Merge branch 'robbert/repair_plainly' into 'gen_proofmode'

Repair the plainly modality

See merge request FP/iris-coq!122

theories/base_logic/upred.v

@@ -486,8 +486,6 @@ Proof.
     intros P QR HP. unseal; split=> n x ? /=. by apply HP, cmra_core_validN.
   - (* bi_persistently P ⊢ bi_persistently (bi_persistently P) *)
     intros P. unseal; split=> n x ?? /=. by rewrite cmra_core_idemp.
-  - (* bi_plainly (bi_persistently P) ⊢ bi_plainly P (ADMISSIBLE) *)
-    intros P. unseal; split=> n  x ?? /=. by rewrite -(core_id_core ε).
   - (* (∀ a, bi_persistently (Ψ a)) ⊢ bi_persistently (∀ a, Ψ a) *)
     by unseal.
   - (* bi_persistently (∃ a, Ψ a) ⊢ ∃ a, bi_persistently (Ψ a) *)
@@ -503,8 +501,8 @@ Qed.
 
 Lemma uPred_sbi_mixin (M : ucmraT) : SbiMixin uPred_ofe_mixin
   uPred_entails uPred_pure uPred_and uPred_or uPred_impl
-  (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_plainly uPred_persistently
-  (@uPred_internal_eq M) uPred_later.
+  (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_wand
+  uPred_plainly uPred_persistently (@uPred_internal_eq M) uPred_later.
 Proof.
   split.
   - (* Contractive sbi_later *)
@@ -525,9 +523,10 @@ Proof.
     by unseal.
   - (* Discrete a → a ≡ b ⊣⊢ ⌜a ≡ b⌝ *)
     intros A a b ?. unseal; split=> n x ?; by apply (discrete_iff n).
-  - (* bi_plainly ((P → Q) ∧ (Q → P)) ⊢ P ≡ Q *)
-    unseal; split=> n x ? /= HPQ; split=> n' x' ? HP;
-    split; eapply HPQ; eauto using @ucmra_unit_least.
+  - (* bi_plainly ((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q *)
+    unseal; split=> n x ? /= HPQ. split=> n' x' ??.
+    move: HPQ=> [] /(_ n' x'); rewrite !left_id=> ?.
+    move=> /(_ n' x'); rewrite !left_id=> ?. naive_solver.
   - (* Next x ≡ Next y ⊢ ▷ (x ≡ y) *)
     by unseal.
   - (* ▷ (x ≡ y) ⊢ Next x ≡ Next y *)

theories/bi/derived_laws.v

@@ -957,8 +957,11 @@ Proof.
 Qed.
 Lemma plainly_persistently P : bi_plainly (bi_persistently P) ⊣⊢ bi_plainly P.
 Proof.
-  apply (anti_symm _); first apply plainly_persistently_1.
-  by rewrite {1}plainly_idemp_2 (plainly_elim_persistently P).
+  apply (anti_symm _).
+  - rewrite -{1}(left_id True%I bi_and (bi_plainly _)) (plainly_emp_intro True%I).
+    rewrite -{2}(persistently_and_emp_elim P).
+    rewrite !and_alt -plainly_forall_2. by apply forall_mono=> -[].
+  - by rewrite {1}plainly_idemp_2 (plainly_elim_persistently P).
 Qed.
 
 Lemma absorbingly_plainly P : bi_absorbingly (bi_plainly P) ⊣⊢ bi_plainly P.
@@ -1080,6 +1083,13 @@ Qed.
 Lemma impl_wand_plainly_2 P Q : (bi_plainly P -∗ Q) ⊢ (bi_plainly P → Q).
 Proof. apply impl_intro_l. by rewrite plainly_and_sep_l_1 wand_elim_r. Qed.
 
+Lemma valid_plainly P : bi_valid (bi_plainly P) ↔ bi_valid P.
+Proof.
+  rewrite /bi_valid. split; intros HP.
+  - by rewrite -(idemp bi_and emp%I) {2}HP plainly_and_emp_elim.
+  - by rewrite (plainly_emp_intro emp%I) HP.
+Qed.
+
 Section plainly_affinely_bi.
   Context `{BiAffine PROP}.
 
@@ -1906,14 +1916,34 @@ Proof.
   rewrite -(internal_eq_refl True%I a) plainly_pure; auto.
 Qed.
 
-Lemma plainly_alt P : bi_plainly P ⊣⊢ P ≡ True.
+Lemma plainly_alt P : bi_plainly P ⊣⊢ bi_affinely P ≡ emp.
+Proof.
+  rewrite -plainly_affinely. apply (anti_symm (⊢)).
+  - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l.
+    + by rewrite affinely_elim_emp left_id.
+    + by rewrite left_id.
+  - rewrite internal_eq_sym (internal_eq_rewrite _ _ bi_plainly).
+    by rewrite -plainly_True_emp plainly_pure True_impl.
+Qed.
+
+Lemma plainly_alt_absorbing P `{!Absorbing P} : bi_plainly P ⊣⊢ P ≡ True.
 Proof.
   apply (anti_symm (⊢)).
-  - rewrite -prop_ext. apply plainly_mono, and_intro; apply impl_intro_r; auto.
+  - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l; auto.
   - rewrite internal_eq_sym (internal_eq_rewrite _ _ bi_plainly).
     by rewrite plainly_pure True_impl.
 Qed.
 
