610 lines
21 KiB
Coq
610 lines
21 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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(** * Various definitions of a subsequences
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Different definitions of a subsequence are given; they are proved
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below to be equivalent, allowing to choose the most convenient at
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any step of a proof.
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*)
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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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Identical elementary properties of all definitions of a subsequence.
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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 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; try induction l; easy.
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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 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. 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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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_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. 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_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. intro H. destruct H. destruct H. destruct H.
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destruct x; 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. destruct H. destruct H. destruct H.
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exists (a::x). exists x0. split. assumption. 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. destruct H. destruct H. exists (false::x).
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split; try apply eq_S; 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. destruct s. apply subsequence3_nil_r.
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destruct H. destruct H. destruct H. exists (a::x0). exists x1. rewrite H.
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split; easy.
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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. 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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apply subsequence2_cons_l. apply IHl with (a := a0).
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exists x. split; inversion H; easy.
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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; 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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inversion H0. exists l0. exists x0.
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inversion H. rewrite H3. split; reflexivity.
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destruct x0.
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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.
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split. reflexivity. inversion H0. rewrite <- app_assoc.
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apply app_inv_head_iff. rewrite app_comm_cons. reflexivity.
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exists nil. exists (x::x0). simpl.
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split; [ rewrite H | 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; destruct H; destruct H.
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destruct x. inversion H0. destruct b. exists (x).
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split; inversion H; try rewrite H2; inversion H0; reflexivity.
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inversion H. assert (subsequence2 l (a::s)).
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exists x. split; assumption.
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apply subsequence2_cons_r with (a := a). assumption.
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exists (true :: x). split. apply eq_S. assumption. 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; easy.
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exists nil. exists l. split; easy.
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Qed.
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(**
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Decidability and equivalence of all definitions of a subsequence.
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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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intro 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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assert({subsequence2 l s} + {~ subsequence2 l s}). apply IHl. 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); [ left | right ];
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rewrite subsequence2_cons_eq; assumption.
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right. 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; inversion H0; easy.
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apply n in H2. contradiction H2.
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Qed.
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Theorem subsequence3_dec {X: Type}:
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(forall x y : X, {x = y} + {x <> y})
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-> forall (l s : list X), { subsequence3 l s } + { ~ subsequence3 l s }.
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Proof.
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intro H. intro l. induction l. intro s. destruct s. left. reflexivity.
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right. apply subsequence3_nil_cons_r.
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intro s. assert({subsequence3 l s} + {~ subsequence3 l s}). apply IHl.
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destruct H0.
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rewrite <- subsequence3_cons_eq with (a := a) in s0.
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apply subsequence3_cons_r in s0. left. assumption.
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destruct s. left. apply subsequence3_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); [ left | right ];
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rewrite subsequence3_cons_eq; assumption.
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right. intro I.
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destruct I. destruct H0. destruct H0. destruct x0.
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inversion H0. rewrite H3 in n0. contradiction n0. reflexivity.
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assert (subsequence3 l (x::s)). exists x2. exists x1.
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inversion H0. split. reflexivity. assumption.
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apply n in H2. contradiction H2.
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Qed.
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Theorem subsequence_eq_def_2 {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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intro s. split; intro H. destruct s. apply subsequence2_nil_r.
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apply subsequence_nil_cons_r in H. contradiction H.
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destruct s. apply subsequence_nil_r.
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apply subsequence2_nil_cons_r in H. contradiction H.
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intro s. destruct s. split; intro H. apply subsequence2_nil_r.
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apply subsequence_nil_r.
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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_2 in s0. left. assumption.
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assumption. rewrite <- subsequence_eq_def_2 in n. right. assumption.
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assumption.
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Qed.
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Theorem subsequence_eq_def_3 {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 <-> subsequence3 l s).
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Proof.
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intro I. intro l. induction l.
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intro s. split; intro H. destruct s. apply subsequence3_nil_r.
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apply subsequence_nil_cons_r in H. contradiction H.
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destruct s. apply subsequence_nil_r.
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apply subsequence3_nil_cons_r in H. contradiction H.
