Commit 83746e0a authored by Michael Sammler's avatar Michael Sammler

some list related lemmas

parent 4ff965b2
......@@ -565,6 +565,9 @@ Lemma list_insert_id l i x : l !! i = Some x → <[i:=x]>l = l.
Proof. revert i. induction l; intros [|i] [=]; f_equal/=; auto. Qed.
Lemma list_insert_ge l i x : length l i <[i:=x]>l = l.
Proof. revert i. induction l; intros [|i] ?; f_equal/=; auto with lia. Qed.
Lemma list_insert_insert l i x y :
<[i:=x]> (<[i:=y]> l) = <[i:=x]> l.
Proof. revert i. induction l; intros [|i]; f_equal/=; auto. Qed.
Lemma list_lookup_other l i x :
length l 1 l !! i = Some x j y, j i l !! j = Some y.
......@@ -955,6 +958,8 @@ Proof.
Qed.
Lemma take_nil n : take n [] =@{list A} [].
Proof. by destruct n. Qed.
Lemma take_S_r l n x : l !! n = Some x take (S n) l = take n l ++ [x].
Proof. revert n. induction l; intros []; naive_solver eauto with f_equal. Qed.
Lemma take_app l k : take (length l) (l ++ k) = l.
Proof. induction l; f_equal/=; auto. Qed.
Lemma take_app_alt l k n : n = length l take n (l ++ k) = l.
......@@ -1012,6 +1017,9 @@ Lemma drop_0 l : drop 0 l = l.
Proof. done. Qed.
Lemma drop_nil n : drop n [] =@{list A} [].
Proof. by destruct n. Qed.
Lemma drop_S l x n :
l !! n = Some x drop n l = x :: drop (S n) l.
Proof. revert n. induction l; intros []; naive_solver. Qed.
Lemma drop_length l n : length (drop n l) = length l - n.
Proof. revert n. by induction l; intros [|i]; f_equal/=. Qed.
Lemma drop_ge l n : length l n drop n l = [].
......@@ -1110,6 +1118,9 @@ Proof. done. Qed.
Lemma replicate_plus n m x :
replicate (n + m) x = replicate n x ++ replicate m x.
Proof. induction n; f_equal/=; auto. Qed.
Lemma replicate_cons_app n x :
x :: replicate n x = replicate n x ++ [x].
Proof. induction n; f_equal/=; eauto. Qed.
Lemma take_replicate n m x : take n (replicate m x) = replicate (min n m) x.
Proof. revert m. by induction n; intros [|?]; f_equal/=. Qed.
Lemma take_replicate_plus n m x : take n (replicate (n + m) x) = replicate n x.
......@@ -3606,6 +3617,35 @@ Section zip_with.
Forall (λ x, y, P y Q (f x y)) l Forall P k
Forall Q (zip_with f l k).
Proof. intros Hl. revert k. induction Hl; destruct 1; simpl in *; auto. Qed.
Lemma elem_of_lookup_zip_with_1 l k (z : C) :
z zip_with f l k i x y, z = f x y l !! i = Some x k !! i = Some y.
Proof.
intros [i Hin]%elem_of_list_lookup. rewrite lookup_zip_with in Hin.
simplify_option_eq; naive_solver.
Qed.
Lemma elem_of_lookup_zip_with_2 l k x y (z : C) i :
l !! i = Some x k !! i = Some y f x y zip_with f l k.
Proof.
intros Hl Hk. rewrite elem_of_list_lookup.
exists i. by rewrite lookup_zip_with, Hl, Hk.
Qed.
Lemma elem_of_lookup_zip_with l k (z : C) :
z zip_with f l k i x y, z = f x y l !! i = Some x k !! i = Some y.
Proof.
naive_solver eauto using
elem_of_lookup_zip_with_1, elem_of_lookup_zip_with_2.
Qed.
Lemma elem_of_zip_with l k (z : C) :
z zip_with f l k x y, z = f x y x l y k.
Proof.
intros ?%elem_of_lookup_zip_with.
naive_solver eauto using elem_of_list_lookup_2.
Qed.
End zip_with.
Lemma zip_with_sublist_alter {A B} (f : A B A) g l k i n l' k' :
......@@ -3644,6 +3684,14 @@ Section zip.
rewrite <-Forall2_same_length. intros Hlk1 Hlk2. revert l1 k1 Hlk1.
induction Hlk2; intros ?? [|??????]; simpl; auto.
Qed.
Lemma elem_of_zip_l x1 x2 l k :
(x1, x2) zip l k x1 l.
Proof. intros ?%elem_of_zip_with. naive_solver. Qed.
Lemma elem_of_zip_r x1 x2 l k :
(x1, x2) zip l k x2 k.
Proof. intros ?%elem_of_zip_with. naive_solver. Qed.
End zip.
Lemma elem_of_zipped_map {A B} (f : list A list A A B) l k x :
......
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