Ltac normalizes :=
match goal with
| [ |- Normalizes _ (respectful _ _) _ ] => class_apply @flip_arrow
| _ => class_apply @flip_atom
end.
Ltac proper_normalization :=
match goal with
| [ _ : normalization_done |- _ ] => fail 1
| [ _ : apply_subrelation |- @Proper _ ?R _ ] =>
let H := fresh "H" in
set(H:=did_normalization) ; class_apply @proper_normalizes_proper
end.
#[global]
Hint Extern 1 (Normalizes _ _ _) => normalizes : typeclass_instances.
#[global]
Hint Extern 6 (@Proper _ _ _) => proper_normalization
: typeclass_instances.
(** When the relation on the domain is symmetric, we can flip the relation on the codomain. Same for binary functions. *)

(* Specifications: *)
Lemma proper_sym_flip :
forall `(Symmetric A R1)`(Proper (A->B) (R1==>R2) f),
Proper (R1==>flip R2) f.
Lemma proper_sym_flip_2 :
forall `(Symmetric A R1)`(Symmetric B R2)`(Proper (A->B->C) (R1==>R2==>R3) f),
Proper (R1==>R2==>flip R3) f.