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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. Qed.
Lemma subsequence_alt_defs {X: Type} : Lemma Subsequence_alt_defs {X: Type} :
forall l s : list X, forall l s : list X,
(exists (l1: list X) (l2 : list (list X)), (exists (l1: list X) (l2 : list (list X)),
length s = length l2 length s = length l2
@ -109,7 +109,7 @@ Proof.
Qed. Qed.
Lemma subsequence_alt_defs2 {X: Type} : Lemma Subsequence_alt_defs2 {X: Type} :
forall l s : list X, forall l s : list X,
(exists (t: list bool), (exists (t: list bool),
length t = length l /\ s = map snd (filter fst (combine t l))) length t = length l /\ s = map snd (filter fst (combine t l)))
@ -134,7 +134,7 @@ Proof.
Qed. Qed.
Theorem subsequence_flat_map {X: Type} : Theorem Subsequence_flat_map {X: Type} :
forall l s : list X, Subsequence l s forall l s : list X, Subsequence l s
<-> exists (l1: list X) (l2 : list (list X)), <-> exists (l1: list X) (l2 : list (list X)),
length s = length l2 length s = length l2
@ -145,19 +145,19 @@ Proof.
reflexivity. destruct H. destruct H. destruct H. apply IHs in H0. reflexivity. destruct H. destruct H. destruct H. apply IHs in H0.
destruct H0. destruct H0. destruct H0. exists x. exists (x1::x2). destruct H0. destruct H0. destruct H0. exists x. exists (x1::x2).
split; simpl; [rewrite H0 | rewrite <- H1]; easy. 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. assumption.
Qed. Qed.
Theorem subsequence_bools {X: Type} : Theorem Subsequence_bools {X: Type} :
forall l s : list X, Subsequence l s forall l s : list X, Subsequence l s
<-> (exists (t: list bool), <-> (exists (t: list bool),
length t = length l /\ s = map snd (filter fst (combine t l))). length t = length l /\ s = map snd (filter fst (combine t l))).
Proof. Proof.
intros l s. split; intro. apply subsequence_alt_defs. intros l s. split; intro. apply Subsequence_alt_defs.
apply subsequence_flat_map. assumption. apply Subsequence_flat_map. assumption.
apply subsequence_alt_defs2. assumption. apply Subsequence_alt_defs2. assumption.
Qed. Qed.
@ -172,19 +172,19 @@ Proof.
apply IHl. destruct H0. apply IHl. destruct H0.
rewrite <- Subsequence_cons_eq with (a := a) in s0. rewrite <- Subsequence_cons_eq with (a := a) in s0.
apply Subsequence_cons_r in s0. rewrite subsequence_bools in s0. apply Subsequence_cons_r in s0. rewrite Subsequence_bools in s0.
left. rewrite subsequence_bools. assumption. left. rewrite Subsequence_bools. assumption.
destruct s. left. apply Subsequence_nil_r. destruct s. left. apply Subsequence_nil_r.
assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e. assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e.
destruct IHl with (s := s); [ left | right ]; destruct IHl with (s := s); [ left | right ];
rewrite Subsequence_cons_eq. assumption. assumption. 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. destruct I. destruct H0. destruct x0.
symmetry in H1. apply nil_cons in H1. contradiction H1. symmetry in H1. apply nil_cons in H1. contradiction H1.
destruct b. inversion H1. rewrite H3 in n0. easy. 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. Qed.