354 lines
11 KiB
Coq
354 lines
11 KiB
Coq
Require Import Nat.
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Require Import PeanoNat.
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Require Import List.
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Import ListNotations.
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Definition subsequence {X: Type} (l s : list X) :=
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exists (l1: list X) (l2 : list (list X)),
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length s = length l2
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/\ l = l1 ++ flat_map (fun e => (fst e) :: (snd e)) (combine s l2).
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Definition subsequence2 {X: Type} (l s : list X) :=
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exists (t: list bool),
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length t = length l /\ s = map snd (filter fst (combine t l)).
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Fixpoint subsequence3 {X: Type} (l s : list X) : Prop :=
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match s with
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| nil => True
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| hd::tl => exists l1 l2, l = l1 ++ (hd::l2) /\ subsequence3 l2 tl
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end.
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(*
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TODO: problème, les éléments de l ne sont pas uniques ; or
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il doit pouvoir y avoir des différences dans la décision de les
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prendre ou non
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Definition subsequence3 (l s : list Type) :=
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exists f, s = fst (partition f l).
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*)
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Theorem subsequence_nil_r {X: Type}: forall (l : list X), subsequence l nil.
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Proof.
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intro l. exists l. exists nil. rewrite app_nil_r. split; easy.
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Qed.
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Theorem subsequence_nil_cons_r {X: Type}: forall (l: list X) (a:X),
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~ subsequence nil (a::l).
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Proof.
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intros l a. unfold not. intro H.
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destruct H. destruct H. destruct H.
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destruct x. rewrite app_nil_l in H0.
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destruct x0. apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
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simpl in H0. apply nil_cons in H0. contradiction H0.
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apply nil_cons in H0. contradiction H0.
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Qed.
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Theorem subsequence2_nil_r {X: Type} : forall (l : list X), subsequence2 l nil.
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Proof.
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intro l.
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exists (repeat false (length l)). rewrite repeat_length.
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split. easy.
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induction l. reflexivity. simpl. assumption.
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Qed.
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Theorem subsequence2_nil_cons_r {X: Type}: forall (l: list X) (a:X),
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~ subsequence2 nil (a::l).
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Proof.
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intros l a. unfold not. intro H. destruct H.
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destruct H. assert (x = nil). destruct x. reflexivity.
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apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
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rewrite H1 in H0. symmetry in H0. apply nil_cons in H0. contradiction H0.
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Qed.
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Theorem subsequence3_nil_r {X: Type} : forall (l : list X), subsequence3 l nil.
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Proof.
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intro l. reflexivity.
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Qed.
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Theorem subsequence3_nil_cons_r {X: Type}: forall (l: list X) (a:X),
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~ subsequence3 nil (a::l).
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Proof.
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intros l a. unfold not. intro H. destruct H. destruct H. destruct H.
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destruct x. simpl in H. apply nil_cons in H. contradiction H.
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simpl in H. apply nil_cons in H. contradiction H.
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Qed.
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Theorem subsequence_cons_l {X: Type}: forall (l s: list X) (a: X),
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subsequence l s -> subsequence (a::l) s.
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Proof.
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intros l s a. intro H.
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destruct H. destruct H. destruct H.
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exists (a::x). exists x0. split. assumption.
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rewrite H0. reflexivity.
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Qed.
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Theorem subsequence2_cons_l {X: Type} : forall (l s: list X) (a: X),
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subsequence2 l s -> subsequence2 (a::l) s.
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Proof.
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intros l s a. intro H.
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destruct H. destruct H. exists (false::x).
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split. simpl. apply eq_S. assumption.
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simpl. assumption.
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Qed.
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Theorem subsequence3_cons_l {X: Type} : forall (l s: list X) (a: X),
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subsequence3 l s -> subsequence3 (a::l) s.
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Proof.
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intros l s a. intro H.
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destruct s. apply subsequence3_nil_r.
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destruct H. destruct H. destruct H.
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exists (a::x0). exists x1. rewrite H.
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split. reflexivity. assumption.
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Qed.
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Theorem subsequence_cons_r {X: Type} : forall (l s: list X) (a: X),
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subsequence l (a::s) -> subsequence l s.
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Proof.
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intros l s a. intro H. destruct H. destruct H. destruct H.
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destruct x0. apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
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exists (x++a::l0). exists x0. split. inversion H. reflexivity. rewrite H0.
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rewrite <- app_assoc. apply app_inv_head_iff. simpl. reflexivity.
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Qed.
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Theorem subsequence2_cons_r {X: Type} : forall (l s: list X) (a: X),
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subsequence2 l (a::s) -> subsequence2 l s.
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Proof.
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intro l. induction l. intros. apply subsequence2_nil_cons_r in H.
