Commit c72d1bb1 authored by Robbert Krebbers's avatar Robbert Krebbers

Prove more primitive version of uPred.eq_refl in model.

parent 5b224d9c
......@@ -494,7 +494,7 @@ Lemma exist_intro {A} {Ψ : A → uPred M} a : Ψ a ⊢ (∃ a, Ψ a).
Proof. unseal; split=> n x ??; by exists a. Qed.
Lemma exist_elim {A} (Φ : A uPred M) Q : ( a, Φ a Q) ( a, Φ a) Q.
Proof. unseal; intros HΦΨ; split=> n x ? [a ?]; by apply HΦΨ with a. Qed.
Lemma eq_refl {A : cofeT} (a : A) P : P (a a).
Lemma eq_refl {A : cofeT} (a : A) : True (a a).
Proof. unseal; by split=> n x ??; simpl. Qed.
Lemma eq_rewrite {A : cofeT} a b (Ψ : A uPred M) P
`{HΨ : n, Proper (dist n ==> dist n) Ψ} : P (a b) P Ψ a P Ψ b.
......@@ -679,10 +679,13 @@ Lemma const_elim_r φ Q R : (φ → Q ⊢ R) → (Q ∧ ■ φ) ⊢ R.
Proof. intros; apply const_elim with φ; eauto. Qed.
Lemma const_equiv (φ : Prop) : φ ( φ) ⊣⊢ True.
Proof. intros; apply (anti_symm _); auto using const_intro. Qed.
Lemma eq_refl' {A : cofeT} (a : A) P : P (a a).
Proof. rewrite (True_intro P). apply eq_refl. Qed.
Hint Resolve eq_refl'.
Lemma equiv_eq {A : cofeT} P (a b : A) : a b P (a b).
Proof. intros ->; apply eq_refl. Qed.
Proof. by intros ->. Qed.
Lemma eq_sym {A : cofeT} (a b : A) : (a b) (b a).
Proof. apply (eq_rewrite a b (λ b, b a)%I); auto using eq_refl. solve_proper. Qed.
Proof. apply (eq_rewrite a b (λ b, b a)%I); auto. solve_proper. Qed.
(* BI connectives *)
Lemma sep_mono P P' Q Q' : P Q P' Q' (P P') (Q Q').
......@@ -898,7 +901,7 @@ Proof.
apply (anti_symm ()); auto using always_elim.
apply (eq_rewrite a b (λ b, (a b))%I); auto.
{ intros n; solve_proper. }
rewrite -(eq_refl a True) always_const; auto.
rewrite -(eq_refl a) always_const; auto.
Qed.
Lemma always_and_sep P Q : ( (P Q)) ⊣⊢ ( (P Q)).
Proof. apply (anti_symm ()); auto using always_and_sep_1. Qed.
......@@ -1174,5 +1177,5 @@ Hint Resolve and_intro and_elim_l' and_elim_r' : I.
Hint Resolve always_mono : I.
Hint Resolve sep_elim_l' sep_elim_r' sep_mono : I.
Hint Immediate True_intro False_elim : I.
Hint Immediate iff_refl eq_refl : I.
Hint Immediate iff_refl eq_refl' : I.
End uPred.
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