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This commit is contained in:
Thomas Baruchel 2023-12-06 08:14:01 +01:00
parent 8e9e85bd71
commit 5db7822d3c
1 changed files with 12 additions and 12 deletions
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24
src/subsequences2.v
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@ -62,7 +62,7 @@ Proof.
Qed.
Lemma subsequence_alt_defs {X: Type} :
Lemma Subsequence_alt_defs {X: Type} :
forall l s : list X,
(exists (l1: list X) (l2 : list (list X)),
length s = length l2
@ -109,7 +109,7 @@ Proof.
Qed.
Lemma subsequence_alt_defs2 {X: Type} :
Lemma Subsequence_alt_defs2 {X: Type} :
forall l s : list X,
(exists (t: list bool),
length t = length l /\ s = map snd (filter fst (combine t l)))
@ -134,7 +134,7 @@ Proof.
Qed.
Theorem subsequence_flat_map {X: Type} :
Theorem Subsequence_flat_map {X: Type} :
forall l s : list X, Subsequence l s
<-> exists (l1: list X) (l2 : list (list X)),
length s = length l2
@ -145,19 +145,19 @@ Proof.
reflexivity. destruct H. destruct H. destruct H. apply IHs in H0.
destruct H0. destruct H0. destruct H0. exists x. exists (x1::x2).
split; simpl; [rewrite H0 | rewrite <- H1]; easy.
intro. apply subsequence_alt_defs in H. apply subsequence_alt_defs2.
intro. apply Subsequence_alt_defs in H. apply Subsequence_alt_defs2.
assumption.
Qed.
Theorem subsequence_bools {X: Type} :
Theorem Subsequence_bools {X: Type} :
forall l s : list X, Subsequence l s
<-> (exists (t: list bool),
length t = length l /\ s = map snd (filter fst (combine t l))).
Proof.
intros l s. split; intro. apply subsequence_alt_defs.
apply subsequence_flat_map. assumption.
apply subsequence_alt_defs2. assumption.
intros l s. split; intro. apply Subsequence_alt_defs.
apply Subsequence_flat_map. assumption.
apply Subsequence_alt_defs2. assumption.
Qed.
@ -172,19 +172,19 @@ Proof.
apply IHl. destruct H0.
rewrite <- Subsequence_cons_eq with (a := a) in s0.
apply Subsequence_cons_r in s0. rewrite subsequence_bools in s0.
left. rewrite subsequence_bools. assumption.
apply Subsequence_cons_r in s0. rewrite Subsequence_bools in s0.
left. rewrite Subsequence_bools. assumption.
destruct s. left. apply Subsequence_nil_r.
assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e.
destruct IHl with (s := s); [ left | right ];
rewrite Subsequence_cons_eq. assumption. assumption.
right. intro I. rewrite subsequence_bools in I.
right. intro I. rewrite Subsequence_bools in I.
destruct I. destruct H0. destruct x0.
symmetry in H1. apply nil_cons in H1. contradiction H1.
destruct b. inversion H1. rewrite H3 in n0. easy.
apply n. rewrite subsequence_bools. exists x0; split; inversion H0; easy.
apply n. rewrite Subsequence_bools. exists x0; split; inversion H0; easy.
Qed.