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Require Import Crypto.Util.FixCoqMistakes.
Require Import Crypto.Util.Logic.
Require Import Coq.Classes.Morphisms Coq.Setoids.Setoid.
Lemma symmetry_iff {T} {R} {Rsym:@Symmetric T R} x y: R x y <-> R y x.
intuition eauto using symmetry.
Qed.
Lemma and_rewrite_l {A B} {RA RB} {Equivalence_RA:Equivalence RA} {Equivalence_RB:Equivalence RB}
(f:A->B) ref P {Proper_P: Proper (RA==>RB==>iff) P} a
: (RB (f a) ref /\ P a (f a)) <-> (RB (f a) ref /\ P a ref).
Proof.
logic; match goal with [H:_|-_] => (rewrite H || rewrite <-H); assumption end.
Qed.
Lemma and_rewriteleft_l {A B} {RA RB} {Equivalence_RA:Equivalence RA} {Equivalence_RB:Equivalence RB}
(f:A->B) ref P {Proper_P: Proper (RA==>RB==>iff) P} a
: (RB ref (f a) /\ P a (f a)) <-> (RB ref (f a) /\ P a ref).
Proof.
logic; match goal with [H:_|-_] => (rewrite H || rewrite <-H); assumption end.
Qed.
Lemma exists_and_equiv_r {A} {RA} {Equivalence_RA:Equivalence RA}
P {Proper_P: Proper (RA==>iff) P} (ref:A)
: (exists a, P a /\ RA a ref) <-> P ref.
Proof.
logic; try match goal with [H:_|-_] => (rewrite H || rewrite <-H); assumption end.
repeat (assumption||reflexivity||econstructor); assumption. (* WHY the last [assumption]?*)
Qed.
Lemma iff_R_R_same_r {T R} {Req:@Equivalence T R} x y ref : R x y -> (R x ref <-> R y ref).
Proof.
intro Hx; rewrite Hx; clear Hx. reflexivity.
Qed.
Global Instance Equivalence_and {A B RA RB}
{Equivalence_RA:@Equivalence A RA}
{Equivalence_RB:@Equivalence B RB}
: Equivalence (fun ab AB => RA (fst ab) (fst AB) /\ RB (snd ab) (snd AB)).
Proof.
destruct Equivalence_RA as [], Equivalence_RB as []; cbv in *|-.
repeat match goal with
| _ => intro
| _ => split
| [p:prod _ _ |- _ ] => destruct p
| [p:and _ _ |- _ ] => destruct p
| _ => progress (cbv [fst snd] in * )
| _ => solve[eauto]
end.
Qed.
Global Instance eq_eta_Reflexive {T} : Reflexive (fun x y : T => x = y) | 1
:= eq_Reflexive.
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