+ +

Library mathcomp.ssreflect.eqtype

+ +

+(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
+ Distributed under the terms of CeCILL-B. *)
+Require Import mathcomp.ssreflect.ssreflect.
+ +
+

+ +

+ 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.
+ +
+Module Equality.
+ +
+Definition axiom T (e : rel T) := ∀ 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}.
+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.
+ +
+Lemma eqP T : Equality.axiom (@eq_op T).
+ +
+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.
+ +
+ +
+Lemma eq_refl (T : eqType) (x : T) : x == x.
+Notation eqxx := eq_refl.
+ +
+Lemma eq_sym (T : eqType) (x y : T) : (x == y ) = (y == x ).
+ +
+Hint Resolve eq_refl eq_sym.
+ +
+Section Contrapositives.
+ +
+Variables (T1 T2 : eqType).
+Implicit Types (A : pred T1) (b : bool) (x : T1) (z : T2).
+ +
+Lemma contraTeq b x y : (x != y → ~~ b ) → b → x = y.
+ +
+Lemma contraNeq b x y : (x != y → b ) → ~~ b → x = y.
+ +
+Lemma contraFeq b x y : (x != y → b ) → b = false → x = y.
+ +
+Lemma contraTneq b x y : (x = y → ~~ b ) → b → x != y.
+ +
+Lemma contraNneq b x y : (x = y → b ) → ~~ b → x != y.
+ +
+Lemma contraFneq b x y : (x = y → b ) → b = false → x != y.
+ +
+Lemma contra_eqN b x y : (b → x != y ) → x = y → ~~ b.
+ +
+Lemma contra_eqF b x y : (b → x != y ) → x = y → b = false.
+ +
+Lemma contra_eqT b x y : (~~ b → x != y ) → x = y → b.
+ +
+Lemma contra_eq z1 z2 x1 x2 : (x1 != x2 → z1 != z2 ) → z1 = z2 → x1 = x2.
+ +
+Lemma contra_neq z1 z2 x1 x2 : (x1 = x2 → z1 = z2 ) → z1 != z2 → x1 != x2.
+ +
+Lemma memPn A x : reflect {in A , ∀ y, y != x } (x \notin A).
+ +
+Lemma memPnC A x : reflect {in A , ∀ y, x != y } (x \notin A).
+ +
+Lemma ifN_eq R x y vT vF : x != y → (if x == y then vT else vF ) = vF :> R.
+ +
+Lemma ifN_eqC R x y vT vF : x != y → (if y == x then vT else vF ) = vF :> R.
+ +
+End Contrapositives.
+ +
+ +
+Theorem eq_irrelevance (T : eqType) x y : ∀ e1 e2 : x = y :> T, e1 = e2.
+ +
+Corollary eq_axiomK (T : eqType) (x : T) : all_equal_to (erefl x).
+ +
+

+ +

+ 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).
+ +
+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.
+ +
+Canonical bool_eqMixin := EqMixin eqbP.
+Canonical bool_eqType := Eval hnf in EqType bool bool_eqMixin.
+ +
+Lemma eqbE : eqb = eq_op.
+ +
+Lemma bool_irrelevance (x y : bool) (E E' : x = y) : E = E'.
+ +
+Lemma negb_add b1 b2 : ~~ (b1 (+) b2 ) = (b1 == b2 ).
+ +
+Lemma negb_eqb b1 b2 : (b1 != b2 ) = b1 (+) b2.
+ +
+Lemma eqb_id b : (b == true ) = b.
+ +
+Lemma eqbF_neg b : (b == false ) = ~~ b.
+ +
+Lemma eqb_negLR b1 b2 : (~~ b1 == b2 ) = (b1 == ~~ b2 ).
+ +
+

