Commit 4ea9c34a authored by Robbert Krebbers's avatar Robbert Krebbers

Misc lemmas for lists.

parent 9d1092a0
......@@ -986,6 +986,19 @@ Proof.
f_equal/=; auto using take_drop with lia.
Qed.
Lemma app_eq_inv l1 l2 k1 k2 :
l1 ++ l2 = k1 ++ k2 →
(∃ k, l1 = k1 ++ k ∧ k2 = k ++ l2) ∨ (∃ k, k1 = l1 ++ k ∧ l2 = k ++ k2).
Proof.
intros Hlk. destruct (decide (length l1 < length k1)).
- right. rewrite <-(take_drop (length l1) k1), <-(assoc_L _) in Hlk.
apply app_inj_1 in Hlk as [Hl1 Hl2]; [|rewrite take_length; lia].
exists (drop (length l1) k1). by rewrite Hl1 at 1; rewrite take_drop.
- left. rewrite <-(take_drop (length k1) l1), <-(assoc_L _) in Hlk.
apply app_inj_1 in Hlk as [Hk1 Hk2]; [|rewrite take_length; lia].
exists (drop (length k1) l1). by rewrite <-Hk1 at 1; rewrite take_drop.
Qed.
(** ** Properties of the [replicate] function *)
Lemma replicate_length n x : length (replicate n x) = n.
Proof. induction n; simpl; auto. Qed.
......@@ -1421,6 +1434,17 @@ Qed.
Lemma elem_of_Permutation l x : x ∈ l → ∃ k, l ≡ₚ x :: k.
Proof. intros [i ?]%elem_of_list_lookup. eauto using delete_Permutation. Qed.
Lemma Permutation_cons_inv l k x :
k ≡ₚ x :: l → ∃ k1 k2, k = k1 ++ x :: k2 ∧ l ≡ₚ k1 ++ k2.
Proof.
intros Hk. assert (∃ i, k !! i = Some x) as [i Hi].
{ apply elem_of_list_lookup. rewrite Hk, elem_of_cons; auto. }
exists (take i k), (drop (S i) k). split.
- by rewrite take_drop_middle.
- rewrite <-delete_take_drop. apply (inj (x ::)).
by rewrite <-Hk, <-(delete_Permutation k) by done.
Qed.
Lemma length_delete l i :
is_Some (l !! i) → length (delete i l) = length l - 1.
Proof.
......@@ -1993,7 +2017,7 @@ Qed.
Lemma submseteq_app_inv_l l1 l2 k : k ++ l1 ⊆+ k ++ l2 → l1 ⊆+ l2.
Proof.
induction k as [|y k IH]; simpl; [done |]. rewrite submseteq_cons_l.
intros (?&E&?). apply Permutation_cons_inv in E. apply IH. by rewrite E.
intros (?&E%(inj _)&?). apply IH. by rewrite E.
Qed.
Lemma submseteq_app_inv_r l1 l2 k : l1 ++ k ⊆+ l2 ++ k → l1 ⊆+ l2.
Proof.
......
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