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intro s. destruct s. split; intro H. apply subsequence3_nil_r.
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apply subsequence_nil_r.
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assert ({a=x} + {a<>x}). apply I. destruct H.
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rewrite e. rewrite subsequence3_cons_eq. rewrite <- IHl.
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rewrite subsequence_cons_eq. split; intro; assumption.
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split; intro H. apply subsequence3_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. destruct H.
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destruct x0. inversion H. rewrite H2 in n. contradiction n. reflexivity.
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exists x2. exists x1. split. inversion H. reflexivity. assumption.
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Qed.
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(**
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Various general properties of a subsequence.
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*)
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Theorem subsequence2_app {X: Type} :
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forall l1 s1 l2 s2 : list X,
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subsequence2 l1 s1 -> subsequence2 l2 s2 -> subsequence2 (l1++l2) (s1++s2).
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Proof.
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intros l1 s1 l2 s2. intros H I.
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destruct H. destruct I. exists (x++x0). destruct H. destruct H0.
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split. rewrite app_length. rewrite app_length.
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rewrite H. rewrite H0. reflexivity.
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rewrite H1. rewrite H2.
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assert (J: forall (t : list bool) (u : list X) (v: list bool) (w : list X),
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length t = length u
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-> combine (t++v) (u++w) = (combine t u) ++ (combine v w)).
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intros t u v w. generalize u. induction t; intro u0; intro K.
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replace u0 with (nil : list X). reflexivity.
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destruct u0. reflexivity. apply O_S in K. contradiction K.
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destruct u0. apply PeanoNat.Nat.neq_succ_0 in K. contradiction K.
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simpl. rewrite IHt. reflexivity. inversion K. reflexivity.
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rewrite J. rewrite filter_app. rewrite map_app. reflexivity.
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assumption.
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Qed.
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Theorem subsequence_trans {X: Type} :
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forall (l1 l2 l3: list X),
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subsequence l1 l2 -> subsequence2 l2 l3 -> subsequence2 l1 l3.
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Proof.
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intros l1 l2 l3. intros H I.
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destruct H. destruct H. destruct H. destruct I. destruct H1.
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exists (
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(repeat false (length x)) ++
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(flat_map (fun e => (fst e) :: (repeat false (length (snd e))))
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(combine x1 x0))).
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split.
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rewrite app_length. rewrite repeat_length.
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rewrite H0. rewrite app_length. rewrite Nat.add_cancel_l.
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rewrite flat_map_length. rewrite flat_map_length.
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assert (forall (u: list X) (v: list (list X)),
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length u = length v
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-> list_sum (map (fun z => length (fst z :: snd z))
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(combine u v))
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= list_sum (map (fun z => S (length z)) v)).
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intros u v. generalize u. induction v; intro u0; intro I.
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apply length_zero_iff_nil in I. rewrite I. reflexivity.
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destruct u0. apply O_S in I. contradiction I.
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simpl. rewrite IHv. reflexivity. inversion I. reflexivity.
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rewrite H3.
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assert (forall (u: list bool) (v: list (list X)),
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length u = length v
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-> list_sum (map (fun z => length (fst z :: repeat false (length (snd z))))
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(combine u v))
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= list_sum (map (fun z => S (length z)) v)).
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intros u v. generalize u.
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induction v; intro u0; intro I; destruct u0. reflexivity.
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apply PeanoNat.Nat.neq_succ_0 in I. contradiction I.
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apply O_S in I. contradiction I.
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simpl. rewrite IHv. rewrite repeat_length. reflexivity.
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inversion I. reflexivity. rewrite H4. reflexivity.
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rewrite H1. rewrite H. reflexivity. assumption.
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assert (K: forall (w v: list X) u,
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filter fst (combine ((repeat false (length w)) ++ u)
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(w ++ v)) = filter fst (combine u v)).
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intro w0. induction w0. reflexivity. apply IHw0.
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rewrite H2. rewrite H0.
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assert (forall (u v: list X) (w: list bool),
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map snd (filter fst (combine ((repeat false (length u)) ++ w) (u ++ v)))
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= map snd (filter fst (combine w v))).
