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diff --git a/theories/ssr/ssreflect.v b/theories/ssr/ssreflect.v new file mode 100644 index 0000000000..bc4a57dedd --- /dev/null +++ b/theories/ssr/ssreflect.v @@ -0,0 +1,656 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2019 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +(* This file is (C) Copyright 2006-2015 Microsoft Corporation and Inria. *) + +(** #<style> .doc { font-family: monospace; white-space: pre; } </style># **) + +Require Import Bool. (* For bool_scope delimiter 'bool'. *) +Require Import ssrmatching. +Declare ML Module "ssreflect_plugin". + + +(** + 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. + nonPropType == an interface for non-Prop Types: a nonPropType coerces + to a Type, and only types that do _not_ have sort + Prop are canonical nonPropType instances. This is + useful for applied views (see mid-file comment). + notProp T == the nonPropType instance for type T. + 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. **) + +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 0, n at level 0, + format "<hidden n >"). +Reserved Notation "T (* n *)" (at level 200, format "T (* n *)"). + +End SsrSyntax. + +Export SsrMatchingSyntax. +Export SsrSyntax. + +(** Save primitive notation that will be overloaded. **) +Local Notation CoqGenericIf c vT vF := (if c then vT else vF) (only parsing). +Local Notation CoqGenericDependentIf c x R vT vF := + (if c as x return R then vT else vF) (only parsing). +Local Notation CoqCast x T := (x : T) (only parsing). + +(** Reserve notation that introduced in this file. **) +Reserved Notation "'if' c 'then' vT 'else' vF" (at level 200, + c, vT, vF at level 200, only parsing). +Reserved Notation "'if' c 'return' R 'then' vT 'else' vF" (at level 200, + c, R, vT, vF at level 200, only parsing). +Reserved Notation "'if' c 'as' x 'return' R 'then' vT 'else' vF" (at level 200, + c, R, vT, vF at level 200, x ident, only parsing). + +Reserved Notation "x : T" (at level 100, right associativity, + format "'[hv' x '/ ' : T ']'"). +Reserved Notation "T : 'Type'" (at level 100, format "T : 'Type'"). +Reserved Notation "P : 'Prop'" (at level 100, format "P : 'Prop'"). + +Reserved Notation "[ 'the' sT 'of' v 'by' f ]" (at level 0, + format "[ 'the' sT 'of' v 'by' f ]"). +Reserved Notation "[ 'the' sT 'of' v ]" (at level 0, + format "[ 'the' sT 'of' v ]"). +Reserved Notation "{ 'type' 'of' c 'for' s }" (at level 0, + format "{ 'type' 'of' c 'for' s }"). + +Reserved Notation "=^~ r" (at level 100, format "=^~ r"). + +Reserved Notation "[ 'unlockable' 'of' C ]" (at level 0, + format "[ 'unlockable' 'of' C ]"). +Reserved Notation "[ 'unlockable' 'fun' C ]" (at level 0, + format "[ 'unlockable' 'fun' C ]"). + +(** + To define notations for tactic in intro patterns. + When "=> /t" is parsed, "t:%ssripat" is actually interpreted. **) +Declare Scope ssripat_scope. +Delimit Scope ssripat_scope with ssripat. + +(** + 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 spurious trailing %%GEN_IF. **) + +Declare Scope general_if_scope. +Delimit Scope general_if_scope with GEN_IF. + +Notation "'if' c 'then' vT 'else' vF" := + (CoqGenericIf c vT vF) (only parsing) : general_if_scope. + +Notation "'if' c 'return' R 'then' vT 'else' vF" := + (CoqGenericDependentIf c c R vT vF) (only parsing) : general_if_scope. + +Notation "'if' c 'as' x 'return' R 'then' vT 'else' vF" := + (CoqGenericDependentIf c x R vT vF) (only parsing) : general_if_scope. + +(** Force boolean interpretation of simple if expressions. **) + +Declare Scope boolean_if_scope. +Delimit Scope boolean_if_scope with BOOL_IF. + +Notation "'if' c 'return' R 'then' vT 'else' vF" := + (if c is true as c in bool return R then vT else vF) : boolean_if_scope. + +Notation "'if' c 'then' vT 'else' vF" := + (if c%bool is true as _ in bool return _ then vT else vF) : boolean_if_scope. + +Notation "'if' c 'as' x 'return' R 'then' vT 'else' vF" := + (if c%bool is true as x in bool return R then vT else vF) : 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. **) +Declare Scope form_scope. +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" := (CoqCast 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'" := (CoqCast T%type Type) (only parsing) : core_scope. +(** Allow similarly Prop annotation for, e.g., rewrite multirules. **) +Notation "P : 'Prop'" := (CoqCast P%type Prop) (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. + +Declare Scope ssr_scope. +Notation "<hidden n >" := (abstract _ n _) : ssr_scope. +Notation "T (* n *)" := (abstract T n abstract_key) : ssr_scope. +Open Scope ssr_scope. + +Register abstract_lock as plugins.ssreflect.abstract_lock. +Register abstract_key as plugins.ssreflect.abstract_key. +Register abstract as plugins.ssreflect.abstract. + +(** Constants for tactic-views **) +Inductive external_view : Type := tactic_view of Type. + +(** + 