Rename pure_equiv -> pure_True.

The old name didn't make much sense. Also now we can have
pure_False too.

base_logic/big_op.v

@@ -260,7 +260,7 @@ Section list.
     revert Φ HΦ. induction l as [|x l IH]=> Φ HΦ.
     { rewrite big_sepL_nil; auto with I. }
     rewrite big_sepL_cons. rewrite -always_and_sep_l; apply and_intro.
-    - by rewrite (forall_elim 0) (forall_elim x) pure_equiv // True_impl.
+    - by rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
     - rewrite -IH. apply forall_intro=> k; by rewrite (forall_elim (S k)).
   Qed.
 
@@ -384,11 +384,11 @@ Section gmap.
     induction m as [|i x m ? IH] using map_ind; [rewrite ?big_sepM_empty; auto|].
     rewrite big_sepM_insert // -always_and_sep_l. apply and_intro.
     - rewrite (forall_elim i) (forall_elim x) lookup_insert.
-      by rewrite pure_equiv // True_impl.
+      by rewrite pure_True // True_impl.
     - rewrite -IH. apply forall_mono=> k; apply forall_mono=> y.
       apply impl_intro_l, pure_elim_l=> ?.
       rewrite lookup_insert_ne; last by intros ?; simplify_map_eq.
-      by rewrite pure_equiv // True_impl.
+      by rewrite pure_True // True_impl.
   Qed.
 
   Lemma big_sepM_impl Φ Ψ m :
@@ -492,12 +492,12 @@ Section gset.
     intros. apply (anti_symm _).
     { apply forall_intro=> x.
       apply impl_intro_l, pure_elim_l=> ?; by apply big_sepS_elem_of. }
-    induction X as [|x m ? IH] using collection_ind_L.
+    induction X as [|x X ? IH] using collection_ind_L.
     { rewrite big_sepS_empty; auto. }
     rewrite big_sepS_insert // -always_and_sep_l. apply and_intro.
-    - by rewrite (forall_elim x) pure_equiv ?True_impl; last set_solver.
+    - by rewrite (forall_elim x) pure_True ?True_impl; last set_solver.
     - rewrite -IH. apply forall_mono=> y. apply impl_intro_l, pure_elim_l=> ?.
-      by rewrite pure_equiv ?True_impl; last set_solver.
+      by rewrite pure_True ?True_impl; last set_solver.
   Qed.
 
   Lemma big_sepS_impl Φ Ψ X :

base_logic/derived.v

@@ -163,6 +163,11 @@ Proof.
   - by rewrite -(left_id True%I uPred_and (_ → _)%I) impl_elim_r.
   - by apply impl_intro_l; rewrite left_id.
 Qed.
+Lemma False_impl P : (False → P) ⊣⊢ True.
+Proof.
+  apply (anti_symm (⊢)); [by auto|].
+  apply impl_intro_l. rewrite left_absorb. auto.
+Qed.
 
 Lemma exists_impl_forall {A} P (Ψ : A → uPred M) :
   ((∃ x : A, Ψ x) → P) ⊣⊢ ∀ x : A, Ψ x → P.
@@ -223,8 +228,11 @@ Lemma pure_elim_l φ Q R : (φ → Q ⊢ R) → ■ φ ∧ Q ⊢ R.
 Proof. intros; apply pure_elim with φ; eauto. Qed.
 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.
+
+Lemma pure_True (φ : Prop) : φ → ■ φ ⊣⊢ True.
 Proof. intros; apply (anti_symm _); auto. Qed.
+Lemma pure_False (φ : Prop) : ¬φ → ■ φ ⊣⊢ False.
+Proof. intros; apply (anti_symm _); eauto using pure_elim. Qed.
 
 Lemma pure_and φ1 φ2 : ■ (φ1 ∧ φ2) ⊣⊢ ■ φ1 ∧ ■ φ2.
 Proof.
@@ -243,7 +251,7 @@ Proof.
   apply (anti_symm _).
   - apply impl_intro_l. rewrite -pure_and. apply pure_mono. naive_solver.
   - rewrite -pure_forall_2. apply forall_intro=> ?.
-    by rewrite -(left_id True uPred_and (_→_))%I (pure_equiv φ1) // impl_elim_r.
+    by rewrite -(left_id True uPred_and (_→_))%I (pure_True φ1) // impl_elim_r.
 Qed.
 Lemma pure_forall {A} (φ : A → Prop) : ■ (∀ x, φ x) ⊣⊢ ∀ x, ■ φ x.
 Proof.
@@ -268,7 +276,7 @@ Proof. apply (internal_eq_rewrite a b (λ b, b ≡ a)%I); auto. solve_proper. Qe
 Lemma pure_impl_forall φ P : (■ φ → P) ⊣⊢ (∀ _ : φ, P).
 Proof.
   apply (anti_symm _).
-  - apply forall_intro=> ?. by rewrite pure_equiv // left_id.
+  - apply forall_intro=> ?. by rewrite pure_True // left_id.
   - apply impl_intro_l, pure_elim_l=> Hφ. by rewrite (forall_elim Hφ).
 Qed.
 Lemma pure_alt φ : ■ φ ⊣⊢ ∃ _ : φ, True.

