp. + +Variable T1 T2 T3 : Type. + +Lemma map_id (s : seq T1) : map id s = s. +Proof. by elim: s => //= x s ->. Qed. + +Lemma eq_map (f1 f2 : T1 -> T2) : f1 =1 f2 -> map f1 =1 map f2. +Proof. by move=> Ef; elim=> //= x s ->; rewrite Ef. Qed. + +Lemma map_comp (f1 : T2 -> T3) (f2 : T1 -> T2) s : + map (f1 \o f2) s = map f1 (map f2 s). +Proof. by elim: s => //= x s ->. Qed. + +Lemma mapK (f1 : T1 -> T2) (f2 : T2 -> T1) : + cancel f1 f2 -> cancel (map f1) (map f2). +Proof. by move=> eq_f12; elim=> //= x s ->; rewrite eq_f12. Qed. + +End MapComp. + +Lemma eq_in_map (T1 : eqType) T2 (f1 f2 : T1 -> T2) (s : seq T1) : + {in s, f1 =1 f2} <-> map f1 s = map f2 s. +Proof. +elim: s => //= x s IHs; split=> [eqf12 | [f12x /IHs eqf12]]; last first. + by move=> y /predU1P[-> | /eqf12]. +rewrite eqf12 ?mem_head //; congr (_ :: _). +by apply/IHs=> y s_y; rewrite eqf12 // mem_behead. +Qed. + +Lemma map_id_in (T : eqType) f (s : seq T) : {in s, f =1 id} -> map f s = s. +Proof. by move/eq_in_map->; apply: map_id. Qed. + +(* Map a partial function *) + +Section Pmap. + +Variables (aT rT : Type) (f : aT -> option rT) (g : rT -> aT). + +Fixpoint pmap s := + if s is x :: s' then let r := pmap s' in oapp (cons^~ r) r (f x) else [::]. + +Lemma map_pK : pcancel g f -> cancel (map g) pmap. +Proof. by move=> gK; elim=> //= x s ->; rewrite gK. Qed. + +Lemma size_pmap s : size (pmap s) = count [eta f] s. +Proof. by elim: s => //= x s <-; case: (f _). Qed. + +Lemma pmapS_filter s : map some (pmap s) = map f (filter [eta f] s). +Proof. by elim: s => //= x s; case fx: (f x) => //= [u] <-; congr (_ :: _). Qed. + +Hypothesis fK : ocancel f g. + +Lemma pmap_filter s : map g (pmap s) = filter [eta f] s. +Proof. by elim: s => //= x s <-; rewrite -{3}(fK x); case: (f _). Qed. + +End Pmap. + +Section EqPmap. + +Variables (aT rT : eqType) (f : aT -> option rT) (g : rT -> aT). + +Lemma eq_pmap (f1 f2 : aT -> option rT) : f1 =1 f2 -> pmap f1 =1 pmap f2. +Proof. by move=> Ef; elim=> //= x s ->; rewrite Ef. Qed. + +Lemma mem_pmap s u : (u \in pmap f s) = (Some u \in map f s). +Proof. by elim: s => //= x s IHs; rewrite in_cons -IHs; case: (f x). Qed. + +Hypothesis fK : ocancel f g. + +Lemma can2_mem_pmap : pcancel g f -> forall s u, (u \in pmap f s) = (g u \in s). +Proof. +by move=> gK s u; rewrite -(mem_map (pcan_inj gK)) pmap_filter // mem_filter gK. +Qed. + +Lemma pmap_uniq s : uniq s -> uniq (pmap f s). +Proof. +by move/(filter_uniq [eta f]); rewrite -(pmap_filter fK); exact: map_uniq. +Qed. + +End EqPmap. + +Section PmapSub. + +Variables (T : Type) (p : pred T) (sT : subType p). + +Lemma size_pmap_sub s : size (pmap (insub : T -> option sT) s) = count p s. +Proof. by rewrite size_pmap (eq_count (isSome_insub _)). Qed. + +End PmapSub. + +Section EqPmapSub. + +Variables (T : eqType) (p : pred T) (sT : subType p). + +Let insT : T -> option sT := insub. + +Lemma mem_pmap_sub s u : (u \in pmap insT s) = (val u \in s). +Proof. apply: (can2_mem_pmap (insubK _)); exact: valK. Qed. + +Lemma pmap_sub_uniq s : uniq s -> uniq (pmap insT s). +Proof. exact: (pmap_uniq (insubK _)). Qed. + +End EqPmapSub. + +(* Index sequence *) + +Fixpoint iota m n := if n is n'.+1 then m :: iota m.+1 n' else [::]. + +Lemma size_iota m n : size (iota m n) = n. +Proof. by elim: n m => //= n IHn m; rewrite IHn. Qed. + +Lemma iota_add m n1 n2 : iota m (n1 + n2) = iota m n1 ++ iota (m + n1) n2. +Proof. +by elim: n1 m => //= [|n1 IHn1] m; rewrite ?addn0 // -addSnnS -IHn1. +Qed. + +Lemma iota_addl m1 m2 n : iota (m1 + m2) n = map (addn m1) (iota m2 n). +Proof. by elim: n m2 => //= n IHn m2; rewrite -addnS IHn. Qed. + +Lemma nth_iota m n i : i < n -> nth 0 (iota m n) i = m + i. +Proof. +by move/subnKC <-; rewrite addSnnS iota_add nth_cat size_iota ltnn subnn. +Qed. + +Lemma mem_iota m n i : (i \in iota m n) = (m <= i) && (i < m + n). +Proof. +elim: n m => [|n IHn] /= m; first by rewrite addn0 ltnNge andbN. +rewrite -addSnnS leq_eqVlt in_cons eq_sym. +by case: eqP => [->|_]; [rewrite leq_addr | exact: IHn]. +Qed. + +Lemma iota_uniq m n : uniq (iota m n). +Proof. by elim: n m => //= n IHn m; rewrite mem_iota ltnn /=. Qed. + +(* Making a sequence of a specific length, using indexes to compute items. *) + +Section MakeSeq. + +Variables (T : Type) (x0 : T). + +Definition mkseq f n : seq T := map f (iota 0 n). + +Lemma size_mkseq f n : size (mkseq f n) = n. +Proof. by rewrite size_map size_iota. Qed. + +Lemma eq_mkseq f g : f =1 g -> mkseq f =1 mkseq g. +Proof. by move=> Efg n; exact: eq_map Efg _. Qed. + +Lemma nth_mkseq f n i : i < n -> nth x0 (mkseq f n) i = f i. +Proof. by move=> Hi; rewrite (nth_map 0) ?nth_iota ?size_iota. Qed. + +Lemma mkseq_nth s : mkseq (nth x0 s) (size s) = s. +Proof. +by apply: (@eq_from_nth _ x0); rewrite size_mkseq // => i Hi; rewrite nth_mkseq. +Qed. + +End MakeSeq. + +Section MakeEqSeq. + +Variable T : eqType. + +Lemma mkseq_uniq (f : nat -> T) n : injective f -> uniq (mkseq f n). +Proof. by move/map_inj_uniq->; apply: iota_uniq. Qed. + +Lemma perm_eq_iotaP {s t : seq T} x0 (It := iota 0 (size t)) : + reflect (exists2 Is, perm_eq Is It & s = map (nth x0 t) Is) (perm_eq s t). +Proof. +apply: (iffP idP) => [Est | [Is eqIst ->]]; last first. + by rewrite -{2}[t](mkseq_nth x0) perm_map. +elim: t => [|x t IHt] in s It Est *. + by rewrite (perm_eq_small _ Est) //; exists [::]. +have /rot_to[k s1 Ds]: x \in s by rewrite (perm_eq_mem Est) mem_head. +have [|Is1 eqIst1 Ds1] := IHt s1; first by rewrite -(perm_cons x) -Ds perm_rot. +exists (rotr k (0 :: map succn Is1)). + by rewrite perm_rot /It /= perm_cons (iota_addl 1) perm_map. +by rewrite map_rotr /= -map_comp -(@eq_map _ _ (nth x0 t)) // -Ds1 -Ds rotK. +Qed. + +End MakeEqSeq. + +Implicit Arguments perm_eq_iotaP [[T] [s] [t]]. + +Section FoldRight. + +Variables (T : Type) (R : Type) (f : T -> R -> R) (z0 : R). + +Fixpoint foldr s := if s is x :: s' then f x (foldr s') else z0. + +End FoldRight. + +Section FoldRightComp. + +Variables (T1 T2 : Type) (h : T1 -> T2). +Variables (R : Type) (f : T2 -> R -> R) (z0 : R). + +Lemma foldr_cat s1 s2 : foldr f z0 (s1 ++ s2) = foldr f (foldr f z0 s2) s1. +Proof. by elim: s1 => //= x s1 ->. Qed. + +Lemma foldr_map s : foldr f z0 (map h s) = foldr (fun x z => f (h x) z) z0 s. +Proof. by elim: s => //= x s ->. Qed. + +End FoldRightComp. + +(* Quick characterization of the null sequence. *) + +Definition sumn := foldr addn 0. + +Lemma sumn_nseq x n : sumn (nseq n x) = x * n. +Proof. by rewrite mulnC; elim: n => //= n ->. Qed. + +Lemma sumn_cat s1 s2 : sumn (s1 ++ s2) = sumn s1 + sumn s2. +Proof. by elim: s1 => //= x s1 ->; rewrite addnA. Qed. + +Lemma natnseq0P s : reflect (s = nseq (size s) 0) (sumn s == 0). +Proof. +apply: (iffP idP) => [|->]; last by rewrite sumn_nseq. +by elim: s => //= x s IHs; rewrite addn_eq0 => /andP[/eqP-> /IHs <-]. +Qed. + +Section FoldLeft. + +Variables (T R : Type) (f : R -> T -> R). + +Fixpoint foldl z s := if s is x :: s' then foldl (f z x) s' else z. + +Lemma foldl_rev z s : foldl z (rev s) = foldr (fun x z => f z x) z s. +Proof. +elim/last_ind: s z => [|s x IHs] z //=. +by rewrite rev_rcons -cats1 foldr_cat -IHs. +Qed. + +Lemma foldl_cat z s1 s2 : foldl z (s1 ++ s2) = foldl (foldl z s1) s2. +Proof. +by rewrite -(revK (s1 ++ s2)) foldl_rev rev_cat foldr_cat -!foldl_rev !revK. +Qed. + +End FoldLeft. + +Section Scan. + +Variables (T1 : Type) (x1 : T1) (T2 : Type) (x2 : T2). +Variables (f : T1 -> T1 -> T2) (g : T1 -> T2 -> T1). + +Fixpoint pairmap x s := if s is y :: s' then f x y :: pairmap y s' else [::]. + +Lemma size_pairmap x s : size (pairmap x s) = size s. +Proof. by elim: s x => //= y s IHs x; rewrite IHs. Qed. + +Lemma pairmap_cat x s1 s2 : + pairmap x (s1 ++ s2) = pairmap x s1 ++ pairmap (last x s1) s2. +Proof. by elim: s1 x => //= y s1 IHs1 x; rewrite IHs1. Qed. + +Lemma nth_pairmap s n : n < size s -> + forall x, nth x2 (pairmap x s) n = f (nth x1 (x :: s) n) (nth x1 s n). +Proof. by elim: s n => [|y s IHs] [|n] //= Hn x; apply: IHs. Qed. + +Fixpoint scanl x s := + if s is y :: s' then let x' := g x y in x' :: scanl x' s' else [::]. + +Lemma size_scanl x s : size (scanl x s) = size s. +Proof. by elim: s x => //= y s IHs x; rewrite IHs. Qed. + +Lemma scanl_cat x s1 s2 : + scanl x (s1 ++ s2) = scanl x s1 ++ scanl (foldl g x s1) s2. +Proof. by elim: s1 x => //= y s1 IHs1 x; rewrite IHs1. Qed. + +Lemma nth_scanl s n : n < size s -> + forall x, nth x1 (scanl x s) n = foldl g x (take n.+1 s). +Proof. by elim: s n => [|y s IHs] [|n] Hn x //=; rewrite ?take0 ?IHs. Qed. + +Lemma scanlK : + (forall x, cancel (g x) (f x)) -> forall x, cancel (scanl x) (pairmap x). +Proof. by move=> Hfg x s; elim: s x => //= y s IHs x; rewrite Hfg IHs. Qed. + +Lemma pairmapK : + (forall x, cancel (f x) (g x)) -> forall x, cancel (pairmap x) (scanl x). +Proof. by move=> Hgf x s; elim: s x => //= y s IHs x; rewrite Hgf IHs. Qed. + +End Scan. + +Prenex Implicits mask map pmap foldr foldl scanl pairmap. + +Section Zip. + +Variables S T : Type. + +Fixpoint zip (s : seq S) (t : seq T) {struct t} := + match s, t with + | x :: s', y :: t' => (x, y) :: zip s' t' + | _, _ => [::] + end. + +Definition unzip1 := map (@fst S T). +Definition unzip2 := map (@snd S T). + +Lemma zip_unzip s : zip (unzip1 s) (unzip2 s) = s. +Proof. by elim: s => [|[x y] s /= ->]. Qed. + +Lemma unzip1_zip s t : size s <= size t -> unzip1 (zip s t) = s. +Proof. by elim: s t => [|x s IHs] [|y t] //= le_s_t; rewrite IHs. Qed. + +Lemma unzip2_zip s t : size t <= size s -> unzip2 (zip s t) = t. +Proof. by elim: s t => [|x s IHs] [|y t] //= le_t_s; rewrite IHs. Qed. + +Lemma size1_zip s t : size s <= size t -> size (zip s t) = size s. +Proof. by elim: s t => [|x s IHs] [|y t] //= Hs; rewrite IHs. Qed. + +Lemma size2_zip s t : size t <= size s -> size (zip s t) = size t. +Proof. by elim: s t => [|x s IHs] [|y t] //= Hs; rewrite IHs. Qed. + +Lemma size_zip s t : size (zip s t) = minn (size s) (size t). +Proof. +by elim: s t => [|x s IHs] [|t2 t] //=; rewrite IHs -add1n addn_minr. +Qed. + +Lemma zip_cat s1 s2 t1 t2 : + size s1 = size t1 -> zip (s1 ++ s2) (t1 ++ t2) = zip s1 t1 ++ zip s2 t2. +Proof. by elim: s1 t1 => [|x s IHs] [|y t] //= [/IHs->]. Qed. + +Lemma nth_zip x y s t i : + size s = size t -> nth (x, y) (zip s t) i = (nth x s i, nth y t i). +Proof. by elim: i s t => [|i IHi] [|y1 s1] [|y2 t] //= [/IHi->]. Qed. + +Lemma nth_zip_cond p s t i : + nth p (zip s t) i + = (if i < size (zip s t) then (nth p.1 s i, nth p.2 t i) else p). +Proof. +rewrite size_zip ltnNge geq_min. +by elim: s t i => [|x s IHs] [|y t] [|i] //=; rewrite ?orbT -?IHs. +Qed. + +Lemma zip_rcons s1 s2 z1 z2 : + size s1 = size s2 -> + zip (rcons s1 z1) (rcons s2 z2) = rcons (zip s1 s2) (z1, z2). +Proof. by move=> eq_sz; rewrite -!cats1 zip_cat //= eq_sz. Qed. + +Lemma rev_zip s1 s2 : + size s1 = size s2 -> rev (zip s1 s2) = zip (rev s1) (rev s2). +Proof. +elim: s1 s2 => [|x s1 IHs] [|y s2] //= [eq_sz]. +by rewrite !rev_cons zip_rcons ?IHs ?size_rev. +Qed. + +End Zip. + +Prenex Implicits zip unzip1 unzip2. + +Section Flatten. + +Variable T : Type. +Implicit Types (s : seq T) (ss : seq (seq T)). + +Definition flatten := foldr cat (Nil T). +Definition shape := map (@size T). +Fixpoint reshape sh s := + if sh is n :: sh' then take n s :: reshape sh' (drop n s) else [::]. + +Lemma size_flatten ss : size (flatten ss) = sumn (shape ss). +Proof. by elim: ss => //= s ss <-; rewrite size_cat. Qed. + +Lemma flatten_cat ss1 ss2 : + flatten (ss1 ++ ss2) = flatten ss1 ++ flatten ss2. +Proof. by elim: ss1 => //= s ss1 ->; rewrite catA. Qed. + +Lemma flattenK ss : reshape (shape ss) (flatten ss) = ss. +Proof. +by elim: ss => //= s ss IHss; rewrite take_size_cat ?drop_size_cat ?IHss. +Qed. + +Lemma reshapeKr sh s : size s <= sumn sh -> flatten (reshape sh s) = s. +Proof. +elim: sh s => [[]|n sh IHsh] //= s sz_s; rewrite IHsh ?cat_take_drop //. +by rewrite size_drop leq_subLR. +Qed. + +Lemma reshapeKl sh s : size s >= sumn sh -> shape (reshape sh s) = sh. +Proof. +elim: sh s => [[]|n sh IHsh] //= s sz_s. +rewrite size_takel; last exact: leq_trans (leq_addr _ _) sz_s. +by rewrite IHsh // -(leq_add2l n) size_drop -maxnE leq_max sz_s orbT. +Qed. + +End Flatten. + +Prenex Implicits flatten shape reshape. + +Section EqFlatten. + +Variables S T : eqType. + +Lemma flattenP (A : seq (seq T)) x : + reflect (exists2 s, s \in A & x \in s) (x \in flatten A). +Proof. +elim: A => /= [|s A /iffP IH_A]; [by right; case | rewrite mem_cat]. +have [s_x|s'x] := @idP (x \in s); first by left; exists s; rewrite ?mem_head. +by apply: IH_A => [[t] | [t /predU1P[->|]]]; exists t; rewrite // mem_behead. +Qed. +Implicit Arguments flattenP [A x]. + +Lemma flatten_mapP (A : S -> seq T) s y : + reflect (exists2 x, x \in s & y \in A x) (y \in flatten (map A s)). +Proof. +apply: (iffP flattenP) => [[_ /mapP[x sx ->]] | [x sx]] Axy; first by exists x. +by exists (A x); rewrite ?map_f. +Qed. + +End EqFlatten. + +Implicit Arguments flattenP [T A x]. +Implicit Arguments flatten_mapP [S T A s y]. + +Lemma perm_undup_count (T : eqType) (s : seq T) : + perm_eq (flatten [seq nseq (count_mem x s) x | x <- undup s]) s. +Proof. +pose N x r := count_mem x (flatten [seq nseq (count_mem y s) y | y <- r]). +apply/allP=> x _; rewrite /= -/(N x _). +have Nx0 r (r'x : x \notin r): N x r = 0. + by apply/count_memPn; apply: contra r'x => /flatten_mapP[y r_y /nseqP[->]]. +have [|s'x] := boolP (x \in s); last by rewrite Nx0 ?mem_undup ?(count_memPn _). +rewrite -mem_undup => /perm_to_rem/catCA_perm_subst->; last first. + by move=> s1 s2 s3; rewrite /N !map_cat !flatten_cat !count_cat addnCA. +rewrite /N /= count_cat -/(N x _) Nx0 ?mem_rem_uniq ?undup_uniq ?inE ?eqxx //. +by rewrite addn0 -{2}(size_nseq (_ s) x) -all_count all_pred1_nseq. +Qed. + +Section AllPairs. + +Variables (S T R : Type) (f : S -> T -> R). +Implicit Types (s : seq S) (t : seq T). + +Definition allpairs s t := foldr (fun x => cat (map (f x) t)) [::] s. + +Lemma size_allpairs s t : size (allpairs s t) = size s * size t. +Proof. by elim: s => //= x s IHs; rewrite size_cat size_map IHs. Qed. + +Lemma allpairs_cat s1 s2 t : + allpairs (s1 ++ s2) t = allpairs s1 t ++ allpairs s2 t. +Proof. by elim: s1 => //= x s1 ->; rewrite catA. Qed. + +End AllPairs. + +Prenex Implicits allpairs. + +Notation "[ 'seq' E | i <- s , j <- t ]" := (allpairs (fun i j => E) s t) + (at level 0, E at level 99, i ident, j ident, + format "[ '[hv' 'seq' E '/ ' | i <- s , '/ ' j <- t ] ']'") + : seq_scope. +Notation "[ 'seq' E | i : T <- s , j : U <- t ]" := + (allpairs (fun (i : T) (j : U) => E) s t) + (at level 0, E at level 99, i ident, j ident, only parsing) : seq_scope. + +Section EqAllPairs. + +Variables S T : eqType. +Implicit Types (R : eqType) (s : seq S) (t : seq T). + +Lemma allpairsP R (f : S -> T -> R) s t z : + reflect (exists p, [/\ p.1 \in s, p.2 \in t & z = f p.1 p.2]) + (z \in allpairs f s t). +Proof. +elim: s => [|x s IHs /=]; first by right=> [[p []]]. +rewrite mem_cat; have [fxt_z | not_fxt_z] := altP mapP. + by left; have [y t_y ->] := fxt_z; exists (x, y); rewrite mem_head. +apply: (iffP IHs) => [] [[x' y] /= [s_x' t_y def_z]]; exists (x', y). + by rewrite !inE predU1r. +by have [def_x' | //] := predU1P s_x'; rewrite def_z def_x' map_f in not_fxt_z. +Qed. + +Lemma mem_allpairs R (f : S -> T -> R) s1 t1 s2 t2 : + s1 =i s2 -> t1 =i t2 -> allpairs f s1 t1 =i allpairs f s2 t2. +Proof. +move=> eq_s eq_t z. +by apply/allpairsP/allpairsP=> [] [p fpz]; exists p; rewrite eq_s eq_t in fpz *. +Qed. + +Lemma allpairs_catr R (f : S -> T -> R) s t1 t2 : + allpairs f s (t1 ++ t2) =i allpairs f s t1 ++ allpairs f s t2. +Proof. +move=> z; rewrite mem_cat. +apply/allpairsP/orP=> [[p [sP1]]|]. + by rewrite mem_cat; case/orP; [left | right]; apply/allpairsP; exists p. +by case=> /allpairsP[p [sp1 sp2 ->]]; exists p; rewrite mem_cat sp2 ?orbT. +Qed. + +Lemma allpairs_uniq R (f : S -> T -> R) s t : + uniq s -> uniq t -> + {in [seq (x, y) | x <- s, y <- t] &, injective (prod_curry f)} -> + uniq (allpairs f s t). +Proof. +move=> Us Ut inj_f; have: all (mem s) s by exact/allP. +elim: {-2}s Us => //= x s1 IHs /andP[s1'x Us1] /andP[sx1 ss1]. +rewrite cat_uniq {}IHs // andbT map_inj_in_uniq ?Ut // => [|y1 y2 *]. + apply/hasPn=> _ /allpairsP[z [s1z tz ->]]; apply/mapP=> [[y ty Dy]]. + suffices [Dz1 _]: (z.1, z.2) = (x, y) by rewrite -Dz1 s1z in s1'x. + apply: inj_f => //; apply/allpairsP; last by exists (x, y). + by have:= allP ss1 _ s1z; exists z. +suffices: (x, y1) = (x, y2) by case. +by apply: inj_f => //; apply/allpairsP; [exists (x, y1) | exists (x, y2)]. +Qed. + +End EqAllPairs. diff --git a/mathcomp/ssreflect/ssrbool.v b/mathcomp/ssreflect/ssrbool.v new file mode 100644 index 0000000..aeaa266 --- /dev/null +++ b/mathcomp/ssreflect/ssrbool.v @@ -0,0 +1,1818 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Require Import ssreflect ssrfun. + +(******************************************************************************) +(* A theory of boolean predicates and operators. A large part of this file is *) +(* concerned with boolean reflection. *) +(* Definitions and notations: *) +(* is_true b == the coercion of b : bool to Prop (:= b = true). *) +(* This is just input and displayed as `b''. *) +(* reflect P b == the reflection inductive predicate, asserting *) +(* that the logical proposition P : prop with the *) +(* formula b : bool. Lemmas asserting reflect P b *) +(* are often referred to as "views". *) +(* iffP, appP, sameP, rwP :: lemmas for direct manipulation of reflection *) +(* views: iffP is used to prove reflection from *) +(* logical equivalence, appP to compose views, and *) +(* sameP and rwP to perform boolean and setoid *) +(* rewriting. *) +(* elimT :: coercion reflect >-> Funclass, which allows the *) +(* direct application of `reflect' views to *) +(* boolean assertions. *) +(* decidable P <-> P is effectively decidable (:= {P} + {~ P}. *) +(* contra, contraL, ... :: contraposition lemmas. *) +(* altP my_viewP :: natural alternative for reflection; given *) +(* lemma myvieP: reflect my_Prop my_formula, *) +(* have [myP | not_myP] := altP my_viewP. *) +(* generates two subgoals, in which my_formula has *) +(* been replaced by true and false, resp., with *) +(* new assumptions myP : my_Prop and *) +(* not_myP: ~~ my_formula. *) +(* Caveat: my_formula must be an APPLICATION, not *) +(* a variable, constant, let-in, etc. (due to the *) +(* poor behaviour of dependent index matching). *) +(* boolP my_formula :: boolean disjunction, equivalent to *) +(* altP (idP my_formula) but circumventing the *) +(* dependent index capture issue; destructing *) +(* boolP my_formula generates two subgoals with *) +(* assumtions my_formula and ~~ myformula. As *) +(* with altP, my_formula must be an application. *) +(* unless C P <-> hP : P may be assumed when proving P. *) +(* := (P -> C) -> C (Pierce's law). *) +(* This is slightly weaker but easier to use than *) +(* P \/ C when P C : Prop. *) +(* classically P <-> hP : P can be assumed when proving is_true b *) +(* := forall b : bool, (P -> b) -> b. *) +(* This is equivalent to ~ (~ P) when P : Prop. *) +(* a && b == the boolean conjunction of a and b. *) +(* a || b == then boolean disjunction of a and b. *) +(* a ==> b == the boolean implication of b by a. *) +(* ~~ a == the boolean negation of a. *) +(* a (+) b == the boolean exclusive or (or sum) of a and b. *) +(* [ /\ P1 , P2 & P3 ] == multiway logical conjunction, up to 5 terms. *) +(* [ \/ P1 , P2 | P3 ] == multiway logical disjunction, up to 4 terms. *) +(* [&& a, b, c & d] == iterated, right associative boolean conjunction *) +(* with arbitrary arity. *) +(* [|| a, b, c | d] == iterated, right associative boolean disjunction *) +(* with arbitrary arity. *) +(* [==> a, b, c => d] == iterated, right associative boolean implication *) +(* with arbitrary arity. *) +(* and3P, ... == specific reflection lemmas for iterated *) +(* connectives. *) +(* andTb, orbAC, ... == systematic names for boolean connective *) +(* properties (see suffix conventions below). *) +(* prop_congr == a tactic to move a boolean equality from *) +(* its coerced form in Prop to the equality *) +(* in bool. *) +(* bool_congr == resolution tactic for blindly weeding out *) +(* like terms from boolean equalities (can fail). *) +(* This file provides a theory of boolean predicates and relations: *) +(* pred T == the type of bool predicates (:= T -> bool). *) +(* simpl_pred T == the type of simplifying bool predicates, using *) +(* the simpl_fun from ssrfun.v. *) +(* rel T == the type of bool relations. *) +(* := T -> pred T or T -> T -> bool. *) +(* simpl_rel T == type of simplifying relations. *) +(* predType == the generic predicate interface, supported for *) +(* for lists and sets. *) +(* pred_class == a coercion class for the predType projection to *) +(* pred; declaring a coercion to pred_class is an *) +(* alternative way of equipping a type with a *) +(* predType structure, which interoperates better *) +(* with coercion subtyping. This is used, e.g., *) +(* for finite sets, so that finite groups inherit *) +(* the membership operation by coercing to sets. *) +(* If P is a predicate the proposition "x satisfies P" can be written *) +(* applicatively as (P x), or using an explicit connective as (x \in P); in *) +(* the latter case we say that P is a "collective" predicate. We use A, B *) +(* rather than P, Q for collective predicates: *) +(* x \in A == x satisfies the (collective) predicate A. *) +(* x \notin A == x doesn't satisfy the (collective) predicate A. *) +(* The pred T type can be used as a generic predicate type for either kind, *) +(* but the two kinds of predicates should not be confused. When a "generic" *) +(* pred T value of one type needs to be passed as the other the following *) +(* conversions should be used explicitly: *) +(* SimplPred P == a (simplifying) applicative equivalent of P. *) +(* mem A == an applicative equivalent of A: *) +(* mem A x simplifies to x \in A. *) +(* Alternatively one can use the syntax for explicit simplifying predicates *) +(* and relations (in the following x is bound in E): *) +(* [pred x | E] == simplifying (see ssrfun) predicate x => E. *) +(* [pred x : T | E] == predicate x => T, with a cast on the argument. *) +(* [pred : T | P] == constant predicate P on type T. *) +(* [pred x | E1 & E2] == [pred x | E1 && E2]; an x : T cast is allowed. *) +(* [pred x in A] == [pred x | x in A]. *) +(* [pred x in A | E] == [pred x | x in A & E]. *) +(* [pred x in A | E1 & E2] == [pred x in A | E1 && E2]. *) +(* [predU A & B] == union of two collective predicates A and B. *) +(* [predI A & B] == intersection of collective predicates A and B. *) +(* [predD A & B] == difference of collective predicates A and B. *) +(* [predC A] == complement of the collective predicate A. *) +(* [preim f of A] == preimage under f of the collective predicate A. *) +(* predU P Q, ... == union, etc of applicative predicates. *) +(* pred0 == the empty predicate. *) +(* predT == the total (always true) predicate. *) +(* if T : predArgType, then T coerces to predT. *) +(* {: T} == T cast to predArgType (e.g., {: bool * nat}) *) +(* In the following, x and y are bound in E: *) +(* [rel x y | E] == simplifying relation x, y => E. *) +(* [rel x y : T | E] == simplifying relation with arguments cast. *) +(* [rel x y in A & B | E] == [rel x y | [&& x \in A, y \in B & E]]. *) +(* [rel x y in A & B] == [rel x y | (x \in A) && (y \in B)]. *) +(* [rel x y in A | E] == [rel x y in A & A | E]. *) +(* [rel x y in A] == [rel x y in A & A]. *) +(* relU R S == union of relations R and S. *) +(* Explicit values of type pred T (i.e., lamdba terms) should always be used *) +(* applicatively, while values of collection types implementing the predType *) +(* interface, such as sequences or sets should always be used as collective *) +(* predicates. Defined constants and functions of type pred T or simpl_pred T *) +(* as well as the explicit simpl_pred T values described below, can generally *) +(* be used either way. Note however that x \in A will not auto-simplify when *) +(* A is an explicit simpl_pred T value; the generic simplification rule inE *) +(* must be used (when A : pred T, the unfold_in rule can be used). Constants *) +(* of type pred T with an explicit simpl_pred value do not auto-simplify when *) +(* used applicatively, but can still be expanded with inE. This behavior can *) +(* be controlled as follows: *) +(* Let A : collective_pred T := [pred x | ... ]. *) +(* The collective_pred T type is just an alias for pred T, but this cast *) +(* stops rewrite inE from expanding the definition of A, thus treating A *) +(* into an abstract collection (unfold_in or in_collective can be used to *) +(* expand manually). *) +(* Let A : applicative_pred T := [pred x | ...]