Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Structures.EqualitiesFacts

Require Import Equalities Bool SetoidList RelationPairs.
Set Implicit Arguments.

# Keys and datas used in the future MMaps

Module KeyDecidableType(D:DecidableType).
Local Open Scope signature_scope.
Definition eqk {elt} : relation (key*elt) := D.eq @@1.
Definition eqke {elt} : relation (key*elt) := D.eq * Logic.eq.
Hint Unfold eqk eqke : core.
eqk, eqke are equalities
Instance eqk_equiv {elt} : Equivalence (@eqk elt) := _.
Instance eqke_equiv {elt} : Equivalence (@eqke elt) := _.
eqke is stricter than eqk
Instance eqke_eqk {elt} : subrelation (@eqke elt) (@eqk elt).
Alternative definitions of eqke and eqk
Lemma eqke_def {elt} k k' (e e':elt) :
eqke (k, e) (k',e') = (D.eq k k' /\ e = e').
Lemma eqke_def' {elt} (p q:key*elt) :
eqke p q = (D.eq (fst p) (fst q) /\ snd p = snd q).
Lemma eqke_1 {elt} k k' (e e':elt) : eqke (k, e) (k',e') -> D.eq k k'.
Lemma eqke_2 {elt} k k' (e e':elt) : eqke (k, e) (k',e') -> e=e'.
Lemma eqk_def {elt} k k' (e e':elt) : eqk (k, e) (k',e') = D.eq k k'.
Lemma eqk_def' {elt} (p q:key*elt) : eqk p q = D.eq (fst p) (fst q).
Lemma eqk_1 {elt} k k' (e e':elt) : eqk (k, e) (k',e') -> D.eq k k'.
Hint Resolve eqke_1 eqke_2 eqk_1 : core.
Lemma InA_eqke_eqk {elt} p (m:list (key*elt)) :
InA eqke p m -> InA eqk p m.
Hint Resolve InA_eqke_eqk : core.
Lemma InA_eqk_eqke {elt} p (m:list (key*elt)) :
InA eqk p m -> exists q, eqk p q /\ InA eqke q m.
Lemma InA_eqk {elt} p q (m:list (key*elt)) :
eqk p q -> InA eqk p m -> InA eqk q m.
Definition MapsTo {elt} (k:key)(e:elt):= InA eqke (k, e).
Definition In {elt} k m := exists e:elt, MapsTo k e m.
Hint Unfold MapsTo In : core.
Lemma In_alt {elt} k (l:list (key*elt)) :
In k l <-> exists e, InA eqk (k, e) l.
Lemma In_alt' {elt} (l:list (key*elt)) k e :
In k l <-> InA eqk (k, e) l.
Lemma In_alt2 {elt} k (l:list (key*elt)) :
In k l <-> Exists (fun p => D.eq k (fst p)) l.
Lemma In_nil {elt} k : In k (@nil (key*elt)) <-> False.
Lemma In_cons {elt} k p (l:list (key*elt)) :
In k (p::l) <-> D.eq k (fst p) \/ In k l.
Instance MapsTo_compat {elt} :
Proper (D.eq==>Logic.eq==>equivlistA eqke==>iff) (@MapsTo elt).
Instance In_compat {elt} : Proper (D.eq==>equivlistA eqk==>iff) (@In elt).
Lemma MapsTo_eq {elt} (l:list (key*elt)) x y e :
D.eq x y -> MapsTo x e l -> MapsTo y e l.
Lemma In_eq {elt} (l:list (key*elt)) x y :
D.eq x y -> In x l -> In y l.
Lemma In_inv {elt} k k' e (l:list (key*elt)) :
In k ((k',e) :: l) -> D.eq k k' \/ In k l.
Lemma In_inv_2 {elt} k k' e e' (l:list (key*elt)) :
InA eqk (k, e) ((k', e') :: l) -> ~ D.eq k k' -> InA eqk (k, e) l.
Lemma In_inv_3 {elt} x x' (l:list (key*elt)) :
InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l.
Hint Extern 2 (eqke ?a ?b) => split : core.
Hint Resolve InA_eqke_eqk : core.
Hint Resolve In_inv_2 In_inv_3 : core.
End KeyDecidableType.

# PairDecidableType

From two decidable types, we can build a new DecidableType over their cartesian product.
Module PairDecidableType(D1 D2:DecidableType) <: DecidableType.
Definition t := (D1.t * D2.t)%type.
Definition eq := (D1.eq * D2.eq)%signature.
Instance eq_equiv : Equivalence eq := _.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
End PairDecidableType.
Similarly for pairs of UsualDecidableType
Module PairUsualDecidableType(D1 D2:UsualDecidableType) <: UsualDecidableType.
Definition t := (D1.t * D2.t)%type.
Definition eq := @eq t.
Instance eq_equiv : Equivalence eq := _.
Definition eq_dec : forall x y, { eq x y }+{ ~eq x y }.
End PairUsualDecidableType.
And also for pairs of UsualDecidableTypeFull
Module PairUsualDecidableTypeFull (D1 D2:UsualDecidableTypeFull)
<: UsualDecidableTypeFull.
Module M := PairUsualDecidableType D1 D2.
Include Backport_DT (M).
Include HasEqDec2Bool.
End PairUsualDecidableTypeFull.