Library Coq.Structures.Equalities

Require Export RelationClasses.
Require Import Bool Morphisms Setoid.

Set Implicit Arguments.

Structure with nothing inside. Used to force a module type T into a module via Nop <+ T. (HACK!)

Module Type Nop.
End Nop.

Structure with just a base type t

Module Type Typ.
Parameter Inline(10) t : Type.
End Typ.

Structure with an equality relation eq

Module Type HasEq (Import T:Typ).
  Parameter Inline(30) eq : t -> t -> Prop.
End HasEq.

Module Type Eq := Typ <+ HasEq.

Module Type EqNotation (Import E:Eq).
  Infix "==" := eq (at level 70, no associativity).
  Notation "x ~= y" := (~eq x y) (at level 70, no associativity).
End EqNotation.

Module Type Eq´ := Eq <+ EqNotation.

Specification of the equality via the Equivalence type class

Module Type IsEq (Import E:Eq).
Declare Instance eq_equiv : Equivalence eq.
End IsEq.

Earlier specification of equality by three separate lemmas.

Module Type IsEqOrig (Import E:Eq´).
  Axiom eq_refl : forall x : t, x ==x.
  Axiom eq_sym : forall x y : t, x ==y -> y ==x.
  Axiom eq_trans : forall x y z : t, x ==y -> y ==z -> x ==z.
  Hint Immediate eq_sym.
  Hint Resolve eq_refl eq_trans.
End IsEqOrig.

Types with decidable equality

Module Type HasEqDec (Import E:Eq´).
Parameter eq_dec : forall x y : t, { x ==y } + { ~ x ==y }.
End HasEqDec.

Boolean Equality

Having eq_dec is the same as having a boolean equality plus a correctness proof.

Module Type HasEqb (Import T:Typ).
  Parameter Inline eqb : t -> t -> bool.
End HasEqb.

Module Type EqbSpec (T:Typ)(X:HasEq T)(Y:HasEqb T).
  Parameter eqb_eq : forall x y, Y.eqb x y = true <-> X.eq x y.
End EqbSpec.

Module Type EqbNotation (T:Typ)(E:HasEqb T).
  Infix "=?" := E.eqb (at level 70, no associativity).
End EqbNotation.

Module Type HasEqBool (E:Eq) := HasEqb E <+ EqbSpec E E.

From these basic blocks, we can build many combinations of static standalone module types.

Module Type EqualityType := Eq <+ IsEq.

Module Type EqualityTypeOrig := Eq <+ IsEqOrig.

Module Type EqualityTypeBoth <: EqualityType <: EqualityTypeOrig
:= Eq <+ IsEq <+ IsEqOrig.

Module Type DecidableType <: EqualityType
:= Eq <+ IsEq <+ HasEqDec.

Module Type DecidableTypeOrig <: EqualityTypeOrig
:= Eq <+ IsEqOrig <+ HasEqDec.

Module Type DecidableTypeBoth <: DecidableType <: DecidableTypeOrig
:= EqualityTypeBoth <+ HasEqDec.

Module Type BooleanEqualityType <: EqualityType
:= Eq <+ IsEq <+ HasEqBool.

Module Type BooleanDecidableType <: DecidableType <: BooleanEqualityType
:= Eq <+ IsEq <+ HasEqDec <+ HasEqBool.

Module Type DecidableTypeFull <: DecidableTypeBoth <: BooleanDecidableType
:= Eq <+ IsEq <+ IsEqOrig <+ HasEqDec <+ HasEqBool.

Same, with notation for eq

Compatibility wrapper from/to the old version of

EqualityType and DecidableType

Module BackportEq (E:Eq)(F:IsEq E) <: IsEqOrig E.
Definition eq_refl := @Equivalence_Reflexive _ _ F.eq_equiv.
Definition eq_sym := @Equivalence_Symmetric _ _ F.eq_equiv.
Definition eq_trans := @Equivalence_Transitive _ _ F.eq_equiv.
End BackportEq.

Module UpdateEq (E:Eq)(F:IsEqOrig E) <: IsEq E.
Instance eq_equiv : Equivalence E.eq.
End UpdateEq.