. *) +(* This cast causes inE to turn x \in A into the applicative A x form; *) +(* A will then have to unfolded explicitly with the /A rule. This will *) +(* also apply to any definition that reduces to A (e.g., Let B := A). *) +(* Canonical A_app_pred := ApplicativePred A. *) +(* This declaration, given after definition of A, similarly causes inE to *) +(* turn x \in A into A x, but in addition allows the app_predE rule to *) +(* turn A x back into x \in A; it can be used for any definition of type *) +(* pred T, which makes it especially useful for ambivalent predicates *) +(* as the relational transitive closure connect, that are used in both *) +(* applicative and collective styles. *) +(* Purely for aesthetics, we provide a subtype of collective predicates: *) +(* qualifier q T == a pred T pretty-printing wrapper. An A : qualifier q T *) +(* coerces to pred_class and thus behaves as a collective *) +(* predicate, but x \in A and x \notin A are displayed as: *) +(* x \is A and x \isn't A when q = 0, *) +(* x \is a A and x \isn't a A when q = 1, *) +(* x \is an A and x \isn't an A when q = 2, respectively. *) +(* [qualify x | P] := Qualifier 0 (fun x => P), constructor for the above. *) +(* [qualify x : T | P], [qualify a x | P], [qualify an X | P], etc. *) +(* variants of the above with type constraints and different *) +(* values of q. *) +(* We provide an internal interface to support attaching properties (such as *) +(* being multiplicative) to predicates: *) +(* pred_key p == phantom type that will serve as a support for properties *) +(* to be attached to p : pred_class; instances should be *) +(* created with Fact/Qed so as to be opaque. *) +(* KeyedPred k_p == an instance of the interface structure that attaches *) +(* (k_p : pred_key P) to P; the structure projection is a *) +(* coercion to pred_class. *) +(* KeyedQualifier k_q == an instance of the interface structure that attaches *) +(* (k_q : pred_key q) to (q : qualifier n T). *) +(* DefaultPredKey p == a default value for pred_key p; the vernacular command *) +(* Import DefaultKeying attaches this key to all predicates *) +(* that are not explicitly keyed. *) +(* Keys can be used to attach properties to predicates, qualifiers and *) +(* generic nouns in a way that allows them to be used transparently. The key *) +(* projection of a predicate property structure such as unsignedPred should *) +(* be a pred_key, not a pred, and corresponding lemmas will have the form *) +(* Lemma rpredN R S (oppS : @opprPred R S) (kS : keyed_pred oppS) : *) +(* {mono -%R: x / x \in kS}. *) +(* Because x \in kS will be displayed as x \in S (or x \is S, etc), the *) +(* canonical instance of opprPred will not normally be exposed (it will also *) +(* be erased by /= simplification). In addition each predicate structure *) +(* should have a DefaultPredKey Canonical instance that simply issues the *) +(* property as a proof obligation (which can be caught by the Prop-irrelevant *) +(* feature of the ssreflect plugin). *) +(* Some properties of predicates and relations: *) +(* A =i B <-> A and B are extensionally equivalent. *) +(* {subset A <= B} <-> A is a (collective) subpredicate of B. *) +(* subpred P Q <-> P is an (applicative) subpredicate or Q. *) +(* subrel R S <-> R is a subrelation of S. *) +(* In the following R is in rel T: *) +(* reflexive R <-> R is reflexive. *) +(* irreflexive R <-> R is irreflexive. *) +(* symmetric R <-> R (in rel T) is symmetric (equation). *) +(* pre_symmetric R <-> R is symmetric (implication). *) +(* antisymmetric R <-> R is antisymmetric. *) +(* total R <-> R is total. *) +(* transitive R <-> R is transitive. *) +(* left_transitive R <-> R is a congruence on its left hand side. *) +(* right_transitive R <-> R is a congruence on its right hand side. *) +(* equivalence_rel R <-> R is an equivalence relation. *) +(* Localization of (Prop) predicates; if P1 is convertible to forall x, Qx, *) +(* P2 to forall x y, Qxy and P3 to forall x y z, Qxyz : *) +(* {for y, P1} <-> Qx{y / x}. *) +(* {in A, P1} <-> forall x, x \in A -> Qx. *) +(* {in A1 & A2, P2} <-> forall x y, x \in A1 -> y \in A2 -> Qxy. *) +(* {in A &, P2} <-> forall x y, x \in A -> y \in A -> Qxy. *) +(* {in A1 & A2 & A3, Q3} <-> forall x y z, *) +(* x \in A1 -> y \in A2 -> z \in A3 -> Qxyz. *) +(* {in A1 & A2 &, Q3} == {in A1 & A2 & A2, Q3}. *) +(* {in A1 && A3, Q3} == {in A1 & A1 & A3, Q3}. *) +(* {in A &&, Q3} == {in A & A & A, Q3}. *) +(* {in A, bijective f} == f has a right inverse in A. *) +(* {on C, P1} == forall x, (f x) \in C -> Qx *) +(* when P1 is also convertible to Pf f. *) +(* {on C &, P2} == forall x y, f x \in C -> f y \in C -> Qxy *) +(* when P2 is also convertible to Pf f. *) +(* {on C, P1' & g} == forall x, (f x) \in cd -> Qx *) +(* when P1' is convertible to Pf f *) +(* and P1' g is convertible to forall x, Qx. *) +(* {on C, bijective f} == f has a right inverse on C. *) +(* This file extends the lemma name suffix conventions of ssrfun as follows: *) +(* A -- associativity, as in andbA : associative andb. *) +(* AC -- right commutativity. *) +(* ACA -- self-interchange (inner commutativity), e.g., *) +(* orbACA : (a || b) || (c || d) = (a || c) || (b || d). *) +(* b -- a boolean argument, as in andbb : idempotent andb. *) +(* C -- commutativity, as in andbC : commutative andb, *) +(* or predicate complement, as in predC. *) +(* CA -- left commutativity. *) +(* D -- predicate difference, as in predD. *) +(* E -- elimination, as in negbEf : ~~ b = false -> b. *) +(* F or f -- boolean false, as in andbF : b && false = false. *) +(* I -- left/right injectivity, as in addbI : right_injective addb, *) +(* or predicate intersection, as in predI. *) +(* l -- a left-hand operation, as andb_orl : left_distributive andb orb. *) +(* N or n -- boolean negation, as in andbN : a && (~~ a) = false. *) +(* P -- a characteristic property, often a reflection lemma, as in *) +(* andP : reflect (a /\ b) (a && b). *) +(* r -- a right-hand operation, as orb_andr : rightt_distributive orb andb. *) +(* T or t -- boolean truth, as in andbT: right_id true andb. *) +(* U -- predicate union, as in predU. *) +(* W -- weakening, as in in1W : {in D, forall x, P} -> forall x, P. *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Reserved Notation "~~ b" (at level 35, right associativity). +Reserved Notation "b ==> c" (at level 55, right associativity). +Reserved Notation "b1 (+) b2" (at level 50, left associativity). +Reserved Notation "x \in A" + (at level 70, format "'[hv' x '/ ' \in A ']'", no associativity). +Reserved Notation "x \notin A" + (at level 70, format "'[hv' x '/ ' \notin A ']'", no associativity). +Reserved Notation "p1 =i p2" + (at level 70, format "'[hv' p1 '/ ' =i p2 ']'", no associativity). + +(* We introduce a number of n-ary "list-style" notations that share a common *) +(* format, namely *) +(* [op arg1, arg2, ... last_separator last_arg] *) +(* This usually denotes a right-associative applications of op, e.g., *) +(* [&& a, b, c & d] denotes a && (b && (c && d)) *) +(* The last_separator must be a non-operator token. Here we use &, | or =>; *) +(* our default is &, but we try to match the intended meaning of op. The *) +(* separator is a workaround for limitations of the parsing engine; the same *) +(* limitations mean the separator cannot be omitted even when last_arg can. *) +(* The Notation declarations are complicated by the separate treatment for *) +(* some fixed arities (binary for bool operators, and all arities for Prop *) +(* operators). *) +(* We also use the square brackets in comprehension-style notations *) +(* [type var separator expr] *) +(* where "type" is the type of the comprehension (e.g., pred) and "separator" *) +(* is | or => . It is important that in other notations a leading square *) +(* bracket [ is always by an operator symbol or a fixed identifier. *) + +Reserved Notation "[ /\ P1 & P2 ]" (at level 0, only parsing). +Reserved Notation "[ /\ P1 , P2 & P3 ]" (at level 0, format + "'[hv' [ /\ '[' P1 , '/' P2 ']' '/ ' & P3 ] ']'"). +Reserved Notation "[ /\ P1 , P2 , P3 & P4 ]" (at level 0, format + "'[hv' [ /\ '[' P1 , '/' P2 , '/' P3 ']' '/ ' & P4 ] ']'"). +Reserved Notation "[ /\ P1 , P2 , P3 , P4 & P5 ]" (at level 0, format + "'[hv' [ /\ '[' P1 , '/' P2 , '/' P3 , '/' P4 ']' '/ ' & P5 ] ']'"). + +Reserved Notation "[ \/ P1 | P2 ]" (at level 0, only parsing). +Reserved Notation "[ \/ P1 , P2 | P3 ]" (at level 0, format + "'[hv' [ \/ '[' P1 , '/' P2 ']' '/ ' | P3 ] ']'"). +Reserved Notation "[ \/ P1 , P2 , P3 | P4 ]" (at level 0, format + "'[hv' [ \/ '[' P1 , '/' P2 , '/' P3 ']' '/ ' | P4 ] ']'"). + +Reserved Notation "[ && b1 & c ]" (at level 0, only parsing). +Reserved Notation "[ && b1 , b2 , .. , bn & c ]" (at level 0, format + "'[hv' [ && '[' b1 , '/' b2 , '/' .. , '/' bn ']' '/ ' & c ] ']'"). + +Reserved Notation "[ || b1 | c ]" (at level 0, only parsing). +Reserved Notation "[ || b1 , b2 , .. , bn | c ]" (at level 0, format + "'[hv' [ || '[' b1 , '/' b2 , '/' .. , '/' bn ']' '/ ' | c ] ']'"). + +Reserved Notation "[ ==> b1 => c ]" (at level 0, only parsing). +Reserved Notation "[ ==> b1 , b2 , .. , bn => c ]" (at level 0, format + "'[hv' [ ==> '[' b1 , '/' b2 , '/' .. , '/' bn ']' '/' => c ] ']'"). + +Reserved Notation "[ 'pred' : T => E ]" (at level 0, format + "'[hv' [ 'pred' : T => '/ ' E ] ']'"). +Reserved Notation "[ 'pred' x => E ]" (at level 0, x at level 8, format + "'[hv' [ 'pred' x => '/ ' E ] ']'"). +Reserved Notation "[ 'pred' x : T => E ]" (at level 0, x at level 8, format + "'[hv' [ 'pred' x : T => '/ ' E ] ']'"). + +Reserved Notation "[ 'rel' x y => E ]" (at level 0, x, y at level 8, format + "'[hv' [ 'rel' x y => '/ ' E ] ']'"). +Reserved Notation "[ 'rel' x y : T => E ]" (at level 0, x, y at level 8, format + "'[hv' [ 'rel' x y : T => '/ ' E ] ']'"). + +(* Shorter delimiter *) +Delimit Scope bool_scope with B. +Open Scope bool_scope. + +(* An alternative to xorb that behaves somewhat better wrt simplification. *) +Definition addb b := if b then negb else id. + +(* Notation for && and || is declared in Init.Datatypes. *) +Notation "~~ b" := (negb b) : bool_scope. +Notation "b ==> c" := (implb b c) : bool_scope. +Notation "b1 (+) b2" := (addb b1 b2) : bool_scope. + +(* Constant is_true b := b = true is defined in Init.Datatypes. *) +Coercion is_true : bool >-> Sortclass. (* Prop *) + +Lemma prop_congr : forall b b' : bool, b = b' -> b = b' :> Prop. +Proof. by move=> b b' ->. Qed. + +Ltac prop_congr := apply: prop_congr. + +(* Lemmas for trivial. *) +Lemma is_true_true : true. Proof. by []. Qed. +Lemma not_false_is_true : ~ false. Proof. by []. Qed. +Lemma is_true_locked_true : locked true. Proof. by unlock. Qed. +Hint Resolve is_true_true not_false_is_true is_true_locked_true. + +(* Shorter names. *) +Definition isT := is_true_true. +Definition notF := not_false_is_true. + +(* Negation lemmas. *) + +(* We generally take NEGATION as the standard form of a false condition: *) +(* negative boolean hypotheses should be of the form ~~ b, rather than ~ b or *) +(* b = false, as much as possible. *) + +Lemma negbT b : b = false -> ~~ b. Proof. by case: b. Qed. +Lemma negbTE b : ~~ b -> b = false. Proof. by case: b. Qed. +Lemma negbF b : (b : bool) -> ~~ b = false. Proof. by case: b. Qed. +Lemma negbFE b : ~~ b = false -> b. Proof. by case: b. Qed. +Lemma negbK : involutive negb. Proof. by case. Qed. +Lemma negbNE b : ~~ ~~ b -> b. Proof. by case: b. Qed. + +Lemma negb_inj : injective negb. Proof. exact: can_inj negbK. Qed. +Lemma negbLR b c : b = ~~ c -> ~~ b = c. Proof. exact: canLR negbK. Qed. +Lemma negbRL b c : ~~ b = c -> b = ~~ c. Proof. exact: canRL negbK. Qed. + +Lemma contra (c b : bool) : (c -> b) -> ~~ b -> ~~ c. +Proof. by case: b => //; case: c. Qed. +Definition contraNN := contra. + +Lemma contraL (c b : bool) : (c -> ~~ b) -> b -> ~~ c. +Proof. by case: b => //; case: c. Qed. +Definition contraTN := contraL. + +Lemma contraR (c b : bool) : (~~ c -> b) -> ~~ b -> c. +Proof. by case: b => //; case: c. Qed. +Definition contraNT := contraR. + +Lemma contraLR (c b : bool) : (~~ c -> ~~ b) -> b -> c. +Proof. by case: b => //; case: c. Qed. +Definition contraTT := contraLR. + +Lemma contraT b : (~~ b -> false) -> b. Proof. by case: b => // ->. Qed. + +Lemma wlog_neg b : (~~ b -> b) -> b. Proof. by case: b => // ->. Qed. + +Lemma contraFT (c b : bool) : (~~ c -> b) -> b = false -> c. +Proof. by move/contraR=> notb_c /negbT. Qed. + +Lemma contraFN (c b : bool) : (c -> b) -> b = false -> ~~ c. +Proof. by move/contra=> notb_notc /negbT. Qed. + +Lemma contraTF (c b : bool) : (c -> ~~ b) -> b -> c = false. +Proof. by move/contraL=> b_notc /b_notc/negbTE. Qed. + +Lemma contraNF (c b : bool) : (c -> b) -> ~~ b -> c = false. +Proof. by move/contra=> notb_notc /notb_notc/negbTE. Qed. + +Lemma contraFF (c b : bool) : (c -> b) -> b = false -> c = false. +Proof. by move/contraFN=> bF_notc /bF_notc/negbTE. Qed. + +(* Coercion of sum-style datatypes into bool, which makes it possible *) +(* to use ssr's boolean if rather than Coq's "generic" if. *) + +Coercion isSome T (u : option T) := if u is Some _ then true else false. + +Coercion is_inl A B (u : A + B) := if u is inl _ then true else false. + +Coercion is_left A B (u : {A} + {B}) := if u is left _ then true else false. + +Coercion is_inleft A B (u : A + {B}) := if u is inleft _ then true else false. + +Prenex Implicits isSome is_inl is_left is_inleft. + +Definition decidable P := {P} + {~ P}. + +(* Lemmas for ifs with large conditions, which allow reasoning about the *) +(* condition without repeating it inside the proof (the latter IS *) +(* preferable when the condition is short). *) +(* Usage : *) +(* if the goal contains (if cond then ...) = ... *) +(* case: ifP => Hcond. *) +(* generates two subgoal, with the assumption Hcond : cond = true/false *) +(* Rewrite if_same eliminates redundant ifs *) +(* Rewrite (fun_if f) moves a function f inside an if *) +(* Rewrite if_arg moves an argument inside a function-valued if *) + +Section BoolIf. + +Variables (A B : Type) (x : A) (f : A -> B) (b : bool) (vT vF : A). + +CoInductive if_spec (not_b : Prop) : bool -> A -> Set := + | IfSpecTrue of b : if_spec not_b true vT + | IfSpecFalse of not_b : if_spec not_b false vF. + +Lemma ifP : if_spec (b = false) b (if b then vT else vF). +Proof. by case def_b: b; constructor. Qed. + +Lemma ifPn : if_spec (~~ b) b (if b then vT else vF). +Proof. by case def_b: b; constructor; rewrite ?def_b. Qed. + +Lemma ifT : b -> (if b then vT else vF) = vT. Proof. by move->. Qed. +Lemma ifF : b = false -> (if b then vT else vF) = vF. Proof. by move->. Qed. +Lemma ifN : ~~ b -> (if b then vT else vF) = vF. Proof. by move/negbTE->. Qed. + +Lemma if_same : (if b then vT else vT) = vT. +Proof. by case b. Qed. + +Lemma if_neg : (if ~~ b then vT else vF) = if b then vF else vT. +Proof. by case b. Qed. + +Lemma fun_if : f (if b then vT else vF) = if b then f vT else f vF. +Proof. by case b. Qed. + +Lemma if_arg (fT fF : A -> B) : + (if b then fT else fF) x = if b then fT x else fF x. +Proof. by case b. Qed. + +(* Turning a boolean "if" form into an application. *) +Definition if_expr := if b then vT else vF. +Lemma ifE : (if b then vT else vF) = if_expr. Proof. by []. Qed. + +End BoolIf. + +(* The reflection predicate. *) + +Inductive reflect (P : Prop) : bool -> Set := + | ReflectT of P : reflect P true + | ReflectF of ~ P : reflect P false. + +(* Core (internal) reflection lemmas, used for the three kinds of views. *) + +Section ReflectCore. + +Variables (P Q : Prop) (b c : bool). + +Hypothesis Hb : reflect P b. + +Lemma introNTF : (if c then ~ P else P) -> ~~ b = c. +Proof. by case c; case Hb. Qed. + +Lemma introTF : (if c then P else ~ P) -> b = c. +Proof. by case c; case Hb. Qed. + +Lemma elimNTF : ~~ b = c -> if c then ~ P else P. +Proof. by move <-; case Hb. Qed. + +Lemma elimTF : b = c -> if c then P else ~ P. +Proof. by move <-; case Hb. Qed. + +Lemma equivPif : (Q -> P) -> (P -> Q) -> if b then Q else ~ Q. +Proof. by case Hb; auto. Qed. + +Lemma xorPif : Q \/ P -> ~ (Q /\ P) -> if b then ~ Q else Q. +Proof. by case Hb => [? _ H ? | ? H _]; case: H. Qed. + +End ReflectCore. + +(* Internal negated reflection lemmas *) +Section ReflectNegCore. + +Variables (P Q : Prop) (b c : bool). +Hypothesis Hb : reflect P (~~ b). + +Lemma introTFn : (if c then ~ P else P) -> b = c. +Proof. by move/(introNTF Hb) <-; case b. Qed. + +Lemma elimTFn : b = c -> if c then ~ P else P. +Proof. by move <-; apply: (elimNTF Hb); case b. Qed. + +Lemma equivPifn : (Q -> P) -> (P -> Q) -> if b then ~ Q else Q. +Proof. rewrite -if_neg; exact: equivPif. Qed. + +Lemma xorPifn : Q \/ P -> ~ (Q /\ P) -> if b then Q else ~ Q. +Proof. rewrite -if_neg; exact: xorPif. Qed. + +End ReflectNegCore. + +(* User-oriented reflection lemmas *) +Section Reflect. + +Variables (P Q : Prop) (b b' c : bool). +Hypotheses (Pb : reflect P b) (Pb' : reflect P (~~ b')). + +Lemma introT : P -> b. Proof. exact: introTF true _. Qed. +Lemma introF : ~ P -> b = false. Proof. exact: introTF false _. Qed. +Lemma introN : ~ P -> ~~ b. Proof. exact: introNTF true _. Qed. +Lemma introNf : P -> ~~ b = false. Proof. exact: introNTF false _. Qed. +Lemma introTn : ~ P -> b'. Proof. exact: introTFn true _. Qed. +Lemma introFn : P -> b' = false. Proof. exact: introTFn false _. Qed. + +Lemma elimT : b -> P. Proof. exact: elimTF true _. Qed. +Lemma elimF : b = false -> ~ P. Proof. exact: elimTF false _. Qed. +Lemma elimN : ~~ b -> ~P. Proof. exact: elimNTF true _. Qed. +Lemma elimNf : ~~ b = false -> P. Proof. exact: elimNTF false _. Qed. +Lemma elimTn : b' -> ~ P. Proof. exact: elimTFn true _. Qed. +Lemma elimFn : b' = false -> P. Proof. exact: elimTFn false _. Qed. + +Lemma introP : (b -> Q) -> (~~ b -> ~ Q) -> reflect Q b. +Proof. by case b; constructor; auto. Qed. + +Lemma iffP : (P -> Q) -> (Q -> P) -> reflect Q b. +Proof. by case: Pb; constructor; auto. Qed. + +Lemma equivP : (P <-> Q) -> reflect Q b. +Proof. by case; exact: iffP. Qed. + +Lemma sumboolP (decQ : decidable Q) : reflect Q decQ. +Proof. by case: decQ; constructor. Qed. + +Lemma appP : reflect Q b -> P -> Q. +Proof. by move=> Qb; move/introT; case: Qb. Qed. + +Lemma sameP : reflect P c -> b = c. +Proof. case; [exact: introT | exact: introF]. Qed. + +Lemma decPcases : if b then P else ~ P. Proof. by case Pb. Qed. + +Definition decP : decidable P. by case: b decPcases; [left | right]. Defined. + +Lemma rwP : P <-> b. Proof. by split; [exact: introT | exact: elimT]. Qed. + +Lemma rwP2 : reflect Q b -> (P <-> Q). +Proof. by move=> Qb; split=> ?; [exact: appP | apply: elimT; case: Qb]. Qed. + +(* Predicate family to reflect excluded middle in bool. *) +CoInductive alt_spec : bool -> Type := + | AltTrue of P : alt_spec true + | AltFalse of ~~ b : alt_spec false. + +Lemma altP : alt_spec b. +Proof. by case def_b: b / Pb; constructor; rewrite ?def_b. Qed. + +End Reflect. + +Hint View for move/ elimTF|3 elimNTF|3 elimTFn|3 introT|2 introTn|2 introN|2. + +Hint View for apply/ introTF|3 introNTF|3 introTFn|3 elimT|2 elimTn|2 elimN|2. + +Hint View for apply// equivPif|3 xorPif|3 equivPifn|3 xorPifn|3. + +(* Allow the direct application of a reflection lemma to a boolean assertion. *) +Coercion elimT : reflect >-> Funclass. + +(* Pierce's law, a weak form of classical reasoning. *) +Definition unless condition property := (property -> condition) -> condition. + +Lemma bind_unless C P {Q} : unless C P -> unless (unless C Q) P. +Proof. by move=> haveP suffPQ suffQ; apply: haveP => /suffPQ; exact. Qed. + +Lemma unless_contra b C : (~~ b -> C) -> unless C b. +Proof. by case: b => [_ haveC | haveC _]; exact: haveC. Qed. + +(* Classical reasoning becomes directly accessible for any bool subgoal. *) +(* Note that we cannot use "unless" here for lack of universe polymorphism. *) +Definition classically P : Prop := forall b : bool, (P -> b) -> b. + +Lemma classicP : forall P : Prop, classically P <-> ~ ~ P. +Proof. +move=> P; split=> [cP nP | nnP [] // nP]; last by case nnP; move/nP. +by have: P -> false; [move/nP | move/cP]. +Qed. + +Lemma classic_bind : forall P Q, + (P -> classically Q) -> (classically P -> classically Q). +Proof. by move=> P Q IH IH_P b IH_Q; apply: IH_P; move/IH; exact. Qed. + +Lemma classic_EM : forall P, classically (decidable P). +Proof. +by move=> P [] // IH; apply IH; right => ?; apply: notF (IH _); left. +Qed. + +Lemma classic_imply : forall P Q, (P -> classically Q) -> classically (P -> Q). +Proof. +move=> P Q IH [] // notPQ; apply notPQ; move/IH=> hQ; case: notF. +by apply: hQ => hQ; case: notF; exact: notPQ. +Qed. + +Lemma classic_pick : forall T P, + classically ({x : T | P x} + (forall x, ~ P x)). +Proof. +move=> T P [] // IH; apply IH; right=> x Px; case: notF. +by apply: IH; left; exists x. +Qed. + +(* List notations for wider connectives; the Prop connectives have a fixed *) +(* width so as to avoid iterated destruction (we go up to width 5 for /\, and *) +(* width 4 for or. The bool connectives have arbitrary widths, but denote *) +(* expressions that associate to the RIGHT. This is consistent with the right *) +(* associativity of list expressions and thus more convenient in most proofs. *) + +Inductive and3 (P1 P2 P3 : Prop) : Prop := And3 of P1 & P2 & P3. + +Inductive and4 (P1 P2 P3 P4 : Prop) : Prop := And4 of P1 & P2 & P3 & P4. + +Inductive and5 (P1 P2 P3 P4 P5 : Prop) : Prop := + And5 of P1 & P2 & P3 & P4 & P5. + +Inductive or3 (P1 P2 P3 : Prop) : Prop := Or31 of P1 | Or32 of P2 | Or33 of P3. + +Inductive or4 (P1 P2 P3 P4 : Prop) : Prop := + Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4. + +Notation "[ /\ P1 & P2 ]" := (and P1 P2) (only parsing) : type_scope. +Notation "[ /\ P1 , P2 & P3 ]" := (and3 P1 P2 P3) : type_scope. +Notation "[ /\ P1 , P2 , P3 & P4 ]" := (and4 P1 P2 P3 P4) : type_scope. +Notation "[ /\ P1 , P2 , P3 , P4 & P5 ]" := (and5 P1 P2 P3 P4 P5) : type_scope. + +Notation "[ \/ P1 | P2 ]" := (or P1 P2) (only parsing) : type_scope. +Notation "[ \/ P1 , P2 | P3 ]" := (or3 P1 P2 P3) : type_scope. +Notation "[ \/ P1 , P2 , P3 | P4 ]" := (or4 P1 P2 P3 P4) : type_scope. + +Notation "[ && b1 & c ]" := (b1 && c) (only parsing) : bool_scope. +Notation "[ && b1 , b2 , .. , bn & c ]" := (b1 && (b2 && .. (bn && c) .. )) + : bool_scope. + +Notation "[ || b1 | c ]" := (b1 || c) (only parsing) : bool_scope. +Notation "[ || b1 , b2 , .. , bn | c ]" := (b1 || (b2 || .. (bn || c) .. )) + : bool_scope. + +Notation "[ ==> b1 , b2 , .. , bn => c ]" := + (b1 ==> (b2 ==> .. (bn ==> c) .. )) : bool_scope. +Notation "[ ==> b1 => c ]" := (b1 ==> c) (only parsing) : bool_scope. + +Section AllAnd. + +Variables (T : Type) (P1 P2 P3 P4 P5 : T -> Prop). +Local Notation a P := (forall x, P x). + +Lemma all_and2 (hP : forall x, [/\ P1 x & P2 x]) : [/\ a P1 & a P2]. +Proof. by split=> x; case: (hP x). Qed. + +Lemma all_and3 (hP : forall x, [/\ P1 x, P2 x & P3 x]) : + [/\ a P1, a P2 & a P3]. +Proof. by split=> x; case: (hP x). Qed. + +Lemma all_and4 (hP : forall x, [/\ P1 x, P2 x, P3 x & P4 x]) : + [/\ a P1, a P2, a P3 & a P4]. +Proof. by split=> x; case: (hP x). Qed. + +Lemma all_and5 (hP : forall x, [/\ P1 x, P2 x, P3 x, P4 x & P5 x]) : + [/\ a P1, a P2, a P3, a P4 & a P5]. +Proof. by split=> x; case: (hP x). Qed. + +End AllAnd. + +Lemma pair_andP P Q : P /\ Q <-> P * Q. Proof. by split; case. Qed. + +Section ReflectConnectives. + +Variable b1 b2 b3 b4 b5 : bool. + +Lemma idP : reflect b1 b1. +Proof. by case b1; constructor. Qed. + +Lemma boolP : alt_spec b1 b1 b1. +Proof. exact: (altP idP). Qed. + +Lemma idPn : reflect (~~ b1) (~~ b1). +Proof. by case b1; constructor. Qed. + +Lemma negP : reflect (~ b1) (~~ b1). +Proof. by case b1; constructor; auto. Qed. + +Lemma negPn : reflect b1 (~~ ~~ b1). +Proof. by case b1; constructor. Qed. + +Lemma negPf : reflect (b1 = false) (~~ b1). +Proof. by case b1; constructor. Qed. + +Lemma andP : reflect (b1 /\ b2) (b1 && b2). +Proof. by case b1; case b2; constructor=> //; case. Qed. + +Lemma and3P : reflect [/\ b1, b2 & b3] [&& b1, b2 & b3]. +Proof. by case b1; case b2; case b3; constructor; try by case. Qed. + +Lemma and4P : reflect [/\ b1, b2, b3 & b4] [&& b1, b2, b3 & b4]. +Proof. by case b1; case b2; case b3; case b4; constructor; try by case. Qed. + +Lemma and5P : reflect [/\ b1, b2, b3, b4 & b5] [&& b1, b2, b3, b4 & b5]. +Proof. +by case b1; case b2; case b3; case b4; case b5; constructor; try by case. +Qed. + +Lemma orP : reflect (b1 \/ b2) (b1 || b2). +Proof. by case b1; case b2; constructor; auto; case. Qed. + +Lemma or3P : reflect [\/ b1, b2 | b3] [|| b1, b2 | b3]. +Proof. +case b1; first by constructor; constructor 1. +case b2; first by constructor; constructor 2. +case b3; first by constructor; constructor 3. +by constructor; case. +Qed. + +Lemma or4P : reflect [\/ b1, b2, b3 | b4] [|| b1, b2, b3 | b4]. +Proof. +case b1; first by constructor; constructor 1. +case b2; first by constructor; constructor 2. +case b3; first by constructor; constructor 3. +case b4; first by constructor; constructor 4. +by constructor; case. +Qed. + +Lemma nandP : reflect (~~ b1 \/ ~~ b2) (~~ (b1 && b2)). +Proof. by case b1; case b2; constructor; auto; case; auto. Qed. + +Lemma norP : reflect (~~ b1 /\ ~~ b2) (~~ (b1 || b2)). +Proof. by case b1; case b2; constructor; auto; case; auto. Qed. + +Lemma implyP : reflect (b1 -> b2) (b1 ==> b2). +Proof. by case b1; case b2; constructor; auto. Qed. + +End ReflectConnectives. + +Implicit Arguments idP [b1]. +Implicit Arguments idPn [b1]. +Implicit Arguments negP [b1]. +Implicit Arguments negPn [b1]. +Implicit Arguments negPf [b1]. +Implicit Arguments andP [b1 b2]. +Implicit Arguments and3P [b1 b2 b3]. +Implicit Arguments and4P [b1 b2 b3 b4]. +Implicit Arguments and5P [b1 b2 b3 b4 b5]. +Implicit Arguments orP [b1 b2]. +Implicit Arguments or3P [b1 b2 b3]. +Implicit Arguments or4P [b1 b2 b3 b4]. +Implicit Arguments nandP [b1 b2]. +Implicit Arguments norP [b1 b2]. +Implicit Arguments implyP [b1 b2]. +Prenex Implicits idP idPn negP negPn negPf. +Prenex Implicits andP and3P and4P and5P orP or3P or4P nandP norP implyP. + +(* Shorter, more systematic names for the boolean connectives laws. *) + +Lemma andTb : left_id true andb. Proof. by []. Qed. +Lemma andFb : left_zero false andb. Proof. by []. Qed. +Lemma andbT : right_id true andb. Proof. by case. Qed. +Lemma andbF : right_zero false andb. Proof. by case. Qed. +Lemma andbb : idempotent andb. Proof. by case. Qed. +Lemma andbC : commutative andb. Proof. by do 2!case. Qed. +Lemma andbA : associative andb. Proof. by do 3!case. Qed. +Lemma andbCA : left_commutative andb. Proof. by do 3!case. Qed. +Lemma andbAC : right_commutative andb. Proof. by do 3!case. Qed. +Lemma andbACA : interchange andb andb. Proof. by do 4!case. Qed. + +Lemma orTb : forall b, true || b. Proof. by []. Qed. +Lemma orFb : left_id false orb. Proof. by []. Qed. +Lemma orbT : forall b, b || true. Proof. by case. Qed. +Lemma orbF : right_id false orb. Proof. by case. Qed. +Lemma orbb : idempotent orb. Proof. by case. Qed. +Lemma orbC : commutative orb. Proof. by do 2!case. Qed. +Lemma orbA : associative orb. Proof. by do 3!case. Qed. +Lemma orbCA : left_commutative orb. Proof. by do 3!case. Qed. +Lemma orbAC : right_commutative orb. Proof. by do 3!case. Qed. +Lemma