2023-10-25 07:22:47 -04:00
|
|
|
Require Import Nat.
|
|
|
|
Require Import PeanoNat.
|
|
|
|
Require Import List.
|
|
|
|
Import ListNotations.
|
|
|
|
|
|
|
|
|
2023-10-30 11:40:11 -04:00
|
|
|
Definition subsequence {X: Type} (l s : list X) :=
|
|
|
|
exists (l1: list X) (l2 : list (list X)),
|
2023-10-25 07:22:47 -04:00
|
|
|
length s = length l2
|
|
|
|
/\ l = l1 ++ flat_map (fun e => (fst e) :: (snd e)) (combine s l2).
|
|
|
|
|
2023-10-30 11:40:11 -04:00
|
|
|
Definition subsequence2 {X: Type} (l s : list X) :=
|
2023-10-25 07:22:47 -04:00
|
|
|
exists (t: list bool),
|
|
|
|
length t = length l /\ s = map snd (filter fst (combine t l)).
|
|
|
|
|
2023-10-30 12:43:24 -04:00
|
|
|
Fixpoint subsequence3 {X: Type} (l s : list X) : Prop :=
|
|
|
|
match s with
|
|
|
|
| nil => True
|
|
|
|
| hd::tl => exists l1 l2, l = l1 ++ (hd::l2) /\ subsequence3 l2 tl
|
|
|
|
end.
|
|
|
|
|
2023-10-29 16:10:19 -04:00
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence_nil_r {X: Type}: forall (l : list X), subsequence l nil.
|
2023-10-29 16:10:19 -04:00
|
|
|
Proof.
|
2023-10-30 03:18:20 -04:00
|
|
|
intro l. exists l. exists nil. rewrite app_nil_r. split; easy.
|
2023-10-29 16:10:19 -04:00
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 14:16:04 -04:00
|
|
|
Theorem subsequence2_nil_r {X: Type} : forall (l : list X), subsequence2 l nil.
|
|
|
|
Proof.
|
|
|
|
intro l.
|
|
|
|
exists (repeat false (length l)). rewrite repeat_length.
|
|
|
|
split. easy.
|
|
|
|
induction l. reflexivity. simpl. assumption.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
|
|
Theorem subsequence3_nil_r {X: Type} : forall (l : list X), subsequence3 l nil.
|
|
|
|
Proof.
|
|
|
|
intro l. reflexivity.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence_nil_cons_r {X: Type}: forall (l: list X) (a:X),
|
2023-10-29 16:10:19 -04:00
|
|
|
~ subsequence nil (a::l).
|
|
|
|
Proof.
|
2023-10-31 04:19:23 -04:00
|
|
|
intros l a. intro H.
|
2023-10-29 16:10:19 -04:00
|
|
|
destruct H. destruct H. destruct H.
|
|
|
|
destruct x. rewrite app_nil_l in H0.
|
|
|
|
destruct x0. apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
|
|
|
|
simpl in H0. apply nil_cons in H0. contradiction H0.
|
|
|
|
apply nil_cons in H0. contradiction H0.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence2_nil_cons_r {X: Type}: forall (l: list X) (a:X),
|
2023-10-29 16:10:19 -04:00
|
|
|
~ subsequence2 nil (a::l).
|
|
|
|
Proof.
|
2023-10-31 04:19:23 -04:00
|
|
|
intros l a. intro H. destruct H.
|
2023-10-29 16:10:19 -04:00
|
|
|
destruct H. assert (x = nil). destruct x. reflexivity.
|
|
|
|
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
|
|
|
|
rewrite H1 in H0. symmetry in H0. apply nil_cons in H0. contradiction H0.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:43:24 -04:00
|
|
|
Theorem subsequence3_nil_cons_r {X: Type}: forall (l: list X) (a:X),
|
|
|
|
~ subsequence3 nil (a::l).
|
|
|
|
Proof.
|
2023-10-31 04:19:23 -04:00
|
|
|
intros l a. intro H. destruct H. destruct H. destruct H.