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contradiction H. intros s a0 . intro H. destruct H. destruct H. destruct x.
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symmetry in H0. apply nil_cons in H0. contradiction H0.
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destruct b. simpl in H0. inversion H0.
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exists (false::x). split; try rewrite <- H; reflexivity.
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simpl in H0. apply subsequence2_cons_l. apply IHl with (a := a0).
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unfold subsequence2. exists (x). split. inversion H.
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reflexivity. assumption.
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Qed.
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Theorem subsequence3_cons_r {X: Type} : forall (l s: list X) (a: X),
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subsequence3 l (a::s) -> subsequence3 l s.
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Proof.
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intros l s a. intro H. simpl in H. destruct H. destruct H.
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destruct s. apply subsequence3_nil_r. destruct H. simpl in H0.
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destruct H0. destruct H0. destruct H0.
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exists (x ++ a::x2). exists x3.
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rewrite H. rewrite H0. split. rewrite <- app_assoc.
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rewrite app_comm_cons. reflexivity. assumption.
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Qed.
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Theorem subsequence_cons_eq {X: Type} : forall (l1 l2: list X) (a: X),
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subsequence (a::l1) (a::l2) <-> subsequence l1 l2.
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Proof.
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intros l s a. split. intro H.
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destruct H. destruct H. destruct H.
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destruct x. destruct x0.
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apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
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simpl in H0. inversion H0. exists l0. exists x0.
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inversion H. rewrite H3. split; reflexivity.
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destruct x0. simpl in H0.
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apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
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exists (x1 ++ (a::l0)). exists x0. inversion H. rewrite H2. split. reflexivity.
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inversion H0. rewrite <- app_assoc. apply app_inv_head_iff.
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rewrite app_comm_cons. reflexivity.
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intro H. destruct H. destruct H. destruct H.
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exists nil. exists (x::x0). simpl. split. rewrite H. reflexivity.
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rewrite H0. reflexivity.
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Qed.
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Theorem subsequence2_cons_eq {X: Type}: forall (l1 l2: list X) (a: X),
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subsequence2 (a::l1) (a::l2) <-> subsequence2 l1 l2.
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Proof.
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intros l s a. split. intro H. unfold subsequence2 in H.
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destruct H. destruct H. destruct x. inversion H0.
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destruct b. exists (x). split. inversion H. rewrite H2. reflexivity. simpl in H0.
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inversion H0. reflexivity. simpl in H0. inversion H.
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assert (subsequence2 l (a::s)). exists x. split; assumption.
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apply subsequence2_cons_r with (a := a). assumption.
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intro H. destruct H. destruct H. exists (true :: x).
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split. simpl. apply eq_S. assumption. simpl. rewrite H0. reflexivity.
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Qed.
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Theorem subsequence3_cons_eq {X: Type} : forall (l1 l2: list X) (a: X),
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subsequence3 (a::l1) (a::l2) <-> subsequence3 l1 l2.
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Proof.
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intros l s a. split. intro H.
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destruct H. destruct H. destruct H.
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destruct x. inversion H. assumption.
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destruct s. apply subsequence3_nil_r.
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destruct H0. destruct H0. destruct H0.
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exists (x1 ++ (a::x3)). exists x4.
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inversion H. rewrite H0. rewrite <- app_assoc.
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rewrite app_comm_cons. split. reflexivity.
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assumption.
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intro H. exists nil. exists l. split. reflexivity. assumption.
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Qed.
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(*
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assert (forall (l s: list Type) t,
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s = map snd (filter fst (combine t l))
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-> length t = length l
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-> count_occ Bool.bool_dec t true = length s).
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intros l s t.
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generalize l s. induction t. intros l0 s0. intros J K.
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simpl in J. rewrite J. reflexivity.
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intros l0 s0. intros J K.
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destruct a. destruct s0. destruct l0.
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apply Nat.neq_succ_0 in K. contradiction K.
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apply nil_cons in J. contradiction J. simpl. apply eq_S.
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destruct l0.
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apply Nat.neq_succ_0 in K. contradiction K.
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apply IHt with (l := l0). inversion J. rewrite <- H1.
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reflexivity. inversion K. reflexivity.
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simpl. destruct l0.
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apply Nat.neq_succ_0 in K. contradiction K.
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apply IHt with (l := l0). inversion J. simpl.
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reflexivity. inversion K. reflexivity.
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*)
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Theorem subsequence2_dec {X: Type}:
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(forall x y : X, {x = y} + {x <> y})
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-> forall (l s : list X), { subsequence2 l s } + { ~ subsequence2 l s }.
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Proof.
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intro H.
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intros l. induction l. intro s. destruct s. left. apply subsequence2_nil_r.
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right. apply subsequence2_nil_cons_r.