orbACA : interchange orb orb. Proof. by do 4!case. Qed. + +Lemma andbN b : b && ~~ b = false. Proof. by case: b. Qed. +Lemma andNb b : ~~ b && b = false. Proof. by case: b. Qed. +Lemma orbN b : b || ~~ b = true. Proof. by case: b. Qed. +Lemma orNb b : ~~ b || b = true. Proof. by case: b. Qed. + +Lemma andb_orl : left_distributive andb orb. Proof. by do 3!case. Qed. +Lemma andb_orr : right_distributive andb orb. Proof. by do 3!case. Qed. +Lemma orb_andl : left_distributive orb andb. Proof. by do 3!case. Qed. +Lemma orb_andr : right_distributive orb andb. Proof. by do 3!case. Qed. + +Lemma andb_idl (a b : bool) : (b -> a) -> a && b = b. +Proof. by case: a; case: b => // ->. Qed. +Lemma andb_idr (a b : bool) : (a -> b) -> a && b = a. +Proof. by case: a; case: b => // ->. Qed. +Lemma andb_id2l (a b c : bool) : (a -> b = c) -> a && b = a && c. +Proof. by case: a; case: b; case: c => // ->. Qed. +Lemma andb_id2r (a b c : bool) : (b -> a = c) -> a && b = c && b. +Proof. by case: a; case: b; case: c => // ->. Qed. + +Lemma orb_idl (a b : bool) : (a -> b) -> a || b = b. +Proof. by case: a; case: b => // ->. Qed. +Lemma orb_idr (a b : bool) : (b -> a) -> a || b = a. +Proof. by case: a; case: b => // ->. Qed. +Lemma orb_id2l (a b c : bool) : (~~ a -> b = c) -> a || b = a || c. +Proof. by case: a; case: b; case: c => // ->. Qed. +Lemma orb_id2r (a b c : bool) : (~~ b -> a = c) -> a || b = c || b. +Proof. by case: a; case: b; case: c => // ->. Qed. + +Lemma negb_and (a b : bool) : ~~ (a && b) = ~~ a || ~~ b. +Proof. by case: a; case: b. Qed. + +Lemma negb_or (a b : bool) : ~~ (a || b) = ~~ a && ~~ b. +Proof. by case: a; case: b. Qed. + +(* Pseudo-cancellation -- i.e, absorbtion *) + +Lemma andbK a b : a && b || a = a. Proof. by case: a; case: b. Qed. +Lemma andKb a b : a || b && a = a. Proof. by case: a; case: b. Qed. +Lemma orbK a b : (a || b) && a = a. Proof. by case: a; case: b. Qed. +Lemma orKb a b : a && (b || a) = a. Proof. by case: a; case: b. Qed. + +(* Imply *) + +Lemma implybT b : b ==> true. Proof. by case: b. Qed. +Lemma implybF b : (b ==> false) = ~~ b. Proof. by case: b. Qed. +Lemma implyFb b : false ==> b. Proof. by []. Qed. +Lemma implyTb b : (true ==> b) = b. Proof. by []. Qed. +Lemma implybb b : b ==> b. Proof. by case: b. Qed. + +Lemma negb_imply a b : ~~ (a ==> b) = a && ~~ b. +Proof. by case: a; case: b. Qed. + +Lemma implybE a b : (a ==> b) = ~~ a || b. +Proof. by case: a; case: b. Qed. + +Lemma implyNb a b : (~~ a ==> b) = a || b. +Proof. by case: a; case: b. Qed. + +Lemma implybN a b : (a ==> ~~ b) = (b ==> ~~ a). +Proof. by case: a; case: b. Qed. + +Lemma implybNN a b : (~~ a ==> ~~ b) = b ==> a. +Proof. by case: a; case: b. Qed. + +Lemma implyb_idl (a b : bool) : (~~ a -> b) -> (a ==> b) = b. +Proof. by case: a; case: b => // ->. Qed. +Lemma implyb_idr (a b : bool) : (b -> ~~ a) -> (a ==> b) = ~~ a. +Proof. by case: a; case: b => // ->. Qed. +Lemma implyb_id2l (a b c : bool) : (a -> b = c) -> (a ==> b) = (a ==> c). +Proof. by case: a; case: b; case: c => // ->. Qed. + +(* Addition (xor) *) + +Lemma addFb : left_id false addb. Proof. by []. Qed. +Lemma addbF : right_id false addb. Proof. by case. Qed. +Lemma addbb : self_inverse false addb. Proof. by case. Qed. +Lemma addbC : commutative addb. Proof. by do 2!case. Qed. +Lemma addbA : associative addb. Proof. by do 3!case. Qed. +Lemma addbCA : left_commutative addb. Proof. by do 3!case. Qed. +Lemma addbAC : right_commutative addb. Proof. by do 3!case. Qed. +Lemma addbACA : interchange addb addb. Proof. by do 4!case. Qed. +Lemma andb_addl : left_distributive andb addb. Proof. by do 3!case. Qed. +Lemma andb_addr : right_distributive andb addb. Proof. by do 3!case. Qed. +Lemma addKb : left_loop id addb. Proof. by do 2!case. Qed. +Lemma addbK : right_loop id addb. Proof. by do 2!case. Qed. +Lemma addIb : left_injective addb. Proof. by do 3!case. Qed. +Lemma addbI : right_injective addb. Proof. by do 3!case. Qed. + +Lemma addTb b : true (+) b = ~~ b. Proof. by []. Qed. +Lemma addbT b : b (+) true = ~~ b. Proof. by case: b. Qed. + +Lemma addbN a b : a (+) ~~ b = ~~ (a (+) b). +Proof. by case: a; case: b. Qed. +Lemma addNb a b : ~~ a (+) b = ~~ (a (+) b). +Proof. by case: a; case: b. Qed. + +Lemma addbP a b : reflect (~~ a = b) (a (+) b). +Proof. by case: a; case: b; constructor. Qed. +Implicit Arguments addbP [a b]. + +(* Resolution tactic for blindly weeding out common terms from boolean *) +(* equalities. When faced with a goal of the form (andb/orb/addb b1 b2) = b3 *) +(* they will try to locate b1 in b3 and remove it. This can fail! *) + +Ltac bool_congr := + match goal with + | |- (?X1 && ?X2 = ?X3) => first + [ symmetry; rewrite -1?(andbC X1) -?(andbCA X1); congr 1 (andb X1); symmetry + | case: (X1); [ rewrite ?andTb ?andbT // | by rewrite ?andbF /= ] ] + | |- (?X1 || ?X2 = ?X3) => first + [ symmetry; rewrite -1?(orbC X1) -?(orbCA X1); congr 1 (orb X1); symmetry + | case: (X1); [ by rewrite ?orbT //= | rewrite ?orFb ?orbF ] ] + | |- (?X1 (+) ?X2 = ?X3) => + symmetry; rewrite -1?(addbC X1) -?(addbCA X1); congr 1 (addb X1); symmetry + | |- (~~ ?X1 = ?X2) => congr 1 negb + end. + +(******************************************************************************) +(* Predicates, i.e., packaged functions to bool. *) +(* - pred T, the basic type for predicates over a type T, is simply an alias *) +(* for T -> bool. *) +(* We actually distinguish two kinds of predicates, which we call applicative *) +(* and collective, based on the syntax used to test them at some x in T: *) +(* - For an applicative predicate P, one uses prefix syntax: *) +(* P x *) +(* Also, most operations on applicative predicates use prefix syntax as *) +(* well (e.g., predI P Q). *) +(* - For a collective predicate A, one uses infix syntax: *) +(* x \in A *) +(* and all operations on collective predicates use infix syntax as well *) +(* (e.g., [predI A & B]). *) +(* There are only two kinds of applicative predicates: *) +(* - pred T, the alias for T -> bool mentioned above *) +(* - simpl_pred T, an alias for simpl_fun T bool with a coercion to pred T *) +(* that auto-simplifies on application (see ssrfun). *) +(* On the other hand, the set of collective predicate types is open-ended via *) +(* - predType T, a Structure that can be used to put Canonical collective *) +(* predicate interpretation on other types, such as lists, tuples, *) +(* finite sets, etc. *) +(* Indeed, we define such interpretations for applicative predicate types, *) +(* which can therefore also be used with the infix syntax, e.g., *) +(* x \in predI P Q *) +(* Moreover these infix forms are convertible to their prefix counterpart *) +(* (e.g., predI P Q x which in turn simplifies to P x && Q x). The converse *) +(* is not true, however; collective predicate types cannot, in general, be *) +(* general, be used applicatively, because of the "uniform inheritance" *) +(* restriction on implicit coercions. *) +(* However, we do define an explicit generic coercion *) +(* - mem : forall (pT : predType), pT -> mem_pred T *) +(* where mem_pred T is a variant of simpl_pred T that preserves the infix *) +(* syntax, i.e., mem A x auto-simplifies to x \in A. *) +(* Indeed, the infix "collective" operators are notation for a prefix *) +(* operator with arguments of type mem_pred T or pred T, applied to coerced *) +(* collective predicates, e.g., *) +(* Notation "x \in A" := (in_mem x (mem A)). *) +(* This prevents the variability in the predicate type from interfering with *) +(* the application of generic lemmas. Moreover this also makes it much easier *) +(* to define generic lemmas, because the simplest type -- pred T -- can be *) +(* used as the type of generic collective predicates, provided one takes care *) +(* not to use it applicatively; this avoids the burden of having to declare a *) +(* different predicate type for each predicate parameter of each section or *) +(* lemma. *) +(* This trick is made possible by the fact that the constructor of the *) +(* mem_pred T type aligns the unification process, forcing a generic *) +(* "collective" predicate A : pred T to unify with the actual collective B, *) +(* which mem has coerced to pred T via an internal, hidden implicit coercion, *) +(* supplied by the predType structure for B. Users should take care not to *) +(* inadvertently "strip" (mem B) down to the coerced B, since this will *) +(* expose the internal coercion: Coq will display a term B x that cannot be *) +(* typed as such. The topredE lemma can be used to restore the x \in B *) +(* syntax in this case. While -topredE can conversely be used to change *) +(* x \in P into P x, it is safer to use the inE and memE lemmas instead, as *) +(* they do not run the risk of exposing internal coercions. As a consequence *) +(* it is better to explicitly cast a generic applicative pred T to simpl_pred *) +(* using the SimplPred constructor, when it is used as a collective predicate *) +(* (see, e.g., Lemma eq_big in bigop). *) +(* We also sometimes "instantiate" the predType structure by defining a *) +(* coercion to the sort of the predPredType structure. This works better for *) +(* types such as {set T} that have subtypes that coerce to them, since the *) +(* same coercion will be inserted by the application of mem. It also lets us *) +(* turn any Type aT : predArgType into the total predicate over that type, *) +(* i.e., fun _: aT => true. This allows us to write, e.g., #|'I_n| for the *) +(* cardinal of the (finite) type of integers less than n. *) +(* Collective predicates have a specific extensional equality, *) +(* - A =i B, *) +(* while applicative predicates use the extensional equality of functions, *) +(* - P =1 Q *) +(* The two forms are convertible, however. *) +(* We lift boolean operations to predicates, defining: *) +(* - predU (union), predI (intersection), predC (complement), *) +(* predD (difference), and preim (preimage, i.e., composition) *) +(* For each operation we define three forms, typically: *) +(* - predU : pred T -> pred T -> simpl_pred T *) +(* - [predU A & B], a Notation for predU (mem A) (mem B) *) +(* - xpredU, a Notation for the lambda-expression inside predU, *) +(* which is mostly useful as an argument of =1, since it exposes the head *) +(* head constant of the expression to the ssreflect matching algorithm. *) +(* The syntax for the preimage of a collective predicate A is *) +(* - [preim f of A] *) +(* Finally, the generic syntax for defining a simpl_pred T is *) +(* - [pred x : T | P(x)], [pred x | P(x)], [pred x in A | P(x)], etc. *) +(* We also support boolean relations, but only the applicative form, with *) +(* types *) +(* - rel T, an alias for T -> pred T *) +(* - simpl_rel T, an auto-simplifying version, and syntax *) +(* [rel x y | P(x,y)], [rel x y in A & B | P(x,y)], etc. *) +(* The notation [rel of fA] can be used to coerce a function returning a *) +(* collective predicate to one returning pred T. *) +(* Finally, note that there is specific support for ambivalent predicates *) +(* that can work in either style, as per this file's head descriptor. *) +(******************************************************************************) + +Definition pred T := T -> bool. + +Identity Coercion fun_of_pred : pred >-> Funclass. + +Definition rel T := T -> pred T. + +Identity Coercion fun_of_rel : rel >-> Funclass. + +Notation xpred0 := (fun _ => false). +Notation xpredT := (fun _ => true). +Notation xpredI := (fun (p1 p2 : pred _) x => p1 x && p2 x). +Notation xpredU := (fun (p1 p2 : pred _) x => p1 x || p2 x). +Notation xpredC := (fun (p : pred _) x => ~~ p x). +Notation xpredD := (fun (p1 p2 : pred _) x => ~~ p2 x && p1 x). +Notation xpreim := (fun f (p : pred _) x => p (f x)). +Notation xrelU := (fun (r1 r2 : rel _) x y => r1 x y || r2 x y). + +Section Predicates. + +Variables T : Type. + +Definition subpred (p1 p2 : pred T) := forall x, p1 x -> p2 x. + +Definition subrel (r1 r2 : rel T) := forall x y, r1 x y -> r2 x y. + +Definition simpl_pred := simpl_fun T bool. +Definition applicative_pred := pred T. +Definition collective_pred := pred T. + +Definition SimplPred (p : pred T) : simpl_pred := SimplFun p. + +Coercion pred_of_simpl (p : simpl_pred) : pred T := fun_of_simpl p. +Coercion applicative_pred_of_simpl (p : simpl_pred) : applicative_pred := + fun_of_simpl p. +Coercion collective_pred_of_simpl (p : simpl_pred) : collective_pred := + fun x => (let: SimplFun f := p in fun _ => f x) x. +(* Note: applicative_of_simpl is convertible to pred_of_simpl, while *) +(* collective_of_simpl is not. *) + +Definition pred0 := SimplPred xpred0. +Definition predT := SimplPred xpredT. +Definition predI p1 p2 := SimplPred (xpredI p1 p2). +Definition predU p1 p2 := SimplPred (xpredU p1 p2). +Definition predC p := SimplPred (xpredC p). +Definition predD p1 p2 := SimplPred (xpredD p1 p2). +Definition preim rT f (d : pred rT) := SimplPred (xpreim f d). + +Definition simpl_rel := simpl_fun T (pred T). + +Definition SimplRel (r : rel T) : simpl_rel := [fun x => r x]. + +Coercion rel_of_simpl_rel (r : simpl_rel) : rel T := fun x y => r x y. + +Definition relU r1 r2 := SimplRel (xrelU r1 r2). + +Lemma subrelUl r1 r2 : subrel r1 (relU r1 r2). +Proof. by move=> *; apply/orP; left. Qed. + +Lemma subrelUr r1 r2 : subrel r2 (relU r1 r2). +Proof. by move=> *; apply/orP; right. Qed. + +CoInductive mem_pred := Mem of pred T. + +Definition isMem pT topred mem := mem = (fun p : pT => Mem [eta topred p]). + +Structure predType := PredType { + pred_sort :> Type; + topred : pred_sort -> pred T; + _ : {mem | isMem topred mem} +}. + +Definition mkPredType pT toP := PredType (exist (@isMem pT toP) _ (erefl _)). + +Canonical predPredType := Eval hnf in @mkPredType (pred T) id. +Canonical simplPredType := Eval hnf in mkPredType pred_of_simpl. +Canonical boolfunPredType := Eval hnf in @mkPredType (T -> bool) id. + +Coercion pred_of_mem mp : pred_sort predPredType := let: Mem p := mp in [eta p]. +Canonical memPredType := Eval hnf in mkPredType pred_of_mem. + +Definition clone_pred U := + fun pT & pred_sort pT -> U => + fun a mP (pT' := @PredType U a mP) & phant_id pT' pT => pT'. + +End Predicates. + +Implicit Arguments pred0 [T]. +Implicit Arguments predT [T]. +Prenex Implicits pred0 predT predI predU predC predD preim relU. + +Notation "[ 'pred' : T | E ]" := (SimplPred (fun _ : T => E%B)) + (at level 0, format "[ 'pred' : T | E ]") : fun_scope. +Notation "[ 'pred' x | E ]" := (SimplPred (fun x => E%B)) + (at level 0, x ident, format "[ 'pred' x | E ]") : fun_scope. +Notation "[ 'pred' x | E1 & E2 ]" := [pred x | E1 && E2 ] + (at level 0, x ident, format "[ 'pred' x | E1 & E2 ]") : fun_scope. +Notation "[ 'pred' x : T | E ]" := (SimplPred (fun x : T => E%B)) + (at level 0, x ident, only parsing) : fun_scope. +Notation "[ 'pred' x : T | E1 & E2 ]" := [pred x : T | E1 && E2 ] + (at level 0, x ident, only parsing) : fun_scope. +Notation "[ 'rel' x y | E ]" := (SimplRel (fun x y => E%B)) + (at level 0, x ident, y ident, format "[ 'rel' x y | E ]") : fun_scope. +Notation "[ 'rel' x y : T | E ]" := (SimplRel (fun x y : T => E%B)) + (at level 0, x ident, y ident, only parsing) : fun_scope. + +Notation "[ 'predType' 'of' T ]" := (@clone_pred _ T _ id _ _ id) + (at level 0, format "[ 'predType' 'of' T ]") : form_scope. + +(* This redundant coercion lets us "inherit" the simpl_predType canonical *) +(* instance by declaring a coercion to simpl_pred. This hack is the only way *) +(* to put a predType structure on a predArgType. We use simpl_pred rather *) +(* than pred to ensure that /= removes the identity coercion. Note that the *) +(* coercion will never be used directly for simpl_pred, since the canonical *) +(* instance should always be resolved. *) + +Notation pred_class := (pred_sort (predPredType _)). +Coercion sort_of_simpl_pred T (p : simpl_pred T) : pred_class := p : pred T. + +(* This lets us use some types as a synonym for their universal predicate. *) +(* Unfortunately, this won't work for existing types like bool, unless we *) +(* redefine bool, true, false and all bool ops. *) +Definition predArgType := Type. +Bind Scope type_scope with predArgType. +Identity Coercion sort_of_predArgType : predArgType >-> Sortclass. +Coercion pred_of_argType (T : predArgType) : simpl_pred T := predT. + +Notation "{ : T }" := (T%type : predArgType) + (at level 0, format "{ : T }") : type_scope. + +(* These must be defined outside a Section because "cooking" kills the *) +(* nosimpl tag. *) + +Definition mem T (pT : predType T) : pT -> mem_pred T := + nosimpl (let: PredType _ _ (exist mem _) := pT return pT -> _ in mem). +Definition in_mem T x mp := nosimpl pred_of_mem T mp x. + +Prenex Implicits mem. + +Coercion pred_of_mem_pred T mp := [pred x : T | in_mem x mp]. + +Definition eq_mem T p1 p2 := forall x : T, in_mem x p1 = in_mem x p2. +Definition sub_mem T p1 p2 := forall x : T, in_mem x p1 -> in_mem x p2. + +Typeclasses Opaque eq_mem. + +Lemma sub_refl T (p : mem_pred T) : sub_mem p p. Proof. by []. Qed. +Implicit Arguments sub_refl [[T] [p]]. + +Notation "x \in A" := (in_mem x (mem A)) : bool_scope. +Notation "x \in A" := (in_mem x (mem A)) : bool_scope. +Notation "x \notin A" := (~~ (x \in A)) : bool_scope. +Notation "A =i B" := (eq_mem (mem A) (mem B)) : type_scope. +Notation "{ 'subset' A <= B }" := (sub_mem (mem A) (mem B)) + (at level 0, A, B at level 69, + format "{ '[hv' 'subset' A '/ ' <= B ']' }") : type_scope. +Notation "[ 'mem' A ]" := (pred_of_simpl (pred_of_mem_pred (mem A))) + (at level 0, only parsing) : fun_scope. +Notation "[ 'rel' 'of' fA ]" := (fun x => [mem (fA x)]) + (at level 0, format "[ 'rel' 'of' fA ]") : fun_scope. +Notation "[ 'predI' A & B ]" := (predI [mem A] [mem B]) + (at level 0, format "[ 'predI' A & B ]") : fun_scope. +Notation "[ 'predU' A & B ]" := (predU [mem A] [mem B]) + (at level 0, format "[ 'predU' A & B ]") : fun_scope. +Notation "[ 'predD' A & B ]" := (predD [mem A] [mem B]) + (at level 0, format "[ 'predD' A & B ]") : fun_scope. +Notation "[ 'predC' A ]" := (predC [mem A]) + (at level 0, format "[ 'predC' A ]") : fun_scope. +Notation "[ 'preim' f 'of' A ]" := (preim f [mem A]) + (at level 0, format "[ 'preim' f 'of' A ]") : fun_scope. + +Notation "[ 'pred' x 'in' A ]" := [pred x | x \in A] + (at level 0, x ident, format "[ 'pred' x 'in' A ]") : fun_scope. +Notation "[ 'pred' x 'in' A | E ]" := [pred x | x \in A & E] + (at level 0, x ident, format "[ 'pred' x 'in' A | E ]") : fun_scope. +Notation "[ 'pred' x 'in' A | E1 & E2 ]" := [pred x | x \in A & E1 && E2 ] + (at level 0, x ident, + format "[ 'pred' x 'in' A | E1 & E2 ]") : fun_scope. +Notation "[ 'rel' x y 'in' A & B | E ]" := + [rel x y | (x \in A) && (y \in B) && E] + (at level 0, x ident, y ident, + format "[ 'rel' x y 'in' A & B | E ]") : fun_scope. +Notation "[ 'rel' x y 'in' A & B ]" := [rel x y | (x \in A) && (y \in B)] + (at level 0, x ident, y ident, + format "[ 'rel' x y 'in' A & B ]") : fun_scope. +Notation "[ 'rel' x y 'in' A | E ]" := [rel x y in A & A | E] + (at level 0, x ident, y ident, + format "[ 'rel' x y 'in' A | E ]") : fun_scope. +Notation "[ 'rel' x y 'in' A ]" := [rel x y in A & A] + (at level 0, x ident, y ident, + format "[ 'rel' x y 'in' A ]") : fun_scope. + +Section simpl_mem. + +Variables (T : Type) (pT : predType T). +Implicit Types (x : T) (p : pred T) (sp : simpl_pred T) (pp : pT). + +(* Bespoke structures that provide fine-grained control over matching the *) +(* various forms of the \in predicate; note in particular the different forms *) +(* of hoisting that are used. We had to work around several bugs in the *) +(* implementation of unification, notably improper expansion of telescope *) +(* projections and overwriting of a variable assignment by a later *) +(* unification (probably due to conversion cache cross-talk). *) +Structure manifest_applicative_pred p := ManifestApplicativePred { + manifest_applicative_pred_value :> pred T; + _ : manifest_applicative_pred_value = p +}. +Definition ApplicativePred p := ManifestApplicativePred (erefl p). +Canonical applicative_pred_applicative sp := + ApplicativePred (applicative_pred_of_simpl sp). + +Structure manifest_simpl_pred p := ManifestSimplPred { + manifest_simpl_pred_value :> simpl_pred T; + _ : manifest_simpl_pred_value = SimplPred p +}. +Canonical expose_simpl_pred p := ManifestSimplPred (erefl (SimplPred p)). + +Structure manifest_mem_pred p := ManifestMemPred { + manifest_mem_pred_value :> mem_pred T; + _ : manifest_mem_pred_value= Mem [eta p] +}. +Canonical expose_mem_pred p := @ManifestMemPred p _ (erefl _). + +Structure applicative_mem_pred p := + ApplicativeMemPred {applicative_mem_pred_value :> manifest_mem_pred p}. +Canonical check_applicative_mem_pred p (ap : manifest_applicative_pred p) mp := + @ApplicativeMemPred ap mp. + +Lemma mem_topred (pp : pT) : mem (topred pp) = mem pp. +Proof. by rewrite /mem; case: pT pp => T1 app1 [mem1 /= ->]. Qed. + +Lemma topredE x (pp : pT) : topred pp x = (x \in pp). +Proof. by rewrite -mem_topred. Qed. + +Lemma app_predE x p (ap : manifest_applicative_pred p) : ap x = (x \in p). +Proof. by case: ap => _ /= ->. Qed. + +Lemma in_applicative x p (amp : applicative_mem_pred p) : in_mem x amp = p x. +Proof. by case: amp => [[_ /= ->]]. Qed. + +Lemma in_collective x p (msp : manifest_simpl_pred p) : + (x \in collective_pred_of_simpl msp) = p x. +Proof. by case: msp => _ /= ->. Qed. + +Lemma in_simpl x p (msp : manifest_simpl_pred p) : + in_mem x (Mem [eta fun_of_simpl (msp : simpl_pred T)]) = p x. +Proof. by case: msp => _ /= ->. Qed. + +(* Because of the explicit eta expansion in the left-hand side, this lemma *) +(* should only be used in a right-to-left direction. The 8.3 hack allowing *) +(* partial right-to-left use does not work with the improved expansion *) +(* heuristics in 8.4. *) +Lemma unfold_in x p : (x \in ([eta p] : pred T)) = p x. +Proof. by []. Qed. + +Lemma simpl_predE p : SimplPred p =1 p. +Proof. by []. Qed. + +Definition inE := (in_applicative, in_simpl, simpl_predE). (* to be extended *) + +Lemma mem_simpl sp : mem sp = sp :> pred T. +Proof. by []. Qed. + +Definition memE := mem_simpl. (* could be extended *) + +Lemma mem_mem (pp : pT) : (mem (mem pp) = mem pp) * (mem [mem pp] = mem pp). +Proof. by rewrite -mem_topred. Qed. + +End simpl_mem. + +(* Qualifiers and keyed predicates. *) + +CoInductive qualifier (q : nat) T := Qualifier of predPredType T. + +Coercion has_quality n T (q : qualifier n T) : pred_class := + fun x => let: Qualifier p := q in p x. +Implicit Arguments has_quality [T]. + +Lemma qualifE n T p x : (x \in @Qualifier n T p) = p x. Proof. by []. Qed. + +Notation "x \is A" := (x \in has_quality 0 A) + (at level 70, no associativity, + format "'[hv' x '/ ' \is A ']'") : bool_scope. +Notation "x \is 'a' A" := (x \in has_quality 1 A) + (at level 70, no associativity, + format "'[hv' x '/ ' \is 'a' A ']'") : bool_scope. +Notation "x \is 'an' A" := (x \in has_quality 2 A) + (at level 70, no associativity, + format "'[hv' x '/ ' \is 'an' A ']'") : bool_scope. +Notation "x \isn't A" := (x \notin has_quality 0 A) + (at level 70, no associativity, + format "'[hv' x '/ ' \isn't A ']'") : bool_scope. +Notation "x \isn't 'a' A" := (x \notin has_quality 1 A) + (at level 70, no associativity, + format "'[hv' x '/ ' \isn't 'a' A ']'") : bool_scope. +Notation "x \isn't 'an' A" := (x \notin has_quality 2 A) + (at level 70, no associativity, + format "'[hv' x '/ ' \isn't 'an' A ']'") : bool_scope. +Notation "[ 'qualify' x | P ]" := (Qualifier 0 (fun x => P%B)) + (at level 0, x at level 99, + format "'[hv' [ 'qualify' x | '/ ' P ] ']'") : form_scope. +Notation "[ 'qualify' x : T | P ]" := (Qualifier 0 (fun x : T => P%B)) + (at level 0, x at level 99, only parsing) : form_scope. +Notation "[ 'qualify' 'a' x | P ]" := (Qualifier 1 (fun x => P%B)) + (at level 0, x at level 99, + format "'[hv' [ 'qualify' 'a' x | '/ ' P ] ']'") : form_scope. +Notation "[ 'qualify' 'a' x : T | P ]" := (Qualifier 1 (fun x : T => P%B)) + (at level 0, x at level 99, only parsing) : form_scope. +Notation "[ 'qualify' 'an' x | P ]" := (Qualifier 2 (fun x => P%B)) + (at level 0, x at level 99, + format "'[hv' [ 'qualify' 'an' x | '/ ' P ] ']'") : form_scope. +Notation "[ 'qualify' 'an' x : T | P ]" := (Qualifier 2 (fun x : T => P%B)) + (at level 0, x at level 99, only parsing) : form_scope. + +(* Keyed predicates: support for property-bearing predicate interfaces. *) + +Section KeyPred. + +Variable T : Type. +CoInductive pred_key (p : predPredType T) := DefaultPredKey. + +Variable p : predPredType T. +Structure keyed_pred (k : pred_key p) := + PackKeyedPred {unkey_pred :> pred_class; _ : unkey_pred =i p}. + +Variable k : pred_key p. +Definition KeyedPred := @PackKeyedPred k p (frefl _). + +Variable k_p : keyed_pred k. +Lemma keyed_predE : k_p =i p. Proof. by case: k_p. Qed. + +(* Instances that strip the mem cast; the first one has "pred_of_mem" as its *) +(* projection head value, while the second has "pred_of_simpl". The latter *) +(* has the side benefit of preempting accidental misdeclarations. *) +(* Note: pred_of_mem is the registered mem >-> pred_class coercion, while *) +(* simpl_of_mem; pred_of_simpl is the mem >-> pred >=> Funclass coercion. We *) +(* must write down the coercions explicitly as the Canonical head constant *) +(* computation does not strip casts !! *) +Canonical keyed_mem := + @PackKeyedPred k (pred_of_mem (mem k_p)) keyed_predE. +Canonical keyed_mem_simpl := + @PackKeyedPred k (pred_of_simpl (mem k_p)) keyed_predE. + +End KeyPred. + +Notation "x \i 'n' S" := (x \in @unkey_pred _ S _ _) + (at level 70, format "'[hv' x '/ ' \i 'n' S ']'") : bool_scope. + +Section KeyedQualifier. + +Variables (T : Type) (n : nat) (q : qualifier n T). + +Structure keyed_qualifier (k : pred_key q) := + PackKeyedQualifier {unkey_qualifier; _ : unkey_qualifier = q}. +Definition KeyedQualifier k := PackKeyedQualifier k (erefl q). +Variables (k : pred_key q) (k_q : keyed_qualifier k). +Fact keyed_qualifier_suproof : unkey_qualifier k_q =i q. +Proof. by case: k_q => /= _ ->. Qed. +Canonical keyed_qualifier_keyed := PackKeyedPred k keyed_qualifier_suproof. + +End KeyedQualifier. + +Notation "x \i 's' A" := (x \i n has_quality 0 A) + (at level 70, format "'[hv' x '/ ' \i 's' A ']'") : bool_scope. +Notation "x \i 's' 'a' A" := (x \i n has_quality 1 A) + (at level 70, format "'[hv' x '/ ' \i 's' 'a' A ']'") : bool_scope. +Notation "x \i 's' 'an' A" := (x \i n has_quality 2 A) + (at level 70, format "'[hv' x '/ ' \i 's' 'an' A ']'") : bool_scope. + +Module DefaultKeying. + +Canonical default_keyed_pred T p := KeyedPred (@DefaultPredKey T p). +Canonical default_keyed_qualifier T n (q : qualifier n T) := + KeyedQualifier (DefaultPredKey q). + +End DefaultKeying. + +(* Skolemizing with conditions. *) + +Lemma all_tag_cond_dep I T (C : pred I) U : + (forall x, T x) -> (forall x, C x -> {y : T x & U x y}) -> + {f : forall x, T x & forall x, C x -> U x (f x)}. +Proof. +move=> f0 fP; apply: all_tag (fun x y => C x -> U x y) _ => x. +by case Cx: (C x); [case/fP: Cx => y; exists y | exists (f0 x)]. +Qed. + +Lemma all_tag_cond I T (C : pred I) U : + T -> (forall x, C x -> {y : T & U x y}) -> + {f : I -> T & forall x, C x -> U x (f x)}. +Proof. by move=> y0; apply: all_tag_cond_dep. Qed. + +Lemma all_sig_cond_dep I T (C : pred I) P : + (forall x, T x) -> (forall x, C x -> {y : T x | P x y}) -> + {f : forall x, T x | forall x, C x -> P x (f x)}. +Proof. by move=> f0 /(all_tag_cond_dep f0)[f]; exists f. Qed. + +Lemma all_sig_cond I T (C : pred I) P : + T -> (forall x, C x -> {y : T | P x y}) -> + {f : I -> T | forall x, C x -> P x (f x)}. +Proof. by move=> y0; apply: all_sig_cond_dep. Qed. + +Section RelationProperties. + +(* Caveat: reflexive should not be used to state lemmas, as auto and trivial *) +(* will not expand the constant. *) + +Variable T : Type. + +Variable R : rel T. + +Definition total := forall x y, R x y || R y x. +Definition transitive := forall y x z, R x y -> R y z -> R x z. + +Definition symmetric := forall x y, R x y = R y x. +Definition antisymmetric := forall x y, R x y && R y x -> x = y. +Definition pre_symmetric := forall x y, R x y -> R y x. + +Lemma symmetric_from_pre : pre_symmetric -> symmetric. +Proof. move=> symR x y; apply/idP/idP; exact: symR. Qed. + +Definition reflexive := forall x, R x x. +Definition irreflexive := forall x, R x x = false. + +Definition left_transitive := forall x y, R x y -> R x =1 R y. +Definition right_transitive := forall x y, R x y -> R^~ x =1 R^~ y. + +Section PER. + +Hypotheses (symR : symmetric) (trR : transitive). + +Lemma sym_left_transitive : left_transitive. +Proof. by move=> x y Rxy z; apply/idP/idP; apply: trR; rewrite // symR. Qed. + +Lemma sym_right_transitive : right_transitive. +Proof. by move=> x y /sym_left_transitive Rxy z; rewrite !