|
2023-10-30 12:43:24 -04:00
|
|
|
destruct x. simpl in H. apply nil_cons in H. contradiction H.
|
|
|
|
simpl in H. apply nil_cons in H. contradiction H.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence_cons_l {X: Type}: forall (l s: list X) (a: X),
|
2023-10-30 05:57:47 -04:00
|
|
|
subsequence l s -> subsequence (a::l) s.
|
|
|
|
Proof.
|
|
|
|
intros l s a. intro H.
|
|
|
|
destruct H. destruct H. destruct H.
|
|
|
|
exists (a::x). exists x0. split. assumption.
|
|
|
|
rewrite H0. reflexivity.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence2_cons_l {X: Type} : forall (l s: list X) (a: X),
|
2023-10-30 01:50:55 -04:00
|
|
|
subsequence2 l s -> subsequence2 (a::l) s.
|
|
|
|
Proof.
|
2023-10-30 03:18:20 -04:00
|
|
|
intros l s a. intro H.
|
2023-10-30 01:50:55 -04:00
|
|
|
destruct H. destruct H. exists (false::x).
|
|
|
|
split. simpl. apply eq_S. assumption.
|
|
|
|
simpl. assumption.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:43:24 -04:00
|
|
|
Theorem subsequence3_cons_l {X: Type} : forall (l s: list X) (a: X),
|
|
|
|
subsequence3 l s -> subsequence3 (a::l) s.
|
|
|
|
Proof.
|
|
|
|
intros l s a. intro H.
|
|
|
|
destruct s. apply subsequence3_nil_r.
|
|
|
|
destruct H. destruct H. destruct H.
|
|
|
|
exists (a::x0). exists x1. rewrite H.
|
|
|
|
split. reflexivity. assumption.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence_cons_r {X: Type} : forall (l s: list X) (a: X),
|
2023-10-30 05:57:47 -04:00
|
|
|
subsequence l (a::s) -> subsequence l s.
|
|
|
|
Proof.
|
|
|
|
intros l s a. intro H. destruct H. destruct H. destruct H.
|
|
|
|
destruct x0. apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
|
|
|
|
exists (x++a::l0). exists x0. split. inversion H. reflexivity. rewrite H0.
|
|
|
|
rewrite <- app_assoc. apply app_inv_head_iff. simpl. reflexivity.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence2_cons_r {X: Type} : forall (l s: list X) (a: X),
|
2023-10-30 01:50:55 -04:00
|
|
|
subsequence2 l (a::s) -> subsequence2 l s.
|
|
|
|
Proof.
|
|
|
|
intro l. induction l. intros. apply subsequence2_nil_cons_r in H.
|
2023-10-30 03:18:20 -04:00
|
|
|
contradiction H. intros s a0 . intro H. destruct H. destruct H. destruct x.
|
2023-10-30 01:50:55 -04:00
|
|
|
symmetry in H0. apply nil_cons in H0. contradiction H0.
|
|
|
|
destruct b. simpl in H0. inversion H0.
|
2023-10-30 03:18:20 -04:00
|
|
|
exists (false::x). split; try rewrite <- H; reflexivity.
|
|
|
|
simpl in H0. apply subsequence2_cons_l. apply IHl with (a := a0).
|
2023-10-31 04:19:23 -04:00
|
|
|
exists (x). split. inversion H.
|
2023-10-30 01:50:55 -04:00
|
|
|
reflexivity. assumption.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:43:24 -04:00
|
|
|
Theorem subsequence3_cons_r {X: Type} : forall (l s: list X) (a: X),
|
|
|
|
subsequence3 l (a::s) -> subsequence3 l s.
|
|
|
|
Proof.
|
|
|
|
intros l s a. intro H. simpl in H. destruct H. destruct H.
|
|
|
|
destruct s. apply subsequence3_nil_r. destruct H. simpl in H0.
|
|
|
|
destruct H0. destruct H0. destruct H0.
|
|
|
|
exists (x ++ a::x2). exists x3.
