### Function `map_seq` to turn a list into a finite map of consequentive elements.

parent 2cf0cd35
 ... ... @@ -137,6 +137,12 @@ Definition map_fold `{FinMapToList K A M} {B} Instance map_filter `{FinMapToList K A M, Insert K A M, Empty M} : Filter (K * A) M := λ P _, map_fold (λ k v m, if decide (P (k,v)) then <[k := v]>m else m) ∅. Fixpoint map_seq `{Insert nat A M, Empty M} (start : nat) (xs : list A) : M := match xs with | [] => ∅ | x :: xs => <[start:=x]> (map_seq (S start) xs) end. (** * Theorems *) Section theorems. Context `{FinMap K M}. ... ... @@ -1911,6 +1917,82 @@ Proof. Qed. End theorems. (** ** The seq operation *) Section map_seq. Context `{FinMap nat M} {A : Type}. Implicit Types x : A. Implicit Types xs : list A. Lemma lookup_map_seq_Some_inv start xs i x : xs !! i = Some x ↔ map_seq (M:=M A) start xs !! (start + i) = Some x. Proof. revert start i. induction xs as [|x' xs IH]; intros start i; simpl. { by rewrite lookup_empty, lookup_nil. } destruct i as [|i]; simpl. { by rewrite Nat.add_0_r, lookup_insert. } rewrite lookup_insert_ne by lia. by rewrite (IH (S start)), Nat.add_succ_r. Qed. Lemma lookup_map_seq_Some start xs i x : map_seq (M:=M A) start xs !! i = Some x ↔ start ≤ i ∧ xs !! (i - start) = Some x. Proof. destruct (decide (start ≤ i)) as [|Hsi]. { rewrite (lookup_map_seq_Some_inv start). replace (start + (i - start)) with i by lia. naive_solver. } split; [|naive_solver]. intros Hi; destruct Hsi. revert start i Hi. induction xs as [|x' xs IH]; intros start i; simpl. { by rewrite lookup_empty. } rewrite lookup_insert_Some; intros [[-> ->]|[? ?%IH]]; lia. Qed. Lemma lookup_map_seq_None start xs i : map_seq (M:=M A) start xs !! i = None ↔ i < start ∨ start + length xs ≤ i. Proof. trans (¬start ≤ i ∨ ¬is_Some (xs !! (i - start))). - rewrite eq_None_not_Some, <-not_and_l. unfold is_Some. setoid_rewrite lookup_map_seq_Some. naive_solver. - rewrite lookup_lt_is_Some. lia. Qed. Lemma lookup_map_seq_0 xs i : map_seq (M:=M A) 0 xs !! i = xs !! i. Proof. apply option_eq; intros x. by rewrite (lookup_map_seq_Some_inv 0). Qed. Lemma map_seq_singleton start x : map_seq (M:=M A) start [x] = {[ start := x ]}. Proof. done. Qed. Lemma map_seq_app_disjoint start xs1 xs2 : map_seq (M:=M A) start xs1 ##ₘ map_seq (start + length xs1) xs2. Proof. apply map_disjoint_spec; intros i x1 x2. rewrite !lookup_map_seq_Some. intros [??%lookup_lt_Some] [??%lookup_lt_Some]; lia. Qed. Lemma map_seq_app start xs1 xs2 : map_seq start (xs1 ++ xs2) =@{M A} map_seq start xs1 ∪ map_seq (start + length xs1) xs2. Proof. revert start. induction xs1 as [|x1 xs1 IH]; intros start; simpl. - by rewrite (left_id_L _ _), Nat.add_0_r. - by rewrite IH, Nat.add_succ_r, !insert_union_singleton_l, (assoc_L _). Qed. Lemma map_seq_cons_disjoint start xs x : map_seq (M:=M A) (S start) xs !! start = None. Proof. rewrite lookup_map_seq_None. lia. Qed. Lemma map_seq_cons start xs x : map_seq start (x :: xs) =@{M A} <[start:=x]> (map_seq (S start) xs). Proof. done. Qed. Lemma map_seq_snoc_disjoint start xs x : map_seq (M:=M A) start xs !! (start+length xs) = None. Proof. rewrite lookup_map_seq_None. lia. Qed. Lemma map_seq_snoc start xs x : map_seq start (xs ++ [x]) =@{M A} <[start+length xs:=x]> (map_seq start xs). Proof. rewrite map_seq_app, map_seq_singleton. by rewrite insert_union_singleton_r by (by rewrite map_seq_snoc_disjoint). Qed. End map_seq. (** * Tactics *) (** The tactic [decompose_map_disjoint] simplifies occurrences of [disjoint] in the hypotheses that involve the empty map [∅], the union [(∪)] or insert ... ...
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