(symR z) Rxy. Qed. + +End PER. + +(* We define the equivalence property with prenex quantification so that it *) +(* can be localized using the {in ..., ..} form defined below. *) + +Definition equivalence_rel := forall x y z, R z z * (R x y -> R x z = R y z). + +Lemma equivalence_relP : equivalence_rel <-> reflexive /\ left_transitive. +Proof. +split=> [eqiR | [Rxx trR] x y z]; last by split=> [|/trR->]. +by split=> [x | x y Rxy z]; [rewrite (eqiR x x x) | rewrite (eqiR x y z)]. +Qed. + +End RelationProperties. + +Lemma rev_trans T (R : rel T) : transitive R -> transitive (fun x y => R y x). +Proof. by move=> trR x y z Ryx Rzy; exact: trR Rzy Ryx. Qed. + +(* Property localization *) + +Notation Local "{ 'all1' P }" := (forall x, P x : Prop) (at level 0). +Notation Local "{ 'all2' P }" := (forall x y, P x y : Prop) (at level 0). +Notation Local "{ 'all3' P }" := (forall x y z, P x y z: Prop) (at level 0). +Notation Local ph := (phantom _). + +Section LocalProperties. + +Variables T1 T2 T3 : Type. + +Variables (d1 : mem_pred T1) (d2 : mem_pred T2) (d3 : mem_pred T3). +Notation Local ph := (phantom Prop). + +Definition prop_for (x : T1) P & ph {all1 P} := P x. + +Lemma forE x P phP : @prop_for x P phP = P x. Proof. by []. Qed. + +Definition prop_in1 P & ph {all1 P} := + forall x, in_mem x d1 -> P x. + +Definition prop_in11 P & ph {all2 P} := + forall x y, in_mem x d1 -> in_mem y d2 -> P x y. + +Definition prop_in2 P & ph {all2 P} := + forall x y, in_mem x d1 -> in_mem y d1 -> P x y. + +Definition prop_in111 P & ph {all3 P} := + forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d3 -> P x y z. + +Definition prop_in12 P & ph {all3 P} := + forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d2 -> P x y z. + +Definition prop_in21 P & ph {all3 P} := + forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d2 -> P x y z. + +Definition prop_in3 P & ph {all3 P} := + forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d1 -> P x y z. + +Variable f : T1 -> T2. + +Definition prop_on1 Pf P & phantom T3 (Pf f) & ph {all1 P} := + forall x, in_mem (f x) d2 -> P x. + +Definition prop_on2 Pf P & phantom T3 (Pf f) & ph {all2 P} := + forall x y, in_mem (f x) d2 -> in_mem (f y) d2 -> P x y. + +End LocalProperties. + +Definition inPhantom := Phantom Prop. +Definition onPhantom T P (x : T) := Phantom Prop (P x). + +Definition bijective_in aT rT (d : mem_pred aT) (f : aT -> rT) := + exists2 g, prop_in1 d (inPhantom (cancel f g)) + & prop_on1 d (Phantom _ (cancel g)) (onPhantom (cancel g) f). + +Definition bijective_on aT rT (cd : mem_pred rT) (f : aT -> rT) := + exists2 g, prop_on1 cd (Phantom _ (cancel f)) (onPhantom (cancel f) g) + & prop_in1 cd (inPhantom (cancel g f)). + +Notation "{ 'for' x , P }" := + (prop_for x (inPhantom P)) + (at level 0, format "{ 'for' x , P }") : type_scope. + +Notation "{ 'in' d , P }" := + (prop_in1 (mem d) (inPhantom P)) + (at level 0, format "{ 'in' d , P }") : type_scope. + +Notation "{ 'in' d1 & d2 , P }" := + (prop_in11 (mem d1) (mem d2) (inPhantom P)) + (at level 0, format "{ 'in' d1 & d2 , P }") : type_scope. + +Notation "{ 'in' d & , P }" := + (prop_in2 (mem d) (inPhantom P)) + (at level 0, format "{ 'in' d & , P }") : type_scope. + +Notation "{ 'in' d1 & d2 & d3 , P }" := + (prop_in111 (mem d1) (mem d2) (mem d3) (inPhantom P)) + (at level 0, format "{ 'in' d1 & d2 & d3 , P }") : type_scope. + +Notation "{ 'in' d1 & & d3 , P }" := + (prop_in21 (mem d1) (mem d3) (inPhantom P)) + (at level 0, format "{ 'in' d1 & & d3 , P }") : type_scope. + +Notation "{ 'in' d1 & d2 & , P }" := + (prop_in12 (mem d1) (mem d2) (inPhantom P)) + (at level 0, format "{ 'in' d1 & d2 & , P }") : type_scope. + +Notation "{ 'in' d & & , P }" := + (prop_in3 (mem d) (inPhantom P)) + (at level 0, format "{ 'in' d & & , P }") : type_scope. + +Notation "{ 'on' cd , P }" := + (prop_on1 (mem cd) (inPhantom P) (inPhantom P)) + (at level 0, format "{ 'on' cd , P }") : type_scope. + +Notation "{ 'on' cd & , P }" := + (prop_on2 (mem cd) (inPhantom P) (inPhantom P)) + (at level 0, format "{ 'on' cd & , P }") : type_scope. + +Notation "{ 'on' cd , P & g }" := + (prop_on1 (mem cd) (Phantom (_ -> Prop) P) (onPhantom P g)) + (at level 0, format "{ 'on' cd , P & g }") : type_scope. + +Notation "{ 'in' d , 'bijective' f }" := (bijective_in (mem d) f) + (at level 0, f at level 8, + format "{ 'in' d , 'bijective' f }") : type_scope. + +Notation "{ 'on' cd , 'bijective' f }" := (bijective_on (mem cd) f) + (at level 0, f at level 8, + format "{ 'on' cd , 'bijective' f }") : type_scope. + +(* Weakening and monotonicity lemmas for localized predicates. *) +(* Note that using these lemmas in backward reasoning will force expansion of *) +(* the predicate definition, as Coq needs to expose the quantifier to apply *) +(* these lemmas. We define a few specialized variants to avoid this for some *) +(* of the ssrfun predicates. *) + +Section LocalGlobal. + +Variables T1 T2 T3 : predArgType. +Variables (D1 : pred T1) (D2 : pred T2) (D3 : pred T3). +Variables (d1 d1' : mem_pred T1) (d2 d2' : mem_pred T2) (d3 d3' : mem_pred T3). +Variables (f f' : T1 -> T2) (g : T2 -> T1) (h : T3). +Variables (P1 : T1 -> Prop) (P2 : T1 -> T2 -> Prop). +Variable P3 : T1 -> T2 -> T3 -> Prop. +Variable Q1 : (T1 -> T2) -> T1 -> Prop. +Variable Q1l : (T1 -> T2) -> T3 -> T1 -> Prop. +Variable Q2 : (T1 -> T2) -> T1 -> T1 -> Prop. + +Hypothesis sub1 : sub_mem d1 d1'. +Hypothesis sub2 : sub_mem d2 d2'. +Hypothesis sub3 : sub_mem d3 d3'. + +Lemma in1W : {all1 P1} -> {in D1, {all1 P1}}. +Proof. by move=> ? ?. Qed. +Lemma in2W : {all2 P2} -> {in D1 & D2, {all2 P2}}. +Proof. by move=> ? ?. Qed. +Lemma in3W : {all3 P3} -> {in D1 & D2 & D3, {all3 P3}}. +Proof. by move=> ? ?. Qed. + +Lemma in1T : {in T1, {all1 P1}} -> {all1 P1}. +Proof. by move=> ? ?; auto. Qed. +Lemma in2T : {in T1 & T2, {all2 P2}} -> {all2 P2}. +Proof. by move=> ? ?; auto. Qed. +Lemma in3T : {in T1 & T2 & T3, {all3 P3}} -> {all3 P3}. +Proof. by move=> ? ?; auto. Qed. + +Lemma sub_in1 (Ph : ph {all1 P1}) : prop_in1 d1' Ph -> prop_in1 d1 Ph. +Proof. move=> allP x /sub1; exact: allP. Qed. + +Lemma sub_in11 (Ph : ph {all2 P2}) : prop_in11 d1' d2' Ph -> prop_in11 d1 d2 Ph. +Proof. move=> allP x1 x2 /sub1 d1x1 /sub2; exact: allP. Qed. + +Lemma sub_in111 (Ph : ph {all3 P3}) : + prop_in111 d1' d2' d3' Ph -> prop_in111 d1 d2 d3 Ph. +Proof. by move=> allP x1 x2 x3 /sub1 d1x1 /sub2 d2x2 /sub3; exact: allP. Qed. + +Let allQ1 f'' := {all1 Q1 f''}. +Let allQ1l f'' h' := {all1 Q1l f'' h'}. +Let allQ2 f'' := {all2 Q2 f''}. + +Lemma on1W : allQ1 f -> {on D2, allQ1 f}. Proof. by move=> ? ?. Qed. + +Lemma on1lW : allQ1l f h -> {on D2, allQ1l f & h}. Proof. by move=> ? ?. Qed. + +Lemma on2W : allQ2 f -> {on D2 &, allQ2 f}. Proof. by move=> ? ?. Qed. + +Lemma on1T : {on T2, allQ1 f} -> allQ1 f. Proof. by move=> ? ?; auto. Qed. + +Lemma on1lT : {on T2, allQ1l f & h} -> allQ1l f h. +Proof. by move=> ? ?; auto. Qed. + +Lemma on2T : {on T2 &, allQ2 f} -> allQ2 f. +Proof. by move=> ? ?; auto. Qed. + +Lemma subon1 (Phf : ph (allQ1 f)) (Ph : ph (allQ1 f)) : + prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph. +Proof. by move=> allQ x /sub2; exact: allQ. Qed. + +Lemma subon1l (Phf : ph (allQ1l f)) (Ph : ph (allQ1l f h)) : + prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph. +Proof. by move=> allQ x /sub2; exact: allQ. Qed. + +Lemma subon2 (Phf : ph (allQ2 f)) (Ph : ph (allQ2 f)) : + prop_on2 d2' Phf Ph -> prop_on2 d2 Phf Ph. +Proof. by move=> allQ x y /sub2=> d2fx /sub2; exact: allQ. Qed. + +Lemma can_in_inj : {in D1, cancel f g} -> {in D1 &, injective f}. +Proof. by move=> fK x y /fK{2}<- /fK{2}<- ->. Qed. + +Lemma canLR_in x y : {in D1, cancel f g} -> y \in D1 -> x = f y -> g x = y. +Proof. by move=> fK D1y ->; rewrite fK. Qed. + +Lemma canRL_in x y : {in D1, cancel f g} -> x \in D1 -> f x = y -> x = g y. +Proof. by move=> fK D1x <-; rewrite fK. Qed. + +Lemma on_can_inj : {on D2, cancel f & g} -> {on D2 &, injective f}. +Proof. by move=> fK x y /fK{2}<- /fK{2}<- ->. Qed. + +Lemma canLR_on x y : {on D2, cancel f & g} -> f y \in D2 -> x = f y -> g x = y. +Proof. by move=> fK D2fy ->; rewrite fK. Qed. + +Lemma canRL_on x y : {on D2, cancel f & g} -> f x \in D2 -> f x = y -> x = g y. +Proof. by move=> fK D2fx <-; rewrite fK. Qed. + +Lemma inW_bij : bijective f -> {in D1, bijective f}. +Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. + +Lemma onW_bij : bijective f -> {on D2, bijective f}. +Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. + +Lemma inT_bij : {in T1, bijective f} -> bijective f. +Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. + +Lemma onT_bij : {on T2, bijective f} -> bijective f. +Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. + +Lemma sub_in_bij (D1' : pred T1) : + {subset D1 <= D1'} -> {in D1', bijective f} -> {in D1, bijective f}. +Proof. +by move=> subD [g' fK g'K]; exists g' => x; move/subD; [exact: fK | exact: g'K]. +Qed. + +Lemma subon_bij (D2' : pred T2) : + {subset D2 <= D2'} -> {on D2', bijective f} -> {on D2, bijective f}. +Proof. +by move=> subD [g' fK g'K]; exists g' => x; move/subD; [exact: fK | exact: g'K]. +Qed. + +End LocalGlobal. + +Lemma sub_in2 T d d' (P : T -> T -> Prop) : + sub_mem d d' -> forall Ph : ph {all2 P}, prop_in2 d' Ph -> prop_in2 d Ph. +Proof. by move=> /= sub_dd'; exact: sub_in11. Qed. + +Lemma sub_in3 T d d' (P : T -> T -> T -> Prop) : + sub_mem d d' -> forall Ph : ph {all3 P}, prop_in3 d' Ph -> prop_in3 d Ph. +Proof. by move=> /= sub_dd'; exact: sub_in111. Qed. + +Lemma sub_in12 T1 T d1 d1' d d' (P : T1 -> T -> T -> Prop) : + sub_mem d1 d1' -> sub_mem d d' -> + forall Ph : ph {all3 P}, prop_in12 d1' d' Ph -> prop_in12 d1 d Ph. +Proof. by move=> /= sub1 sub; exact: sub_in111. Qed. + +Lemma sub_in21 T T3 d d' d3 d3' (P : T -> T -> T3 -> Prop) : + sub_mem d d' -> sub_mem d3 d3' -> + forall Ph : ph {all3 P}, prop_in21 d' d3' Ph -> prop_in21 d d3 Ph. +Proof. by move=> /= sub sub3; exact: sub_in111. Qed. + +Lemma equivalence_relP_in T (R : rel T) (A : pred T) : + {in A & &, equivalence_rel R} + <-> {in A, reflexive R} /\ {in A &, forall x y, R x y -> {in A, R x =1 R y}}. +Proof. +split=> [eqiR | [Rxx trR] x y z *]; last by split=> [|/trR-> //]; exact: Rxx. +by split=> [x Ax|x y Ax Ay Rxy z Az]; [rewrite (eqiR x x) | rewrite (eqiR x y)]. +Qed. + +Section MonoHomoMorphismTheory. + +Variables (aT rT sT : Type) (f : aT -> rT) (g : rT -> aT). +Variables (aP : pred aT) (rP : pred rT) (aR : rel aT) (rR : rel rT). + +Lemma monoW : {mono f : x / aP x >-> rP x} -> {homo f : x / aP x >-> rP x}. +Proof. by move=> hf x ax; rewrite hf. Qed. + +Lemma mono2W : + {mono f : x y / aR x y >-> rR x y} -> {homo f : x y / aR x y >-> rR x y}. +Proof. by move=> hf x y axy; rewrite hf. Qed. + +Hypothesis fgK : cancel g f. + +Lemma homoRL : + {homo f : x y / aR x y >-> rR x y} -> forall x y, aR (g x) y -> rR x (f y). +Proof. by move=> Hf x y /Hf; rewrite fgK. Qed. + +Lemma homoLR : + {homo f : x y / aR x y >-> rR x y} -> forall x y, aR x (g y) -> rR (f x) y. +Proof. by move=> Hf x y /Hf; rewrite fgK. Qed. + +Lemma homo_mono : + {homo f : x y / aR x y >-> rR x y} -> {homo g : x y / rR x y >-> aR x y} -> + {mono g : x y / rR x y >-> aR x y}. +Proof. +move=> mf mg x y; case: (boolP (rR _ _))=> [/mg //|]. +by apply: contraNF=> /mf; rewrite !fgK. +Qed. + +Lemma monoLR : + {mono f : x y / aR x y >-> rR x y} -> forall x y, rR (f x) y = aR x (g y). +Proof. by move=> mf x y; rewrite -{1}[y]fgK mf. Qed. + +Lemma monoRL : + {mono f : x y / aR x y >-> rR x y} -> forall x y, rR x (f y) = aR (g x) y. +Proof. by move=> mf x y; rewrite -{1}[x]fgK mf. Qed. + +Lemma can_mono : + {mono f : x y / aR x y >-> rR x y} -> {mono g : x y / rR x y >-> aR x y}. +Proof. by move=> mf x y /=; rewrite -mf !fgK. Qed. + +End MonoHomoMorphismTheory. + +Section MonoHomoMorphismTheory_in. + +Variables (aT rT sT : predArgType) (f : aT -> rT) (g : rT -> aT). +Variable (aD : pred aT). +Variable (aP : pred aT) (rP : pred rT) (aR : rel aT) (rR : rel rT). + +Notation rD := [pred x | g x \in aD]. + +Lemma monoW_in : + {in aD &, {mono f : x y / aR x y >-> rR x y}} -> + {in aD &, {homo f : x y / aR x y >-> rR x y}}. +Proof. by move=> hf x y hx hy axy; rewrite hf. Qed. + +Lemma mono2W_in : + {in aD, {mono f : x / aP x >-> rP x}} -> + {in aD, {homo f : x / aP x >-> rP x}}. +Proof. by move=> hf x hx ax; rewrite hf. Qed. + +Hypothesis fgK_on : {on aD, cancel g & f}. + +Lemma homoRL_in : + {in aD &, {homo f : x y / aR x y >-> rR x y}} -> + {in rD & aD, forall x y, aR (g x) y -> rR x (f y)}. +Proof. by move=> Hf x y hx hy /Hf; rewrite fgK_on //; apply. Qed. + +Lemma homoLR_in : + {in aD &, {homo f : x y / aR x y >-> rR x y}} -> + {in aD & rD, forall x y, aR x (g y) -> rR (f x) y}. +Proof. by move=> Hf x y hx hy /Hf; rewrite fgK_on //; apply. Qed. + +Lemma homo_mono_in : + {in aD &, {homo f : x y / aR x y >-> rR x y}} -> + {in rD &, {homo g : x y / rR x y >-> aR x y}} -> + {in rD &, {mono g : x y / rR x y >-> aR x y}}. +Proof. +move=> mf mg x y hx hy; case: (boolP (rR _ _))=> [/mg //|]; first exact. +by apply: contraNF=> /mf; rewrite !fgK_on //; apply. +Qed. + +Lemma monoLR_in : + {in aD &, {mono f : x y / aR x y >-> rR x y}} -> + {in aD & rD, forall x y, rR (f x) y = aR x (g y)}. +Proof. by move=> mf x y hx hy; rewrite -{1}[y]fgK_on // mf. Qed. + +Lemma monoRL_in : + {in aD &, {mono f : x y / aR x y >-> rR x y}} -> + {in rD & aD, forall x y, rR x (f y) = aR (g x) y}. +Proof. by move=> mf x y hx hy; rewrite -{1}[x]fgK_on // mf. Qed. + +Lemma can_mono_in : + {in aD &, {mono f : x y / aR x y >-> rR x y}} -> + {in rD &, {mono g : x y / rR x y >-> aR x y}}. +Proof. by move=> mf x y hx hy /=; rewrite -mf // !fgK_on. Qed. + +End MonoHomoMorphismTheory_in. diff --git a/mathcomp/ssreflect/ssreflect.v b/mathcomp/ssreflect/ssreflect.v new file mode 100644 index 0000000..815a0fb --- /dev/null +++ b/mathcomp/ssreflect/ssreflect.v @@ -0,0 +1,420 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Require Import Bool. (* For bool_scope delimiter 'bool'. *) +Require Import ssrmatching. +Declare ML Module "ssreflect". +Set SsrAstVersion. + +(******************************************************************************) +(* This file is the Gallina part of the ssreflect plugin implementation. *) +(* Files that use the ssreflect plugin should always Require ssreflect and *) +(* either Import ssreflect or Import ssreflect.SsrSyntax. *) +(* Part of the contents of this file is technical and will only interest *) +(* advanced developers; in addition the following are defined: *) +(* [the str of v by f] == the Canonical s : str such that f s = v. *) +(* [the str of v] == the Canonical s : str that coerces to v. *) +(* argumentType c == the T such that c : forall x : T, P x. *) +(* returnType c == the R such that c : T -> R. *) +(* {type of c for s} == P s where c : forall x : T, P x. *) +(* phantom T v == singleton type with inhabitant Phantom T v. *) +(* phant T == singleton type with inhabitant Phant v. *) +(* =^~ r == the converse of rewriting rule r (e.g., in a *) +(* rewrite multirule). *) +(* unkeyed t == t, but treated as an unkeyed matching pattern by *) +(* the ssreflect matching algorithm. *) +(* nosimpl t == t, but on the right-hand side of Definition C := *) +(* nosimpl disables expansion of C by /=. *) +(* locked t == t, but locked t is not convertible to t. *) +(* locked_with k t == t, but not convertible to t or locked_with k' t *) +(* unless k = k' (with k : unit). Coq type-checking *) +(* will be much more efficient if locked_with with a *) +(* bespoke k is used for sealed definitions. *) +(* unlockable v == interface for sealed constant definitions of v. *) +(* Unlockable def == the unlockable that registers def : C = v. *) +(* [unlockable of C] == a clone for C of the canonical unlockable for the *) +(* definition of C (e.g., if it uses locked_with). *) +(* [unlockable fun C] == [unlockable of C] with the expansion forced to be *) +(* an explicit lambda expression. *) +(* -> The usage pattern for ADT operations is: *) +(* Definition foo_def x1 .. xn := big_foo_expression. *) +(* Fact foo_key : unit. Proof. by []. Qed. *) +(* Definition foo := locked_with foo_key foo_def. *) +(* Canonical foo_unlockable := [unlockable fun foo]. *) +(* This minimizes the comparison overhead for foo, while still allowing *) +(* rewrite unlock to expose big_foo_expression. *) +(* More information about these definitions and their use can be found in the *) +(* ssreflect manual, and in specific comments below. *) +(******************************************************************************) + +Global Set Asymmetric Patterns. + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Module SsrSyntax. + +(* Declare Ssr keywords: 'is' 'of' '//' '/=' and '//='. We also declare the *) +(* parsing level 8, as a workaround for a notation grammar factoring problem. *) +(* Arguments of application-style notations (at level 10) should be declared *) +(* at level 8 rather than 9 or the camlp5 grammar will not factor properly. *) + +Reserved Notation "(* x 'is' y 'of' z 'isn't' // /= //= *)" (at level 8). +Reserved Notation "(* 69 *)" (at level 69). + +(* Non ambiguous keyword to check if the SsrSyntax module is imported *) +Reserved Notation "(* Use to test if 'SsrSyntax_is_Imported' *)" (at level 8). + +Reserved Notation "<hidden n >" (at level 200). +Reserved Notation "T (* n *)" (at level 200, format "T (* n *)"). + +End SsrSyntax. + +Export SsrMatchingSyntax. +Export SsrSyntax. + +(* Make the general "if" into a notation, so that we can override it below. *) +(* The notations are "only parsing" because the Coq decompiler will not *) +(* recognize the expansion of the boolean if; using the default printer *) +(* avoids a spurrious trailing %GEN_IF. *) + +Delimit Scope general_if_scope with GEN_IF. + +Notation "'if' c 'then' v1 'else' v2" := + (if c then v1 else v2) + (at level 200, c, v1, v2 at level 200, only parsing) : general_if_scope. + +Notation "'if' c 'return' t 'then' v1 'else' v2" := + (if c return t then v1 else v2) + (at level 200, c, t, v1, v2 at level 200, only parsing) : general_if_scope. + +Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" := + (if c as x return t then v1 else v2) + (at level 200, c, t, v1, v2 at level 200, x ident, only parsing) + : general_if_scope. + +(* Force boolean interpretation of simple if expressions. *) + +Delimit Scope boolean_if_scope with BOOL_IF. + +Notation "'if' c 'return' t 'then' v1 'else' v2" := + (if c%bool is true in bool return t then v1 else v2) : boolean_if_scope. + +Notation "'if' c 'then' v1 'else' v2" := + (if c%bool is true in bool return _ then v1 else v2) : boolean_if_scope. + +Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" := + (if c%bool is true as x in bool return t then v1 else v2) : boolean_if_scope. + +Open Scope boolean_if_scope. + +(* To allow a wider variety of notations without reserving a large number of *) +(* of identifiers, the ssreflect library systematically uses "forms" to *) +(* enclose complex mixfix syntax. A "form" is simply a mixfix expression *) +(* enclosed in square brackets and introduced by a keyword: *) +(* [keyword ... ] *) +(* Because the keyword follows a bracket it does not need to be reserved. *) +(* Non-ssreflect libraries that do not respect the form syntax (e.g., the Coq *) +(* Lists library) should be loaded before ssreflect so that their notations *) +(* do not mask all ssreflect forms. *) +Delimit Scope form_scope with FORM. +Open Scope form_scope. + +(* Allow overloading of the cast (x : T) syntax, put whitespace around the *) +(* ":" symbol to avoid lexical clashes (and for consistency with the parsing *) +(* precedence of the notation, which binds less tightly than application), *) +(* and put printing boxes that print the type of a long definition on a *) +(* separate line rather than force-fit it at the right margin. *) +Notation "x : T" := (x : T) + (at level 100, right associativity, + format "'[hv' x '/ ' : T ']'") : core_scope. + +(* Allow the casual use of notations like nat * nat for explicit Type *) +(* declarations. Note that (nat * nat : Type) is NOT equivalent to *) +(* (nat * nat)%type, whose inferred type is legacy type "Set". *) +Notation "T : 'Type'" := (T%type : Type) + (at level 100, only parsing) : core_scope. +(* Allow similarly Prop annotation for, e.g., rewrite multirules. *) +Notation "P : 'Prop'" := (P%type : Prop) + (at level 100, only parsing) : core_scope. + +(* Constants for abstract: and [: name ] intro pattern *) +Definition abstract_lock := unit. +Definition abstract_key := tt. + +Definition abstract (statement : Type) (id : nat) (lock : abstract_lock) := + let: tt := lock in statement. + +Notation "<hidden n >" := (abstract _ n _). +Notation "T (* n *)" := (abstract T n abstract_key). + +(* Syntax for referring to canonical structures: *) +(* [the struct_type of proj_val by proj_fun] *) +(* This form denotes the Canonical instance s of the Structure type *) +(* struct_type whose proj_fun projection is proj_val, i.e., such that *) +(* proj_fun s = proj_val. *) +(* Typically proj_fun will be A record field accessors of struct_type, but *) +(* this need not be the case; it can be, for instance, a field of a record *) +(* type to which struct_type coerces; proj_val will likewise be coerced to *) +(* the return type of proj_fun. In all but the simplest cases, proj_fun *) +(* should be eta-expanded to allow for the insertion of implicit arguments. *) +(* In the common case where proj_fun itself is a coercion, the "by" part *) +(* can be omitted entirely; in this case it is inferred by casting s to the *) +(* inferred type of proj_val. Obviously the latter can be fixed by using an *) +(* explicit cast on proj_val, and it is highly recommended to do so when the *) +(* return type intended for proj_fun is "Type", as the type inferred for *) +(* proj_val may vary because of sort polymorphism (it could be Set or Prop). *) +(* Note when using the [the _ of _] form to generate a substructure from a *) +(* telescopes-style canonical hierarchy (implementing inheritance with *) +(* coercions), one should always project or coerce the value to the BASE *) +(* structure, because Coq will only find a Canonical derived structure for *) +(* the Canonical base structure -- not for a base structure that is specific *) +(* to proj_value. *) + +Module TheCanonical. + +CoInductive put vT sT (v1 v2 : vT) (s : sT) := Put. + +Definition get vT sT v s (p : @put vT sT v v s) := let: Put := p in s. + +Definition get_by vT sT of sT -> vT := @get vT sT. + +End TheCanonical. + +Import TheCanonical. (* Note: no export. *) + +Notation "[ 'the' sT 'of' v 'by' f ]" := + (@get_by _ sT f _ _ ((fun v' (s : sT) => Put v' (f s) s) v _)) + (at level 0, only parsing) : form_scope. + +Notation "[ 'the' sT 'of' v ]" := (get ((fun s : sT => Put v (*coerce*)s s) _)) + (at level 0, only parsing) : form_scope. + +(* The following are "format only" versions of the above notations. Since Coq *) +(* doesn't provide this facility, we fake it by splitting the "the" keyword. *) +(* We need to do this to prevent the formatter from being be thrown off by *) +(* application collapsing, coercion insertion and beta reduction in the right *) +(* hand side of the notations above. *) + +Notation "[ 'th' 'e' sT 'of' v 'by' f ]" := (@get_by _ sT f v _ _) + (at level 0, format "[ 'th' 'e' sT 'of' v 'by' f ]") : form_scope. + +Notation "[ 'th' 'e' sT 'of' v ]" := (@get _ sT v _ _) + (at level 0, format "[ 'th' 'e' sT 'of' v ]") : form_scope. + +(* We would like to recognize +Notation "[ 'th' 'e' sT 'of' v : 'Type' ]" := (@get Type sT v _ _) + (at level 0, format "[ 'th' 'e' sT 'of' v : 'Type' ]") : form_scope. +*) + +(* Helper notation for canonical structure inheritance support. *) +(* This is a workaround for the poor interaction between delta reduction and *) +(* canonical projections in Coq's unification algorithm, by which transparent *) +(* definitions hide canonical instances, i.e., in *) +(* Canonical a_type_struct := @Struct a_type ... *) +(* Definition my_type := a_type. *) +(* my_type doesn't effectively inherit the struct structure from a_type. Our *) +(* solution is to redeclare the instance as follows *) +(* Canonical my_type_struct := Eval hnf in [struct of my_type]. *) +(* The special notation [str of _] must be defined for each Strucure "str" *) +(* with constructor "Str", typically as follows *) +(* Definition clone_str s := *) +(* let: Str _ x y ... z := s return {type of Str for s} -> str in *) +(* fun k => k _ x y ... z. *) +(* Notation "[ 'str' 'of' T 'for' s ]" := (@clone_str s (@Str T)) *) +(* (at level 0, format "[ 'str' 'of' T 'for' s ]") : form_scope. *) +(* Notation "[ 'str' 'of' T ]" := (repack_str (fun x => @Str T x)) *) +(* (at level 0, format "[ 'str' 'of' T ]") : form_scope. *) +(* The notation for the match return predicate is defined below; the eta *) +(* expansion in the second form serves both to distinguish it from the first *) +(* and to avoid the delta reduction problem. *) +(* There are several variations on the notation and the definition of the *) +(* the "clone" function, for telescopes, mixin classes, and join (multiple *) +(* inheritance) classes. We describe a different idiom for clones in ssrfun; *) +(* it uses phantom types (see below) and static unification; see fintype and *) +(* ssralg for examples. *) + +Definition argumentType T P & forall x : T, P x := T. +Definition dependentReturnType T P & forall x : T, P x := P. +Definition returnType aT rT & aT -> rT := rT. + +Notation "{ 'type' 'of' c 'for' s }" := (dependentReturnType c s) + (at level 0, format "{ 'type' 'of' c 'for' s }") : type_scope. + +(* A generic "phantom" type (actually, a unit type with a phantom parameter). *) +(* This type can be used for type definitions that require some Structure *) +(* on one of their parameters, to allow Coq to infer said structure so it *) +(* does not have to be supplied explicitly or via the "[the _ of _]" notation *) +(* (the latter interacts poorly with other Notation). *) +(* The definition of a (co)inductive type with a parameter p : p_type, that *) +(* needs to use the operations of a structure *) +(* Structure p_str : Type := p_Str {p_repr :> p_type; p_op : p_repr -> ...