|
|
|
|
rewrite H. rewrite H0. split. rewrite <- app_assoc.
|
|
|
|
rewrite app_comm_cons. reflexivity. assumption.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence_cons_eq {X: Type} : forall (l1 l2: list X) (a: X),
|
2023-10-30 07:33:00 -04:00
|
|
|
subsequence (a::l1) (a::l2) <-> subsequence l1 l2.
|
|
|
|
Proof.
|
2023-10-31 04:07:36 -04:00
|
|
|
intros l s a. split; intro H; destruct H; destruct H; destruct H.
|
2023-10-30 07:33:00 -04:00
|
|
|
destruct x. destruct x0.
|
|
|
|
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
|
2023-10-31 04:19:23 -04:00
|
|
|
inversion H0. exists l0. exists x0.
|
2023-10-30 11:40:11 -04:00
|
|
|
inversion H. rewrite H3. split; reflexivity.
|
2023-10-31 04:07:36 -04:00
|
|
|
destruct x0.
|
2023-10-30 07:33:00 -04:00
|
|
|
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
|
2023-10-30 12:07:08 -04:00
|
|
|
exists (x1 ++ (a::l0)). exists x0. inversion H. rewrite H2. split. reflexivity.
|
2023-10-30 07:33:00 -04:00
|
|
|
inversion H0. rewrite <- app_assoc. apply app_inv_head_iff.
|
|
|
|
rewrite app_comm_cons. reflexivity.
|
|
|
|
|
2023-10-31 04:07:36 -04:00
|
|
|
exists nil. exists (x::x0). simpl.
|
|
|
|
split; [ rewrite H | rewrite H0 ]; reflexivity.
|
2023-10-30 07:33:00 -04:00
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence2_cons_eq {X: Type}: forall (l1 l2: list X) (a: X),
|
2023-10-29 16:10:19 -04:00
|
|
|
subsequence2 (a::l1) (a::l2) <-> subsequence2 l1 l2.
|
|
|
|
Proof.
|
2023-10-31 04:03:37 -04:00
|
|
|
intros l s a. split; intro H; destruct H; destruct H.
|
|
|
|
destruct x. inversion H0.
|
2023-10-31 04:07:36 -04:00
|
|
|
destruct b. exists (x). split. inversion H. rewrite H2. reflexivity.
|
2023-10-30 02:48:09 -04:00
|
|
|
inversion H0. reflexivity. simpl in H0. inversion H.
|
|
|
|
assert (subsequence2 l (a::s)). exists x. split; assumption.
|
|
|
|
apply subsequence2_cons_r with (a := a). assumption.
|
2023-10-31 04:03:37 -04:00
|
|
|
exists (true :: x).
|
2023-10-31 04:07:36 -04:00
|
|
|
split. apply eq_S. assumption. rewrite H0. reflexivity.
|
2023-10-30 02:51:59 -04:00
|
|
|
Qed.
|
2023-10-30 12:43:24 -04:00
|
|
|
|
|
|
|
|
|
|
|
Theorem subsequence3_cons_eq {X: Type} : forall (l1 l2: list X) (a: X),
|
|
|
|
subsequence3 (a::l1) (a::l2) <-> subsequence3 l1 l2.
|
|
|
|
Proof.
|
2023-10-31 04:07:36 -04:00
|
|
|
intros l s a. split; intro H.
|
2023-10-30 12:43:24 -04:00
|
|
|
destruct H. destruct H. destruct H.
|
|
|
|
destruct x. inversion H. assumption.
|
|
|
|
destruct s. apply subsequence3_nil_r.
|
|
|
|
destruct H0. destruct H0. destruct H0.
|
|
|
|
exists (x1 ++ (a::x3)). exists x4.
|
|
|
|
inversion H. rewrite H0. rewrite <- app_assoc.
|
2023-10-30 14:16:04 -04:00
|
|
|
rewrite app_comm_cons. split. reflexivity. assumption.
|
2023-10-31 04:07:36 -04:00
|
|
|
exists nil. exists l. split. reflexivity. assumption.