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. + +#[universes(template)] +Variant 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. *) + +Local Arguments get_by _%type_scope _%type_scope _ _ _ _. + +Notation "[ 'the' sT 'of' v 'by' f ]" := + (@get_by _ sT f _ _ ((fun v' (s : sT) => Put v' (f s) s) v _)) + (only parsing) : form_scope. + +Notation "[ 'the' sT 'of' v ]" := (get ((fun s : sT => Put v (*coerce*) s s) _)) + (only parsing) : form_scope. + +(** + The following are "format only" versions of the above notations. + 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 "[ 'the' sT 'of' v 'by' f ]" := (@get_by _ sT f v _ _) + (only printing) : form_scope. + +Notation "[ 'the' sT 'of' v ]" := (@get _ sT v _ _) + (only printing) : form_scope. + +(** + We would like to recognize +Notation " #[# 'the' sT 'of' v : 'Type' #]#" := (@get Type sT v _ _) + (at level 0, format " #[# 'the' 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) : 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. **) + +#[universes(template)] +Variant phantom T (p : T) := Phantom. +Arguments phantom : clear implicits. +Arguments Phantom : clear implicits. +#[universes(template)] +Variant phant (p : Type) := Phant. + +(** Internal tagging used by the implementation of the ssreflect elim. **) + +Definition protect_term (A : Type) (x : A) : A := x. + +Register protect_term as plugins.ssreflect.protect_term. + +(** + The ssreflect idiom for a non-keyed pattern: + - unkeyed t will 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, respectively, 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) : 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 compares 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. + +Register master_key as plugins.ssreflect.master_key. +Register locked as plugins.ssreflect.locked. + +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 ]. + +(** Quicker done tactic not including split, syntax: /0/ **) +Ltac ssrdone0 := + trivial; hnf; intros; solve + [ do ![solve [trivial | apply: sym_equal; trivial] + | discriminate | contradiction ] + | case not_locked_false_eq_true; assumption + | match goal with H : ~ _ |- _ => solve [case H; trivial] end ]. + +(** To unlock opaque constants. **) +#[universes(template)] +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 _)) : form_scope. + +Notation "[ 'unlockable' 'fun' C ]" := + (@Unlockable _ (fun _ => _) C (unlock _)) : 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. +Arguments ssr_have Plemma [Pgoal]. + +Definition ssr_have_let Pgoal Plemma step + (rest : let x : Plemma := step in Pgoal) : Pgoal := rest. +Arguments ssr_have_let [Pgoal]. + +Register ssr_have as plugins.ssreflect.ssr_have. +Register ssr_have_let as plugins.ssreflect.ssr_have_let. + +Definition ssr_suff Plemma Pgoal step (rest : Plemma) : Pgoal := step rest. +Arguments ssr_suff Plemma [Pgoal]. + +Definition ssr_wlog := ssr_suff. +Arguments ssr_wlog Plemma [Pgoal]. + +Register ssr_suff as plugins.ssreflect.ssr_suff. +Register ssr_wlog as plugins.ssreflect.ssr_wlog. + +(** 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. +Arguments ssr_congr_arrow : clear implicits. + +Register nary_congruence as plugins.ssreflect.nary_congruence. +Register ssr_congr_arrow as plugins.ssreflect.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. + +(** + To focus non-ssreflect tactics on a subterm, eg vm_compute. + Usage: + elim/abstract_context: (pattern) => G defG. + vm_compute; rewrite {}defG {G}. + Note that vm_cast are not stored in the proof term + for reductions occurring in the context, hence + set here := pattern; vm_compute in (value of here) + blows up at Qed time. **) +Lemma abstract_context T (P : T -> Type) x : + (forall Q, Q = P -> Q x) -> P x. +Proof. by move=> /(_ P); apply. Qed. + +(*****************************************************************************) +(* Material for under/over (to rewrite under binders using "context lemmas") *) + +Require Export ssrunder. + +Hint Extern 0 (@Under_rel.Over_rel _ _ _ _) => + solve [ apply: Under_rel.over_rel_done ] : core. +Hint Resolve Under_rel.over_rel_done : core. + +Register Under_rel.Under_rel as plugins.ssreflect.Under_rel. +Register Under_rel.Under_rel_from_rel as plugins.ssreflect.Under_rel_from_rel. + +(** Closing rewrite rule *) +Definition over := over_rel. + +(** Closing tactic *) +Ltac over := + by [ apply: Under_rel.under_rel_done + | rewrite over + ]. + +(** Convenience rewrite rule to unprotect evars, e.g., to instantiate + them in another way than with reflexivity. *) +Definition UnderE := Under_relE. + +(*****************************************************************************) + +(** An interface for non-Prop types; used to avoid improper instantiation + of polymorphic lemmas with on-demand implicits when they are used as views. + For example: Some_inj {T} : forall x y : T, Some x = Some y -> x = y. + Using move/Some_inj on a goal