Commit 965852ff authored by Ralf Jung's avatar Ralf Jung

derive internal_eq_rewrite_contractive inside the logic

parent 7b00cda2
Pipeline #3210 passed with stage
in 10 minutes and 41 seconds
......@@ -770,6 +770,13 @@ Section later.
Proof. intros [|n] x y Hxy; [done|]; apply Hxy; lia. Qed.
Global Instance Later_inj n : Inj (dist n) (dist (S n)) (@Next A).
Proof. by intros x y. Qed.
Lemma later_car_compose_ne {B : ofeT} (f : A B) :
(Contractive f) n, Proper (dist n ==> dist n) (f later_car).
Proof.
move=> Hcont n [x] [y] /= Hxy. apply Hcont. intros m Hmn.
destruct n; first by (exfalso; omega). eapply dist_le', Hxy. omega.
Qed.
End later.
Arguments laterC : clear implicits.
......
......@@ -287,6 +287,14 @@ Lemma equiv_internal_eq {A : ofeT} P (a b : A) : a ≡ b → P ⊢ a ≡ b.
Proof. by intros ->. Qed.
Lemma internal_eq_sym {A : ofeT} (a b : A) : a b b a.
Proof. apply (internal_eq_rewrite a b (λ b, b a)%I); auto. solve_proper. Qed.
Lemma internal_eq_rewrite_contractive {A : ofeT} a b (Ψ : A uPred M) P
{HΨ : Contractive Ψ} : (P (a b)) (P Ψ a) P Ψ b.
Proof.
rewrite -later_equivI. intros Heq.
change ((P (Ψ later_car) (Next a)) P (Ψ later_car) (Next b)).
apply internal_eq_rewrite; last done.
exact: later_car_compose_ne.
Qed.
Lemma pure_impl_forall φ P : (⌜φ⌝ P) ( _ : φ, P).
Proof.
......
......@@ -375,14 +375,6 @@ Proof.
- by symmetry; apply Hab with x.
- by apply Ha.
Qed.
Lemma internal_eq_rewrite_contractive {A : ofeT} a b (Ψ : A uPred M) P
{HΨ : Contractive Ψ} : (P (a b)) (P Ψ a) P Ψ b.
Proof.
unseal; intros Hab Ha; split=> n x ??. apply HΨ with n a; auto.
- destruct n; intros m ?; first omega. apply (dist_le n); last omega.
symmetry. by destruct Hab as [Hab]; eapply (Hab (S n)).
- by apply Ha.
Qed.
(* BI connectives *)
Lemma sep_mono P P' Q Q' : (P Q) (P' Q') P P' Q Q'.
......@@ -565,7 +557,7 @@ Proof. by unseal. Qed.
Lemma prod_validI {A B : cmraT} (x : A * B) : x x.1 x.2.
Proof. by unseal. Qed.
(* Later *)
(* Type-level Later *)
Lemma later_equivI {A : ofeT} (x y : A) : Next x Next y (x y).
Proof. by unseal. Qed.
......
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