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Thomas Baruchel 2023-10-21 15:17:49 +02:00
parent d018b6d30a
commit 33fde1883f
1 changed files with 4 additions and 10 deletions
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14
src/permutations.v
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@ -152,10 +152,9 @@ Proof.
Qed.
Lemma perm_max_insert : forall (l1 l2: list nat),
permutation (l1 ++ l2) -> permutation (l1 ++ [ length (l1++l2) ] ++ l2).
permutation (l1 ++ l2) <-> permutation (l1 ++ [ length (l1++l2) ] ++ l2).
Proof.
intros l1 l2. intro H.
(* first part of the proof *)
intros l1 l2. split; intro H.
unfold permutation. intro k. intro I.
rewrite app_length in I. simpl in I. rewrite Nat.add_succ_r in I.
rewrite Nat.lt_succ_r in I. rewrite <- app_length in I.
@ -163,12 +162,6 @@ Proof.
apply in_app_iff in H0. destruct H0. rewrite in_app_iff. left.
assumption. rewrite in_app_iff. right. apply in_cons. assumption.
rewrite <- H0. rewrite in_app_iff. right. apply in_eq.
Qed.
Lemma perm_max_insert_alt : forall (l1 l2: list nat),
permutation (l1 ++ [ length (l1++l2) ] ++ l2) -> permutation (l1 ++ l2).
Proof.
intros l1 l2. intro H.
assert (I: incl (seq 0 (length (l1++l2))) (l1++l2)).
assert (incl ([length (l1++l2)]++l2++l1) (l1++[length (l1++l2)]++l2)).
@ -193,7 +186,8 @@ Proof.
easy. apply Add_app.
unfold incl in I. unfold permutation.
intro k. intro. apply I. apply in_seq. lia. (* TODO *)
intro k. intro. apply I. apply in_seq.
split. apply Nat.le_0_l. assumption.
Qed.