# Library iris.algebra.cmra_big_op

From stdpp Require Import gmap gmultiset.
From iris.algebra Require Export big_op cmra.
Set Default Proof Using "Type*".

Option
Lemma big_opL_None {M : cmraT} {A} (f : nat A option M) l :
([^op list] kx 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)).
Qed.
Lemma big_opM_None {M : cmraT} `{Countable K} {A} (f : K A option M) m :
([^op map] kx 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.
{ 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 :
([^op set] x X, f x) = None x, x X f x = None.
Proof.
induction X as [|x X ? IH] using set_ind_L; [done|].
rewrite -equiv_None big_opS_insert // equiv_None op_None IH. set_solver.
Qed.
Lemma big_opMS_None {M : cmraT} `{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_disj_union big_opMS_singleton equiv_None op_None IH.
set_solver.
Qed.