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Require Import Coq.Classes.Equivalence.
Require Import Coq.Program.Program.
Import Relation_Definitions.
Import Morphisms.
Require Setoid.
Obligation Tactic := program_simpl ; simpl_relation.
Generalizable Variables A eqA.
Lemma bla : forall `{ ! @Equivalence A (eqA : relation A) } x y, eqA x y -> eqA y x.
Proof.
intros.
rewrite H0.
reflexivity.
Defined.
Lemma bla' : forall `{ ! @Equivalence A (eqA : relation A) } x y, eqA x y -> eqA y x.
Proof.
intros.
(* Need delta on [relation] to unify with the right lemmas. *)
rewrite <- H0.
reflexivity.
Qed.
Axiom euclid : nat -> { x : nat | x > 0 } -> nat.
Definition eq_proj {A} {s : A -> Prop} : relation (sig s) :=
fun x y => `x = `y.
#[export] Program Instance foo {A : Type} {s : A -> Prop} : @Equivalence (sig s) eq_proj.
Next Obligation.
Proof.
cbv in *;congruence.
Qed.
#[export] Instance bar : Proper (eq ==> eq_proj ==> eq) euclid.
Proof.
Admitted.
Goal forall (x : nat) (y : nat | y > 0) (z : nat | z > 0), eq_proj y z -> euclid x y = euclid x z.
Proof.
intros.
(* Breaks if too much delta in unification *)
rewrite H.
reflexivity.
Qed.
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