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intro s. assert({subsequence2 l s} + {~ subsequence2 l s}). apply IHl.
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destruct H0.
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rewrite <- subsequence2_cons_eq with (a := a) in s0.
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apply subsequence2_cons_r in s0. left. assumption.
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destruct s. left. apply subsequence2_nil_r.
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assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e.
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destruct IHl with (s := s).
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left. rewrite subsequence2_cons_eq. assumption.
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right. rewrite subsequence2_cons_eq. assumption.
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right. unfold not in n. unfold not. intro I.
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destruct I. destruct H0. destruct x0.
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symmetry in H1. apply nil_cons in H1. contradiction H1.
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destruct b.
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inversion H1. rewrite H3 in n0. contradiction n0. reflexivity.
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assert (subsequence2 l (x::s)). exists x0. split.
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inversion H0. reflexivity. assumption. apply n in H2. contradiction H2.
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Qed.
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Theorem subsequence_eq_def {X: Type} :
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(forall (x y : X), {x = y} + {x <> y})
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-> (forall l s : list X, subsequence l s <-> subsequence2 l s).
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Proof.
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intro I. intro l. induction l.
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(* first part of the induction *)
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intro s. unfold subsequence. unfold subsequence2.
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split. exists nil. split. reflexivity. simpl.
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destruct H. destruct H. destruct H.
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assert (x = nil). destruct x. reflexivity. simpl in H0.
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apply nil_cons in H0. contradiction H0. rewrite H1 in H0.
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simpl in H0. assert (combine s x0 = nil).
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destruct (combine s x0). reflexivity. simpl in H0.
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apply nil_cons in H0. contradiction H0.
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destruct x0. destruct s. reflexivity. simpl in H.
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apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
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destruct s. reflexivity. simpl in H2.
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symmetry in H2. apply nil_cons in H2. contradiction H2.
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exists nil. exists nil. destruct s. simpl. easy.
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destruct H. destruct H.
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assert (x0 = nil). destruct x0. reflexivity. simpl in H.
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apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
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rewrite H1 in H0. simpl in H0.
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symmetry in H0. apply nil_cons in H0. contradiction H0.
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(* second part of the induction *)
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intro s. destruct s. unfold subsequence. unfold subsequence2. split.
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exists (repeat false (S (length l))). rewrite repeat_length.
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split. easy. simpl.
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assert (forall u,
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(nil: list X)
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= map snd (filter fst (combine (repeat false (length u)) u))).
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intro u. induction u. reflexivity. simpl. assumption. apply H0.
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exists (a::l). exists (nil). simpl. split; try rewrite app_nil_r; reflexivity.
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(* deux cas : a = n ou non *)
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assert ({a=x} + {a<>x}). apply I. destruct H.
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rewrite e. rewrite subsequence2_cons_eq. rewrite <- IHl.
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rewrite subsequence_cons_eq. split; intro; assumption.
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split; intro H. apply subsequence2_cons_l. apply IHl.
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destruct H. destruct H. destruct H.
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destruct x0. destruct x1.
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apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
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simpl in H0. inversion H0. rewrite H2 in n. contradiction n.
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reflexivity. inversion H0.
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exists x2. exists x1. split. assumption.
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reflexivity.
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apply subsequence_cons_l. apply IHl.
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destruct H. destruct H.
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destruct x0. simpl in H0.
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symmetry in H0. apply nil_cons in H0. contradiction H0.
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destruct b. simpl in H0. inversion H0. rewrite H2 in n.
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contradiction n. reflexivity.
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exists x0. inversion H. inversion H0.
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split; reflexivity.
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Qed.
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Theorem subsequence_dec {X: Type}:
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(forall x y : X, {x = y} + {x <> y})
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-> forall (l s : list X), { subsequence l s } + { ~ subsequence l s }.
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Proof.
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intro H. intros l s.
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assert ({ subsequence2 l s } + { ~ subsequence2 l s }).
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apply subsequence2_dec. assumption. destruct H0.
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rewrite <- subsequence_eq_def in s0. left. assumption.
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assumption. rewrite <- subsequence_eq_def in n. right. assumption.
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assumption.
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Qed.
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Example test1: subsequence [1;2;3;4;5] [1;3;5].
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Proof.
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unfold subsequence.
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exists [].
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exists [[2];[4];[]]. simpl. easy.
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Qed.
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Example test2: subsequence [1;2;3;4;5] [2;4].
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Proof.
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unfold subsequence.
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exists [1].
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exists [[3];[5]]. simpl. easy.
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Qed.
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Example test3: subsequence [1;2;3;4;5] [1;3;5].
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Proof.
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rewrite subsequence_eq_def.
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exists [true; false; true; false; true].
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split; reflexivity.
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apply Nat.eq_dec.
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Qed.
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