theories/algebra/cmra_big_op.v → iris/algebra/cmra_big_op.v

-From iris.algebra Require Export big_op cmra.
 From stdpp Require Import gmap gmultiset.
-Set Default Proof Using "Type*".
+From iris.algebra Require Export big_op cmra.
+From iris.prelude Require Import options.
 
 (** Option *)
-Lemma big_opL_None {M : cmraT} {A} (f : nat → A → option M) l :
+Lemma big_opL_None {M : cmra} {A} (f : nat → A → option M) l :
   ([^op list] k↦x ∈ l, f k x) = None ↔ ∀ k x, l !! k = Some x → f k x = None.
 Proof.
   revert f. induction l as [|x l IH]=> f //=. rewrite op_None IH. split.
   - intros [??] [|k] y ?; naive_solver.
-  - intros Hl. split. by apply (Hl 0). intros k. apply (Hl (S k)).
+  - intros Hl. split; [by apply (Hl 0)|]. intros k. apply (Hl (S k)).
 Qed.
-Lemma big_opM_None {M : cmraT} `{Countable K} {A} (f : K → A → option M) m :
+Lemma big_opM_None {M : cmra} `{Countable K} {A} (f : K → A → option M) m :
   ([^op map] k↦x ∈ m, f k x) = None ↔ ∀ k x, m !! k = Some x → f k x = None.
 Proof.
-  induction m as [|i x m ? IH] using map_ind=> //=.
-  rewrite -equiv_None big_opM_insert // equiv_None op_None IH. split.
+  induction m as [|i x m ? IH] using map_ind=> /=.
+  { by rewrite big_opM_empty. }
+  rewrite -None_equiv_eq big_opM_insert // None_equiv_eq op_None IH. split.
   { intros [??] k y. rewrite lookup_insert_Some; naive_solver. }
   intros Hm; split.
   - apply (Hm i). by simplify_map_eq.
   - intros k y ?. apply (Hm k). by simplify_map_eq.
 Qed.
-Lemma big_opS_None {M : cmraT} `{Countable A} (f : A → option M) X :
+Lemma big_opS_None {M : cmra} `{Countable A} (f : A → option M) X :
   ([^op set] x ∈ X, f x) = None ↔ ∀ x, x ∈ X → f x = None.
 Proof.
-  induction X as [|x X ? IH] using collection_ind_L; [done|].
-  rewrite -equiv_None big_opS_insert // equiv_None op_None IH. set_solver.
+  induction X as [|x X ? IH] using set_ind_L.
+  { by rewrite big_opS_empty. }
+  rewrite -None_equiv_eq big_opS_insert // None_equiv_eq op_None IH. set_solver.
 Qed.
-Lemma big_opMS_None {M : cmraT} `{Countable A} (f : A → option M) X :
+Lemma big_opMS_None {M : cmra} `{Countable A} (f : A → option M) X :
   ([^op mset] x ∈ X, f x) = None ↔ ∀ x, x ∈ X → f x = None.
 Proof.
   induction X as [|x X IH] using gmultiset_ind.
-  { rewrite big_opMS_empty. set_solver. }
-  rewrite -equiv_None big_opMS_union big_opMS_singleton equiv_None op_None IH.
-  set_solver.
-Qed.
\ No newline at end of file
+  { rewrite big_opMS_empty. multiset_solver. }
+  rewrite -None_equiv_eq big_opMS_disj_union big_opMS_singleton None_equiv_eq op_None IH.
+  multiset_solver.
+Qed.