base_logic/lib/own.v

@@ -95,7 +95,7 @@ Proof.
       first (eapply alloc_updateP_strong', cmra_transport_valid, Ha);
       naive_solver.
   - apply exist_elim=>m; apply pure_elim_l=>-[γ [Hfresh ->]].
-    by rewrite !own_eq /own_def -(exist_intro γ) pure_equiv // left_id.
+    by rewrite !own_eq /own_def -(exist_intro γ) pure_True // left_id.
 Qed.
 Lemma own_alloc a : ✓ a → True ==∗ ∃ γ, own γ a.
 Proof.

heap_lang/heap.v

@@ -88,7 +88,7 @@ Section heap.
     l ↦{q1} v1 ∗ l ↦{q2} v2 ⊣⊢ v1 = v2 ∧ l ↦{q1+q2} v1.
   Proof.
     destruct (decide (v1 = v2)) as [->|].
-    { by rewrite heap_mapsto_op_eq pure_equiv // left_id. }
+    { by rewrite heap_mapsto_op_eq pure_True // left_id. }
     apply (anti_symm (⊢)); last by apply pure_elim_l.
     rewrite heap_mapsto_eq -auth_own_op auth_own_valid discrete_valid.
     eapply pure_elim; [done|] =>  /=.

proofmode/coq_tactics.v

@@ -152,10 +152,10 @@ Proof.
   rewrite /envs_lookup /of_envs=>?; apply pure_elim_sep_l=> Hwf.
   destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=.
   - rewrite (env_lookup_perm Γp) //= always_sep.
-    rewrite pure_equiv // left_id.
+    rewrite pure_True // left_id.
     cancel [□ P]%I. apply wand_intro_l. solve_sep_entails.
   - destruct (Γs !! i) eqn:?; simplify_eq/=.
-    rewrite (env_lookup_perm Γs) //=. rewrite pure_equiv // left_id.
+    rewrite (env_lookup_perm Γs) //=. rewrite pure_True // left_id.
     cancel [P]. apply wand_intro_l. solve_sep_entails.
 Qed.
 
@@ -398,7 +398,7 @@ Proof.
 Qed.
 
 Lemma tac_pure_revert Δ φ Q : (Δ ⊢ ■ φ → Q) → (φ → Δ ⊢ Q).
-Proof. intros HΔ ?. by rewrite HΔ pure_equiv // left_id. Qed.
+Proof. intros HΔ ?. by rewrite HΔ pure_True // left_id. Qed.
 
 (** * Later *)
 Class IntoLaterEnv (Γ1 Γ2 : env (uPred M)) :=
@@ -548,7 +548,7 @@ Lemma tac_specialize_assert_pure Δ Δ' j q R P1 P2 φ Q :
   φ → (Δ' ⊢ Q) → Δ ⊢ Q.
 Proof.
   intros. rewrite envs_simple_replace_sound //; simpl.
-  rewrite right_id (into_wand R) -(from_pure P1) pure_equiv //.
+  rewrite right_id (into_wand R) -(from_pure P1) pure_True //.
   by rewrite wand_True wand_elim_r.
 Qed.
 
@@ -745,7 +745,7 @@ Qed.
 (** * Framing *)
 Lemma tac_frame_pure Δ (φ : Prop) P Q :
   φ → Frame (■ φ) P Q → (Δ ⊢ Q) → Δ ⊢ P.
-Proof. intros ?? ->. by rewrite -(frame (■ φ) P) pure_equiv // left_id. Qed.
+Proof. intros ?? ->. by rewrite -(frame (■ φ) P) pure_True // left_id. Qed.
 
 Lemma tac_frame Δ Δ' i p R P Q :
   envs_lookup_delete i Δ = Some (p, R, Δ') → Frame R P Q →