Remove uPred_big_and, it only use just complicates stuff.

algebra/upred_big_op.v

@@ -4,7 +4,7 @@ Import uPred.
 
 (** We define the following big operators:
 
-- The operators [ [★] Ps ] and [ [∧] Ps ] fold [★] and [∧] over the list [Ps].
+- The operator [ [★] Ps ] folds [★] over the list [Ps].
   This operator is not a quantifier, so it binds strongly.
 - The operator [ [★ list] k ↦ x ∈ l, P ] asserts that [P] holds separately for
   each element [x] at position [x] in the list [l]. This operator is a
@@ -18,10 +18,6 @@ Import uPred.
 
 (** * Big ops over lists *)
 (* These are the basic building blocks for other big ops *)
-Fixpoint uPred_big_and {M} (Ps : list (uPred M)) : uPred M :=
-  match Ps with [] => True | P :: Ps => P ∧ uPred_big_and Ps end%I.
-Instance: Params (@uPred_big_and) 1.
-Notation "'[∧]' Ps" := (uPred_big_and Ps) (at level 20) : uPred_scope.
 Fixpoint uPred_big_sep {M} (Ps : list (uPred M)) : uPred M :=
   match Ps with [] => True | P :: Ps => P ★ uPred_big_sep Ps end%I.
 Instance: Params (@uPred_big_sep) 1.
@@ -75,28 +71,13 @@ Implicit Types Ps Qs : list (uPred M).
 Implicit Types A : Type.
 
 (** ** Generic big ops over lists of upreds *)
-Global Instance big_and_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_and M).
-Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
 Global Instance big_sep_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_sep M).
 Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
-
-Global Instance big_and_ne n : Proper (dist n ==> dist n) (@uPred_big_and M).
-Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
 Global Instance big_sep_ne n : Proper (dist n ==> dist n) (@uPred_big_sep M).
 Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
-
-Global Instance big_and_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_and M).
-Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
 Global Instance big_sep_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_sep M).
 Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed.
 
-Global Instance big_and_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_and M).
-Proof.
-  induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
-  - by rewrite IH.
-  - by rewrite !assoc (comm _ P).
-  - etrans; eauto.
-Qed.
 Global Instance big_sep_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_sep M).
 Proof.
   induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
@@ -105,31 +86,17 @@ Proof.
   - etrans; eauto.
 Qed.
 
-Lemma big_and_app Ps Qs : [∧] (Ps ++ Qs) ⊣⊢ [∧] Ps ∧ [∧] Qs.
-Proof. induction Ps as [|?? IH]; by rewrite /= ?left_id -?assoc ?IH. Qed.
 Lemma big_sep_app Ps Qs : [★] (Ps ++ Qs) ⊣⊢ [★] Ps ★ [★] Qs.
 Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed.
 
-Lemma big_and_contains Ps Qs : Qs `contains` Ps → [∧] Ps ⊢ [∧] Qs.
-Proof.
-  intros [Ps' ->]%contains_Permutation. by rewrite big_and_app and_elim_l.
-Qed.
 Lemma big_sep_contains Ps Qs : Qs `contains` Ps → [★] Ps ⊢ [★] Qs.
 Proof.
   intros [Ps' ->]%contains_Permutation. by rewrite big_sep_app sep_elim_l.
 Qed.
-
-Lemma big_sep_and Ps : [★] Ps ⊢ [∧] Ps.
-Proof. by induction Ps as [|P Ps IH]; simpl; auto with I. Qed.
-
-Lemma big_and_elem_of Ps P : P ∈ Ps → [∧] Ps ⊢ P.
-Proof. induction 1; simpl; auto with I. Qed.
 Lemma big_sep_elem_of Ps P : P ∈ Ps → [★] Ps ⊢ P.
 Proof. induction 1; simpl; auto with I. Qed.
 
 (** ** Persistence *)
-Global Instance big_and_persistent Ps : PersistentL Ps → PersistentP ([∧] Ps).
-Proof. induction 1; apply _. Qed.
 Global Instance big_sep_persistent Ps : PersistentL Ps → PersistentP ([★] Ps).
 Proof. induction 1; apply _. Qed.
 
@@ -157,8 +124,6 @@ Proof.
 Qed.
 
 (** ** Timelessness *)
-Global Instance big_and_timeless Ps : TimelessL Ps → TimelessP ([∧] Ps).
-Proof. induction 1; apply _. Qed.
 Global Instance big_sep_timeless Ps : TimelessL Ps → TimelessP ([★] Ps).
 Proof. induction 1; apply _. Qed.
 

proofmode/coq_tactics.v

@@ -19,7 +19,7 @@ Record envs_wf {M} (Δ : envs M) := {
 }.
 
