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Thomas Baruchel 2023-10-30 19:27:09 +01:00
parent f0705b24ae
commit 49649d6fd0
1 changed files with 38 additions and 29 deletions
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67
src/subsequences.v
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@ -282,35 +282,14 @@ Theorem subsequence_eq_def_2 {X: Type} :
-> (forall l s : list X, subsequence l s <-> subsequence2 l s).
Proof.
intro I. intro l. induction l.
(* first part of the induction *)
intro s. unfold subsequence. unfold subsequence2.
split. exists nil. split. reflexivity. simpl.
destruct H. destruct H. destruct H.
assert (x = nil). destruct x. reflexivity. simpl in H0.
apply nil_cons in H0. contradiction H0. rewrite H1 in H0.
simpl in H0. assert (combine s x0 = nil).
destruct (combine s x0). reflexivity. simpl in H0.
apply nil_cons in H0. contradiction H0.
destruct x0. destruct s. reflexivity. simpl in H.
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
destruct s. reflexivity. simpl in H2.
symmetry in H2. apply nil_cons in H2. contradiction H2.
exists nil. exists nil. destruct s. simpl. easy.
destruct H. destruct H.
assert (x0 = nil). destruct x0. reflexivity. simpl in H.
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
rewrite H1 in H0. simpl in H0.
symmetry in H0. apply nil_cons in H0. contradiction H0.
(* second part of the induction *)
intro s. destruct s. unfold subsequence. unfold subsequence2. split.
exists (repeat false (S (length l))). rewrite repeat_length.
split. easy. simpl.
assert (forall u,
(nil: list X)
= map snd (filter fst (combine (repeat false (length u)) u))).
intro u. induction u. reflexivity. simpl. assumption. apply H0.
exists (a::l). exists (nil). simpl. split; try rewrite app_nil_r; reflexivity.
(* deux cas : a = n ou non *)
intro s. split; intro H. destruct s. apply subsequence2_nil_r.
apply subsequence_nil_cons_r in H. contradiction H.
destruct s. apply subsequence_nil_r.
apply subsequence2_nil_cons_r in H. contradiction H.
intro s. destruct s. split; intro H. apply subsequence2_nil_r.
apply subsequence_nil_r.
assert ({a=x} + {a<>x}). apply I. destruct H.
rewrite e. rewrite subsequence2_cons_eq. rewrite <- IHl.
rewrite subsequence_cons_eq. split; intro; assumption.
@ -340,6 +319,36 @@ Theorem subsequence_eq_def_3 {X: Type} :
-> (forall l s : list X, subsequence l s <-> subsequence3 l s).
Proof.
intro I. intro l. induction l.
intro s. split; intro H. destruct s. apply subsequence3_nil_r.
apply subsequence_nil_cons_r in H. contradiction H.
destruct s. apply subsequence_nil_r.
apply subsequence3_nil_cons_r in H. contradiction H.
intro s. destruct s. split; intro H. apply subsequence3_nil_r.
apply subsequence_nil_r.
assert ({a=x} + {a<>x}). apply I. destruct H.
rewrite e. rewrite subsequence2_cons_eq. rewrite <- IHl.
rewrite subsequence_cons_eq. split; intro; assumption.
split; intro H. apply subsequence2_cons_l. apply IHl.
destruct H. destruct H. destruct H.
destruct x0. destruct x1.
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
simpl in H0. inversion H0. rewrite H2 in n. contradiction n.
reflexivity. inversion H0.
exists x2. exists x1. split. assumption.
reflexivity.
apply subsequence_cons_l. apply IHl.
destruct H. destruct H.
destruct x0. simpl in H0.
symmetry in H0. apply nil_cons in H0. contradiction H0.
destruct b. simpl in H0. inversion H0. rewrite H2 in n.
contradiction n. reflexivity.
exists x0. inversion H. inversion H0.
split; reflexivity.
Admitted.