} *) +(* should be given as *) +(* Inductive indt_type (p : p_str) := Indt ... . *) +(* Definition indt_of (p : p_str) & phantom p_type p := indt_type p. *) +(* Notation "{ 'indt' p }" := (indt_of (Phantom p)). *) +(* Definition indt p x y ... z : {indt p} := @Indt p x y ... z. *) +(* Notation "[ 'indt' x y ... z ]" := (indt x y ... z). *) +(* That is, the concrete type and its constructor should be shadowed by *) +(* definitions that use a phantom argument to infer and display the true *) +(* value of p (in practice, the "indt" constructor often performs additional *) +(* functions, like "locking" the representation -- see below). *) +(* We also define a simpler version ("phant" / "Phant") of phantom for the *) +(* common case where p_type is Type. *) + +CoInductive phantom T (p : T) := Phantom. +Implicit Arguments phantom []. +Implicit Arguments Phantom []. +CoInductive phant (p : Type) := Phant. + +(* Internal tagging used by the implementation of the ssreflect elim. *) + +Definition protect_term (A : Type) (x : A) : A := x. + +(* The ssreflect idiom for a non-keyed pattern: *) +(* - unkeyed t wiil match any subterm that unifies with t, regardless of *) +(* whether it displays the same head symbol as t. *) +(* - unkeyed t a b will match any application of a term f unifying with t, *) +(* to two arguments unifying with with a and b, repectively, regardless of *) +(* apparent head symbols. *) +(* - unkeyed x where x is a variable will match any subterm with the same *) +(* type as x (when x would raise the 'indeterminate pattern' error). *) + +Notation unkeyed x := (let flex := x in flex). + +(* Ssreflect converse rewrite rule rule idiom. *) +Definition ssr_converse R (r : R) := (Logic.I, r). +Notation "=^~ r" := (ssr_converse r) (at level 100) : form_scope. + +(* Term tagging (user-level). *) +(* The ssreflect library uses four strengths of term tagging to restrict *) +(* convertibility during type checking: *) +(* nosimpl t simplifies to t EXCEPT in a definition; more precisely, given *) +(* Definition foo := nosimpl bar, foo (or foo t') will NOT be expanded by *) +(* the /= and //= switches unless it is in a forcing context (e.g., in *) +(* match foo t' with ... end, foo t' will be reduced if this allows the *) +(* match to be reduced). Note that nosimpl bar is simply notation for a *) +(* a term that beta-iota reduces to bar; hence rewrite /foo will replace *) +(* foo by bar, and rewrite -/foo will replace bar by foo. *) +(* CAVEAT: nosimpl should not be used inside a Section, because the end of *) +(* section "cooking" removes the iota redex. *) +(* locked t is provably equal to t, but is not convertible to t; 'locked' *) +(* provides support for selective rewriting, via the lock t : t = locked t *) +(* Lemma, and the ssreflect unlock tactic. *) +(* locked_with k t is equal but not convertible to t, much like locked t, *) +(* but supports explicit tagging with a value k : unit. This is used to *) +(* mitigate a flaw in the term comparison heuristic of the Coq kernel, *) +(* which treats all terms of the form locked t as equal and conpares their *) +(* arguments recursively, leading to an exponential blowup of comparison. *) +(* For this reason locked_with should be used rather than locked when *) +(* defining ADT operations. The unlock tactic does not support locked_with *) +(* but the unlock rewrite rule does, via the unlockable interface. *) +(* we also use Module Type ascription to create truly opaque constants, *) +(* because simple expansion of constants to reveal an unreducible term *) +(* doubles the time complexity of a negative comparison. Such opaque *) +(* constants can be expanded generically with the unlock rewrite rule. *) +(* See the definition of card and subset in fintype for examples of this. *) + +Notation nosimpl t := (let: tt := tt in t). + +Lemma master_key : unit. Proof. exact tt. Qed. +Definition locked A := let: tt := master_key in fun x : A => x. + +Lemma lock A x : x = locked x :> A. Proof. unlock; reflexivity. Qed. + +(* Needed for locked predicates, in particular for eqType's. *) +Lemma not_locked_false_eq_true : locked false <> true. +Proof. unlock; discriminate. Qed. + +(* The basic closing tactic "done". *) +Ltac done := + trivial; hnf; intros; solve + [ do ![solve [trivial | apply: sym_equal; trivial] + | discriminate | contradiction | split] + | case not_locked_false_eq_true; assumption + | match goal with H : ~ _ |- _ => solve [case H; trivial] end ]. + +(* To unlock opaque constants. *) +Structure unlockable T v := Unlockable {unlocked : T; _ : unlocked = v}. +Lemma unlock T x C : @unlocked T x C = x. Proof. by case: C. Qed. + +Notation "[ 'unlockable' 'of' C ]" := (@Unlockable _ _ C (unlock _)) + (at level 0, format "[ 'unlockable' 'of' C ]") : form_scope. + +Notation "[ 'unlockable' 'fun' C ]" := (@Unlockable _ (fun _ => _) C (unlock _)) + (at level 0, format "[ 'unlockable' 'fun' C ]") : form_scope. + +(* Generic keyed constant locking. *) + +(* The argument order ensures that k is always compared before T. *) +Definition locked_with k := let: tt := k in fun T x => x : T. + +(* This can be used as a cheap alternative to cloning the unlockable instance *) +(* below, but with caution as unkeyed matching can be expensive. *) +Lemma locked_withE T k x : unkeyed (locked_with k x) = x :> T. +Proof. by case: k. Qed. + +(* Intensionaly, this instance will not apply to locked u. *) +Canonical locked_with_unlockable T k x := + @Unlockable T x (locked_with k x) (locked_withE k x). + +(* More accurate variant of unlock, and safer alternative to locked_withE. *) +Lemma unlock_with T k x : unlocked (locked_with_unlockable k x) = x :> T. +Proof. exact: unlock. Qed. + +(* The internal lemmas for the have tactics. *) + +Definition ssr_have Plemma Pgoal (step : Plemma) rest : Pgoal := rest step. +Implicit Arguments ssr_have [Pgoal]. + +Definition ssr_have_let Pgoal Plemma step + (rest : let x : Plemma := step in Pgoal) : Pgoal := rest. +Implicit Arguments ssr_have_let [Pgoal]. + +Definition ssr_suff Plemma Pgoal step (rest : Plemma) : Pgoal := step rest. +Implicit Arguments ssr_suff [Pgoal]. + +Definition ssr_wlog := ssr_suff. +Implicit Arguments ssr_wlog [Pgoal]. + +(* Internal N-ary congruence lemmas for the congr tactic. *) + +Fixpoint nary_congruence_statement (n : nat) + : (forall B, (B -> B -> Prop) -> Prop) -> Prop := + match n with + | O => fun k => forall B, k B (fun x1 x2 : B => x1 = x2) + | S n' => + let k' A B e (f1 f2 : A -> B) := + forall x1 x2, x1 = x2 -> (e (f1 x1) (f2 x2) : Prop) in + fun k => forall A, nary_congruence_statement n' (fun B e => k _ (k' A B e)) + end. + +Lemma nary_congruence n (k := fun B e => forall y : B, (e y y : Prop)) : + nary_congruence_statement n k. +Proof. +have: k _ _ := _; rewrite {1}/k. +elim: n k => [|n IHn] k k_P /= A; first exact: k_P. +by apply: IHn => B e He; apply: k_P => f x1 x2 <-. +Qed. + +Lemma ssr_congr_arrow Plemma Pgoal : Plemma = Pgoal -> Plemma -> Pgoal. +Proof. by move->. Qed. +Implicit Arguments ssr_congr_arrow []. + +(* View lemmas that don't use reflection. *) + +Section ApplyIff. + +Variables P Q : Prop. +Hypothesis eqPQ : P <-> Q. + +Lemma iffLR : P -> Q. Proof. by case: eqPQ. Qed. +Lemma iffRL : Q -> P. Proof. by case: eqPQ. Qed. + +Lemma iffLRn : ~P -> ~Q. Proof. by move=> nP tQ; case: nP; case: eqPQ tQ. Qed. +Lemma iffRLn : ~Q -> ~P. Proof. by move=> nQ tP; case: nQ; case: eqPQ tP. Qed. + +End ApplyIff. + +Hint View for move/ iffLRn|2 iffRLn|2 iffLR|2 iffRL|2. +Hint View for apply/ iffRLn|2 iffLRn|2 iffRL|2 iffLR|2. + diff --git a/mathcomp/ssreflect/ssrfun.v b/mathcomp/ssreflect/ssrfun.v new file mode 100644 index 0000000..f83724e --- /dev/null +++ b/mathcomp/ssreflect/ssrfun.v @@ -0,0 +1,885 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Require Import ssreflect. + +(******************************************************************************) +(* This file contains the basic definitions and notations for working with *) +(* functions. The definitions provide for: *) +(* *) +(* - Pair projections: *) +(* p.1 == first element of a pair *) +(* p.2 == second element of a pair *) +(* These notations also apply to p : P /\ Q, via an and >-> pair coercion. *) +(* *) +(* - Simplifying functions, beta-reduced by /= and simpl: *) +(* [fun : T => E] == constant function from type T that returns E *) +(* [fun x => E] == unary function *) +(* [fun x : T => E] == unary function with explicit domain type *) +(* [fun x y => E] == binary function *) +(* [fun x y : T => E] == binary function with common domain type *) +(* [fun (x : T) y => E] \ *) +(* [fun (x : xT) (y : yT) => E] | == binary function with (some) explicit, *) +(* [fun x (y : T) => E] / independent domain types for each argument *) +(* *) +(* - Partial functions using option type: *) +(* oapp f d ox == if ox is Some x returns f x, d otherwise *) +(* odflt d ox == if ox is Some x returns x, d otherwise *) +(* obind f ox == if ox is Some x returns f x, None otherwise *) +(* omap f ox == if ox is Some x returns Some (f x), None otherwise *) +(* *) +(* - Singleton types: *) +(* all_equal_to x0 == x0 is the only value in its type, so any such value *) +(* can be rewritten to x0. *) +(* *) +(* - A generic wrapper type: *) +(* wrapped T == the inductive type with values Wrap x for x : T. *) +(* unwrap w == the projection of w : wrapped T on T. *) +(* wrap x == the canonical injection of x : T into wrapped T; it is *) +(* equivalent to Wrap x, but is declared as a (default) *) +(* Canonical Structure, which lets the Coq HO unification *) +(* automatically expand x into unwrap (wrap x). The delta *) +(* reduction of wrap x to Wrap can be exploited to *) +(* introduce controlled nondeterminism in Canonical *) +(* Structure inference, as in the implementation of *) +(* the mxdirect predicate in matrix.v. *) +(* *) +(* - Sigma types: *) +(* tag w == the i of w : {i : I & T i}. *) +(* tagged w == the T i component of w : {i : I & T i}. *) +(* Tagged T x == the {i : I & T i} with component x : T i. *) +(* tag2 w == the i of w : {i : I & T i & U i}. *) +(* tagged2 w == the T i component of w : {i : I & T i & U i}. *) +(* tagged2' w == the U i component of w : {i : I & T i & U i}. *) +(* Tagged2 T U x y == the {i : I & T i} with components x : T i and y : U i. *) +(* sval u == the x of u : {x : T | P x}. *) +(* s2val u == the x of u : {x : T | P x & Q x}. *) +(* The properties of sval u, s2val u are given by lemmas svalP, s2valP, and *) +(* s2valP'. We provide coercions sigT2 >-> sigT and sig2 >-> sig >-> sigT. *) +(* A suite of lemmas (all_sig, ...) let us skolemize sig, sig2, sigT, sigT2 *) +(* and pair, e.g., *) +(* have /all_sig[f fP] (x : T): {y : U | P y} by ... *) +(* yields an f : T -> U such that fP : forall x, P (f x). *) +(* - Identity functions: *) +(* id == NOTATION for the explicit identity function fun x => x. *) +(* @id T == notation for the explicit identity at type T. *) +(* idfun == an expression with a head constant, convertible to id; *) +(* idfun x simplifies to x. *) +(* @idfun T == the expression above, specialized to type T. *) +(* phant_id x y == the function type phantom _ x -> phantom _ y. *) +(* *** In addition to their casual use in functional programming, identity *) +(* functions are often used to trigger static unification as part of the *) +(* construction of dependent Records and Structures. For example, if we need *) +(* a structure sT over a type T, we take as arguments T, sT, and a "dummy" *) +(* function T -> sort sT: *) +(* Definition foo T sT & T -> sort sT := ... *) +(* We can avoid specifying sT directly by calling foo (@id T), or specify *) +(* the call completely while still ensuring the consistency of T and sT, by *) +(* calling @foo T sT idfun. The phant_id type allows us to extend this trick *) +(* to non-Type canonical projections. It also allows us to sidestep *) +(* dependent type constraints when building explicit records, e.g., given *) +(* Record r := R {x; y : T(x)}. *) +(* if we need to build an r from a given y0 while inferring some x0, such *) +(* that y0 : T(x0), we pose *) +(* Definition mk_r .. y .. (x := ...) y' & phant_id y y' := R x y'. *) +(* Calling @mk_r .. y0 .. id will cause Coq to use y' := y0, while checking *) +(* the dependent type constraint y0 : T(x0). *) +(* *) +(* - Extensional equality for functions and relations (i.e. functions of two *) +(* arguments): *) +(* f1 =1 f2 == f1 x is equal to f2 x for all x. *) +(* f1 =1 f2 :> A == ... and f2 is explicitly typed. *) +(* f1 =2 f2 == f1 x y is equal to f2 x y for all x y. *) +(* f1 =2 f2 :> A == ... and f2 is explicitly typed. *) +(* *) +(* - Composition for total and partial functions: *) +(* f^~ y == function f with second argument specialised to y, *) +(* i.e., fun x => f x y *) +(* CAVEAT: conditional (non-maximal) implicit arguments *) +(* of f are NOT inserted in this context *) +(* @^~ x == application at x, i.e., fun f => f x *) +(* [eta f] == the explicit eta-expansion of f, i.e., fun x => f x *) +(* CAVEAT: conditional (non-maximal) implicit arguments *) +(* of f are NOT inserted in this context. *) +(* fun=> v := the constant function fun _ => v. *) +(* f1 \o f2 == composition of f1 and f2. *) +(* Note: (f1 \o f2) x simplifies to f1 (f2 x). *) +(* f1 \; f2 == categorical composition of f1 and f2. This expands to *) +(* to f2 \o f1 and (f1 \; f2) x simplifies to f2 (f1 x). *) +(* pcomp f1 f2 == composition of partial functions f1 and f2. *) +(* *) +(* - Reserved notation for various arithmetic and algebraic operations: *) +(* e.[a1, ..., a_n] evaluation (e.g., polynomials). *) +(* e`_i indexing (number list, integer pi-part). *) +(* x^-1 inverse (group, field). *) +(* x *+ n, x *- n integer multiplier (modules and rings). *) +(* x ^+ n, x ^- n integer exponent (groups and rings). *) +(* x *: A, A :* x external product (scaling/module product in rings, *) +(* left/right cosets in groups). *) +(* A :&: B intersection (of sets, groups, subspaces, ...). *) +(* A :|: B, a |: B union, union with a singleton (of sets). *) +(* A :\: B, A :\ b relative complement (of sets, subspaces, ...). *) +(* <<A>>, <[a]> generated group/subspace, generated cycle/line. *) +(* 'C[x], 'C_A[x] point centralisers (in groups and F-algebras). *) +(* 'C(A), 'C_B(A) centralisers (in groups and matrix and F_algebras). *) +(* 'Z(A) centers (in groups and matrix and F-algebras). *) +(* m %/ d, m %% d Euclidean division and remainder (nat, polynomials). *) +(* d %| m Euclidean divisibility (nat, polynomial). *) +(* m = n %[mod d] equality mod d (also defined for <>, ==, and !=). *) +(* e^`(n) nth formal derivative (groups, polynomials). *) +(* e^`() simple formal derivative (polynomials only). *) +(* `|x| norm, absolute value, distance (rings, int, nat). *) +(* x <= y ?= iff C x is less than y, and equal iff C holds (nat, rings). *) +(* x <= y :> T, etc cast comparison (rings, all comparison operators). *) +(* [rec a1, ..., an] standard shorthand for hidden recursor (see prime.v). *) +(* The interpretation of these notations is not defined here, but the *) +(* declarations help maintain consistency across the library. *) +(* *) +(* - Properties of functions: *) +(* injective f <-> f is injective. *) +(* cancel f g <-> g is a left inverse of f / f is a right inverse of g. *) +(* pcancel f g <-> g is a left inverse of f where g is partial. *) +(* ocancel f g <-> g is a left inverse of f where f is partial. *) +(* bijective f <-> f is bijective (has a left and right inverse). *) +(* involutive f <-> f is involutive. *) +(* *) +(* - Properties for operations. *) +(* left_id e op <-> e is a left identity for op (e op x = x). *) +(* right_id e op <-> e is a right identity for op (x op e = x). *) +(* left_inverse e inv op <-> inv is a left inverse for op wrt identity e, *) +(* i.e., (inv x) op x = e. *) +(* right_inverse e inv op <-> inv is a right inverse for op wrt identity e *) +(* i.e., x op (i x) = e. *) +(* self_inverse e op <-> each x is its own op-inverse (x op x = e). *) +(* idempotent op <-> op is idempotent for op (x op x = x). *) +(* associative op <-> op is associative, i.e., *) +(* x op (y op z) = (x op y) op z. *) +(* commutative op <-> op is commutative (x op y = y op x). *) +(* left_commutative op <-> op is left commutative, i.e., *) +(* x op (y op z) = y op (x op z). *) +(* right_commutative op <-> op is right commutative, i.e., *) +(* (x op y) op z = (x op z) op y. *) +(* left_zero z op <-> z is a left zero for op (z op x = z). *) +(* right_zero z op <-> z is a right zero for op (x op z = z). *) +(* left_distributive op1 op2 <-> op1 distributes over op2 to the left: *) +(* (x op2 y) op1 z = (x op1 z) op2 (y op1 z). *) +(* right_distributive op1 op2 <-> op distributes over add to the right: *) +(* x op1 (y op2 z) = (x op1 z) op2 (x op1 z). *) +(* interchange op1 op2 <-> op1 and op2 satisfy an interchange law: *) +(* (x op2 y) op1 (z op2 t) = (x op1 z) op2 (y op1 t). *) +(* Note that interchange op op is a commutativity property. *) +(* left_injective op <-> op is injective in its left argument: *) +(* x op y = z op y -> x = z. *) +(* right_injective op <-> op is injective in its right argument: *) +(* x op y = x op z -> y = z. *) +(* left_loop inv op <-> op, inv obey the inverse loop left axiom: *) +(* (inv x) op (x op y) = y for all x, y, i.e., *) +(* op (inv x) is always a left inverse of op x *) +(* rev_left_loop inv op <-> op, inv obey the inverse loop reverse left *) +(* axiom: x op ((inv x) op y) = y, for all x, y. *) +(* right_loop inv op <-> op, inv obey the inverse loop right axiom: *) +(* (x op y) op (inv y) = x for all x, y. *) +(* rev_right_loop inv op <-> op, inv obey the inverse loop reverse right *) +(* axiom: (x op y) op (inv y) = x for all x, y. *) +(* Note that familiar "cancellation" identities like x + y - y = x or *) +(* x - y + x = x are respectively instances of right_loop and rev_right_loop *) +(* The corresponding lemmas will use the K and NK/VK suffixes, respectively. *) +(* *) +(* - Morphisms for functions and relations: *) +(* {morph f : x / a >-> r} <-> f is a morphism with respect to functions *) +(* (fun x => a) and (fun x => r); if r == R[x], *) +(* this states that f a = R[f x] for all x. *) +(* {morph f : x / a} <-> f is a morphism with respect to the *) +(* function expression (fun x => a). This is *) +(* shorthand for {morph f : x / a >-> a}; note *) +(* that the two instances of a are often *) +(* interpreted at different types. *) +(* {morph f : x y / a >-> r} <-> f is a morphism with respect to functions *) +(* (fun x y => a) and (fun x y => r). *) +(* {morph f : x y / a} <-> f is a morphism with respect to the *) +(* function expression (fun x y => a). *) +(* {homo f : x / a >-> r} <-> f is a homomorphism with respect to the *) +(* predicates (fun x => a) and (fun x => r); *) +(* if r == R[x], this states that a -> R[f x] *) +(* for all x. *) +(* {homo f : x / a} <-> f is a homomorphism with respect to the *) +(* predicate expression (fun x => a). *) +(* {homo f : x y / a >-> r} <-> f is a homomorphism with respect to the *) +(* relations (fun x y => a) and (fun x y => r). *) +(* {homo f : x y / a} <-> f is a homomorphism with respect to the *) +(* relation expression (fun x y => a). *) +(* {mono f : x / a >-> r} <-> f is monotone with respect to projectors *) +(* (fun x => a) and (fun x => r); if r == R[x], *) +(* this states that R[f x] = a for all x. *) +(* {mono f : x / a} <-> f is monotone with respect to the projector *) +(* expression (fun x => a). *) +(* {mono f : x y / a >-> r} <-> f is monotone with respect to relators *) +(* (fun x y => a) and (fun x y => r). *) +(* {mono f : x y / a} <-> f is monotone with respect to the relator *) +(* expression (fun x y => a). *) +(* *) +(* The file also contains some basic lemmas for the above concepts. *) +(* Lemmas relative to cancellation laws use some abbreviated suffixes: *) +(* K - a cancellation rule like esymK : cancel (@esym T x y) (@esym T y x). *) +(* LR - a lemma moving an operation from the left hand side of a relation to *) +(* the right hand side, like canLR: cancel g f -> x = g y -> f x = y. *) +(* RL - a lemma moving an operation from the right to the left, e.g., canRL. *) +(* Beware that the LR and RL orientations refer to an "apply" (back chaining) *) +(* usage; when using the same lemmas with "have" or "move" (forward chaining) *) +(* the directions will be reversed!. *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Delimit Scope fun_scope with FUN. +Open Scope fun_scope. + +(* Notations for argument transpose *) +Notation "f ^~ y" := (fun x => f x y) + (at level 10, y at level 8, no associativity, format "f ^~ y") : fun_scope. +Notation "@^~ x" := (fun f => f x) + (at level 10, x at level 8, no associativity, format "@^~ x") : fun_scope. + +Delimit Scope pair_scope with PAIR. +Open Scope pair_scope. + +(* Notations for pair/conjunction projections *) +Notation "p .1" := (fst p) + (at level 2, left associativity, format "p .1") : pair_scope. +Notation "p .2" := (snd p) + (at level 2, left associativity, format "p .2") : pair_scope. + +Coercion pair_of_and P Q (PandQ : P /\ Q) := (proj1 PandQ, proj2 PandQ). + +Definition all_pair I T U (w : forall i : I, T i * U i) := + (fun i => (w i).1, fun i => (w i).2). + +(* Reserved notation for evaluation *) +Reserved Notation "e .[ x ]" + (at level 2, left associativity, format "e .[ x ]"). + +Reserved Notation "e .[ x1 , x2 , .. , xn ]" (at level 2, left associativity, + format "e '[ ' .