From 5c785a10a18dbaa2cad808174a4cf7919f844433 Mon Sep 17 00:00:00 2001
From: Robbert Krebbers <mail@robbertkrebbers.nl>
Date: Thu, 15 Aug 2013 10:06:59 +0200
Subject: [PATCH] Prove better correspondence between Permutation and contains.
---
theories/finite.v | 2 +-
theories/list.v | 36 +++++++++++++++++++++---------------
2 files changed, 22 insertions(+), 16 deletions(-)
diff --git a/theories/finite.v b/theories/finite.v
index 6c3a5f55..62a1d6c7 100644
--- a/theories/finite.v
+++ b/theories/finite.v
@@ -75,7 +75,7 @@ Qed.
Lemma finite_injective_Permutation `{Finite A} `{Finite B} (f : A → B)
`{!Injective (=) (=) f} : card A = card B → f <$> enum A ≡ₚ enum B.
Proof.
- intros. apply contains_Permutation.
+ intros. apply contains_Permutation_length_eq.
* by rewrite fmap_length.
* by apply finite_injective_contains.
Qed.
diff --git a/theories/list.v b/theories/list.v
index 8c243763..2628ae45 100644
--- a/theories/list.v
+++ b/theories/list.v
@@ -1693,22 +1693,26 @@ Lemma Permutation_contains l1 l2 : l1 ≡ₚ l2 → l1 `contains` l2.
Proof. induction 1; econstructor; eauto. Qed.
Lemma sublist_contains l1 l2 : l1 `sublist` l2 → l1 `contains` l2.
Proof. induction 1; constructor; auto. Qed.
-Lemma contains_Permutation_alt l1 l2 :
+Lemma contains_Permutation l1 l2 : l1 `contains` l2 → ∃ k, l2 ≡ₚ l1 ++ k.
+Proof.
+ induction 1 as
+ [|x y l ? [k Hk]| |x l1 l2 ? [k Hk]|l1 l2 l3 ? [k Hk] ? [k' Hk']].
+ * by eexists [].
+ * exists k. by rewrite Hk.
+ * eexists []. rewrite (right_id_L [] (++)). by constructor.
+ * exists (x :: k). by rewrite Hk, Permutation_middle.
+ * exists (k ++ k'). by rewrite Hk', Hk, (associative_L (++)).
+Qed.
+Lemma contains_Permutation_length_le l1 l2 :
length l2 ≤ length l1 → l1 `contains` l2 → l1 ≡ₚ l2.
Proof.
- intros Hl21 Hl12. revert Hl21. elim Hl12; clear l1 l2 Hl12; simpl.
- * constructor.
- * constructor; auto with lia.
- * constructor; auto with lia.
- * intros x l1 l2 ? IH ?. feed specialize IH; [lia|].
- apply Permutation_length in IH. lia.
- * intros l1 l2 l3 Hl12 ? Hl23 ?.
- apply contains_length in Hl12. apply contains_length in Hl23.
- transitivity l2; auto with lia.
-Qed.
-Lemma contains_Permutation l1 l2 :
+ intros Hl21 Hl12. destruct (contains_Permutation l1 l2) as [[|??] Hk]; auto.
+ * by rewrite Hk, (right_id_L [] (++)).
+ * rewrite Hk, app_length in Hl21; simpl in Hl21; lia.
+Qed.
+Lemma contains_Permutation_length_eq l1 l2 :
length l2 = length l1 → l1 `contains` l2 → l1 ≡ₚ l2.
-Proof. intro. apply contains_Permutation_alt. lia. Qed.
+Proof. intro. apply contains_Permutation_length_le. lia. Qed.
Global Instance: Proper ((≡ₚ) ==> (≡ₚ) ==> iff) (@contains A).
Proof.
@@ -1719,7 +1723,7 @@ Proof.
transitivity k2. done. by apply Permutation_contains.
Qed.
Global Instance: AntiSymmetric (≡ₚ) (@contains A).
-Proof. red. auto using contains_Permutation_alt, contains_length. Qed.
+Proof. red. auto using contains_Permutation_length_le, contains_length. Qed.
Lemma contains_take l i : take i l `contains` l.
Proof. auto using sublist_take, sublist_contains. Qed.
@@ -1839,7 +1843,9 @@ Proof. by apply contains_inserts_l, contains_inserts_r. Qed.
Lemma Permutation_alt l1 l2 :
l1 ≡ₚ l2 ↔ length l1 = length l2 ∧ l1 `contains` l2.
Proof.
- split. by intros Hl; rewrite Hl. intros [??]; auto using contains_Permutation.
+ split.
+ * by intros Hl; rewrite Hl.
+ * intros [??]; auto using contains_Permutation_length_eq.
Qed.
Lemma NoDup_contains l k : NoDup l → (∀ x, x ∈ l → x ∈ k) → l `contains` k.
--
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