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intro u. induction u; intros v w. reflexivity.
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apply IHu. rewrite H3.
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assert (forall (u: list bool) (v: list X) (w: list (list X)),
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length u = length w
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-> length u = length v
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-> filter fst (combine
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(flat_map
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(fun e => fst e:: (repeat false (length (snd e))))
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(combine u w))
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(flat_map (fun e => fst e :: snd e) (combine v w)))
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= filter fst (combine u v)).
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intros u v w. generalize u. generalize v.
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induction w; intros v0 u0; intros I J.
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apply length_zero_iff_nil in I. rewrite I. reflexivity.
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destruct u0. reflexivity. destruct v0.
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apply PeanoNat.Nat.neq_succ_0 in J; contradiction J.
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destruct b; simpl; rewrite K; rewrite IHw.
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reflexivity. inversion I; reflexivity. inversion J; reflexivity.
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reflexivity. inversion I; reflexivity. inversion J; reflexivity.
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rewrite H4; try rewrite H1; try rewrite H; reflexivity.
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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_2.
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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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Theorem subsequence0_eq_def_2 {X: Type} :
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forall l s : list X, subsequence l s -> subsequence2 l s.
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Proof.
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intros l s. intro H. destruct H. destruct H. destruct H.
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exists (
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(repeat false (length x)) ++
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(flat_map (fun e => true :: (repeat false (length e))) x0)).
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split.
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rewrite H0. rewrite app_length. rewrite app_length. rewrite repeat_length.
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rewrite Nat.add_cancel_l. rewrite flat_map_length. rewrite flat_map_length.
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assert (forall v (u: list X),
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length u = length v
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-> map (fun e => length (fst e :: snd e)) (combine u v)
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= map (fun e => length (true :: repeat false (length e))) v).
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intro v. induction v; intro u; intro I.
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apply length_zero_iff_nil in I. rewrite I. reflexivity.
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destruct u. apply O_S in I. contradiction I.
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simpl. rewrite IHv. rewrite repeat_length. reflexivity.
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inversion I. reflexivity.
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rewrite H1. reflexivity. inversion H. reflexivity. rewrite H0.
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assert (forall (u: list X) v w,
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filter fst (combine (repeat false (length u) ++ v) (u ++ w))
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= filter fst (combine v w)).
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intro u. induction u; intros v w. reflexivity. simpl. apply IHu.
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assert (forall (v: list (list X)) (u : list X),
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length u = length v
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-> u = map snd (filter fst (combine
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(flat_map (fun e => true:: repeat false (length e)) v)
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(flat_map (fun e => fst e :: snd e) (combine u v))))).
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intro v. induction v; intro u; intro I.
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apply length_zero_iff_nil in I. rewrite I. reflexivity.
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destruct u. apply O_S in I. contradiction I.
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simpl. rewrite H1. rewrite <- IHv. reflexivity. inversion I.
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reflexivity. rewrite H1. rewrite <- H2. reflexivity.
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inversion H. reflexivity.
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Qed.
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Theorem subsequence0_eq_def_3 {X: Type} :
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forall l s : list X, subsequence2 l s -> subsequence3 l s.
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Proof.
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intros l s. intro H. destruct H. destruct H.
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assert (I: forall u (v w: list X),
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length u = length w
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-> v = map snd (filter fst (combine u w))
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-> subsequence3 w v).
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intro u. induction u; intros v w; intros I J.
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rewrite J. apply subsequence3_nil_r.
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destruct v. apply subsequence3_nil_r.
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destruct w. apply Nat.neq_succ_0 in I. contradiction I.
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destruct a. exists nil. exists w.
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inversion J. apply IHu in H3.
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split. reflexivity. inversion J. rewrite <- H5. assumption.
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inversion I. reflexivity. simpl in J.
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assert (subsequence3 w (x0::v)). apply IHu.
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inversion I. reflexivity. assumption.
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apply subsequence3_cons_l. assumption.
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apply I with (u := x); assumption.