|
2023-10-30 12:43:24 -04:00
|
|
|
Qed.
|
2023-10-30 02:48:09 -04:00
|
|
|
|
|
|
|
|
2023-10-30 02:51:59 -04:00
|
|
|
(*
|
2023-10-29 16:10:19 -04:00
|
|
|
assert (forall (l s: list Type) t,
|
|
|
|
s = map snd (filter fst (combine t l))
|
|
|
|
-> length t = length l
|
|
|
|
-> count_occ Bool.bool_dec t true = length s).
|
|
|
|
intros l s t.
|
|
|
|
|
|
|
|
generalize l s. induction t. intros l0 s0. intros J K.
|
|
|
|
simpl in J. rewrite J. reflexivity.
|
|
|
|
intros l0 s0. intros J K.
|
|
|
|
destruct a. destruct s0. destruct l0.
|
|
|
|
apply Nat.neq_succ_0 in K. contradiction K.
|
|
|
|
apply nil_cons in J. contradiction J. simpl. apply eq_S.
|
|
|
|
|
|
|
|
destruct l0.
|
|
|
|
apply Nat.neq_succ_0 in K. contradiction K.
|
|
|
|
|
|
|
|
apply IHt with (l := l0). inversion J. rewrite <- H1.
|
|
|
|
reflexivity. inversion K. reflexivity.
|
|
|
|
|
|
|
|
simpl. destruct l0.
|
|
|
|
apply Nat.neq_succ_0 in K. contradiction K.
|
|
|
|
|
|
|
|
apply IHt with (l := l0). inversion J. simpl.
|
|
|
|
reflexivity. inversion K. reflexivity.
|
2023-10-30 02:51:59 -04:00
|
|
|
*)
|
2023-10-30 01:50:55 -04:00
|
|
|
|
2023-10-29 16:10:19 -04:00
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
Theorem subsequence2_dec {X: Type}:
|
|
|
|
(forall x y : X, {x = y} + {x <> y})
|
|
|
|
-> forall (l s : list X), { subsequence2 l s } + { ~ subsequence2 l s }.
|
2023-10-29 16:10:19 -04:00
|
|
|
Proof.
|
|
|
|
intro H.
|
2023-10-31 04:19:23 -04:00
|
|
|
intro l. induction l; intro s. destruct s. left. apply subsequence2_nil_r.
|
2023-10-29 16:10:19 -04:00
|
|
|
right. apply subsequence2_nil_cons_r.
|
|
|
|
|
2023-10-31 04:19:23 -04:00
|
|
|
assert({subsequence2 l s} + {~ subsequence2 l s}). apply IHl.
|
2023-10-29 16:10:19 -04:00
|
|
|
destruct H0.
|
|
|
|
|
2023-10-30 05:30:22 -04:00
|
|
|
rewrite <- subsequence2_cons_eq with (a := a) in s0.
|
|
|
|
apply subsequence2_cons_r in s0. left. assumption.
|
2023-10-29 16:10:19 -04:00
|
|
|
|
2023-10-30 05:30:22 -04:00
|
|
|
destruct s. left. apply subsequence2_nil_r.
|
2023-10-30 12:07:08 -04:00
|
|
|
assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e.
|
2023-10-31 04:19:23 -04:00
|
|
|
destruct IHl with (s := s); [ left | right ];
|
|
|
|
rewrite subsequence2_cons_eq; assumption.
|
2023-10-30 05:30:22 -04:00
|
|
|
|
2023-10-31 04:19:23 -04:00
|
|
|
right. intro I.
|
2023-10-30 12:07:08 -04:00
|
|
|
destruct I. destruct H0. destruct x0.
|
2023-10-30 05:30:22 -04:00
|
|
|
symmetry in H1. apply nil_cons in H1. contradiction H1.
|
|
|
|
destruct b.
|
|
|
|
inversion H1. rewrite H3 in n0. contradiction n0. reflexivity.