[ x1 , '/' x2 , '/' .. , '/' xn ] ']'"). + +(* Reserved notation for subscripting and superscripting *) +Reserved Notation "s `_ i" (at level 3, i at level 2, left associativity, + format "s `_ i"). +Reserved Notation "x ^-1" (at level 3, left associativity, format "x ^-1"). + +(* Reserved notation for integer multipliers and exponents *) +Reserved Notation "x *+ n" (at level 40, left associativity). +Reserved Notation "x *- n" (at level 40, left associativity). +Reserved Notation "x ^+ n" (at level 29, left associativity). +Reserved Notation "x ^- n" (at level 29, left associativity). + +(* Reserved notation for external multiplication. *) +Reserved Notation "x *: A" (at level 40). +Reserved Notation "A :* x" (at level 40). + +(* Reserved notation for set-theretic operations. *) +Reserved Notation "A :&: B" (at level 48, left associativity). +Reserved Notation "A :|: B" (at level 52, left associativity). +Reserved Notation "a |: A" (at level 52, left associativity). +Reserved Notation "A :\: B" (at level 50, left associativity). +Reserved Notation "A :\ b" (at level 50, left associativity). + +(* Reserved notation for generated structures *) +Reserved Notation "<< A >>" (at level 0, format "<< A >>"). +Reserved Notation "<[ a ] >" (at level 0, format "<[ a ] >"). + +(* Reserved notation for centralisers and centers. *) +Reserved Notation "''C' [ x ]" (at level 8, format "''C' [ x ]"). +Reserved Notation "''C_' A [ x ]" + (at level 8, A at level 2, format "''C_' A [ x ]"). +Reserved Notation "''C' ( A )" (at level 8, format "''C' ( A )"). +Reserved Notation "''C_' B ( A )" + (at level 8, B at level 2, format "''C_' B ( A )"). +Reserved Notation "''Z' ( A )" (at level 8, format "''Z' ( A )"). +(* Compatibility with group action centraliser notation. *) +Reserved Notation "''C_' ( A ) [ x ]" (at level 8, only parsing). +Reserved Notation "''C_' ( B ) ( A )" (at level 8, only parsing). + +(* Reserved notation for Euclidean division and divisibility. *) +Reserved Notation "m %/ d" (at level 40, no associativity). +Reserved Notation "m %% d" (at level 40, no associativity). +Reserved Notation "m %| d" (at level 70, no associativity). +Reserved Notation "m = n %[mod d ]" (at level 70, n at next level, + format "'[hv ' m '/' = n '/' %[mod d ] ']'"). +Reserved Notation "m == n %[mod d ]" (at level 70, n at next level, + format "'[hv ' m '/' == n '/' %[mod d ] ']'"). +Reserved Notation "m <> n %[mod d ]" (at level 70, n at next level, + format "'[hv ' m '/' <> n '/' %[mod d ] ']'"). +Reserved Notation "m != n %[mod d ]" (at level 70, n at next level, + format "'[hv ' m '/' != n '/' %[mod d ] ']'"). + +(* Reserved notation for derivatives. *) +Reserved Notation "a ^` ()" (at level 8, format "a ^` ()"). +Reserved Notation "a ^` ( n )" (at level 8, format "a ^` ( n )"). + +(* Reserved notation for absolute value. *) +Reserved Notation "`| x |" (at level 0, x at level 99, format "`| x |"). + +(* Reserved notation for conditional comparison *) +Reserved Notation "x <= y ?= 'iff' c" (at level 70, y, c at next level, + format "x '[hv' <= y '/' ?= 'iff' c ']'"). + +(* Reserved notation for cast comparison. *) +Reserved Notation "x <= y :> T" (at level 70, y at next level). +Reserved Notation "x >= y :> T" (at level 70, y at next level, only parsing). +Reserved Notation "x < y :> T" (at level 70, y at next level). +Reserved Notation "x > y :> T" (at level 70, y at next level, only parsing). +Reserved Notation "x <= y ?= 'iff' c :> T" (at level 70, y, c at next level, + format "x '[hv' <= y '/' ?= 'iff' c :> T ']'"). + +(* Complements on the option type constructor, used below to *) +(* encode partial functions. *) + +Module Option. + +Definition apply aT rT (f : aT -> rT) x u := if u is Some y then f y else x. + +Definition default T := apply (fun x : T => x). + +Definition bind aT rT (f : aT -> option rT) := apply f None. + +Definition map aT rT (f : aT -> rT) := bind (fun x => Some (f x)). + +End Option. + +Notation oapp := Option.apply. +Notation odflt := Option.default. +Notation obind := Option.bind. +Notation omap := Option.map. +Notation some := (@Some _) (only parsing). + +(* Shorthand for some basic equality lemmas. *) + +Notation erefl := refl_equal. +Notation ecast i T e x := (let: erefl in _ = i := e return T in x). +Definition esym := sym_eq. +Definition nesym := sym_not_eq. +Definition etrans := trans_eq. +Definition congr1 := f_equal. +Definition congr2 := f_equal2. +(* Force at least one implicit when used as a view. *) +Prenex Implicits esym nesym. + +(* A predicate for singleton types. *) +Definition all_equal_to T (x0 : T) := forall x, unkeyed x = x0. + +Lemma unitE : all_equal_to tt. Proof. by case. Qed. + +(* A generic wrapper type *) + +Structure wrapped T := Wrap {unwrap : T}. +Canonical wrap T x := @Wrap T x. + +Prenex Implicits unwrap wrap Wrap. + +(* Syntax for defining auxiliary recursive function. *) +(* Usage: *) +(* Section FooDefinition. *) +(* Variables (g1 : T1) (g2 : T2). (globals) *) +(* Fixoint foo_auxiliary (a3 : T3) ... := *) +(* body, using [rec e3, ...] for recursive calls *) +(* where "[ 'rec' a3 , a4 , ... ]" := foo_auxiliary. *) +(* Definition foo x y .. := [rec e1, ...]. *) +(* + proofs about foo *) +(* End FooDefinition. *) + +Reserved Notation "[ 'rec' a0 ]" + (at level 0, format "[ 'rec' a0 ]"). +Reserved Notation "[ 'rec' a0 , a1 ]" + (at level 0, format "[ 'rec' a0 , a1 ]"). +Reserved Notation "[ 'rec' a0 , a1 , a2 ]" + (at level 0, format "[ 'rec' a0 , a1 , a2 ]"). +Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 ]" + (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 ]"). +Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 ]" + (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 ]"). +Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 ]" + (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 ]"). +Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 ]" + (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 ]"). +Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 ]" + (at level 0, + format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 ]"). +Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ]" + (at level 0, + format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ]"). +Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 ]" + (at level 0, + format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 ]"). + +(* Definitions and notation for explicit functions with simplification, *) +(* i.e., which simpl and /= beta expand (this is complementary to nosimpl). *) + +Section SimplFun. + +Variables aT rT : Type. + +CoInductive simpl_fun := SimplFun of aT -> rT. + +Definition fun_of_simpl f := fun x => let: SimplFun lam := f in lam x. + +Coercion fun_of_simpl : simpl_fun >-> Funclass. + +End SimplFun. + +Notation "[ 'fun' : T => E ]" := (SimplFun (fun _ : T => E)) + (at level 0, + format "'[hv' [ 'fun' : T => '/ ' E ] ']'") : fun_scope. + +Notation "[ 'fun' x => E ]" := (SimplFun (fun x => E)) + (at level 0, x ident, + format "'[hv' [ 'fun' x => '/ ' E ] ']'") : fun_scope. + +Notation "[ 'fun' x : T => E ]" := (SimplFun (fun x : T => E)) + (at level 0, x ident, only parsing) : fun_scope. + +Notation "[ 'fun' x y => E ]" := (fun x => [fun y => E]) + (at level 0, x ident, y ident, + format "'[hv' [ 'fun' x y => '/ ' E ] ']'") : fun_scope. + +Notation "[ 'fun' x y : T => E ]" := (fun x : T => [fun y : T => E]) + (at level 0, x ident, y ident, only parsing) : fun_scope. + +Notation "[ 'fun' ( x : T ) y => E ]" := (fun x : T => [fun y => E]) + (at level 0, x ident, y ident, only parsing) : fun_scope. + +Notation "[ 'fun' x ( y : T ) => E ]" := (fun x => [fun y : T => E]) + (at level 0, x ident, y ident, only parsing) : fun_scope. + +Notation "[ 'fun' ( x : xT ) ( y : yT ) => E ]" := + (fun x : xT => [fun y : yT => E]) + (at level 0, x ident, y ident, only parsing) : fun_scope. + +(* For delta functions in eqtype.v. *) +Definition SimplFunDelta aT rT (f : aT -> aT -> rT) := [fun z => f z z]. + +(* Extensional equality, for unary and binary functions, including syntactic *) +(* sugar. *) + +Section ExtensionalEquality. + +Variables A B C : Type. + +Definition eqfun (f g : B -> A) : Prop := forall x, f x = g x. + +Definition eqrel (r s : C -> B -> A) : Prop := forall x y, r x y = s x y. + +Lemma frefl f : eqfun f f. Proof. by []. Qed. +Lemma fsym f g : eqfun f g -> eqfun g f. Proof. by move=> eq_fg x. Qed. + +Lemma ftrans f g h : eqfun f g -> eqfun g h -> eqfun f h. +Proof. by move=> eq_fg eq_gh x; rewrite eq_fg. Qed. + +Lemma rrefl r : eqrel r r. Proof. by []. Qed. + +End ExtensionalEquality. + +Typeclasses Opaque eqfun. +Typeclasses Opaque eqrel. + +Hint Resolve frefl rrefl. + +Notation "f1 =1 f2" := (eqfun f1 f2) + (at level 70, no associativity) : fun_scope. +Notation "f1 =1 f2 :> A" := (f1 =1 (f2 : A)) + (at level 70, f2 at next level, A at level 90) : fun_scope. +Notation "f1 =2 f2" := (eqrel f1 f2) + (at level 70, no associativity) : fun_scope. +Notation "f1 =2 f2 :> A" := (f1 =2 (f2 : A)) + (at level 70, f2 at next level, A at level 90) : fun_scope. + +Section Composition. + +Variables A B C : Type. + +Definition funcomp u (f : B -> A) (g : C -> B) x := let: tt := u in f (g x). +Definition catcomp u g f := funcomp u f g. +Local Notation comp := (funcomp tt). + +Definition pcomp (f : B -> option A) (g : C -> option B) x := obind f (g x). + +Lemma eq_comp f f' g g' : f =1 f' -> g =1 g' -> comp f g =1 comp f' g'. +Proof. by move=> eq_ff' eq_gg' x; rewrite /= eq_gg' eq_ff'. Qed. + +End Composition. + +Notation comp := (funcomp tt). +Notation "@ 'comp'" := (fun A B C => @funcomp A B C tt). +Notation "f1 \o f2" := (comp f1 f2) + (at level 50, format "f1 \o '/ ' f2") : fun_scope. +Notation "f1 \; f2" := (catcomp tt f1 f2) + (at level 60, right associativity, format "f1 \; '/ ' f2") : fun_scope. + +Notation "[ 'eta' f ]" := (fun x => f x) + (at level 0, format "[ 'eta' f ]") : fun_scope. + +Notation "'fun' => E" := (fun _ => E) (at level 200, only parsing) : fun_scope. + +Notation id := (fun x => x). +Notation "@ 'id' T" := (fun x : T => x) + (at level 10, T at level 8, only parsing) : fun_scope. + +Definition id_head T u x : T := let: tt := u in x. +Definition explicit_id_key := tt. +Notation idfun := (id_head tt). +Notation "@ 'idfun' T " := (@id_head T explicit_id_key) + (at level 10, T at level 8, format "@ 'idfun' T") : fun_scope. + +Definition phant_id T1 T2 v1 v2 := phantom T1 v1 -> phantom T2 v2. + +(* Strong sigma types. *) + +Section Tag. + +Variables (I : Type) (i : I) (T_ U_ : I -> Type). + +Definition tag := projS1. +Definition tagged : forall w, T_(tag w) := @projS2 I [eta T_]. +Definition Tagged x := @existS I [eta T_] i x. + +Definition tag2 (w : @sigT2 I T_ U_) := let: existT2 i _ _ := w in i. +Definition tagged2 w : T_(tag2 w) := let: existT2 _ x _ := w in x. +Definition tagged2' w : U_(tag2 w) := let: existT2 _ _ y := w in y. +Definition Tagged2 x y := @existS2 I [eta T_] [eta U_] i x y. + +End Tag. + +Implicit Arguments Tagged [I i]. +Implicit Arguments Tagged2 [I i]. +Prenex Implicits tag tagged Tagged tag2 tagged2 tagged2' Tagged2. + +Coercion tag_of_tag2 I T_ U_ (w : @sigT2 I T_ U_) := + Tagged (fun i => T_ i * U_ i)%type (tagged2 w, tagged2' w). + +Lemma all_tag I T U : + (forall x : I, {y : T x & U x y}) -> + {f : forall x, T x & forall x, U x (f x)}. +Proof. by move=> fP; exists (fun x => tag (fP x)) => x; case: (fP x). Qed. + +Lemma all_tag2 I T U V : + (forall i : I, {y : T i & U i y & V i y}) -> + {f : forall i, T i & forall i, U i (f i) & forall i, V i (f i)}. +Proof. by case/all_tag=> f /all_pair[]; exists f. Qed. + +(* Refinement types. *) + +(* Prenex Implicits and renaming. *) +Notation sval := (@proj1_sig _ _). +Notation "@ 'sval'" := (@proj1_sig) (at level 10, format "@ 'sval'"). + +Section Sig. + +Variables (T : Type) (P Q : T -> Prop). + +Lemma svalP (u : sig P) : P (sval u). Proof. by case: u. Qed. + +Definition s2val (u : sig2 P Q) := let: exist2 x _ _ := u in x. + +Lemma s2valP u : P (s2val u). Proof. by case: u. Qed. + +Lemma s2valP' u : Q (s2val u). Proof. by case: u. Qed. + +End Sig. + +Prenex Implicits svalP s2val s2valP s2valP'. + +Coercion tag_of_sig I P (u : @sig I P) := Tagged P (svalP u). + +Coercion sig_of_sig2 I P Q (u : @sig2 I P Q) := + exist (fun i => P i /\ Q i) (s2val u) (conj (s2valP u) (s2valP' u)). + +Lemma all_sig I T P : + (forall x : I, {y : T x | P x y}) -> + {f : forall x, T x | forall x, P x (f x)}. +Proof. by case/all_tag=> f; exists f. Qed. + +Lemma all_sig2 I T P Q : + (forall x : I, {y : T x | P x y & Q x y}) -> + {f : forall x, T x | forall x, P x (f x) & forall x, Q x (f x)}. +Proof. by case/all_sig=> f /all_pair[]; exists f. Qed. + +Section Morphism. + +Variables (aT rT sT : Type) (f : aT -> rT). + +(* Morphism property for unary and binary functions *) +Definition morphism_1 aF rF := forall x, f (aF x) = rF (f x). +Definition morphism_2 aOp rOp := forall x y, f (aOp x y) = rOp (f x) (f y). + +(* Homomorphism property for unary and binary relations *) +Definition homomorphism_1 (aP rP : _ -> Prop) := forall x, aP x -> rP (f x). +Definition homomorphism_2 (aR rR : _ -> _ -> Prop) := + forall x y, aR x y -> rR (f x) (f y). + +(* Stability property for unary and binary relations *) +Definition monomorphism_1 (aP rP : _ -> sT) := forall x, rP (f x) = aP x. +Definition monomorphism_2 (aR rR : _ -> _ -> sT) := + forall x y, rR (f x) (f y) = aR x y. + +End Morphism. + +Notation "{ 'morph' f : x / a >-> r }" := + (morphism_1 f (fun x => a) (fun x => r)) + (at level 0, f at level 99, x ident, + format "{ 'morph' f : x / a >-> r }") : type_scope. + +Notation "{ 'morph' f : x / a }" := + (morphism_1 f (fun x => a) (fun x => a)) + (at level 0, f at level 99, x ident, + format "{ 'morph' f : x / a }") : type_scope. + +Notation "{ 'morph' f : x y / a >-> r }" := + (morphism_2 f (fun x y => a) (fun x y => r)) + (at level 0, f at level 99, x ident, y ident, + format "{ 'morph' f : x y / a >-> r }") : type_scope. + +Notation "{ 'morph' f : x y / a }" := + (morphism_2 f (fun x y => a) (fun x y => a)) + (at level 0, f at level 99, x ident, y ident, + format "{ 'morph' f : x y / a }") : type_scope. + +Notation "{ 'homo' f : x / a >-> r }" := + (homomorphism_1 f (fun x => a) (fun x => r)) + (at level 0, f at level 99, x ident, + format "{ 'homo' f : x / a >-> r }") : type_scope. + +Notation "{ 'homo' f : x / a }" := + (homomorphism_1 f (fun x => a) (fun x => a)) + (at level 0, f at level 99, x ident, + format "{ 'homo' f : x / a }") : type_scope. + +Notation "{ 'homo' f : x y / a >-> r }" := + (homomorphism_2 f (fun x y => a) (fun x y => r)) + (at level 0, f at level 99, x ident, y ident, + format "{ 'homo' f : x y / a >-> r }") : type_scope. + +Notation "{ 'homo' f : x y / a }" := + (homomorphism_2 f (fun x y => a) (fun x y => a)) + (at level 0, f at level 99, x ident, y ident, + format "{ 'homo' f : x y / a }") : type_scope. + +Notation "{ 'homo' f : x y /~ a }" := + (homomorphism_2 f (fun y x => a) (fun x y => a)) + (at level 0, f at level 99, x ident, y ident, + format "{ 'homo' f : x y /~ a }") : type_scope. + +Notation "{ 'mono' f : x / a >-> r }" := + (monomorphism_1 f (fun x => a) (fun x => r)) + (at level 0, f at level 99, x ident, + format "{ 'mono' f : x / a >-> r }") : type_scope. + +Notation "{ 'mono' f : x / a }" := + (monomorphism_1 f (fun x => a) (fun x => a)) + (at level 0, f at level 99, x ident, + format "{ 'mono' f : x / a }") : type_scope. + +Notation "{ 'mono' f : x y / a >-> r }" := + (monomorphism_2 f (fun x y => a) (fun x y => r)) + (at level 0, f at level 99, x ident, y ident, + format "{ 'mono' f : x y / a >-> r }") : type_scope. + +Notation "{ 'mono' f : x y / a }" := + (monomorphism_2 f (fun x y => a) (fun x y => a)) + (at level 0, f at level 99, x ident, y ident, + format "{ 'mono' f : x y / a }") : type_scope. + +Notation "{ 'mono' f : x y /~ a }" := + (monomorphism_2 f (fun y x => a) (fun x y => a)) + (at level 0, f at level 99, x ident, y ident, + format "{ 'mono' f : x y /~ a }") : type_scope. + +(* In an intuitionistic setting, we have two degrees of injectivity. The *) +(* weaker one gives only simplification, and the strong one provides a left *) +(* inverse (we show in `fintype' that they coincide for finite types). *) +(* We also define an intermediate version where the left inverse is only a *) +(* partial function. *) + +Section Injections. + +(* rT must come first so we can use @ to mitigate the Coq 1st order *) +(* unification bug (e..g., Coq can't infer rT from a "cancel" lemma). *) +Variables (rT aT : Type) (f : aT -> rT). + +Definition injective := forall x1 x2, f x1 = f x2 -> x1 = x2. + +Definition cancel g := forall x, g (f x) = x. + +Definition pcancel g := forall x, g (f x) = Some x. + +Definition ocancel (g : aT -> option rT) h := forall x, oapp h x (g x) = x. + +Lemma can_pcan g : cancel g -> pcancel (fun y => Some (g y)). +Proof. by move=> fK x; congr (Some _). Qed. + +Lemma pcan_inj g : pcancel g -> injective. +Proof. by move=> fK x y /(congr1 g); rewrite !fK => [[]]. Qed. + +Lemma can_inj g : cancel g -> injective. +Proof. by move/can_pcan; exact: pcan_inj. Qed. + +Lemma canLR g x y : cancel g -> x = f y -> g x = y. +Proof. by move=> fK ->. Qed. + +Lemma canRL g x y : cancel g -> f x = y -> x = g y. +Proof. by move=> fK <-. Qed. + +End Injections. + +Lemma Some_inj {T} : injective (@Some T). Proof. by move=> x y []. Qed. + +(* cancellation lemmas for dependent type casts. *) +Lemma esymK T x y : cancel (@esym T x y) (@esym T y x). +Proof. by case: y /. Qed. + +Lemma etrans_id T x y (eqxy : x = y :> T) : etrans (erefl x) eqxy = eqxy. +Proof. by case: y / eqxy. Qed. + +Section InjectionsTheory. + +Variables (A B C : Type) (f g : B -> A) (h : C -> B). + +Lemma inj_id : injective (@id A). +Proof. by []. Qed. + +Lemma inj_can_sym f' : cancel f f' -> injective f' -> cancel f' f. +Proof. move=> fK injf' x; exact: injf'. Qed. + +Lemma inj_comp : injective f -> injective h -> injective (f \o h). +Proof. move=> injf injh x y /injf; exact: injh. Qed. + +Lemma can_comp f' h' : cancel f f' -> cancel h h' -> cancel (f \o h) (h' \o f'). +Proof. by move=> fK hK x; rewrite /= fK hK. Qed. + +Lemma pcan_pcomp f' h' : + pcancel f f' -> pcancel h h' -> pcancel (f \o h) (pcomp h' f'). +Proof. by move=> fK hK x; rewrite /pcomp fK /= hK. Qed. + +Lemma eq_inj : injective f -> f =1 g -> injective g. +Proof. by move=> injf eqfg x y; rewrite -2!eqfg; exact: injf. Qed. + +Lemma eq_can f' g' : cancel f f' -> f =1 g -> f' =1 g' -> cancel g g'. +Proof. by move=> fK eqfg eqfg' x; rewrite -eqfg -eqfg'. Qed. + +Lemma inj_can_eq f' : cancel f f' -> injective f' -> cancel g f' -> f =1 g. +Proof. by move=> fK injf' gK x; apply: injf'; rewrite fK. Qed. + +End InjectionsTheory. + +Section Bijections. + +Variables (A B : Type) (f : B -> A). + +CoInductive bijective : Prop := Bijective g of cancel f g & cancel g f. + +Hypothesis bijf : bijective. + +Lemma bij_inj : injective f. +Proof. by case: bijf => g fK _; apply: can_inj fK. Qed. + +Lemma bij_can_sym f' : cancel f' f <-> cancel f f'. +Proof. +split=> fK; first exact: inj_can_sym fK bij_inj. +by case: bijf => h _ hK x; rewrite -[x]hK fK. +Qed. + +Lemma bij_can_eq f' f'' : cancel f f' -> cancel f f'' -> f' =1 f''. +Proof. +by move=> fK fK'; apply: (inj_can_eq _ bij_inj); apply/bij_can_sym. +Qed. + +End Bijections. + +Section BijectionsTheory. + +Variables (A B C : Type) (f : B -> A) (h : C -> B). + +Lemma eq_bij : bijective f -> forall g, f =1 g -> bijective g. +Proof. by case=> f' fK f'K g eqfg; exists f'; eapply eq_can; eauto. Qed. + +Lemma bij_comp : bijective f -> bijective h -> bijective (f \o h). +Proof. +by move=> [f' fK f'K] [h' hK h'K]; exists (h' \o f'); apply: can_comp; auto. +Qed. + +Lemma bij_can_bij : bijective f -> forall f', cancel f f' -> bijective f'. +Proof. by move=> bijf; exists f; first by apply/(bij_can_sym bijf). Qed. + +End BijectionsTheory. + +Section Involutions. + +Variables (A : Type) (f : A -> A). + +Definition involutive := cancel f f. + +Hypothesis Hf : involutive. + +Lemma inv_inj : injective f. Proof. exact: can_inj Hf. Qed. +Lemma inv_bij : bijective f. Proof. by exists f. Qed. + +End Involutions. + +Section OperationProperties. + +Variables S T R : Type. + +Section SopTisR. +Implicit Type op : S -> T -> R. +Definition left_inverse e inv op := forall x, op (inv x) x = e. +Definition right_inverse e inv op := forall x, op x (inv x) = e. +Definition left_injective op := forall x, injective (op^~ x). +Definition right_injective op := forall y, injective (op y). +End SopTisR. + + +Section SopTisS. +Implicit Type op : S -> T -> S. +Definition right_id e op := forall x, op x e = x. +Definition left_zero z op := forall x, op z x = z. +Definition right_commutative op := forall x y z, op (op x y) z = op (op x z) y. +Definition left_distributive op add := + forall x y z, op (add x y) z = add (op x z) (op y z). +Definition right_loop inv op := forall y, cancel (op^~ y) (op^~ (inv y)). +Definition rev_right_loop inv op := forall y, cancel (op^~ (inv y)) (op^~ y). +End SopTisS. + +Section SopTisT. +Implicit Type op : S -> T -> T. +Definition left_id e op := forall x, op e x = x. +Definition right_zero z op := forall x, op x z = z. +Definition left_commutative op := forall x y z, op x (op y z) = op y (op x z). +Definition right_distributive op add := + forall x y z, op x (add y z) = add (op x y) (op x z). +Definition left_loop inv op := forall x, cancel (op x) (op (inv x)). +Definition rev_left_loop inv op := forall x, cancel (op (inv x)) (op x). +End SopTisT. + +Section SopSisT. +Implicit Type op : S -> S -> T. +Definition self_inverse e op := forall x, op x x = e. +Definition commutative op := forall x y, op x y = op y x. +End SopSisT. + +Section SopSisS. +Implicit Type op : S -> S -> S. +Definition idempotent op := forall x, op x x = x. +Definition associative op := forall x y z, op x (op y z) = op (op x y) z. +Definition interchange op1 op2 := + forall x y z t, op1 (op2 x y) (op2 z t) = op2 (op1 x z) (op1 y t). +End SopSisS. + +End OperationProperties. + + + + + + + + + + diff --git a/mathcomp/ssreflect/ssrmatching.v b/mathcomp/ssreflect/ssrmatching.v new file mode 100644 index 0000000..a48b121 --- /dev/null +++ b/mathcomp/ssreflect/ssrmatching.v @@ -0,0 +1,27 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Declare ML Module "ssreflect". + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +Module SsrMatchingSyntax. + +(* Reserve the notation for rewrite patterns so that the user is not allowed *) +(* to declare it at a different level. *) +Reserved Notation "( a 'in' b )" (at level 0). +Reserved Notation "( a 'as' b )" (at level 0). +Reserved Notation "( a 'in' b 'in' c )" (at level 0). +Reserved Notation "( a 'as' b 'in' c )" (at level 0). + +(* Notation to define shortcuts for the "X in t" part of a pattern. *) +Notation "( X 'in' t )" := (_ : fun X => t) : ssrpatternscope. +Delimit Scope ssrpatternscope with pattern. + +(* Some shortcuts for recurrent "X in t" parts. *) +Notation RHS := (X in _ = X)%pattern. +Notation LHS := (X in X = _)%pattern. + +End SsrMatchingSyntax. + +Export SsrMatchingSyntax. diff --git a/mathcomp/ssreflect/ssrnat.v b/mathcomp/ssreflect/ssrnat.v new file mode 100644 index 0000000..bd54e57 --- /dev/null +++ b/mathcomp/ssreflect/ssrnat.v @@ -0,0 +1,1598 @@ +(* (c) Copyright Microsoft Corporation and Inria. All rights reserved. *) +Require Import ssreflect ssrfun ssrbool eqtype. +Require Import BinNat. +Require BinPos Ndec. +Require Export Ring. + +(******************************************************************************) +(* A version of arithmetic on nat (natural numbers) that is better suited to *) +(* small scale reflection than the Coq Arith library. It contains an *) +(* extensive equational theory (including, e.g., the AGM inequality), as well *) +(* as support for the ring tactic, and congruence tactics. *) +(* The following operations and notations are provided: *) +(* *) +(* successor and predecessor *) +(* n.+1, n.+2, n.+3, n.+4 and n.-1, n.-2 *) +(* this frees the names "S" and "pred" *) +(* *) +(* basic arithmetic *) +(* m + n, m - n, m * n *) +(* Important: m - n denotes TRUNCATED subtraction: m - n = 0 if m <= n. *) +(* The definitions use the nosimpl tag to prevent undesirable computation *) +(* computation during simplification, but remain compatible with the ones *) +(* provided in the Coq.Init.Peano prelude. *) +(* For computation, a module NatTrec rebinds all arithmetic notations *) +(* to less convenient but also less inefficient tail-recursive functions; *) +(* the auxiliary functions used by these versions are flagged with %Nrec. *) +(* Also, there is support for input and output of large nat values. *) +(* Num 3 082 241 inputs the number 3082241 *) +(* [Num of n] outputs the value n *) +(* There are coercions num >-> BinNat.N >-> nat; ssrnat rebinds the scope *) +(* delimiter for BinNat.N to %num, as it uses the shorter %N for its own *) +(* notations (Peano notations are flagged with %coq_nat). *) +(* *) +(* doubling, halving, and parity *) +(* n.*2, n./2, odd n, uphalf n, with uphalf n = n.+1./2 *) +(* bool coerces to nat so we can write, e.g., n = odd n + n./2.*2. *) +(* *) +(* iteration *) +(* iter n f x0 == f ( .. (f x0)) *) +(* iteri n g x0 == g n.-1 (g ... (g 0 x0)) *) +(* iterop n op x x0 == op x (... op x x) (n x's) or x0 if n = 0 *) +(* *) +(* exponentiation, factorial *) +(* m ^ n, n`! *) +(* m ^ 1 is convertible to m, and m ^ 2 to m * m *) +(* *) +(* comparison *) +(* m <= n, m < n, m >= n, m > n, m == n, m <= n <= p, etc., *) +(* comparisons are BOOLEAN operators, and m == n is the generic eqType *) +(* operation. *) +(* Most compatibility lemmas are stated as boolean equalities; this keeps *) +(* the size of the library down. All the inequalities refer to the same *) +(* constant "leq"; in particular m < n is identical to m.+1 <= n. *) +(* *) +(* conditionally strict inequality `leqif' *) +(* m <= n ?= iff condition == (m <= n) and ((m == n) = condition) *) +(* This is actually a pair of boolean equalities, so rewriting with an *) +(* `leqif' lemma can affect several kinds of comparison. The transitivity *) +(* lemma for leqif aggregates the conditions, allowing for arguments of *) +(* the form ``m <= n <= p <= m, so equality holds throughout''. *) +(* *) +(* maximum and minimum *) +(* maxn m n, minn m n *) +(* Note that maxn m n = m + (m - n), due to the truncating subtraction. *) +(* Absolute difference (linear distance) between nats is defined in the int *) +(* library (in the int.IntDist sublibrary), with the syntax `|m - n|. The *) +(* '-' in this notation is the signed integer difference. *) +(* *) +(* countable choice *) +(* ex_minn : forall P : pred nat, (exists n, P n) -> nat *) +(* This returns the smallest n such that P n holds. *) +(* ex_maxn : forall (P : pred nat) m, *) +(* (exists n, P n) -> (forall n, P n -> n <= m) -> nat *) +(* This returns the largest n such that P n holds (given an explicit upper *) +(* bound). *) +(* *) +(* This file adds the following suffix conventions to those documented in *) +(* ssrbool.v and eqtype.v: *) +(* A (infix) -- conjunction, as in *) +(* ltn_neqAle : (m < n) = (m != n) && (m <= n). *) +(* B -- subtraction, as in subBn : (m - n) - p = m - (n + p). *) +(* D -- addition, as in mulnDl : (m + n) * p = m * p + n * p. *) +(* M -- multiplication, as in expnMn : (m * n) ^ p = m ^ p * n ^ p. *) +(* p (prefix) -- positive, as in *) +(* eqn_pmul2l : m > 0 -> (m * n1 == m * n2) = (n1 == n2). *) +(* P -- greater than 1, as in *) +(* ltn_Pmull : 1 < n -> 0 < m -> m < n * m. *) +(* S -- successor, as in addSn : n.