Commit 92a22d63 by Robbert Krebbers

### Some stuff about cmras and finite maps.

parent f079e680
 ... ... @@ -94,11 +94,12 @@ Qed. Section map. Context `{FinMap K M}. Existing Instances map_dist map_compl map_cofe. Instance map_op `{Op A} : Op (M A) := merge op. Instance map_unit `{Unit A} : Unit (M A) := fmap unit. Instance map_valid `{Valid A} : Valid (M A) := λ m, ∀ i, ✓ (m !! i). Instance map_validN `{ValidN A} : ValidN (M A) := λ n m, ∀ i, ✓{n} (m!!i). Instance map_minus `{Minus A} : Minus (M A) := merge minus. Global Instance map_op `{Op A} : Op (M A) := merge op. Global Instance map_unit `{Unit A} : Unit (M A) := fmap unit. Global Instance map_valid `{Valid A} : Valid (M A) := λ m, ∀ i, ✓ (m !! i). Global Instance map_validN `{ValidN A} : ValidN (M A) := λ n m, ∀ i, ✓{n} (m!!i). Global Instance map_minus `{Minus A} : Minus (M A) := merge minus. Lemma lookup_op `{Op A} m1 m2 i : (m1 ⋅ m2) !! i = m1 !! i ⋅ m2 !! i. Proof. by apply lookup_merge. Qed. Lemma lookup_minus `{Minus A} m1 m2 i : (m1 ⩪ m2) !! i = m1 !! i ⩪ m2 !! i. ... ... @@ -121,7 +122,7 @@ Section map. * intros Hm; exists (m2 ⩪ m1); intros i. by rewrite lookup_op, lookup_minus, cmra_op_minus. Qed. Instance map_cmra `{CMRA A} : CMRA (M A). Global Instance map_cmra `{CMRA A} : CMRA (M A). Proof. split. * apply _. ... ... @@ -145,13 +146,13 @@ Section map. * intros x y n; rewrite map_includedN_spec; intros ? i. by rewrite lookup_op, lookup_minus, cmra_op_minus by done. Qed. Instance map_ra_empty `{RA A} : RAEmpty (M A). Global Instance map_ra_empty `{RA A} : RAEmpty (M A). Proof. split. * by intros ?; rewrite lookup_empty. * by intros m i; simpl; rewrite lookup_op, lookup_empty; destruct (m !! i). Qed. Instance map_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (M A). Global Instance map_cmra_extend `{CMRA A, !CMRAExtend A} : CMRAExtend (M A). Proof. intros n m m1 m2 Hm Hm12. assert (∀ i, m !! i ={n}= m1 !! i ⋅ m2 !! i) as Hm12' ... ... @@ -188,6 +189,12 @@ Section map. intros ?; apply (map_ra_insert_valid_timeless _ _ _ _ _); simpl. by rewrite lookup_empty. Qed. Lemma map_insert_valid `{ValidN A} (m : M A) n i x : ✓{n} x → ✓{n} m → ✓{n} (<[i:=x]>m). Proof. by intros ?? j; destruct (decide (i = j)); simplify_map_equality. Qed. Lemma map_insert_op `{Op A} (m1 m2 : M A) i x : m2 !! i = None → <[i:=x]>(m1 ⋅ m2) = <[i:=x]>m1 ⋅ m2. Proof. by intros Hi; apply (insert_merge_l _); rewrite Hi. Qed. Definition mapRA (A : cmraT) : cmraT := CMRAT (M A). Global Instance map_fmap_cmra_monotone `{CMRA A, CMRA B} (f : A → B) `{!CMRAMonotone f} : CMRAMonotone (fmap f : M A → M B). ... ... @@ -205,9 +212,29 @@ Section map. Global Instance mapRA_map_monotone {A B : cmraT} (f : A -n> B) `{!CMRAMonotone f} : CMRAMonotone (mapRA_map f) := _. End map. Arguments mapRA {_} _ {_ _ _ _ _ _ _ _ _} _. Section map_dom. Context `{FinMapDom K M D, Fresh K D, !FreshSpec K D}. Lemma map_dom_op `{Op A} (m1 m2: M A) : dom D (m1 ⋅ m2) ≡ dom D m1 ∪ dom D m2. Proof. apply elem_of_equiv; intros i; rewrite elem_of_union, !elem_of_dom. unfold is_Some; setoid_rewrite lookup_op. destruct (m1 !! i), (m2 !! i); naive_solver. Qed. Lemma map_update_alloc `{CMRA A} (m : M A) x : ✓ x → m ⇝: λ m', ∃ i, m' = <[i:=x]>m ∧ m !! i = None. Proof. intros ? mf n Hm. set (i := fresh (dom D (m ⋅ mf))). assert (i ∉ dom D m ∧ i ∉ dom D mf) as [??]. { rewrite <-not_elem_of_union, <-map_dom_op; apply is_fresh. } exists (<[i:=x]>m); split; [exists i; split; [done|]|]. * by apply not_elem_of_dom. * rewrite <-map_insert_op by (by apply not_elem_of_dom). by apply map_insert_valid; [apply cmra_valid_validN|]. Qed. End map_dom. Canonical Structure natmapRA := mapRA natmap. Definition natmapRA_map {A B : cmraT} (f : A -n> B) : natmapRA A -n> natmapRA B := mapRA_map f. ... ...
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