Commit e06423f4 by Robbert Krebbers

### Prove more later properties in the logic.

parent bd15a981
 ... ... @@ -731,12 +731,6 @@ Lemma pure_elim_r φ Q R : (φ → Q ⊢ R) → Q ∧ ■ φ ⊢ R. Proof. intros; apply pure_elim with φ; eauto. Qed. Lemma pure_equiv (φ : Prop) : φ → ■ φ ⊣⊢ True. Proof. intros; apply (anti_symm _); auto using pure_intro. Qed. Lemma pure_alt φ : ■ φ ⊣⊢ ∃ _ : φ, True. Proof. apply (anti_symm _). - eapply pure_elim; eauto=> H. rewrite -(exist_intro H); auto. - by apply exist_elim, pure_intro. Qed. Lemma eq_refl' {A : cofeT} (a : A) P : P ⊢ a ≡ a. Proof. rewrite (True_intro P). apply eq_refl. Qed. ... ... @@ -750,6 +744,23 @@ Proof. apply (eq_rewrite P Q (λ Q, P ↔ Q))%I; first solve_proper; auto using iff_refl. Qed. Lemma pure_alt φ : ■ φ ⊣⊢ ∃ _ : φ, True. Proof. apply (anti_symm _). - eapply pure_elim; eauto=> H. rewrite -(exist_intro H); auto. - by apply exist_elim, pure_intro. Qed. Lemma and_alt P Q : P ∧ Q ⊣⊢ ∀ b : bool, if b then P else Q. Proof. apply (anti_symm _); first apply forall_intro=> -[]; auto. apply and_intro. by rewrite (forall_elim true). by rewrite (forall_elim false). Qed. Lemma or_alt P Q : P ∨ Q ⊣⊢ ∃ b : bool, if b then P else Q. Proof. apply (anti_symm _); last apply exist_elim=> -[]; auto. apply or_elim. by rewrite -(exist_intro true). by rewrite -(exist_intro false). Qed. (* BI connectives *) Lemma sep_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ★ P' ⊢ Q ★ Q'. Proof. ... ... @@ -1020,11 +1031,7 @@ Proof. unseal; split=> n x ? HP; induction n as [|n IH]; [by apply HP|]. apply HP, IH, uPred_closed with (S n); eauto using cmra_validN_S. Qed. Lemma later_and P Q : ▷ (P ∧ Q) ⊣⊢ ▷ P ∧ ▷ Q. Proof. unseal; split=> -[|n] x; by split. Qed. Lemma later_or P Q : ▷ (P ∨ Q) ⊣⊢ ▷ P ∨ ▷ Q. Proof. unseal; split=> -[|n] x; simpl; tauto. Qed. Lemma later_forall {A} (Φ : A → uPred M) : (▷ ∀ a, Φ a) ⊣⊢ (∀ a, ▷ Φ a). Lemma later_forall_2 {A} (Φ : A → uPred M) : (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a. Proof. unseal; by split=> -[|n] x. Qed. Lemma later_exist_false {A} (Φ : A → uPred M) : (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a). ... ... @@ -1059,13 +1066,17 @@ Proof. intros P Q; apply later_mono. Qed. Lemma later_intro P : P ⊢ ▷ P. Proof. rewrite -(and_elim_l (▷ P) P) -(löb (▷ P ∧ P)) later_and. apply impl_intro_l; auto. rewrite -(and_elim_l (▷ P) P) -(löb (▷ P ∧ P)). apply impl_intro_l. by rewrite {1}(and_elim_r (▷ P)). Qed. Lemma later_True : ▷ True ⊣⊢ True. Proof. apply (anti_symm (⊢)); auto using later_intro. Qed. Lemma later_impl P Q : ▷ (P → Q) ⊢ ▷ P → ▷ Q. Proof. apply impl_intro_l; rewrite -later_and; eauto using impl_elim. Qed. Lemma later_forall {A} (Φ : A → uPred M) : (▷ ∀ a, Φ a) ⊣⊢ (∀ a, ▷ Φ a). Proof. apply (anti_symm _); auto using later_forall_2. apply forall_intro=> x. by rewrite (forall_elim x). Qed. Lemma later_exist `{Inhabited A} (Φ : A → uPred M) : ▷ (∃ a, Φ a) ⊣⊢ (∃ a, ▷ Φ a). Proof. ... ... @@ -1073,6 +1084,12 @@ Proof. rewrite later_exist_false. apply or_elim; last done. rewrite -(exist_intro inhabitant); auto. Qed. Lemma later_and P Q : ▷ (P ∧ Q) ⊣⊢ ▷ P ∧ ▷ Q. Proof. rewrite !and_alt later_forall. by apply forall_proper=> -[]. Qed. Lemma later_or P Q : ▷ (P ∨ Q) ⊣⊢ ▷ P ∨ ▷ Q. Proof. rewrite !or_alt later_exist. by apply exist_proper=> -[]. Qed. Lemma later_impl P Q : ▷ (P → Q) ⊢ ▷ P → ▷ Q. Proof. apply impl_intro_l; rewrite -later_and; eauto using impl_elim. Qed. Lemma later_wand P Q : ▷ (P -★ Q) ⊢ ▷ P -★ ▷ Q. Proof. apply wand_intro_r; rewrite -later_sep; eauto using wand_elim_l. Qed. Lemma later_iff P Q : ▷ (P ↔ Q) ⊢ ▷ P ↔ ▷ Q. ... ...
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