 Coercion of_envs {M} (Δ : envs M) : uPred M :=
-  (■ envs_wf Δ ★ □ [∧] env_persistent Δ ★ [★] env_spatial Δ)%I.
+  (■ envs_wf Δ ★ □ [★] env_persistent Δ ★ [★] env_spatial Δ)%I.
 Instance: Params (@of_envs) 1.
 
 Record envs_Forall2 {M} (R : relation (uPred M)) (Δ1 Δ2 : envs M) : Prop := {
@@ -110,7 +110,7 @@ Implicit Types Δ : envs M.
 Implicit Types P Q : uPred M.
 
 Lemma of_envs_def Δ :
-  of_envs Δ = (■ envs_wf Δ ★ □ [∧] env_persistent Δ ★ [★] env_spatial Δ)%I.
+  of_envs Δ = (■ envs_wf Δ ★ □ [★] env_persistent Δ ★ [★] env_spatial Δ)%I.
 Proof. done. Qed.
 
 Lemma envs_lookup_delete_Some Δ Δ' i p P :
@@ -127,13 +127,13 @@ Lemma envs_lookup_sound Δ i p P :
 Proof.
   rewrite /envs_lookup /envs_delete /of_envs=>?; apply pure_elim_sep_l=> Hwf.
   destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=.
-  - rewrite (env_lookup_perm Γp) //= always_and_sep always_sep.
-    ecancel [□ [∧] _; □ P; [★] _]%I; apply pure_intro.
+  - rewrite (env_lookup_perm Γp) //= always_sep.
+    ecancel [□ [★] _; □ P; [★] Γs]%I; apply pure_intro.
     destruct Hwf; constructor;
       naive_solver eauto using env_delete_wf, env_delete_fresh.
   - destruct (Γs !! i) eqn:?; simplify_eq/=.
     rewrite (env_lookup_perm Γs) //=.
-    ecancel [□ [∧] _; P; [★] _]%I; apply pure_intro.
+    ecancel [□ [★] _; P; [★] (env_delete _ _)]%I; apply pure_intro.
     destruct Hwf; constructor;
       naive_solver eauto using env_delete_wf, env_delete_fresh.
 Qed.
@@ -151,7 +151,7 @@ Lemma envs_lookup_split Δ i p P :
 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_and_sep always_sep.
+  - rewrite (env_lookup_perm Γp) //= always_sep.
     rewrite pure_equiv // left_id.
     cancel [□ P]%I. apply wand_intro_l. solve_sep_entails.
   - destruct (Γs !! i) eqn:?; simplify_eq/=.
@@ -188,7 +188,7 @@ Proof.
   - apply sep_intro_True_l; [apply pure_intro|].
     + destruct Hwf; constructor; simpl; eauto using Esnoc_wf.
       intros j; case_decide; naive_solver.
-    + by rewrite always_and_sep always_sep assoc.
+    + by rewrite always_sep assoc.
   - apply sep_intro_True_l; [apply pure_intro|].
     + destruct Hwf; constructor; simpl; eauto using Esnoc_wf.
       intros j; case_decide; naive_solver.
@@ -206,8 +206,7 @@ Proof.
       intros j. apply (env_app_disjoint _ _ _ j) in Happ.
       naive_solver eauto using env_app_fresh.
     + rewrite (env_app_perm _ _ Γp') //.
-      rewrite big_and_app always_and_sep always_sep (big_sep_and Γ).
-      solve_sep_entails.
+      rewrite big_sep_app always_sep. solve_sep_entails.
   - destruct (env_app Γ Γp) eqn:Happ,
       (env_app Γ Γs) as [Γs'|] eqn:?; simplify_eq/=.
     apply wand_intro_l, sep_intro_True_l; [apply pure_intro|].
@@ -230,8 +229,7 @@ Proof.
       intros j. apply (env_app_disjoint _ _ _ j) in Happ.
       destruct (decide (i = j)); try naive_solver eauto using env_replace_fresh.
     + rewrite (env_replace_perm _ _ Γp') //.
-      rewrite big_and_app always_and_sep always_sep (big_sep_and Γ).
-      solve_sep_entails.
+      rewrite big_sep_app always_sep. solve_sep_entails.
   - destruct (env_app Γ Γp) eqn:Happ,
       (env_replace i Γ Γs) as [Γs'|] eqn:?; simplify_eq/=.
     apply wand_intro_l, sep_intro_True_l; [apply pure_intro|].
@@ -428,7 +426,7 @@ Proof.
   repeat apply sep_mono; try apply always_mono.
   - rewrite -later_intro; apply pure_mono; destruct 1; constructor;
       naive_solver eauto using env_Forall2_wf, env_Forall2_fresh.
-  - induction Hp; rewrite /= ?later_and; auto using and_mono, later_intro.
+  - induction Hp; rewrite /= ?later_sep; auto using sep_mono, later_intro.
   - induction Hs; rewrite /= ?later_sep; auto using sep_mono, later_intro.
 Qed.