Coq.Setoids.Setoid

# Module Coq.Setoids.Setoid

``Require` `Export` Setoid_replace.

`Section` Setoid.

`Variable` A : Type.
`Variable` Aeq : A -> A -> Prop.

`Record` Setoid_Theory : Prop :=
{ Seq_refl : (x:A) (Aeq x x);
Seq_sym : (x,y:A) (Aeq x y) -> (Aeq y x);
Seq_trans : (x,y,z:A) (Aeq x y) -> (Aeq y z) -> (Aeq x z)
}.

`End` Setoid.

`Definition` Prop_S : (Setoid_Theory Prop iff).
Split; [Exact iff_refl | Exact iff_sym | Exact iff_trans].
`Save`.

`Add` Setoid Prop iff Prop_S.

`Hint` prop_set : setoid := Resolve (Seq_refl Prop iff Prop_S).
`Hint` prop_set : setoid := Resolve (Seq_sym Prop iff Prop_S).
`Hint` prop_set : setoid := Resolve (Seq_trans Prop iff Prop_S).

`Add` Morphism or : or_ext.
Intros.
Inversion H1.
Left.
Inversion H.
Apply (H3 H2).

Right.
Inversion H0.
Apply (H3 H2).
`Save`.

`Add` Morphism and : and_ext.
Intros.
Inversion H1.
Split.
Inversion H.
Apply (H4 H2).

Inversion H0.
Apply (H4 H3).
`Save`.

`Add` Morphism not : not_ext.
Red ; Intros.
Apply H0.
Inversion H.
Apply (H3 H1).
`Save`.

`Definition` fleche [A,B:Prop] := A -> B.

`Add` Morphism fleche : fleche_ext.
Unfold fleche.
Intros.
Inversion H0.
Inversion H.
Apply (H3 (H1 (H6 H2))).
`Save`.

`

Index