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Thomas Baruchel 2023-12-05 08:38:00 +01:00
parent 3a679921ae
commit e95c1f9a3c

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@ -83,7 +83,65 @@ Proof.
Qed. Qed.
Lemma permutation_mapping_app {X: Type} :
forall (u1 u2 v1 v2 base : list X),
permutation_mapping (u1 ++ u2) (v1 ++ v2) base
-> length u1 = length v1 ->
permutation_mapping u1 v1 base /\
permutation_mapping u2 v2 base.
Proof.
intros u1 u2 v1 v2 base. intros H I.
destruct H as [p]. destruct H as [l]. destruct H. destruct H0.
rewrite map_app in H0. rewrite map_app in H1.
assert (K: length u1 <= length (map (nth_error base) l)).
rewrite <- H0. rewrite app_length. rewrite map_length. apply Nat.le_add_r.
rewrite map_length in K.
rewrite <- firstn_skipn with (n := length u1) in H0.
rewrite <- firstn_skipn with (n := length u1) in H1.
assert (forall (a b c d: list (option X)),
a ++ b = c ++ d -> length a = length c -> a=c /\ b = d).
intro a. induction a; intros b c d; intros J1 J2. symmetry in J2.
rewrite length_zero_iff_nil in J2. rewrite J2. split. reflexivity.
rewrite J2 in J1. assumption.
destruct c. inversion J2.
split; inversion J1; apply IHa in H4; destruct H4.
rewrite H2. reflexivity. inversion J2. reflexivity.
assumption. inversion J2. reflexivity.
apply H2 in H0. apply H2 in H1.
destruct H0. destruct H1.
split; exists p.
- exists (firstn (length u1) l). split. assumption.
rewrite <- firstn_map. rewrite <- firstn_map.
split; assumption.
- exists (skipn (length u1) l). split. assumption.
rewrite <- skipn_map. rewrite <- skipn_map.
split; assumption.
- rewrite map_length. rewrite firstn_length_le. rewrite I. reflexivity.
rewrite map_length. assumption.
- rewrite map_length. rewrite firstn_length_le. rewrite I. reflexivity.
rewrite map_length. assumption.
Qed.
Lemma permutation_mapping_base_cons {X: Type} :
forall (u v base : list X) a,
permutation_mapping u v base -> permutation_mapping u v (a::base).
Proof.
intros u v base a. intro H. destruct H as [p]. destruct H as [l].
destruct H. destruct H0. exists (a::p). exists (map S l).
split. apply perm_skip. assumption.
assert (forall x (y: list X),
map (nth_error (a :: y)) (map S x) = map (nth_error y) x).
intro x. induction x; intro y. reflexivity. simpl. rewrite IHx.
reflexivity.
split; rewrite H2; assumption.
Qed.