+1 + m = (n + m).+1. *) +(* V (infix) -- disjunction, as in *) +(* leq_eqVlt : (m <= n) = (m == n) || (m < n). *) +(* X - exponentiation, as in lognX : logn p (m ^ n) = logn p m * n in *) +(* file prime.v (the suffix is not used in ths file). *) +(* Suffixes that abbreviate operations (D, B, M and X) are used to abbreviate *) +(* second-rank operations in equational lemma names that describe left-hand *) +(* sides (e.g., mulnDl); they are not used to abbreviate the main operation *) +(* of relational lemmas (e.g., leq_add2l). *) +(* For the asymmetrical exponentiation operator expn (m ^ n) a right suffix *) +(* indicates an operation on the exponent, e.g., expnM : m ^ (n1 * n2) = ...; *) +(* a trailing "n" is used to indicate the left operand, e.g., *) +(* expnMn : (m1 * m2) ^ n = ... The operands of other operators are selected *) +(* using the l/r suffixes. *) +(******************************************************************************) + +Set Implicit Arguments. +Unset Strict Implicit. +Unset Printing Implicit Defensive. + +(* Declare legacy Arith operators in new scope. *) + +Delimit Scope coq_nat_scope with coq_nat. + +Notation "m + n" := (plus m n) : coq_nat_scope. +Notation "m - n" := (minus m n) : coq_nat_scope. +Notation "m * n" := (mult m n) : coq_nat_scope. +Notation "m <= n" := (le m n) : coq_nat_scope. +Notation "m < n" := (lt m n) : coq_nat_scope. +Notation "m >= n" := (ge m n) : coq_nat_scope. +Notation "m > n" := (gt m n) : coq_nat_scope. + +(* Rebind scope delimiters, reserving a scope for the "recursive", *) +(* i.e., unprotected version of operators. *) + +Delimit Scope N_scope with num. +Delimit Scope nat_scope with N. +Delimit Scope nat_rec_scope with Nrec. + +(* Postfix notation for the successor and predecessor functions. *) +(* SSreflect uses "pred" for the generic predicate type, and S as *) +(* a local bound variable. *) + +Notation succn := Datatypes.S. +Notation predn := Peano.pred. + +Notation "n .+1" := (succn n) (at level 2, left associativity, + format "n .+1") : nat_scope. +Notation "n .+2" := n.+1.+1 (at level 2, left associativity, + format "n .+2") : nat_scope. +Notation "n .+3" := n.+2.+1 (at level 2, left associativity, + format "n .+3") : nat_scope. +Notation "n .+4" := n.+2.+2 (at level 2, left associativity, + format "n .+4") : nat_scope. + +Notation "n .-1" := (predn n) (at level 2, left associativity, + format "n .-1") : nat_scope. +Notation "n .-2" := n.-1.-1 (at level 2, left associativity, + format "n .-2") : nat_scope. + +Lemma succnK : cancel succn predn. Proof. by []. Qed. +Lemma succn_inj : injective succn. Proof. by move=> n m []. Qed. + +(* Predeclare postfix doubling/halving operators. *) + +Reserved Notation "n .*2" (at level 2, format "n .*2"). +Reserved Notation "n ./2" (at level 2, format "n ./2"). + +(* Canonical comparison and eqType for nat. *) + +Fixpoint eqn m n {struct m} := + match m, n with + | 0, 0 => true + | m'.+1, n'.+1 => eqn m' n' + | _, _ => false + end. + +Lemma eqnP : Equality.axiom eqn. +Proof. +move=> n m; apply: (iffP idP) => [|<-]; last by elim n. +by elim: n m => [|n IHn] [|m] //= /IHn->. +Qed. + +Canonical nat_eqMixin := EqMixin eqnP. +Canonical nat_eqType := Eval hnf in EqType nat nat_eqMixin. + +Implicit Arguments eqnP [x y]. +Prenex Implicits eqnP. + +Lemma eqnE : eqn = eq_op. Proof. by []. Qed. + +Lemma eqSS m n : (m.+1 == n.+1) = (m == n). Proof. by []. Qed. + +Lemma nat_irrelevance (x y : nat) (E E' : x = y) : E = E'. +Proof. exact: eq_irrelevance. Qed. + +(* Protected addition, with a more systematic set of lemmas. *) + +Definition addn_rec := plus. +Notation "m + n" := (addn_rec m n) : nat_rec_scope. + +Definition addn := nosimpl addn_rec. +Notation "m + n" := (addn m n) : nat_scope. + +Lemma addnE : addn = addn_rec. Proof. by []. Qed. + +Lemma plusE : plus = addn. Proof. by []. Qed. + +Lemma add0n : left_id 0 addn. Proof. by []. Qed. +Lemma addSn m n : m.+1 + n = (m + n).+1. Proof. by []. Qed. +Lemma add1n n : 1 + n = n.+1. Proof. by []. Qed. + +Lemma addn0 : right_id 0 addn. Proof. by move=> n; apply/eqP; elim: n. Qed. + +Lemma addnS m n : m + n.+1 = (m + n).+1. Proof. by elim: m. Qed. + +Lemma addSnnS m n : m.+1 + n = m + n.+1. Proof. by rewrite addnS. Qed. + +Lemma addnCA : left_commutative addn. +Proof. by move=> m n p; elim: m => //= m; rewrite addnS => <-. Qed. + +Lemma addnC : commutative addn. +Proof. by move=> m n; rewrite -{1}[n]addn0 addnCA addn0. Qed. + +Lemma addn1 n : n + 1 = n.+1. Proof. by rewrite addnC. Qed. + +Lemma addnA : associative addn. +Proof. by move=> m n p; rewrite (addnC n) addnCA addnC. Qed. + +Lemma addnAC : right_commutative addn. +Proof. by move=> m n p; rewrite -!addnA (addnC n). Qed. + +Lemma addnACA : interchange addn addn. +Proof. by move=> m n p q; rewrite -!addnA (addnCA n). Qed. + +Lemma addn_eq0 m n : (m + n == 0) = (m == 0) && (n == 0). +Proof. by case: m; case: n. Qed. + +Lemma eqn_add2l p m n : (p + m == p + n) = (m == n). +Proof. by elim: p. Qed. + +Lemma eqn_add2r p m n : (m + p == n + p) = (m == n). +Proof. by rewrite -!(addnC p) eqn_add2l. Qed. + +Lemma addnI : right_injective addn. +Proof. by move=> p m n Heq; apply: eqP; rewrite -(eqn_add2l p) Heq eqxx. Qed. + +Lemma addIn : left_injective addn. +Proof. move=> p m n; rewrite -!(addnC p); apply addnI. Qed. + +Lemma addn2 m : m + 2 = m.+2. Proof. by rewrite addnC. Qed. +Lemma add2n m : 2 + m = m.+2. Proof. by []. Qed. +Lemma addn3 m : m + 3 = m.+3. Proof. by rewrite addnC. Qed. +Lemma add3n m : 3 + m = m.+3. Proof. by []. Qed. +Lemma addn4 m : m + 4 = m.+4. Proof. by rewrite addnC. Qed. +Lemma add4n m : 4 + m = m.+4. Proof. by []. Qed. + +(* Protected, structurally decreasing subtraction, and basic lemmas. *) +(* Further properties depend on ordering conditions. *) + +Definition subn_rec := minus. +Notation "m - n" := (subn_rec m n) : nat_rec_scope. + +Definition subn := nosimpl subn_rec. +Notation "m - n" := (subn m n) : nat_scope. + +Lemma subnE : subn = subn_rec. Proof. by []. Qed. +Lemma minusE : minus = subn. Proof. by []. Qed. + +Lemma sub0n : left_zero 0 subn. Proof. by []. Qed. +Lemma subn0 : right_id 0 subn. Proof. by case. Qed. +Lemma subnn : self_inverse 0 subn. Proof. by elim. Qed. + +Lemma subSS n m : m.+1 - n.+1 = m - n. Proof. by []. Qed. +Lemma subn1 n : n - 1 = n.-1. Proof. by case: n => [|[]]. Qed. +Lemma subn2 n : (n - 2)%N = n.-2. Proof. by case: n => [|[|[]]]. Qed. + +Lemma subnDl p m n : (p + m) - (p + n) = m - n. +Proof. by elim: p. Qed. + +Lemma subnDr p m n : (m + p) - (n + p) = m - n. +Proof. by rewrite -!(addnC p) subnDl. Qed. + +Lemma addKn n : cancel (addn n) (subn^~ n). +Proof. by move=> m; rewrite /= -{2}[n]addn0 subnDl subn0. Qed. + +Lemma addnK n : cancel (addn^~ n) (subn^~ n). +Proof. by move=> m; rewrite /= (addnC m) addKn. Qed. + +Lemma subSnn n : n.+1 - n = 1. +Proof. exact (addnK n 1). Qed. + +Lemma subnDA m n p : n - (m + p) = (n - m) - p. +Proof. by elim: m n => [|m IHm] [|n]; try exact (IHm n). Qed. + +Lemma subnAC : right_commutative subn. +Proof. by move=> m n p; rewrite -!subnDA addnC. Qed. + +Lemma subnS m n : m - n.+1 = (m - n).-1. +Proof. by rewrite -addn1 subnDA subn1. Qed. + +Lemma subSKn m n : (m.+1 - n).-1 = m - n. +Proof. by rewrite -subnS. Qed. + +(* Integer ordering, and its interaction with the other operations. *) + +Definition leq m n := m - n == 0. + +Notation "m <= n" := (leq m n) : nat_scope. +Notation "m < n" := (m.+1 <= n) : nat_scope. +Notation "m >= n" := (n <= m) (only parsing) : nat_scope. +Notation "m > n" := (n < m) (only parsing) : nat_scope. + +(* For sorting, etc. *) +Definition geq := [rel m n | m >= n]. +Definition ltn := [rel m n | m < n]. +Definition gtn := [rel m n | m > n]. + +Notation "m <= n <= p" := ((m <= n) && (n <= p)) : nat_scope. +Notation "m < n <= p" := ((m < n) && (n <= p)) : nat_scope. +Notation "m <= n < p" := ((m <= n) && (n < p)) : nat_scope. +Notation "m < n < p" := ((m < n) && (n < p)) : nat_scope. + +Lemma ltnS m n : (m < n.+1) = (m <= n). Proof. by []. Qed. +Lemma leq0n n : 0 <= n. Proof. by []. Qed. +Lemma ltn0Sn n : 0 < n.+1. Proof. by []. Qed. +Lemma ltn0 n : n < 0 = false. Proof. by []. Qed. +Lemma leqnn n : n <= n. Proof. by elim: n. Qed. +Hint Resolve leqnn. +Lemma ltnSn n : n < n.+1. Proof. by []. Qed. +Lemma eq_leq m n : m = n -> m <= n. Proof. by move->. Qed. +Lemma leqnSn n : n <= n.+1. Proof. by elim: n. Qed. +Hint Resolve leqnSn. +Lemma leq_pred n : n.-1 <= n. Proof. by case: n => /=. Qed. +Lemma leqSpred n : n <= n.-1.+1. Proof. by case: n => /=. Qed. + +Lemma ltn_predK m n : m < n -> n.-1.+1 = n. +Proof. by case: n. Qed. + +Lemma prednK n : 0 < n -> n.-1.+1 = n. +Proof. exact: ltn_predK. Qed. + +Lemma leqNgt m n : (m <= n) = ~~ (n < m). +Proof. by elim: m n => [|m IHm] [|n] //; exact: IHm n. Qed. + +Lemma ltnNge m n : (m < n) = ~~ (n <= m). +Proof. by rewrite leqNgt. Qed. + +Lemma ltnn n : n < n = false. +Proof. by rewrite ltnNge leqnn. Qed. + +Lemma leqn0 n : (n <= 0) = (n == 0). Proof. by case: n. Qed. +Lemma lt0n n : (0 < n) = (n != 0). Proof. by case: n. Qed. +Lemma lt0n_neq0 n : 0 < n -> n != 0. Proof. by case: n. Qed. +Lemma eqn0Ngt n : (n == 0) = ~~ (n > 0). Proof. by case: n. Qed. +Lemma neq0_lt0n n : (n == 0) = false -> 0 < n. Proof. by case: n. Qed. +Hint Resolve lt0n_neq0 neq0_lt0n. + +Lemma eqn_leq m n : (m == n) = (m <= n <= m). +Proof. elim: m n => [|m IHm] [|n] //; exact: IHm n. Qed. + +Lemma anti_leq : antisymmetric leq. +Proof. by move=> m n; rewrite -eqn_leq => /eqP. Qed. + +Lemma neq_ltn m n : (m != n) = (m < n) || (n < m). +Proof. by rewrite eqn_leq negb_and orbC -!ltnNge. Qed. + +Lemma gtn_eqF m n : m < n -> n == m = false. +Proof. by rewrite eqn_leq (leqNgt n) => ->. Qed. + +Lemma ltn_eqF m n : m < n -> m == n = false. +Proof. by move/gtn_eqF; rewrite eq_sym. Qed. + +Lemma leq_eqVlt m n : (m <= n) = (m == n) || (m < n). +Proof. elim: m n => [|m IHm] [|n] //; exact: IHm n. Qed. + +Lemma ltn_neqAle m n : (m < n) = (m != n) && (m <= n). +Proof. by rewrite ltnNge leq_eqVlt negb_or -leqNgt eq_sym. Qed. + +Lemma leq_trans n m p : m <= n -> n <= p -> m <= p. +Proof. by elim: n m p => [|i IHn] [|m] [|p] //; exact: IHn m p. Qed. + +Lemma leq_ltn_trans n m p : m <= n -> n < p -> m < p. +Proof. move=> Hmn; exact: leq_trans. Qed. + +Lemma ltnW m n : m < n -> m <= n. +Proof. exact: leq_trans. Qed. +Hint Resolve ltnW. + +Lemma leqW m n : m <= n -> m <= n.+1. +Proof. by move=> le_mn; exact: ltnW. Qed. + +Lemma ltn_trans n m p : m < n -> n < p -> m < p. +Proof. by move=> lt_mn /ltnW; exact: leq_trans. Qed. + +Lemma leq_total m n : (m <= n) || (m >= n). +Proof. by rewrite -implyNb -ltnNge; apply/implyP; exact: ltnW. Qed. + +(* Link to the legacy comparison predicates. *) + +Lemma leP m n : reflect (m <= n)%coq_nat (m <= n). +Proof. +apply: (iffP idP); last by elim: n / => // n _ /leq_trans->. +elim: n => [|n IHn]; first by case: m. +by rewrite leq_eqVlt ltnS => /predU1P[<- // | /IHn]; right. +Qed. +Implicit Arguments leP [m n]. + +Lemma le_irrelevance m n le_mn1 le_mn2 : le_mn1 = le_mn2 :> (m <= n)%coq_nat. +Proof. +elim: {n}n.+1 {-1}n (erefl n.+1) => // n IHn _ [<-] in le_mn1 le_mn2 *. +pose def_n2 := erefl n; transitivity (eq_ind _ _ le_mn2 _ def_n2) => //. +move def_n1: {1 4 5 7}n le_mn1 le_mn2 def_n2 => n1 le_mn1. +case: n1 / le_mn1 def_n1 => [|n1 le_mn1] def_n1 [|n2 le_mn2] def_n2. +- by rewrite [def_n2]eq_axiomK. +- by move/leP: (le_mn2); rewrite -{1}def_n2 ltnn. +- by move/leP: (le_mn1); rewrite {1}def_n2 ltnn. +case: def_n2 (def_n2) => ->{n2} def_n2 in le_mn2 *. +by rewrite [def_n2]eq_axiomK /=; congr le_S; exact: IHn. +Qed. + +Lemma ltP m n : reflect (m < n)%coq_nat (m < n). +Proof. exact leP. Qed. +Implicit Arguments ltP [m n]. + +Lemma lt_irrelevance m n lt_mn1 lt_mn2 : lt_mn1 = lt_mn2 :> (m < n)%coq_nat. +Proof. exact: (@le_irrelevance m.+1). Qed. + +(* Comparison predicates. *) + +CoInductive leq_xor_gtn m n : bool -> bool -> Set := + | LeqNotGtn of m <= n : leq_xor_gtn m n true false + | GtnNotLeq of n < m : leq_xor_gtn m n false true. + +Lemma leqP m n : leq_xor_gtn m n (m <= n) (n < m). +Proof. +by rewrite ltnNge; case le_mn: (m <= n); constructor; rewrite // ltnNge le_mn. +Qed. + +CoInductive ltn_xor_geq m n : bool -> bool -> Set := + | LtnNotGeq of m < n : ltn_xor_geq m n false true + | GeqNotLtn of n <= m : ltn_xor_geq m n true false. + +Lemma ltnP m n : ltn_xor_geq m n (n <= m) (m < n). +Proof. by rewrite -(ltnS n); case: leqP; constructor. Qed. + +CoInductive eqn0_xor_gt0 n : bool -> bool -> Set := + | Eq0NotPos of n = 0 : eqn0_xor_gt0 n true false + | PosNotEq0 of n > 0 : eqn0_xor_gt0 n false true. + +Lemma posnP n : eqn0_xor_gt0 n (n == 0) (0 < n). +Proof. by case: n; constructor. Qed. + +CoInductive compare_nat m n : bool -> bool -> bool -> Set := + | CompareNatLt of m < n : compare_nat m n true false false + | CompareNatGt of m > n : compare_nat m n false true false + | CompareNatEq of m = n : compare_nat m n false false true. + +Lemma ltngtP m n : compare_nat m n (m < n) (n < m) (m == n). +Proof. +rewrite ltn_neqAle eqn_leq; case: ltnP; first by constructor. +by rewrite leq_eqVlt orbC; case: leqP; constructor; first exact/eqnP. +Qed. + +(* Monotonicity lemmas *) + +Lemma leq_add2l p m n : (p + m <= p + n) = (m <= n). +Proof. by elim: p. Qed. + +Lemma ltn_add2l p m n : (p + m < p + n) = (m < n). +Proof. by rewrite -addnS; exact: leq_add2l. Qed. + +Lemma leq_add2r p m n : (m + p <= n + p) = (m <= n). +Proof. by rewrite -!(addnC p); exact: leq_add2l. Qed. + +Lemma ltn_add2r p m n : (m + p < n + p) = (m < n). +Proof. exact: leq_add2r p m.+1 n. Qed. + +Lemma leq_add m1 m2 n1 n2 : m1 <= n1 -> m2 <= n2 -> m1 + m2 <= n1 + n2. +Proof. +by move=> le_mn1 le_mn2; rewrite (@leq_trans (m1 + n2)) ?leq_add2l ?leq_add2r. +Qed. + +Lemma leq_addr m n : n <= n + m. +Proof. by rewrite -{1}[n]addn0 leq_add2l. Qed. + +Lemma leq_addl m n : n <= m + n. +Proof. by rewrite addnC leq_addr. Qed. + +Lemma ltn_addr m n p : m < n -> m < n + p. +Proof. by move/leq_trans=> -> //; exact: leq_addr. Qed. + +Lemma ltn_addl m n p : m < n -> m < p + n. +Proof. by move/leq_trans=> -> //; exact: leq_addl. Qed. + +Lemma addn_gt0 m n : (0 < m + n) = (0 < m) || (0 < n). +Proof. by rewrite !lt0n -negb_and addn_eq0. Qed. + +Lemma subn_gt0 m n : (0 < n - m) = (m < n). +Proof. by elim: m n => [|m IHm] [|n] //; exact: IHm n. Qed. + +Lemma subn_eq0 m n : (m - n == 0) = (m <= n). +Proof. by []. Qed. + +Lemma leq_subLR m n p : (m - n <= p) = (m <= n + p). +Proof. by rewrite -subn_eq0 -subnDA. Qed. + +Lemma leq_subr m n : n - m <= n. +Proof. by rewrite leq_subLR leq_addl. Qed. + +Lemma subnKC m n : m <= n -> m + (n - m) = n. +Proof. by elim: m n => [|m IHm] [|n] // /(IHm n) {2}<-. Qed. + +Lemma subnK m n : m <= n -> (n - m) + m = n. +Proof. by rewrite addnC; exact: subnKC. Qed. + +Lemma addnBA m n p : p <= n -> m + (n - p) = m + n - p. +Proof. by move=> le_pn; rewrite -{2}(subnK le_pn) addnA addnK. Qed. + +Lemma subnBA m n p : p <= n -> m - (n - p) = m + p - n. +Proof. by move=> le_pn; rewrite -{2}(subnK le_pn) subnDr. Qed. + +Lemma subKn m n : m <= n -> n - (n - m) = m. +Proof. by move/subnBA->; rewrite addKn. Qed. + +Lemma subSn m n : m <= n -> n.+1 - m = (n - m).+1. +Proof. by rewrite -add1n => /addnBA <-. Qed. + +Lemma subnSK m n : m < n -> (n - m.+1).+1 = n - m. +Proof. by move/subSn. Qed. + +Lemma leq_sub2r p m n : m <= n -> m - p <= n - p. +Proof. +by move=> le_mn; rewrite leq_subLR (leq_trans le_mn) // -leq_subLR. +Qed. + +Lemma leq_sub2l p m n : m <= n -> p - n <= p - m. +Proof. +rewrite -(leq_add2r (p - m)) leq_subLR. +by apply: leq_trans; rewrite -leq_subLR. +Qed. + +Lemma leq_sub m1 m2 n1 n2 : m1 <= m2 -> n2 <= n1 -> m1 - n1 <= m2 - n2. +Proof. by move/(leq_sub2r n1)=> le_m12 /(leq_sub2l m2); apply: leq_trans. Qed. + +Lemma ltn_sub2r p m n : p < n -> m < n -> m - p < n - p. +Proof. by move/subnSK <-; exact: (@leq_sub2r p.+1). Qed. + +Lemma ltn_sub2l p m n : m < p -> m < n -> p - n < p - m. +Proof. by move/subnSK <-; exact: leq_sub2l. Qed. + +Lemma ltn_subRL m n p : (n < p - m) = (m + n < p). +Proof. by rewrite !ltnNge leq_subLR. Qed. + +(* Eliminating the idiom for structurally decreasing compare and subtract. *) +Lemma subn_if_gt T m n F (E : T) : + (if m.+1 - n is m'.+1 then F m' else E) = (if n <= m then F (m - n) else E). +Proof. +by case: leqP => [le_nm | /eqnP-> //]; rewrite -{1}(subnK le_nm) -addSn addnK. +Qed. + +(* Max and min. *) + +Definition maxn m n := if m < n then n else m. + +Definition minn m n := if m < n then m else n. + +Lemma max0n : left_id 0 maxn. Proof. by case. Qed. +Lemma maxn0 : right_id 0 maxn. Proof. by []. Qed. + +Lemma maxnC : commutative maxn. +Proof. by move=> m n; rewrite /maxn; case ltngtP. Qed. + +Lemma maxnE m n : maxn m n = m + (n - m). +Proof. by rewrite /maxn addnC; case: leqP => [/eqnP-> | /ltnW/subnK]. Qed. + +Lemma maxnAC : right_commutative maxn. +Proof. by move=> m n p; rewrite !maxnE -!addnA !subnDA -!maxnE maxnC. Qed. + +Lemma maxnA : associative maxn. +Proof. by move=> m n p; rewrite !(maxnC m) maxnAC. Qed. + +Lemma maxnCA : left_commutative maxn. +Proof. by move=> m n p; rewrite !maxnA (maxnC m). Qed. + +Lemma maxnACA : interchange maxn maxn. +Proof. by move=> m n p q; rewrite -!maxnA (maxnCA n). Qed. + +Lemma maxn_idPl {m n} : reflect (maxn m n = m) (m >= n). +Proof. by rewrite -subn_eq0 -(eqn_add2l m) addn0 -maxnE; apply: eqP. Qed. + +Lemma maxn_idPr {m n} : reflect (maxn m n = n) (m <= n). +Proof. by rewrite maxnC; apply: maxn_idPl. Qed. + +Lemma maxnn : idempotent maxn. +Proof. by move=> n; apply/maxn_idPl. Qed. + +Lemma leq_max m n1 n2 : (m <= maxn n1 n2) = (m <= n1) || (m <= n2). +Proof. +without loss le_n21: n1 n2 / n2 <= n1. + by case/orP: (leq_total n2 n1) => le_n12; last rewrite maxnC orbC; apply. +by rewrite (maxn_idPl le_n21) orb_idr // => /leq_trans->. +Qed. +Lemma leq_maxl m n : m <= maxn m n. Proof. by rewrite leq_max leqnn. Qed. +Lemma leq_maxr m n : n <= maxn m n. Proof. by rewrite maxnC leq_maxl. Qed. + +Lemma gtn_max m n1 n2 : (m > maxn n1 n2) = (m > n1) && (m > n2). +Proof. by rewrite !ltnNge leq_max negb_or. Qed. + +Lemma geq_max m n1 n2 : (m >= maxn n1 n2) = (m >= n1) && (m >= n2). +Proof. by rewrite -ltnS gtn_max. Qed. + +Lemma maxnSS m n : maxn m.+1 n.+1 = (maxn m n).+1. +Proof. by rewrite !maxnE. Qed. + +Lemma addn_maxl : left_distributive addn maxn. +Proof. by move=> m1 m2 n; rewrite !maxnE subnDr addnAC. Qed. + +Lemma addn_maxr : right_distributive addn maxn. +Proof. by move=> m n1 n2; rewrite !(addnC m) addn_maxl. Qed. + +Lemma min0n : left_zero 0 minn. Proof. by case. Qed. +Lemma minn0 : right_zero 0 minn. Proof. by []. Qed. + +Lemma minnC : commutative minn. +Proof. by move=> m n; rewrite /minn; case ltngtP. Qed. + +Lemma addn_min_max m n : minn m n + maxn m n = m + n. +Proof. by rewrite /minn /maxn; case: ltngtP => // [_|->] //; exact: addnC. Qed. + +Lemma minnE m n : minn m n = m - (m - n). +Proof. by rewrite -(subnDl n) -maxnE -addn_min_max addnK minnC. Qed. + +Lemma minnAC : right_commutative minn. +Proof. +by move=> m n p; rewrite !minnE -subnDA subnAC -maxnE maxnC maxnE subnAC subnDA. +Qed. + +Lemma minnA : associative minn. +Proof. by move=> m n p; rewrite minnC minnAC (minnC n). Qed. + +Lemma minnCA : left_commutative minn. +Proof. by move=> m n p; rewrite !minnA (minnC n). Qed. + +Lemma minnACA : interchange minn minn. +Proof. by move=> m n p q; rewrite -!minnA (minnCA n). Qed. + +Lemma minn_idPl {m n} : reflect (minn m n = m) (m <= n). +Proof. +rewrite (sameP maxn_idPr eqP) -(eqn_add2l m) eq_sym -addn_min_max eqn_add2r. +exact: eqP. +Qed. + +Lemma minn_idPr {m n} : reflect (minn m n = n) (m >= n). +Proof. by rewrite minnC; apply: minn_idPl. Qed. + +Lemma minnn : idempotent minn. +Proof. by move=> n; apply/minn_idPl. Qed. + +Lemma leq_min m n1 n2 : (m <= minn n1 n2) = (m <= n1) && (m <= n2). +Proof. +wlog le_n21: n1 n2 / n2 <= n1. + by case/orP: (leq_total n2 n1) => ?; last rewrite minnC andbC; auto. +by rewrite /minn ltnNge le_n21 /= andbC; case: leqP => // /leq_trans->. +Qed. + +Lemma gtn_min m n1 n2 : (m > minn n1 n2) = (m > n1) || (m > n2). +Proof. by rewrite !ltnNge leq_min negb_and. Qed. + +Lemma geq_min m n1 n2 : (m >= minn n1 n2) = (m >= n1) || (m >= n2). +Proof. by rewrite -ltnS gtn_min. Qed. + +Lemma geq_minl m n : minn m n <= m. Proof. by rewrite geq_min leqnn. Qed. +Lemma geq_minr m n : minn m n <= n. Proof. by rewrite minnC geq_minl. Qed. + +Lemma addn_minr : right_distributive addn minn. +Proof. by move=> m1 m2 n; rewrite !minnE subnDl addnBA ?leq_subr. Qed. + +Lemma addn_minl : left_distributive addn minn. +Proof. by move=> m1 m2 n; rewrite -!(addnC n) addn_minr. Qed. + +Lemma minnSS m n : minn m.+1 n.+1 = (minn m n).+1. +Proof. by rewrite -(addn_minr 1). Qed. + +(* Quasi-cancellation (really, absorption) lemmas *) +Lemma maxnK m n : minn (maxn m n) m = m. +Proof. exact/minn_idPr/leq_maxl. Qed. + +Lemma maxKn m n : minn n (maxn m n) = n. +Proof. exact/minn_idPl/leq_maxr. Qed. + +Lemma minnK m n : maxn (minn m n) m = m. +Proof. exact/maxn_idPr/geq_minl. Qed. + +Lemma minKn m n : maxn n (minn m n) = n. +Proof. exact/maxn_idPl/geq_minr. Qed. + +(* Distributivity. *) +Lemma maxn_minl : left_distributive maxn minn. +Proof. +move=> m1 m2 n; wlog le_m21: m1 m2 / m2 <= m1. + move=> IH; case/orP: (leq_total m2 m1) => /IH //. + by rewrite minnC [in R in _ = R]minnC. +rewrite (minn_idPr le_m21); apply/esym/minn_idPr. +by rewrite geq_max leq_maxr leq_max le_m21. +Qed. + +Lemma maxn_minr : right_distributive maxn minn. +Proof. by move=> m n1 n2; rewrite !(maxnC m) maxn_minl. Qed. + +Lemma minn_maxl : left_distributive minn maxn. +Proof. +by move=> m1 m2 n; rewrite maxn_minr !maxn_minl -minnA maxnn (maxnC _ n) !maxnK. +Qed. + +Lemma minn_maxr : right_distributive minn maxn. +Proof. by move=> m n1 n2; rewrite !(minnC m) minn_maxl. Qed. + +(* Getting a concrete value from an abstract existence proof. *) + +Section ExMinn. + +Variable P : pred nat. +Hypothesis exP : exists n, P n. + +Inductive acc_nat i : Prop := AccNat0 of P i | AccNatS of acc_nat i.+1. + +Lemma find_ex_minn : {m | P m & forall n, P n -> n >= m}. +Proof. +have: forall n, P n -> n >= 0 by []. +have: acc_nat 0. + case exP => n; rewrite -(addn0 n); elim: n 0 => [|n IHn] j; first by left. + rewrite addSnnS; right; exact: IHn. +move: 0; fix 2 => m IHm m_lb; case Pm: (P m); first by exists m. +apply: find_ex_minn m.+1 _ _ => [|n Pn]; first by case: IHm; rewrite ?Pm. +by rewrite ltn_neqAle m_lb //; case: eqP Pm => // -> /idP[]. +Qed. + +Definition ex_minn := s2val find_ex_minn. + +Inductive ex_minn_spec : nat -> Type := + ExMinnSpec m of P m & (forall n, P n -> n >= m) : ex_minn_spec m. + +Lemma ex_minnP : ex_minn_spec ex_minn. +Proof. by rewrite /ex_minn; case: find_ex_minn. Qed. + +End ExMinn. + +Section ExMaxn. + +Variables (P : pred nat) (m : nat). +Hypotheses (exP : exists i, P i) (ubP : forall i, P i -> i <= m). + +Lemma ex_maxn_subproof : exists i, P (m - i). +Proof. by case: exP => i Pi; exists (m - i); rewrite subKn ?ubP. Qed. + +Definition ex_maxn := m - ex_minn ex_maxn_subproof. + +CoInductive ex_maxn_spec : nat -> Type := + ExMaxnSpec i of P i & (forall j, P j -> j <= i) : ex_maxn_spec i. + +Lemma ex_maxnP : ex_maxn_spec ex_maxn. +Proof. +rewrite /ex_maxn; case: ex_minnP => i Pmi min_i; split=> // j Pj. +have le_i_mj: i <= m - j by rewrite min_i // subKn // ubP. +rewrite -subn_eq0 subnBA ?(leq_trans le_i_mj) ?leq_subr //. +by rewrite addnC -subnBA ?ubP. +Qed. + +End ExMaxn. + +Lemma eq_ex_minn P Q exP exQ : P =1 Q -> @ex_minn P exP = @ex_minn Q exQ. +Proof. +move=> eqPQ; case: ex_minnP => m1 Pm1 m1_lb; case: ex_minnP => m2 Pm2 m2_lb. +by apply/eqP; rewrite eqn_leq m1_lb (m2_lb, eqPQ) // -eqPQ. +Qed. + +Lemma eq_ex_maxn (P Q : pred nat) m n exP ubP exQ ubQ : + P =1 Q -> @ex_maxn P m exP ubP = @ex_maxn Q n exQ ubQ. +Proof. +move=> eqPQ; case: ex_maxnP => i Pi max_i; case: ex_maxnP => j Pj max_j. +by apply/eqP; rewrite eqn_leq max_i ?eqPQ // max_j -?eqPQ. +Qed. + +Section Iteration. + +Variable T : Type. +Implicit Types m n : nat. +Implicit Types x y : T. + +Definition iter n f x := + let fix loop m := if m is i.+1 then f (loop i) else x in loop n. + +Definition iteri n f x := + let fix loop m := if m is i.+1 then f i (loop i) else x in loop n. + +Definition iterop n op x := + let f i y := if i is 0 then x else op x y in iteri n f. + +Lemma iterSr n f x : iter n.+1 f x = iter n f (f x). +Proof. by elim: n => //= n <-. Qed. + +Lemma iterS n f x : iter n.+1 f x = f (iter n f x). Proof. by []. Qed. + +Lemma iter_add n m f x : iter (n + m) f x = iter n f (iter m f x). +Proof. by elim: n => //= n ->. Qed. + +Lemma iteriS n f x : iteri n.+1 f x = f n (iteri n f x). +Proof. by []. Qed. + +Lemma iteropS idx n op x : iterop n.+1 op x idx = iter n (op x) x. +Proof. by elim: n => //= n ->. Qed. + +Lemma eq_iter f f' : f =1 f' -> forall n, iter n f =1 iter n f'. +Proof. by move=> eq_f n x; elim: n => //= n ->; rewrite eq_f. Qed. + +Lemma eq_iteri f f' : f =2 f' -> forall n, iteri n f =1 iteri n f'. +Proof. by move=> eq_f n x; elim: n => //= n ->; rewrite eq_f. Qed. + +Lemma eq_iterop n op op' : op =2 op' -> iterop n op =2 iterop n op'. +Proof. by move=> eq_op x; apply: eq_iteri; case. Qed. + +End Iteration. + +Lemma iter_succn m n : iter n succn m = m + n. +Proof. by elim: n => //= n ->. Qed. + +Lemma iter_succn_0 n : iter n succn 0 = n. +Proof. exact: iter_succn. Qed. + +Lemma iter_predn m n : iter n predn m = m - n. +Proof. by elim: n m => /= [|n IHn] m; rewrite ?subn0 // IHn subnS. Qed. + +(* Multiplication. *) + +Definition muln_rec := mult. +Notation "m * n" := (muln_rec m n) : nat_rec_scope. + +Definition muln := nosimpl muln_rec. +Notation "m * n" := (muln m n) : nat_scope. + +Lemma multE : mult = muln. Proof. by []. Qed. +Lemma mulnE : muln = muln_rec. Proof. by []. Qed. + +Lemma mul0n : left_zero 0 muln. Proof. by []. Qed. +Lemma muln0 : right_zero 0 muln. Proof. by elim. Qed. +Lemma mul1n : left_id 1 muln. Proof. exact: addn0. Qed. +Lemma mulSn m n : m.