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
assert (subsequence2 l (x::s)). exists x0. split.
|
2023-10-30 05:30:22 -04:00
|
|
|
inversion H0. reflexivity. assumption. apply n in H2. contradiction H2.
|
|
|
|
Qed.
|
2023-10-29 16:10:19 -04:00
|
|
|
|
|
|
|
|
2023-10-30 14:16:04 -04:00
|
|
|
Theorem subsequence3_dec {X: Type}:
|
|
|
|
(forall x y : X, {x = y} + {x <> y})
|
|
|
|
-> forall (l s : list X), { subsequence3 l s } + { ~ subsequence3 l s }.
|
|
|
|
Proof.
|
|
|
|
intro H. intro l. induction l. intro s. destruct s. left. reflexivity.
|
|
|
|
right. apply subsequence3_nil_cons_r.
|
|
|
|
|
|
|
|
intro s. assert({subsequence3 l s} + {~ subsequence3 l s}). apply IHl.
|
|
|
|
destruct H0.
|
|
|
|
|
|
|
|
rewrite <- subsequence3_cons_eq with (a := a) in s0.
|
|
|
|
apply subsequence3_cons_r in s0. left. assumption.
|
|
|
|
|
|
|
|
destruct s. left. apply subsequence3_nil_r.
|
|
|
|
assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e.
|
2023-10-31 04:19:23 -04:00
|
|
|
destruct IHl with (s := s); [ left | right ];
|
|
|
|
rewrite subsequence3_cons_eq; assumption.
|
2023-10-30 14:16:04 -04:00
|
|
|
|
2023-10-31 04:19:23 -04:00
|
|
|
right. intro I.
|
2023-10-30 14:16:04 -04:00
|
|
|
destruct I. destruct H0. destruct H0. destruct x0.
|
|
|
|
inversion H0. rewrite H3 in n0. contradiction n0. reflexivity.
|
|
|
|
|
|
|
|
assert (subsequence3 l (x::s)). exists x2. exists x1.
|
|
|
|
inversion H0. split. reflexivity. assumption.
|
|
|
|
apply n in H2. contradiction H2.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
|
|
|
Theorem subsequence_eq_def_2 {X: Type} :
|
2023-10-30 12:07:08 -04:00
|
|
|
(forall (x y : X), {x = y} + {x <> y})
|
|
|
|
-> (forall l s : list X, subsequence l s <-> subsequence2 l s).
|
2023-10-25 07:22:47 -04:00
|
|
|
Proof.
|
2023-10-30 05:57:47 -04:00
|
|
|
intro I. intro l. induction l.
|
2023-10-30 14:27:09 -04:00
|
|
|
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.
|
|
|
|
|
2023-10-30 12:07:08 -04:00
|
|
|
assert ({a=x} + {a<>x}). apply I. destruct H.
|
2023-10-30 05:57:47 -04:00
|
|
|
rewrite e. rewrite subsequence2_cons_eq. rewrite <- IHl.
|
2023-10-30 07:34:36 -04:00
|
|
|
rewrite subsequence_cons_eq. split; intro; assumption.
|
2023-10-25 07:22:47 -04:00
|
|
|
|
2023-10-30 07:57:00 -04:00
|
|
|
split; intro H. apply subsequence2_cons_l. apply IHl.
|
|
|
|
destruct H. destruct H. destruct H.
|
2023-10-30 12:07:08 -04:00
|
|
|
destruct x0. destruct x1.
|
2023-10-30 07:57:00 -04:00
|
|
|
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
|
|
|
|
simpl in H0. inversion H0. rewrite H2 in n. contradiction n.
|
|
|
|
reflexivity. inversion H0.
|
2023-10-30 12:07:08 -04:00
|
|
|
exists x2. exists x1. split. assumption.
|
2023-10-30 07:57:00 -04:00
|
|
|
reflexivity.
|
|
|
|
|
|
|
|
apply subsequence_cons_l. apply IHl.
|
|
|
|
destruct H. destruct H.