+Lemma plainly_True_alt P : bi_plainly (True -∗ P) ⊣⊢ P ≡ True.
+Proof.
+  apply (anti_symm (⊢)).
+  - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l; auto.
+    by rewrite wand_elim_r.
+  - rewrite internal_eq_sym (internal_eq_rewrite _ _
+      (λ Q, bi_plainly (True -∗ Q)) ltac:(solve_proper)).
+    by rewrite -entails_wand // -(plainly_emp_intro True%I) True_impl.
+Qed.
+
 Global Instance internal_eq_absorbing {A : ofeT} (x y : A) :
   Absorbing (x ≡ y : PROP)%I.
 Proof. by rewrite /Absorbing absorbingly_internal_eq. Qed.

theories/bi/interface.v

@@ -129,8 +129,6 @@ Section bi_mixin.
     (* In the ordered RA model: `core` is idempotent *)
     bi_mixin_persistently_idemp_2 P :
       bi_persistently P ⊢ bi_persistently (bi_persistently P);
-    bi_mixin_plainly_persistently_1 P :
-      bi_plainly (bi_persistently P) ⊢ bi_plainly P;
 
     (* In the ordered RA model [P ⊢ persisently emp] (which can currently still
     be derived from the plainly axioms, which will be removed): `ε ≼ core x` *)
@@ -160,7 +158,7 @@ Section bi_mixin.
     sbi_mixin_fun_ext {A} {B : A → ofeT} (f g : ofe_fun B) : (∀ x, f x ≡ g x) ⊢ f ≡ g;
     sbi_mixin_sig_eq {A : ofeT} (P : A → Prop) (x y : sig P) : `x ≡ `y ⊢ x ≡ y;
     sbi_mixin_discrete_eq_1 {A : ofeT} (a b : A) : Discrete a → a ≡ b ⊢ ⌜a ≡ b⌝;
-    sbi_mixin_prop_ext P Q : bi_plainly ((P → Q) ∧ (Q → P)) ⊢
+    sbi_mixin_prop_ext P Q : bi_plainly ((P -∗ Q) ∧ (Q -∗ P)) ⊢
       sbi_internal_eq (OfeT PROP prop_ofe_mixin) P Q;
 
     (* Later *)
@@ -263,8 +261,8 @@ Structure sbi := Sbi {
                          sbi_forall sbi_exist sbi_sep sbi_wand sbi_plainly
                          sbi_persistently;
   sbi_sbi_mixin : SbiMixin sbi_ofe_mixin sbi_entails sbi_pure sbi_and sbi_or
-                           sbi_impl sbi_forall sbi_exist sbi_sep sbi_plainly
-                           sbi_persistently sbi_internal_eq sbi_later;
+                           sbi_impl sbi_forall sbi_exist sbi_sep sbi_wand
+                           sbi_plainly sbi_persistently sbi_internal_eq sbi_later;
 }.
 
 Instance: Params (@sbi_later) 1.
@@ -453,9 +451,6 @@ Proof. eapply bi_mixin_persistently_mono, bi_bi_mixin. Qed.
 Lemma persistently_idemp_2 P :
   bi_persistently P ⊢ bi_persistently (bi_persistently P).
 Proof. eapply bi_mixin_persistently_idemp_2, bi_bi_mixin. Qed.
-Lemma plainly_persistently_1 P :
-  bi_plainly (bi_persistently P) ⊢ bi_plainly P.
-Proof. eapply (bi_mixin_plainly_persistently_1 bi_entails), bi_bi_mixin. Qed.
 
 Lemma persistently_forall_2 {A} (Ψ : A → PROP) :
   (∀ a, bi_persistently (Ψ a)) ⊢ bi_persistently (∀ a, Ψ a).
@@ -493,7 +488,7 @@ Lemma discrete_eq_1 {A : ofeT} (a b : A) :
   Discrete a → a ≡ b ⊢ (⌜a ≡ b⌝ : PROP).
 Proof. eapply sbi_mixin_discrete_eq_1, sbi_sbi_mixin. Qed.
 
-Lemma prop_ext P Q : bi_plainly ((P → Q) ∧ (Q → P)) ⊢ P ≡ Q.
+Lemma prop_ext P Q : bi_plainly ((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q.
 Proof. eapply (sbi_mixin_prop_ext _ bi_entails), sbi_sbi_mixin. Qed.
 
 (* Later *)

theories/bi/monpred.v

@@ -326,7 +326,6 @@ Proof.
   - intros P Q. split=> i. apply bi.sep_elim_l, _.
   - intros P Q [?]. split=> i /=. by f_equiv.
   - intros P. split=> i. by apply bi.persistently_idemp_2.
-  - intros P. split=> i /=. by setoid_rewrite bi.plainly_persistently_1.
   - intros A Ψ. split=> i. by apply bi.persistently_forall_2.
   - intros A Ψ. split=> i. by apply bi.persistently_exist_1.
   - intros P Q. split=> i. apply bi.sep_elim_l, _.
@@ -346,8 +345,8 @@ Context (I : biIndex) (PROP : sbi).
 Lemma monPred_sbi_mixin :
   SbiMixin (PROP:=monPred I PROP) monPred_ofe_mixin monPred_entails monPred_pure
            monPred_and monPred_or monPred_impl monPred_forall monPred_exist
-           monPred_sep monPred_plainly monPred_persistently monPred_internal_eq
-           monPred_later.
+           monPred_sep monPred_wand monPred_plainly monPred_persistently
+           monPred_internal_eq monPred_later.
 Proof.
   split; unseal.
   - intros n P Q HPQ. split=> i /=.