+1 * n = n + m * n. Proof. by []. Qed. +Lemma mulSnr m n : m.+1 * n = m * n + n. Proof. exact: addnC. Qed. + +Lemma mulnS m n : m * n.+1 = m + m * n. +Proof. by elim: m => // m; rewrite !mulSn !addSn addnCA => ->. Qed. +Lemma mulnSr m n : m * n.+1 = m * n + m. +Proof. by rewrite addnC mulnS. Qed. + +Lemma iter_addn m n p : iter n (addn m) p = m * n + p. +Proof. by elim: n => /= [|n ->]; rewrite ?muln0 // mulnS addnA. Qed. + +Lemma iter_addn_0 m n : iter n (addn m) 0 = m * n. +Proof. by rewrite iter_addn addn0. Qed. + +Lemma muln1 : right_id 1 muln. +Proof. by move=> n; rewrite mulnSr muln0. Qed. + +Lemma mulnC : commutative muln. +Proof. +by move=> m n; elim: m => [|m]; rewrite (muln0, mulnS) // mulSn => ->. +Qed. + +Lemma mulnDl : left_distributive muln addn. +Proof. by move=> m1 m2 n; elim: m1 => //= m1 IHm; rewrite -addnA -IHm. Qed. + +Lemma mulnDr : right_distributive muln addn. +Proof. by move=> m n1 n2; rewrite !(mulnC m) mulnDl. Qed. + +Lemma mulnBl : left_distributive muln subn. +Proof. +move=> m n [|p]; first by rewrite !muln0. +by elim: m n => // [m IHm] [|n] //; rewrite mulSn subnDl -IHm. +Qed. + +Lemma mulnBr : right_distributive muln subn. +Proof. by move=> m n p; rewrite !(mulnC m) mulnBl. Qed. + +Lemma mulnA : associative muln. +Proof. by move=> m n p; elim: m => //= m; rewrite mulSn mulnDl => ->. Qed. + +Lemma mulnCA : left_commutative muln. +Proof. by move=> m n1 n2; rewrite !mulnA (mulnC m). Qed. + +Lemma mulnAC : right_commutative muln. +Proof. by move=> m n p; rewrite -!mulnA (mulnC n). Qed. + +Lemma mulnACA : interchange muln muln. +Proof. by move=> m n p q; rewrite -!mulnA (mulnCA n). Qed. + +Lemma muln_eq0 m n : (m * n == 0) = (m == 0) || (n == 0). +Proof. by case: m n => // m [|n] //=; rewrite muln0. Qed. + +Lemma muln_eq1 m n : (m * n == 1) = (m == 1) && (n == 1). +Proof. by case: m n => [|[|m]] [|[|n]] //; rewrite muln0. Qed. + +Lemma muln_gt0 m n : (0 < m * n) = (0 < m) && (0 < n). +Proof. by case: m n => // m [|n] //=; rewrite muln0. Qed. + +Lemma leq_pmull m n : n > 0 -> m <= n * m. +Proof. by move/prednK <-; exact: leq_addr. Qed. + +Lemma leq_pmulr m n : n > 0 -> m <= m * n. +Proof. by move/leq_pmull; rewrite mulnC. Qed. + +Lemma leq_mul2l m n1 n2 : (m * n1 <= m * n2) = (m == 0) || (n1 <= n2). +Proof. by rewrite {1}/leq -mulnBr muln_eq0. Qed. + +Lemma leq_mul2r m n1 n2 : (n1 * m <= n2 * m) = (m == 0) || (n1 <= n2). +Proof. by rewrite -!(mulnC m) leq_mul2l. Qed. + +Lemma leq_mul m1 m2 n1 n2 : m1 <= n1 -> m2 <= n2 -> m1 * m2 <= n1 * n2. +Proof. +move=> le_mn1 le_mn2; apply (@leq_trans (m1 * n2)). + by rewrite leq_mul2l le_mn2 orbT. +by rewrite leq_mul2r le_mn1 orbT. +Qed. + +Lemma eqn_mul2l m n1 n2 : (m * n1 == m * n2) = (m == 0) || (n1 == n2). +Proof. by rewrite eqn_leq !leq_mul2l -orb_andr -eqn_leq. Qed. + +Lemma eqn_mul2r m n1 n2 : (n1 * m == n2 * m) = (m == 0) || (n1 == n2). +Proof. by rewrite eqn_leq !leq_mul2r -orb_andr -eqn_leq. Qed. + +Lemma leq_pmul2l m n1 n2 : 0 < m -> (m * n1 <= m * n2) = (n1 <= n2). +Proof. by move/prednK=> <-; rewrite leq_mul2l. Qed. +Implicit Arguments leq_pmul2l [m n1 n2]. + +Lemma leq_pmul2r m n1 n2 : 0 < m -> (n1 * m <= n2 * m) = (n1 <= n2). +Proof. by move/prednK <-; rewrite leq_mul2r. Qed. +Implicit Arguments leq_pmul2r [m n1 n2]. + +Lemma eqn_pmul2l m n1 n2 : 0 < m -> (m * n1 == m * n2) = (n1 == n2). +Proof. by move/prednK <-; rewrite eqn_mul2l. Qed. +Implicit Arguments eqn_pmul2l [m n1 n2]. + +Lemma eqn_pmul2r m n1 n2 : 0 < m -> (n1 * m == n2 * m) = (n1 == n2). +Proof. by move/prednK <-; rewrite eqn_mul2r. Qed. +Implicit Arguments eqn_pmul2r [m n1 n2]. + +Lemma ltn_mul2l m n1 n2 : (m * n1 < m * n2) = (0 < m) && (n1 < n2). +Proof. by rewrite lt0n !ltnNge leq_mul2l negb_or. Qed. + +Lemma ltn_mul2r m n1 n2 : (n1 * m < n2 * m) = (0 < m) && (n1 < n2). +Proof. by rewrite lt0n !ltnNge leq_mul2r negb_or. Qed. + +Lemma ltn_pmul2l m n1 n2 : 0 < m -> (m * n1 < m * n2) = (n1 < n2). +Proof. by move/prednK <-; rewrite ltn_mul2l. Qed. +Implicit Arguments ltn_pmul2l [m n1 n2]. + +Lemma ltn_pmul2r m n1 n2 : 0 < m -> (n1 * m < n2 * m) = (n1 < n2). +Proof. by move/prednK <-; rewrite ltn_mul2r. Qed. +Implicit Arguments ltn_pmul2r [m n1 n2]. + +Lemma ltn_Pmull m n : 1 < n -> 0 < m -> m < n * m. +Proof. by move=> lt1n m_gt0; rewrite -{1}[m]mul1n ltn_pmul2r. Qed. + +Lemma ltn_Pmulr m n : 1 < n -> 0 < m -> m < m * n. +Proof. by move=> lt1n m_gt0; rewrite mulnC ltn_Pmull. Qed. + +Lemma ltn_mul m1 m2 n1 n2 : m1 < n1 -> m2 < n2 -> m1 * m2 < n1 * n2. +Proof. +move=> lt_mn1 lt_mn2; apply (@leq_ltn_trans (m1 * n2)). + by rewrite leq_mul2l orbC ltnW. +by rewrite ltn_pmul2r // (leq_trans _ lt_mn2). +Qed. + +Lemma maxn_mulr : right_distributive muln maxn. +Proof. by case=> // m n1 n2; rewrite /maxn (fun_if (muln _)) ltn_pmul2l. Qed. + +Lemma maxn_mull : left_distributive muln maxn. +Proof. by move=> m1 m2 n; rewrite -!(mulnC n) maxn_mulr. Qed. + +Lemma minn_mulr : right_distributive muln minn. +Proof. by case=> // m n1 n2; rewrite /minn (fun_if (muln _)) ltn_pmul2l. Qed. + +Lemma minn_mull : left_distributive muln minn. +Proof. by move=> m1 m2 n; rewrite -!(mulnC n) minn_mulr. Qed. + +(* Exponentiation. *) + +Definition expn_rec m n := iterop n muln m 1. +Notation "m ^ n" := (expn_rec m n) : nat_rec_scope. +Definition expn := nosimpl expn_rec. +Notation "m ^ n" := (expn m n) : nat_scope. + +Lemma expnE : expn = expn_rec. Proof. by []. Qed. + +Lemma expn0 m : m ^ 0 = 1. Proof. by []. Qed. +Lemma expn1 m : m ^ 1 = m. Proof. by []. Qed. +Lemma expnS m n : m ^ n.+1 = m * m ^ n. Proof. by case: n; rewrite ?muln1. Qed. +Lemma expnSr m n : m ^ n.+1 = m ^ n * m. Proof. by rewrite mulnC expnS. Qed. + +Lemma iter_muln m n p : iter n (muln m) p = m ^ n * p. +Proof. by elim: n => /= [|n ->]; rewrite ?mul1n // expnS mulnA. Qed. + +Lemma iter_muln_1 m n : iter n (muln m) 1 = m ^ n. +Proof. by rewrite iter_muln muln1. Qed. + +Lemma exp0n n : 0 < n -> 0 ^ n = 0. Proof. by case: n => [|[]]. Qed. + +Lemma exp1n n : 1 ^ n = 1. +Proof. by elim: n => // n; rewrite expnS mul1n. Qed. + +Lemma expnD m n1 n2 : m ^ (n1 + n2) = m ^ n1 * m ^ n2. +Proof. by elim: n1 => [|n1 IHn]; rewrite !(mul1n, expnS) // IHn mulnA. Qed. + +Lemma expnMn m1 m2 n : (m1 * m2) ^ n = m1 ^ n * m2 ^ n. +Proof. by elim: n => // n IHn; rewrite !expnS IHn -!mulnA (mulnCA m2). Qed. + +Lemma expnM m n1 n2 : m ^ (n1 * n2) = (m ^ n1) ^ n2. +Proof. +elim: n1 => [|n1 IHn]; first by rewrite exp1n. +by rewrite expnD expnS expnMn IHn. +Qed. + +Lemma expnAC m n1 n2 : (m ^ n1) ^ n2 = (m ^ n2) ^ n1. +Proof. by rewrite -!expnM mulnC. Qed. + +Lemma expn_gt0 m n : (0 < m ^ n) = (0 < m) || (n == 0). +Proof. +by case: m => [|m]; elim: n => //= n IHn; rewrite expnS // addn_gt0 IHn. +Qed. + +Lemma expn_eq0 m e : (m ^ e == 0) = (m == 0) && (e > 0). +Proof. by rewrite !eqn0Ngt expn_gt0 negb_or -lt0n. Qed. + +Lemma ltn_expl m n : 1 < m -> n < m ^ n. +Proof. +move=> m_gt1; elim: n => //= n; rewrite -(leq_pmul2l (ltnW m_gt1)) expnS. +by apply: leq_trans; exact: ltn_Pmull. +Qed. + +Lemma leq_exp2l m n1 n2 : 1 < m -> (m ^ n1 <= m ^ n2) = (n1 <= n2). +Proof. +move=> m_gt1; elim: n1 n2 => [|n1 IHn] [|n2] //; last 1 first. +- by rewrite !expnS leq_pmul2l ?IHn // ltnW. +- by rewrite expn_gt0 ltnW. +by rewrite leqNgt (leq_trans m_gt1) // expnS leq_pmulr // expn_gt0 ltnW. +Qed. + +Lemma ltn_exp2l m n1 n2 : 1 < m -> (m ^ n1 < m ^ n2) = (n1 < n2). +Proof. by move=> m_gt1; rewrite !ltnNge leq_exp2l. Qed. + +Lemma eqn_exp2l m n1 n2 : 1 < m -> (m ^ n1 == m ^ n2) = (n1 == n2). +Proof. by move=> m_gt1; rewrite !eqn_leq !leq_exp2l. Qed. + +Lemma expnI m : 1 < m -> injective (expn m). +Proof. by move=> m_gt1 e1 e2 /eqP; rewrite eqn_exp2l // => /eqP. Qed. + +Lemma leq_pexp2l m n1 n2 : 0 < m -> n1 <= n2 -> m ^ n1 <= m ^ n2. +Proof. by case: m => [|[|m]] // _; [rewrite !exp1n | rewrite leq_exp2l]. Qed. + +Lemma ltn_pexp2l m n1 n2 : 0 < m -> m ^ n1 < m ^ n2 -> n1 < n2. +Proof. by case: m => [|[|m]] // _; [rewrite !exp1n | rewrite ltn_exp2l]. Qed. + +Lemma ltn_exp2r m n e : e > 0 -> (m ^ e < n ^ e) = (m < n). +Proof. +move=> e_gt0; apply/idP/idP=> [|ltmn]. + rewrite !ltnNge; apply: contra => lemn. + by elim: e {e_gt0} => // e IHe; rewrite !expnS leq_mul. +by elim: e e_gt0 => // [[|e] IHe] _; rewrite ?expn1 // ltn_mul // IHe. +Qed. + +Lemma leq_exp2r m n e : e > 0 -> (m ^ e <= n ^ e) = (m <= n). +Proof. by move=> e_gt0; rewrite leqNgt ltn_exp2r // -leqNgt. Qed. + +Lemma eqn_exp2r m n e : e > 0 -> (m ^ e == n ^ e) = (m == n). +Proof. by move=> e_gt0; rewrite !eqn_leq !leq_exp2r. Qed. + +Lemma expIn e : e > 0 -> injective (expn^~ e). +Proof. by move=> e_gt1 m n /eqP; rewrite eqn_exp2r // => /eqP. Qed. + +(* Factorial. *) + +Fixpoint fact_rec n := if n is n'.+1 then n * fact_rec n' else 1. + +Definition factorial := nosimpl fact_rec. + +Notation "n `!" := (factorial n) (at level 2, format "n `!") : nat_scope. + +Lemma factE : factorial = fact_rec. Proof. by []. Qed. + +Lemma fact0 : 0`! = 1. Proof. by []. Qed. + +Lemma factS n : (n.+1)`! = n.+1 * n`!. Proof. by []. Qed. + +Lemma fact_gt0 n : n`! > 0. +Proof. by elim: n => //= n IHn; rewrite muln_gt0. Qed. + +(* Parity and bits. *) + +Coercion nat_of_bool (b : bool) := if b then 1 else 0. + +Lemma leq_b1 (b : bool) : b <= 1. Proof. by case: b. Qed. + +Lemma addn_negb (b : bool) : ~~ b + b = 1. Proof. by case: b. Qed. + +Lemma eqb0 (b : bool) : (b == 0 :> nat) = ~~ b. Proof. by case: b. Qed. + +Lemma eqb1 (b : bool) : (b == 1 :> nat) = b. Proof. by case: b. Qed. + +Lemma lt0b (b : bool) : (b > 0) = b. Proof. by case: b. Qed. + +Lemma sub1b (b : bool) : 1 - b = ~~ b. Proof. by case: b. Qed. + +Lemma mulnb (b1 b2 : bool) : b1 * b2 = b1 && b2. +Proof. by case: b1; case: b2. Qed. + +Lemma mulnbl (b : bool) n : b * n = (if b then n else 0). +Proof. by case: b; rewrite ?mul1n. Qed. + +Lemma mulnbr (b : bool) n : n * b = (if b then n else 0). +Proof. by rewrite mulnC mulnbl. Qed. + +Fixpoint odd n := if n is n'.+1 then ~~ odd n' else false. + +Lemma oddb (b : bool) : odd b = b. Proof. by case: b. Qed. + +Lemma odd_add m n : odd (m + n) = odd m (+) odd n. +Proof. by elim: m => [|m IHn] //=; rewrite -addTb IHn addbA addTb. Qed. + +Lemma odd_sub m n : n <= m -> odd (m - n) = odd m (+) odd n. +Proof. +by move=> le_nm; apply: (@canRL bool) (addbK _) _; rewrite -odd_add subnK. +Qed. + +Lemma odd_opp i m : odd m = false -> i < m -> odd (m - i) = odd i. +Proof. by move=> oddm lt_im; rewrite (odd_sub (ltnW lt_im)) oddm. Qed. + +Lemma odd_mul m n : odd (m * n) = odd m && odd n. +Proof. by elim: m => //= m IHm; rewrite odd_add -addTb andb_addl -IHm. Qed. + +Lemma odd_exp m n : odd (m ^ n) = (n == 0) || odd m. +Proof. by elim: n => // n IHn; rewrite expnS odd_mul {}IHn orbC; case odd. Qed. + +(* Doubling. *) + +Fixpoint double_rec n := if n is n'.+1 then n'.*2%Nrec.+2 else 0 +where "n .*2" := (double_rec n) : nat_rec_scope. + +Definition double := nosimpl double_rec. +Notation "n .*2" := (double n) : nat_scope. + +Lemma doubleE : double = double_rec. Proof. by []. Qed. + +Lemma double0 : 0.*2 = 0. Proof. by []. Qed. + +Lemma doubleS n : n.+1.*2 = n.*2.+2. Proof. by []. Qed. + +Lemma addnn n : n + n = n.*2. +Proof. by apply: eqP; elim: n => // n IHn; rewrite addnS. Qed. + +Lemma mul2n m : 2 * m = m.*2. +Proof. by rewrite mulSn mul1n addnn. Qed. + +Lemma muln2 m : m * 2 = m.*2. +Proof. by rewrite mulnC mul2n. Qed. + +Lemma doubleD m n : (m + n).*2 = m.*2 + n.*2. +Proof. by rewrite -!addnn -!addnA (addnCA n). Qed. + +Lemma doubleB m n : (m - n).*2 = m.*2 - n.*2. +Proof. elim: m n => [|m IHm] [|n] //; exact: IHm n. Qed. + +Lemma leq_double m n : (m.*2 <= n.*2) = (m <= n). +Proof. by rewrite /leq -doubleB; case (m - n). Qed. + +Lemma ltn_double m n : (m.*2 < n.*2) = (m < n). +Proof. by rewrite 2!ltnNge leq_double. Qed. + +Lemma ltn_Sdouble m n : (m.*2.+1 < n.*2) = (m < n). +Proof. by rewrite -doubleS leq_double. Qed. + +Lemma leq_Sdouble m n : (m.*2 <= n.*2.+1) = (m <= n). +Proof. by rewrite leqNgt ltn_Sdouble -leqNgt. Qed. + +Lemma odd_double n : odd n.*2 = false. +Proof. by rewrite -addnn odd_add addbb. Qed. + +Lemma double_gt0 n : (0 < n.*2) = (0 < n). +Proof. by case: n. Qed. + +Lemma double_eq0 n : (n.*2 == 0) = (n == 0). +Proof. by case: n. Qed. + +Lemma doubleMl m n : (m * n).*2 = m.*2 * n. +Proof. by rewrite -!mul2n mulnA. Qed. + +Lemma doubleMr m n : (m * n).*2 = m * n.*2. +Proof. by rewrite -!muln2 mulnA. Qed. + +(* Halving. *) + +Fixpoint half (n : nat) : nat := if n is n'.+1 then uphalf n' else n +with uphalf (n : nat) : nat := if n is n'.+1 then n'./2.+1 else n +where "n ./2" := (half n) : nat_scope. + +Lemma doubleK : cancel double half. +Proof. by elim=> //= n ->. Qed. + +Definition half_double := doubleK. +Definition double_inj := can_inj doubleK. + +Lemma uphalf_double n : uphalf n.*2 = n. +Proof. by elim: n => //= n ->. Qed. + +Lemma uphalf_half n : uphalf n = odd n + n./2. +Proof. by elim: n => //= n ->; rewrite addnA addn_negb. Qed. + +Lemma odd_double_half n : odd n + n./2.*2 = n. +Proof. +by elim: n => //= n {3}<-; rewrite uphalf_half doubleD; case (odd n). +Qed. + +Lemma half_bit_double n (b : bool) : (b + n.*2)./2 = n. +Proof. by case: b; rewrite /= (half_double, uphalf_double). Qed. + +Lemma halfD m n : (m + n)./2 = (odd m && odd n) + (m./2 + n./2). +Proof. +rewrite -{1}[n]odd_double_half addnCA -{1}[m]odd_double_half -addnA -doubleD. +by do 2!case: odd; rewrite /= ?add0n ?half_double ?uphalf_double. +Qed. + +Lemma half_leq m n : m <= n -> m./2 <= n./2. +Proof. by move/subnK <-; rewrite halfD addnA leq_addl. Qed. + +Lemma half_gt0 n : (0 < n./2) = (1 < n). +Proof. by case: n => [|[]]. Qed. + +Lemma odd_geq m n : odd n -> (m <= n) = (m./2.*2 <= n). +Proof. +move=> odd_n; rewrite -{1}[m]odd_double_half -[n]odd_double_half odd_n. +by case: (odd m); rewrite // leq_Sdouble ltnS leq_double. +Qed. + +Lemma odd_ltn m n : odd n -> (n < m) = (n < m./2.*2). +Proof. by move=> odd_n; rewrite !ltnNge odd_geq. Qed. + +Lemma odd_gt0 n : odd n -> n > 0. Proof. by case: n. Qed. + +Lemma odd_gt2 n : odd n -> n > 1 -> n > 2. +Proof. by move=> odd_n n_gt1; rewrite odd_geq. Qed. + +(* Squares and square identities. *) + +Lemma mulnn m : m * m = m ^ 2. +Proof. by rewrite !expnS muln1. Qed. + +Lemma sqrnD m n : (m + n) ^ 2 = m ^ 2 + n ^ 2 + 2 * (m * n). +Proof. +rewrite -!mulnn mul2n mulnDr !mulnDl (mulnC n) -!addnA. +by congr (_ + _); rewrite addnA addnn addnC. +Qed. + +Lemma sqrn_sub m n : n <= m -> (m - n) ^ 2 = m ^ 2 + n ^ 2 - 2 * (m * n). +Proof. +move/subnK=> def_m; rewrite -{2}def_m sqrnD -addnA addnAC. +by rewrite -2!addnA addnn -mul2n -mulnDr -mulnDl def_m addnK. +Qed. + +Lemma sqrnD_sub m n : n <= m -> (m + n) ^ 2 - 4 * (m * n) = (m - n) ^ 2. +Proof. +move=> le_nm; rewrite -[4]/(2 * 2) -mulnA mul2n -addnn subnDA. +by rewrite sqrnD addnK sqrn_sub. +Qed. + +Lemma subn_sqr m n : m ^ 2 - n ^ 2 = (m - n) * (m + n). +Proof. by rewrite mulnBl !mulnDr addnC (mulnC m) subnDl !mulnn. Qed. + +Lemma ltn_sqr m n : (m ^ 2 < n ^ 2) = (m < n). +Proof. by rewrite ltn_exp2r. Qed. + +Lemma leq_sqr m n : (m ^ 2 <= n ^ 2) = (m <= n). +Proof. by rewrite leq_exp2r. Qed. + +Lemma sqrn_gt0 n : (0 < n ^ 2) = (0 < n). +Proof. exact: (ltn_sqr 0). Qed. + +Lemma eqn_sqr m n : (m ^ 2 == n ^ 2) = (m == n). +Proof. by rewrite eqn_exp2r. Qed. + +Lemma sqrn_inj : injective (expn ^~ 2). +Proof. exact: expIn. Qed. + +(* Almost strict inequality: an inequality that is strict unless some *) +(* specific condition holds, such as the Cauchy-Schwartz or the AGM *) +(* inequality (we only prove the order-2 AGM here; the general one *) +(* requires sequences). *) +(* We formalize the concept as a rewrite multirule, that can be used *) +(* both to rewrite the non-strict inequality to true, and the equality *) +(* to the specific condition (for strict inequalities use the ltn_neqAle *) +(* lemma); in addition, the conditional equality also coerces to a *) +(* non-strict one. *) + +Definition leqif m n C := ((m <= n) * ((m == n) = C))%type. + +Notation "m <= n ?= 'iff' C" := (leqif m n C) : nat_scope. + +Coercion leq_of_leqif m n C (H : m <= n ?= iff C) := H.1 : m <= n. + +Lemma leqifP m n C : reflect (m <= n ?= iff C) (if C then m == n else m < n). +Proof. +rewrite ltn_neqAle; apply: (iffP idP) => [|lte]; last by rewrite !lte; case C. +by case C => [/eqP-> | /andP[/negPf]]; split=> //; exact: eqxx. +Qed. + +Lemma leqif_refl m C : reflect (m <= m ?= iff C) C. +Proof. by apply: (iffP idP) => [-> | <-] //; split; rewrite ?eqxx. Qed. + +Lemma leqif_trans m1 m2 m3 C12 C23 : + m1 <= m2 ?= iff C12 -> m2 <= m3 ?= iff C23 -> m1 <= m3 ?= iff C12 && C23. +Proof. +move=> ltm12 ltm23; apply/leqifP; rewrite -ltm12. +case eqm12: (m1 == m2). + by rewrite (eqP eqm12) ltn_neqAle !ltm23 andbT; case C23. +by rewrite (@leq_trans m2) ?ltm23 // ltn_neqAle eqm12 ltm12. +Qed. + +Lemma mono_leqif f : {mono f : m n / m <= n} -> + forall m n C, (f m <= f n ?= iff C) = (m <= n ?= iff C). +Proof. by move=> f_mono m n C; rewrite /leqif !eqn_leq !f_mono. Qed. + +Lemma leqif_geq m n : m <= n -> m <= n ?= iff (m >= n). +Proof. by move=> lemn; split=> //; rewrite eqn_leq lemn. Qed. + +Lemma leqif_eq m n : m <= n -> m <= n ?= iff (m == n). +Proof. by []. Qed. + +Lemma geq_leqif a b C : a <= b ?= iff C -> (b <= a) = C. +Proof. by case=> le_ab; rewrite eqn_leq le_ab. Qed. + +Lemma ltn_leqif a b C : a <= b ?= iff C -> (a < b) = ~~ C. +Proof. by move=> le_ab; rewrite ltnNge (geq_leqif le_ab). Qed. + +Lemma leqif_add m1 n1 C1 m2 n2 C2 : + m1 <= n1 ?= iff C1 -> m2 <= n2 ?= iff C2 -> + m1 + m2 <= n1 + n2 ?= iff C1 && C2. +Proof. +rewrite -(mono_leqif (leq_add2r m2)) -(mono_leqif (leq_add2l n1) m2). +exact: leqif_trans. +Qed. + +Lemma leqif_mul m1 n1 C1 m2 n2 C2 : + m1 <= n1 ?= iff C1 -> m2 <= n2 ?= iff C2 -> + m1 * m2 <= n1 * n2 ?= iff (n1 * n2 == 0) || (C1 && C2). +Proof. +move=> le1 le2; case: posnP => [n12_0 | ]. + rewrite n12_0; move/eqP: n12_0 {le1 le2}le1.1 le2.1; rewrite muln_eq0. + by case/orP=> /eqP->; case: m1 m2 => [|m1] [|m2] // _ _; + rewrite ?muln0; exact/leqif_refl. +rewrite muln_gt0 => /andP[n1_gt0 n2_gt0]. +have [m2_0 | m2_gt0] := posnP m2. + apply/leqifP; rewrite -le2 andbC eq_sym eqn_leq leqNgt m2_0 muln0. + by rewrite muln_gt0 n1_gt0 n2_gt0. +have mono_n1 := leq_pmul2l n1_gt0; have mono_m2 := leq_pmul2r m2_gt0. +rewrite -(mono_leqif mono_m2) in le1; rewrite -(mono_leqif mono_n1) in le2. +exact: leqif_trans le1 le2. +Qed. + +Lemma nat_Cauchy m n : 2 * (m * n) <= m ^ 2 + n ^ 2 ?= iff (m == n). +Proof. +wlog le_nm: m n / n <= m. + by case: (leqP m n); auto; rewrite eq_sym addnC (mulnC m); auto. +apply/leqifP; case: ifP => [/eqP-> | ne_mn]; first by rewrite mulnn addnn mul2n. +by rewrite -subn_gt0 -sqrn_sub // sqrn_gt0 subn_gt0 ltn_neqAle eq_sym ne_mn. +Qed. + +Lemma nat_AGM2 m n : 4 * (m * n) <= (m + n) ^ 2 ?= iff (m == n). +Proof. +rewrite -[4]/(2 * 2) -mulnA mul2n -addnn sqrnD; apply/leqifP. +by rewrite ltn_add2r eqn_add2r ltn_neqAle !nat_Cauchy; case: ifP => ->. +Qed. + +(* Support for larger integers. The normal definitions of +, - and even *) +(* IO are unsuitable for Peano integers larger than 2000 or so because *) +(* they are not tail-recursive. We provide a workaround module, along *) +(* with a rewrite multirule to change the tailrec operators to the *) +(* normal ones. We handle IO via the NatBin module, but provide our *) +(* own (more efficient) conversion functions. *) + +Module NatTrec. + +(* Usage: *) +(* Import NatTrec. *) +(* in section definining functions, rebinds all *) +(* non-tail recursive operators. *) +(* rewrite !trecE. *) +(* in the correctness proof, restores operators *) + +Fixpoint add m n := if m is m'.+1 then m' + n.+1 else n +where "n + m" := (add n m) : nat_scope. + +Fixpoint add_mul m n s := if m is m'.+1 then add_mul m' n (n + s) else s. + +Definition mul m n := if m is m'.+1 then add_mul m' n n else 0. + +Notation "n * m" := (mul n m) : nat_scope. + +Fixpoint mul_exp m n p := if n is n'.+1 then mul_exp m n' (m * p) else p. + +Definition exp m n := if n is n'.+1 then mul_exp m n' m else 1. + +Notation "n ^ m" := (exp n m) : nat_scope. + +Notation Local oddn := odd. +Fixpoint odd n := if n is n'.+2 then odd n' else eqn n 1. + +Notation Local doublen := double. +Definition double n := if n is n'.+1 then n' + n.+1 else 0. +Notation "n .*2" := (double n) : nat_scope. + +Lemma addE : add =2 addn. +Proof. by elim=> //= n IHn m; rewrite IHn addSnnS. Qed. + +Lemma doubleE : double =1 doublen. +Proof. by case=> // n; rewrite -addnn -addE. Qed. + +Lemma add_mulE n m s : add_mul n m s = addn (muln n m) s. +Proof. by elim: n => //= n IHn in m s *; rewrite IHn addE addnCA addnA. Qed. + +Lemma mulE : mul =2 muln. +Proof. by case=> //= n m; rewrite add_mulE addnC. Qed. + +Lemma mul_expE m n p : mul_exp m n p = muln (expn m n) p. +Proof. +by elim: n => [|n IHn] in p *; rewrite ?mul1n //= expnS IHn mulE mulnCA mulnA. +Qed. + +Lemma expE : exp =2 expn. +Proof. by move=> m [|n] //=; rewrite mul_expE expnS mulnC. Qed. + +Lemma oddE : odd =1 oddn. +Proof. +move=> n; rewrite -{1}[n]odd_double_half addnC. +by elim: n./2 => //=; case (oddn n). +Qed. + +Definition trecE := (addE, (doubleE, oddE), (mulE, add_mulE, (expE, mul_expE))). + +End NatTrec. + +Notation natTrecE := NatTrec.trecE. + +Lemma eq_binP : Equality.axiom Ndec.Neqb. +Proof. +move=> p q; apply: (iffP idP) => [|<-]; last by case: p => //; elim. +by case: q; case: p => //; elim=> [p IHp|p IHp|] [q|q|] //=; case/IHp=> ->. +Qed. + +Canonical bin_nat_eqMixin := EqMixin eq_binP. +Canonical bin_nat_eqType := Eval hnf in EqType N bin_nat_eqMixin. + +Section NumberInterpretation. + +Import BinPos. + +Section Trec. + +Import NatTrec. + +Fixpoint nat_of_pos p0 := + match p0 with + | xO p => (nat_of_pos p).*2 + | xI p => (nat_of_pos p).*2.+1 + | xH => 1 + end. + +End Trec. + +Coercion Local nat_of_pos : positive >-> nat. + +Coercion nat_of_bin b := if b is Npos p then p : nat else 0. + +Fixpoint pos_of_nat n0 m0 := + match n0, m0 with + | n.+1, m.+2 => pos_of_nat n m + | n.+1, 1 => xO (pos_of_nat n n) + | n.+1, 0 => xI (pos_of_nat n n) + | 0, _ => xH + end. + +Definition bin_of_nat n0 := if n0 is n.+1 then Npos (pos_of_nat n n) else 0%num. + +Lemma bin_of_natK : cancel bin_of_nat nat_of_bin. +Proof. +have sub2nn n : n.*2 - n = n by rewrite -addnn addKn. +case=> //= n; rewrite -{3}[n]sub2nn. +by elim: n {2 4}n => // m IHm [|[|n]] //=; rewrite IHm // natTrecE sub2nn. +Qed. + +Lemma nat_of_binK : cancel nat_of_bin bin_of_nat. +Proof. +case=> //=; elim=> //= p; case: (nat_of_pos p) => //= n [<-]. + by rewrite natTrecE !addnS {2}addnn; elim: {1 3}n. +by rewrite natTrecE addnS /= addnS {2}addnn; elim: {1 3}n. +Qed. + +Lemma nat_of_succ_gt0 p : Psucc p = p.+1 :> nat. +Proof. by elim: p => //= p ->; rewrite !natTrecE. Qed. + +Lemma nat_of_addn_gt0 p q : (p + q)%positive = p + q :> nat. +Proof. +apply: @fst _ (Pplus_carry p q = (p + q).+1 :> nat) _. +elim: p q => [p IHp|p IHp|] [q|q|] //=; rewrite !natTrecE //; + by rewrite ?IHp ?nat_of_succ_gt0 ?(doubleS, doubleD, addn1, addnS). +Qed. + +Lemma nat_of_add_bin b1 b2 : (b1 + b2)%num = b1 + b2 :> nat. +Proof. case: b1 b2 => [|p] [|q] //=; exact: nat_of_addn_gt0. Qed. + +Lemma nat_of_mul_bin b1 b2 : (b1 * b2)%num = b1 * b2 :> nat. +Proof. +case: b1 b2 => [|p] [|q] //=; elim: p => [p IHp|p IHp|] /=; + by rewrite ?(mul1n, nat_of_addn_gt0, mulSn) //= !natTrecE IHp doubleMl. +Qed. + +Lemma nat_of_exp_bin n (b : N) : n ^ b = pow_N 1 muln n b. +Proof. +case: b => [|p] /=; first exact: expn0. +by elim: p => //= p <-; rewrite natTrecE mulnn -expnM muln2 ?expnS. +Qed. + +End NumberInterpretation. + +(* Big(ger) nat IO; usage: *) +(* Num 1 072 399 *) +(* to create large numbers for test cases *) +(* Eval compute in [Num of some expression] *) +(* to display the resut of an expression that *) +(* returns a larger integer. *) + +Record number : Type := Num {bin_of_number :> N}. + +Definition extend_number (nn : number) m := Num (nn * 1000 + bin_of_nat m). + +Coercion extend_number : number >-> Funclass. + +Canonical number_subType := [newType for bin_of_number]. +Definition number_eqMixin := Eval hnf in [eqMixin of number by <:]. +Canonical number_eqType := Eval hnf in EqType number number_eqMixin. + +Notation "[ 'Num' 'of' e ]" := (Num (bin_of_nat e)) + (at level 0, format "[ 'Num' 'of' e ]") : nat_scope. + +(* Interface to ring/ring_simplify tactics *) + +Lemma nat_semi_ring : semi_ring_theory 0 1 addn muln (@eq _). +Proof. exact: mk_srt add0n addnC addnA mul1n mul0n mulnC mulnA mulnDl. Qed. + +Lemma nat_semi_morph : + semi_morph 0 1 addn muln (@eq _) 0%num 1%num Nplus Nmult pred1 nat_of_bin. +Proof. +by move: nat_of_add_bin nat_of_mul_bin; split=> //= m n; move/eqP->. +Qed. + +Lemma nat_power_theory : power_theory 1 muln (@eq _) nat_of_bin expn. +Proof. split; exact: nat_of_exp_bin. Qed. + +(* Interface to the ring tactic machinery. *) + +Fixpoint pop_succn e := if e is e'.+1 then fun n => pop_succn e' n.+1 else id. + +Ltac pop_succn e := eval lazy beta iota delta [pop_succn] in (pop_succn e 1). + +Ltac nat_litteral e := + match pop_succn e with + | ?n.+1 => constr: (bin_of_nat n) + | _ => NotConstant + end. + +Ltac succn_to_add := + match goal with + | |- context G [?e.+1] => + let x := fresh "NatLit0" in + match pop_succn e with + | ?n.+1 => pose x := n.+1; let G' := context G [x] in change G' + | _ ?e' ?n => pose x := n; let G' := context G [x + e'] in change G' + end; succn_to_add; rewrite {}/x + | _ => idtac + end. + +Add Ring nat_ring_ssr : nat_semi_ring (morphism nat_semi_morph, + constants [nat_litteral], preprocess [succn_to_add], + power_tac nat_power_theory [nat_litteral]). + +(* A congruence tactic, similar to the boolean one, along with an .+1/+ *) +(* normalization tactic. *) + + +Ltac nat_norm := + succn_to_add; rewrite ?add0n ?addn0 -?addnA ?(addSn, addnS, add0n, addn0). + +Ltac nat_congr := first + [ apply: (congr1 succn _) + | apply: (congr1 predn _) + | apply: (congr1 (addn _) _) + | apply: (congr1 (subn _) _) + | apply: (congr1 (addn^~ _) _) + | match goal with |- (?X1 + ?X2 = ?X3) => + symmetry; + rewrite -1?(addnC X1) -?(addnCA X1); + apply: (congr1 (addn X1) _); + symmetry + end ]. |