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Set Implicit Arguments.
Unset Strict Implicit.
Require Import Coq.Classes.Equivalence.
Section Equiv.
Context [ Equivalence A eqA ].
Variables x y z w : A.
Goal eqA x y -> eqA y x.
intros H ; clrewrite H.
refl.
Qed.
Tactic Notation "simpl" "*" := auto || relation_tac.
Goal eqA x y -> eqA y x /\ True.
intros H ; clrewrite H.
split ; simpl*.
Qed.
Goal eqA x y -> eqA y x /\ eqA x x.
intros H ; clrewrite H.
split ; simpl*.
Qed.
Goal eqA x y -> eqA y z -> eqA x y.
intros H.
clrewrite H.
intro. refl.
Qed.
Goal eqA x y -> eqA z y -> eqA x y.
intros H.
clrewrite <- H at 2.
clrewrite <- H at 1.
intro. refl.
Qed.
Opaque complement.
Print iff_inverse_impl_binary_morphism.
Goal eqA x y -> eqA x y -> eqA x y.
intros H.
clrewrite H.
intro. refl.
Qed.
Goal eqA x y -> eqA x y -> eqA x y.
intros H.
clrewrite <- H.
refl.
Qed.
Goal eqA x y -> True /\ True /\ False /\ eqA x x -> True /\ True /\ False /\ eqA x y.
Proof.
intros.
clrewrite <- H.
apply H0.
Qed.
End Equiv.
Section Trans.
Context [ Transitive A R ].
Variables x y z w : A.
Tactic Notation "simpl" "*" := auto || relation_tac.
(* Typeclasses eauto := debug. *)
Goal R x y -> R y x -> R y y -> R x x.
Proof with auto.
intros H H' H''.
clrewrite <- H' at 2.
clrewrite H at 1...
Qed.
Goal R x y -> R y z -> R x z.
intros H.
clrewrite H.
refl.
Qed.
Goal R x y -> R z y -> R x y.
intros H.
clrewrite <- H at 2.
intro.
clrewrite H at 1.
Abort.
Goal R x y -> True /\ R y z -> True /\ R x z.
Proof.
intros.
clrewrite H.
apply H0.
Qed.
Goal R x y -> True /\ True /\ False /\ R y z -> True /\ True /\ False /\ R x z.
Proof.
intros.
clrewrite H.
apply H0.
Qed.
End Trans.
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