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Qed.
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(*
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destruct H. destruct H.
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(* z contient la position des booléens dans x *)
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pose (z := map snd (filter fst (combine x (seq 0 (length x))))).
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destruct s. exists l. exists nil. rewrite app_nil_r. split; easy.
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assert (0 < length z). unfold z. rewrite map_length.
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assert (forall v (u: list X) (w: list nat),
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length u = length v -> length u = length w
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-> length (filter fst (combine v w))
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= length (map snd (filter fst (combine v u)))).
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intro v. induction v; intros u w; intros I J. reflexivity.
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destruct w. apply length_zero_iff_nil in J. rewrite J in I.
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apply O_S in I. contradiction I.
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destruct u. apply O_S in J. contradiction J.
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destruct a; simpl. apply eq_S. apply IHv.
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inversion I. reflexivity. inversion J. reflexivity.
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apply IHv.
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inversion I. reflexivity. inversion J. reflexivity.
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assert (length (x0::s) = length (map snd (filter fst (combine x l)))).
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rewrite H0. reflexivity. rewrite <- H1 with (w := seq 0 (length x)) in H2.
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rewrite <- H2. apply Nat.lt_0_succ. inversion H. reflexivity.
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rewrite H. rewrite seq_length. reflexivity.
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exists (firstn (hd 0 z) l).
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(* z2 contient la position du true suivant *)
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pose (z2 := (skipn 1 z) ++ [ length l ]).
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(* z3 contient la longueur de chaque bloc true; false; false; ... *)
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pose (z3 := map (fun e => (snd e) - (fst e)) (combine z z2)).
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exists (map (fun e => firstn ((snd e) -1) (skipn (S (fst e)) l))
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(combine z z3)).
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assert (length (x0::s) = length z). unfold z. rewrite H0.
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rewrite map_length. rewrite map_length. rewrite H.
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assert (forall u (v: list X) (w: list nat),
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length v = length w
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-> length (filter fst (combine u v))
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= length (filter fst (combine u w))).
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intro u. induction u; intros v w; intro I. reflexivity.
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destruct v. symmetry in I. apply length_zero_iff_nil in I.
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rewrite I. reflexivity. destruct w. apply Nat.neq_succ_0 in I.
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contradiction I. destruct a; simpl; rewrite IHu with (w := w).
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inversion I; reflexivity. inversion I; reflexivity. reflexivity.
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inversion I; reflexivity. rewrite H2 with (w := seq 0 (length l)).
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reflexivity. rewrite seq_length. reflexivity.
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unfold z3. unfold z2. split. destruct z. apply Nat.nlt_0_r in H1.
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contradiction H1.
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rewrite map_length. rewrite combine_length. rewrite map_length.
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rewrite combine_length. rewrite app_length. rewrite Nat.add_1_r.
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simpl. apply eq_S. rewrite Nat.min_id. rewrite Nat.min_id.
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inversion H2. reflexivity. unfold z. rewrite H0.
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assert (forall (u: list bool) (v: list X),
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length u = length v
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-> v =
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firstn (hd 0 (map snd (filter fst (combine u (seq 0 (length u)))))) v ++
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flat_map (fun e : X * list X => fst e :: snd e)
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(combine (map snd (filter fst (combine u v)))
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(map (fun e : nat * nat => firstn (snd e - 1) (skipn (S (fst e)) v))
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(combine (map snd (filter fst (combine u (seq 0 (length u)))))
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(map (fun e : nat * nat => snd e - fst e)
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(combine (map snd (filter fst (combine u (seq 0 (length u)))))
|
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(skipn 1
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(map snd (filter fst (combine u (seq 0 (length u))))) ++
|
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[length v]))))))).
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intro u. induction u; intro v; intro I.
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|
symmetry in I. apply length_zero_iff_nil in I. assumption.
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|
destruct v. rewrite app_nil_r. rewrite firstn_nil. reflexivity.
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|
destruct a.
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|
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replace
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(map snd (filter fst (combine (true :: u) (seq 0 (length (true :: u))))))
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with
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*)
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