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Diffstat (limited to 'theories/Sorting/Permutation.v')
| -rw-r--r-- | theories/Sorting/Permutation.v | 26 |
1 files changed, 26 insertions, 0 deletions
diff --git a/theories/Sorting/Permutation.v b/theories/Sorting/Permutation.v index 45fb48ad5d..2bf54baef3 100644 --- a/theories/Sorting/Permutation.v +++ b/theories/Sorting/Permutation.v @@ -535,6 +535,32 @@ Proof. now apply Permutation_cons_inv with x. Qed. +Hypothesis eq_dec : forall x y : A, {x = y}+{x <> y}. + +Lemma Permutation_count_occ l1 l2 : + Permutation l1 l2 <-> forall x, count_occ eq_dec l1 x = count_occ eq_dec l2 x. +Proof. + split. + - induction 1 as [ | y l1 l2 HP IHP | y z l | l1 l2 l3 HP1 IHP1 HP2 IHP2 ]; + cbn; intros a; auto. + + now rewrite IHP. + + destruct (eq_dec y a); destruct (eq_dec z a); auto. + + now rewrite IHP1, IHP2. + - revert l2; induction l1 as [|y l1 IHl1]; cbn; intros l2 Hocc. + + replace l2 with (@nil A); auto. + symmetry; apply (count_occ_inv_nil eq_dec); intuition. + + assert (exists l2' l2'', l2 = l2' ++ y :: l2'') as [l2' [l2'' ->]]. + { specialize (Hocc y). + destruct (eq_dec y y); intuition. + apply in_split, (count_occ_In eq_dec). + rewrite <- Hocc; apply Nat.lt_0_succ. } + apply Permutation_cons_app, IHl1. + intros z; specialize (Hocc z); destruct (eq_dec y z) as [Heq | Hneq]. + * rewrite (count_occ_elt_eq _ _ _ Heq) in Hocc. + now injection Hocc. + * now rewrite (count_occ_elt_neq _ _ _ Hneq) in Hocc. + Qed. + End Permutation_properties. Section Permutation_map. |