Commit 668baa2b authored by Robbert Krebbers's avatar Robbert Krebbers

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 ## 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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