|
2023-10-30 12:07:08 -04:00
|
|
|
destruct x0. simpl in H0.
|
2023-10-30 07:57:00 -04:00
|
|
|
symmetry in H0. apply nil_cons in H0. contradiction H0.
|
|
|
|
destruct b. simpl in H0. inversion H0. rewrite H2 in n.
|
|
|
|
contradiction n. reflexivity.
|
2023-10-30 12:07:08 -04:00
|
|
|
exists x0. inversion H. inversion H0.
|
2023-10-30 07:57:00 -04:00
|
|
|
split; reflexivity.
|
|
|
|
Qed.
|
2023-10-25 07:22:47 -04:00
|
|
|
|
|
|
|
|
2023-10-31 04:03:37 -04:00
|
|
|
Theorem subsequence_dec {X: Type}:
|
|
|
|
(forall x y : X, {x = y} + {x <> y})
|
|
|
|
-> forall (l s : list X), { subsequence l s } + { ~ subsequence l s }.
|
|
|
|
Proof.
|
|
|
|
intro H. intros l s.
|
|
|
|
assert ({ subsequence2 l s } + { ~ subsequence2 l s }).
|
|
|
|
apply subsequence2_dec. assumption. destruct H0.
|
|
|
|
rewrite <- subsequence_eq_def_2 in s0. left. assumption.
|
|
|
|
assumption. rewrite <- subsequence_eq_def_2 in n. right. assumption.
|
|
|
|
assumption.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
|
2023-10-30 14:16:04 -04:00
|
|
|
Theorem subsequence_eq_def_3 {X: Type} :
|
|
|
|
(forall (x y : X), {x = y} + {x <> y})
|
|
|
|
-> (forall l s : list X, subsequence l s <-> subsequence3 l s).
|
|
|
|
Proof.
|
|
|
|
intro I. intro l. induction l.
|
2023-10-30 14:27:09 -04:00
|
|
|
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.
|
2023-10-30 14:34:21 -04:00
|
|
|
rewrite e. rewrite subsequence3_cons_eq. rewrite <- IHl.
|
2023-10-30 14:27:09 -04:00
|
|
|
rewrite subsequence_cons_eq. split; intro; assumption.
|
|
|
|
|
2023-10-30 14:34:21 -04:00
|
|
|
split; intro H. apply subsequence3_cons_l. apply IHl.
|
2023-10-30 14:27:09 -04:00
|
|
|
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.
|
2023-10-30 14:34:21 -04:00
|
|
|
destruct H. destruct H. destruct H.
|
|
|
|
|
|
|
|
destruct x0. inversion H. rewrite H2 in n. contradiction n. reflexivity.
|
|
|
|
exists x2. exists x1. split. inversion H. reflexivity. assumption.
|
|
|
|
Qed.
|
2023-10-30 14:16:04 -04:00
|
|
|
|
|
|
|
|
2023-10-25 07:22:47 -04:00
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Example test1: subsequence [1;2;3;4;5] [1;3;5].
|
|
|
|
Proof.
|
|
|
|
unfold subsequence.
|
|
|
|
exists [].
|
|
|
|
exists [[2];[4];[]]. simpl. easy.
|
|
|
|
Qed.
|
|
|
|
|
|
|
|
Example test2: subsequence [1;2;3;4;5] [2;4].
|
|
|
|
Proof.
|
|
|
|
unfold subsequence.
|
|
|
|
exists [1].
|
|
|
|
exists [[3];[5]]. simpl. easy.
|
|
|
|
Qed.
|
2023-10-30 12:07:08 -04:00
|
|
|
|
|
|
|
Example test3: subsequence [1;2;3;4;5] [1;3;5].
|
|
|
|
Proof.
|
2023-10-31 04:03:37 -04:00
|
|
|
rewrite subsequence_eq_def_2.
|
2023-10-30 12:11:08 -04:00
|
|
|
exists [true; false; true; false; true].
|
|
|
|
split; reflexivity.
|
|
|
|
apply Nat.eq_dec.
|
|
|
|
Qed.
|