Module Backport_ET (E:EqualityType) <: EqualityTypeBoth
:= E <+ BackportEq.

Module Update_ET (E:EqualityTypeOrig) <: EqualityTypeBoth
:= E <+ UpdateEq.

Module Backport_DT (E:DecidableType) <: DecidableTypeBoth
:= E <+ BackportEq.

Module Update_DT (E:DecidableTypeOrig) <: DecidableTypeBoth
:= E <+ UpdateEq.

Having eq_dec is equivalent to having eqb and its spec.

Module HasEqDec2Bool (E:Eq)(F:HasEqDec E) <: HasEqBool E.
Definition eqb x y := if F.eq_dec x y then true else false.
Lemma eqb_eq : forall x y, eqb x y = true <-> E.eq x y.
End HasEqDec2Bool.

Module HasEqBool2Dec (E:Eq)(F:HasEqBool E) <: HasEqDec E.
Lemma eq_dec : forall x y, {E.eq x y }+{~E.eq x y }.
End HasEqBool2Dec.

Module Dec2Bool (E:DecidableType) <: BooleanDecidableType
:= E <+ HasEqDec2Bool.

Module Bool2Dec (E:BooleanEqualityType) <: BooleanDecidableType
:= E <+ HasEqBool2Dec.

Some properties of boolean equality

Module BoolEqualityFacts (Import E : BooleanEqualityType´).

eqb is compatible with eq

Instance eqb_compat : Proper (E.eq ==> E.eq ==> Logic.eq) eqb.

Alternative specification of eqb based on reflect.

Lemma eqb_spec x y : reflect (x ==y) (x =? y).

Negated form of eqb_eq

Lemma eqb_neq x y : (x =? y ) = false <-> x ~= y.

Basic equality laws for eqb

Lemma eqb_refl x : (x =? x ) = true.

Lemma eqb_sym x y : (x =? y ) = (y =? x ).

Transitivity is a particular case of eqb_compat

End BoolEqualityFacts.

UsualDecidableType

A particular case of DecidableType where the equality is the usual one of Coq.

Module Type HasUsualEq (Import T:Typ) <: HasEq T.
Definition eq := @Logic.eq t.
End HasUsualEq.

Module Type UsualEq <: Eq := Typ <+ HasUsualEq.

Module Type UsualIsEq (E:UsualEq) <: IsEq E.
Definition eq_equiv : Equivalence E.eq := eq_equivalence.
End UsualIsEq.

Module Type UsualIsEqOrig (E:UsualEq) <: IsEqOrig E.
Definition eq_refl := @Logic.eq_refl E.t.
Definition eq_sym := @Logic.eq_sym E.t.
Definition eq_trans := @Logic.eq_trans E.t.
End UsualIsEqOrig.

Module Type UsualEqualityType <: EqualityType
:= UsualEq <+ UsualIsEq.

Module Type UsualDecidableType <: DecidableType
:= UsualEq <+ UsualIsEq <+ HasEqDec.

Module Type UsualDecidableTypeOrig <: DecidableTypeOrig
:= UsualEq <+ UsualIsEqOrig <+ HasEqDec.

Module Type UsualDecidableTypeBoth <: DecidableTypeBoth
:= UsualEq <+ UsualIsEq <+ UsualIsEqOrig <+ HasEqDec.

Module Type UsualBoolEq := UsualEq <+ HasEqBool.

Module Type UsualDecidableTypeFull <: DecidableTypeFull
:= UsualEq <+ UsualIsEq <+ UsualIsEqOrig <+ HasEqDec <+ HasEqBool.

Some shortcuts for easily building a UsualDecidableType

Module Type MiniDecidableType.
Include Typ.
Parameter eq_dec : forall x y : t, {x =y }+{~x =y }.
End MiniDecidableType.

Module Make_UDT (M:MiniDecidableType) <: UsualDecidableTypeBoth
:= M <+ HasUsualEq <+ UsualIsEq <+ UsualIsEqOrig.

Module Make_UDTF (M:UsualBoolEq) <: UsualDecidableTypeFull
:= M <+ UsualIsEq <+ UsualIsEqOrig <+ HasEqBool2Dec.