of the form Some n = Some 0 will fail: + SSReflect will interpret the view as @Some_inj ?T _top_assumption_ + since this is the well-typed application of the view with the minimal + number of inserted evars (taking ?T := Some n = Some 0), and then will + later complain that it cannot erase _top_assumption_ after having + abstracted the viewed assumption. Making x and y maximal implicits + would avoid this and force the intended @Some_inj nat x y _top_assumption_ + interpretation, but is undesirable as it makes it harder to use Some_inj + with the many SSReflect and MathComp lemmas that have an injectivity + premise. Specifying {T : nonPropType} solves this more elegantly, as then + (?T : Type) no longer unifies with (Some n = Some 0), which has sort Prop. + **) + +Module NonPropType. + +(** Implementation notes: + We rely on three interface Structures: + - test_of r, the middle structure, performs the actual check: it has two + canonical instances whose 'condition' projection are maybeProj (?P : Prop) + and tt, and which set r := true and r := false, respectively. Unifying + condition (?t : test_of ?r) with maybeProj T will thus set ?r to true if + T is in Prop as the test_Prop T instance will apply, and otherwise simplify + maybeProp T to tt and use the test_negative instance and set ?r to false. + - call_of c r sets up a call to test_of on condition c with expected result r. + It has a default instance for its 'callee' projection to Type, which + sets c := maybeProj T and r := false when unifying with a type T. + - type is a telescope on call_of c r, which checks that unifying test_of ?r1 + with c indeed sets ?r1 := r; the type structure bundles the 'test' instance + and its 'result' value along with its call_of c r projection. The default + instance essentially provides eta-expansion for 'type'. This is only + essential for the first 'result' projection to bool; using the instance + for other projection merely avoids spurious delta expansions that would + spoil the notProp T notation. + In detail, unifying T =~= ?S with ?S : nonPropType, i.e., + (1) T =~= @callee (@condition (result ?S) (test ?S)) (result ?S) (frame ?S) + first uses the default call instance with ?T := T to reduce (1) to + (2a) @condition (result ?S) (test ?S) =~= maybeProp T + (3) result ?S =~= false + (4) frame ?S =~= call T + along with some trivial universe-related checks which are irrelevant here. + Then the unification tries to use the test_Prop instance to reduce (2a) to + (6a) result ?S =~= true + (7a) ?P =~= T with ?P : Prop + (8a) test ?S =~= test_Prop ?P + Now the default 'check' instance with ?result := true resolves (6a) as + (9a) ?S := @check true ?test ?frame + Then (7a) can be solved precisely if T has sort at most (hence exactly) Prop, + and then (8a) is solved by the check instance, yielding ?test := test_Prop T, + and completing the solution of (2a), and _committing_ to it. But now (3) is + inconsistent with (9a), and this makes the entire problem (1) fails. + If on the othe hand T does not have sort Prop then (7a) fails and the + unification resorts to delta expanding (2a), which gives + (2b) @condition (result ?S) (test ?S) =~= tt + which is then reduced, using the test_negative instance, to + (6b) result ?S =~= false + (8b) test ?S =~= test_negative + Both are solved using the check default instance, as in the (2a) branch, giving + (9b) ?S := @check false test_negative ?frame + Then (3) and (4) are similarly soved using check, giving the final assignment + (9) ?S := notProp T + Observe that we _must_ perform the actual test unification on the arguments + of the initial canonical instance, and not on the instance itself as we do + in mathcomp/matrix and mathcomp/vector, because we want the unification to + fail when T has sort Prop. If both the test_of _and_ the result check + unifications were done as part of the structure telescope then the latter + would be a sub-problem of the former, and thus failing the check would merely + make the test_of unification backtrack and delta-expand and we would not get + failure. + **) + +Structure call_of (condition : unit) (result : bool) := Call {callee : Type}. +Definition maybeProp (T : Type) := tt. +Definition call T := Call (maybeProp T) false T. + +Structure test_of (result : bool) := Test {condition :> unit}. +Definition test_Prop (P : Prop) := Test true (maybeProp P). +Definition test_negative := Test false tt. + +Structure type := + Check {result : bool; test : test_of result; frame : call_of test result}. +Definition check result test frame := @Check result test frame. + +Module Exports. +Canonical call. +Canonical test_Prop. +Canonical test_negative. +Canonical check. +Notation nonPropType := type. +Coercion callee : call_of >-> Sortclass. +Coercion frame : type >-> call_of. +Notation notProp T := (@check false test_negative (call T)). +End Exports. + +End NonPropType. +Export NonPropType.Exports. |