+ +

+ 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.
+ +
+Variables (T2 : eqType) (x y : T) (z u : T2) (b : bool).
+ +
+Lemma predU1P : reflect (x = y ∨ b) ((x == y ) || b).
+ +
+Lemma pred2P : reflect (x = y ∨ z = u) ((x == y ) || (z == u )).
+ +
+Lemma predD1P : reflect (x ≠ y ∧ b) ((x != y ) && b).
+ +
+Lemma predU1l : x = y → (x == y ) || b.
+ +
+Lemma predU1r : b → (x == y ) || b.
+ +
+Lemma eqVneq : {x = y } + {x != y }.
+ +
+End EqPred.
+ +
+ +
+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 → ∀ x y, (f x == f y ) = (x == y ).
+ +
+Lemma can_eq : cancel f g → ∀ x y, (f x == f y ) = (x == y ).
+ +
+Lemma bij_eq : bijective f → ∀ x y, (f x == f y ) = (x == y ).
+ +
+Lemma can2_eq : cancel f g → cancel g f → ∀ x y, (f x == y ) = (x == g y ).
+ +
+Lemma inj_in_eq :
+ {in D &, injective f } → {in D &, ∀ x y, (f x == f y ) = (x == y )}.
+ +
+Lemma can_in_eq :
+ {in D , cancel f g } → {in D &, ∀ x y, (f x == f y ) = (x == y )}.
+ +
+End Exo.
+ +
+Section Endo.
+ +
+Variable T : eqType.
+ +
+Definition frel f := [rel x y : T | f x == y ].
+ +
+Lemma inv_eq f : involutive f → ∀ x y : T, (f x == y ) = (x == f y ).
+ +
+Lemma eq_frel f f' : f =1 f' → frel f =2 frel f'.
+ +
+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)).
+ +
+Lemma invariant_inj : injective h → invariant f (h \o k) =1 invariant f k.
+ +
+End EqFun.
+ +
+ +
+

+ +

+ 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.
+ +
+ +
+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.
+ +
+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 := ∀ x y : T, decidable (x = y).
+ +
+Hypothesis Hcompare : comparable.
+ +
+Definition compareb x y : bool := Hcompare x y.
+ +
+Lemma compareP : Equality.axiom compareb.
+ +
+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 : ∀ x, P x → sub_sort;
+  _ : ∀ K (_ : ∀ x Px, K (@Sub x Px)) u, K u;
+  _ : ∀ x Px, val (@Sub x Px) = x
+}.
+ +
+Lemma vrefl : ∀ x, P x → x = x.
+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.
+ +
+Lemma SubK x Px : @val sT (Sub x Px) = x.
+ +
+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).
+ +
+Lemma insubT x Px : insub x = Some (Sub x Px).
+ +
+Lemma insubF x : P x = false → insub x = None.
+ +
+Lemma insubN x : ~~ P x → insub x = None.
+ +
+Lemma isSome_insub : ([eta insub ] : pred T ) =1 P.
+ +
+Lemma insubK : ocancel insub (@val _).
+ +
+Lemma valP (u : sT) : P (val u).
+ +
+Lemma valK : pcancel (@val _) insub.
+ +
+Lemma val_inj : injective (@val sT).
+ +
+Lemma valKd u0 : cancel (@val _) (insubd u0).
+ +
+Lemma val_insubd u0 x : val (insubd u0 x) = if P x then x else val u0.
+ +
+Lemma insubdK u0 : {in P , cancel (insubd u0) (@val _)}.
+ +
+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.
+ +
+End SubType.
+ +
+ +
+ +
+ +
+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'.
+ +
+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).
+ +
+Lemma innew_val T nT : cancel val (@innew T nT).
+ +
+

+ +

+ 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 : ∀ u : sig P, P (sval u).
+ +
+Definition s2val (u : sig2 P Q) := let: exist2 x _ _ := u in x.
+ +
+Lemma s2valP u : P (s2val u).
+ +
+Lemma s2valP' u : Q (s2val u).
+ +
+End SigProj.
+ +
+ +
+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).
+ +
+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).
+ +
+Lemma val_eqP : ev_ax sT val.
+ +
+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 ).
+ +
+End SubEqType.
+ +
+ +
+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.
+ +
+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 _.
+ +
+Lemma xpair_eqE (x1 y1 : T1) (x2 y2 : T2) :
+  ((x1 , x2 ) == (y1 , y2 )) = ((x1 == y1 ) && (x2 == y2 )).
+ +
+Lemma pair_eq1 (u v : T1 × T2) : u == v → u .1 == v .1.
+ +
+Lemma pair_eq2 (u v : T1 × T2) : u == v → u .2 == v .2.
+ +
+End ProdEqType.
+ +
+ +
+ +
+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.
+ +
+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_ : ∀ u, T_(tag u) := @projS2 I [eta T_].
+Definition Tagged I i T_ x := @existS I [eta T_] i x.
+ +
+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.
+ +
+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.
+ +
+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.
+ +
+Lemma eq_tag u v : u == v → tag u = tag v.
+ +
+Lemma eq_Tagged u x :(u == Tagged _ x ) = (tagged u == x ).
+ +
+End TagEqType.
+ +
+ +
+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.
+ +
+Canonical sum_eqMixin := EqMixin sum_eqP.
+Canonical sum_eqType := Eval hnf in EqType (T1 + T2) sum_eqMixin.
+ +
+Lemma sum_eqE : sum_eq = eq_op.
+ +
+End SumEqType.
+ +
+