Section eqit_eq.
(** *** Properties of relation transformers. *)
Context {E : Type -> Type} {R : Type}.
Local Notation eqit := (@eqit E R R eq).
Global Instance Reflexive_eqitF_eq b1 b2 (sim : itree E R -> itree E R -> Prop)
: Reflexive sim -> Reflexive (eqitF eq b1 b2 id sim).
Global Instance Symmetric_eqitF_eq b (sim : itree E R -> itree E R -> Prop)
: Symmetric sim -> Symmetric (eqitF eq b b id sim).
Global Instance Reflexive_eqit__eq b1 b2 (sim : itree E R -> itree E R -> Prop)
: Reflexive sim -> Reflexive (eqit_ eq b1 b2 id sim).
Global Instance Reflexive_eqit_eq b1 b2 : Reflexive (eqit b1 b2).
Global Instance Symmetric_eqit_eq b : Symmetric (eqit b b).
(** *** Congruence properties *)
Global Instance eqit_observe b1 b2:
Proper (eqit b1 b2 ==> going (eqit b1 b2)) (@observe E R).
Global Instance eqit_tauF b1 b2:
Proper (eqit b1 b2 ==> going (eqit b1 b2)) (@TauF E R _).
Global Instance eqit_VisF b1 b2 {u} (e: E u) :
Proper (pointwise_relation _ (eqit b1 b2) ==> going (eqit b1 b2)) (VisF e).
Global Instance observing_sub_eqit l r :
subrelation (observing eq) (eqit l r).
(** ** Eta-expansion *)

Lemma eqit_inv_vis {E R1 R2 RR} b1 b2 {u} (e1 e2: E u) (k1: u -> itree E R1) (k2: u -> itree E R2) :
eqit RR b1 b2 (Vis e1 k1) (Vis e2 k2) -> e1 = e2 /\ (forall u, eqit RR b1 b2 (k1 u) (k2 u)).