281 lines
8.3 KiB
Coq
281 lines
8.3 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 (l s : list Type) :=
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exists (l1: list Type) (l2 : list (list Type)),
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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 (l s : list Type) :=
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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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Theorem subsequence_nil_r : forall (l : list Type), subsequence l nil.
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Proof.
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intro l. unfold subsequence. exists l. exists nil. rewrite app_nil_r.
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split; easy.
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Qed.
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Theorem subsequence_nil_cons_r : forall (l: list Type) (a:Type),
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~ subsequence nil (a::l).
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Proof.
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intros l a. unfold subsequence. 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 : forall (l : list Type), subsequence2 l nil.
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Proof.
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intro l. unfold subsequence2.
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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 : forall (l: list Type) (a:Type),
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~ subsequence2 nil (a::l).
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Proof.
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intros l a. unfold subsequence2. 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 subsequence2_cons_eq : forall (l1 l2: list Type) (a: Type),
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subsequence2 (a::l1) (a::l2) <-> subsequence2 l1 l2.
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Proof.
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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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(* fin de la démonstration de H *)
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intros l1 l2 a. split; intro I.
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unfold subsequence2 in I. destruct I. destruct H0. destruct x.
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simpl in H0. symmetry in H0.
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apply PeanoNat.Nat.neq_succ_0 in H0. contradiction H0.
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destruct b. simpl in H0. inversion H1. rewrite <- H3.
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unfold subsequence2. exists x. split. inversion H0. reflexivity.
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assumption.
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simpl in H1. unfold subsequence2. exists x. split. inversion H0.
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reflexivity.
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assert (count_occ Bool.bool_dec x true = length (a::l2)).
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apply H with (l := l1). assumption. inversion H0. assumption.
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Theorem subsequence2_dec :
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(forall x y : Type, {x = y} + {x <> y})
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-> forall (l s : list Type), { 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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left. unfold subsequence2. unfold subsequence2 in s0. destruct s0.
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destruct H0. exists (false::x). split. simpl. rewrite H0. reflexivity.
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simpl. assumption.
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destruct s. left. apply subsequence2_nil_r.
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assert ({T=a}+{T<>a}). apply H. destruct H0.
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rewrite e.
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intro l0. destruct l0. right. apply subsequence2_nil_cons_r.
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assert ({ subsequence2 l0 s } + { ~ subsequence2 l0 s }). apply IHs.
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Theorem subsequence2_dec :
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(forall x y : Type, {x = y} + {x <> y})
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-> forall (l s : list Type), { subsequence2 l s } + { ~ subsequence2 l s }.
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Proof.
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intro H.
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intros l s. generalize l. induction s. left. apply subsequence2_nil_r.
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intro l0. destruct l0. right. apply subsequence2_nil_cons_r.
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assert ({ subsequence2 l0 s } + { ~ subsequence2 l0 s }). apply IHs.
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assert ({T=a}+{T<>a}). apply H. destruct H0; destruct H1.
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left. unfold subsequence2. unfold subsequence2 in s0. destruct s0.
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destruct H0. exists (true::x). simpl. rewrite H0. split. reflexivity.
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rewrite e. rewrite H1. reflexivity.
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(*
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assert ({In a l0}+{~ In a l0}). apply In_dec. assumption.
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destruct H0. apply In_split in i. destruct i.
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*)
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Theorem subsequence_dec : forall (l s : list Type),
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{ subsequence l s } + { ~ subsequence l s }.
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Proof.
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intros l s. generalize l. induction s. left. apply subsequence_nil_r.
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intro l0. destruct l0. right. apply subsequence_nil_cons_r.
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intros l s. induction s. left. apply subsequence_nil_r.
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destruct l. right. apply subsequence_nil_cons_r.
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unfold subsequence2. destruct s. simpl.
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left. exists nil. easy. right.
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unfold not. intro H. destruct H. destruct H.
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rewrite combine_nil in H0.
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symmetry in H0. apply nil_cons in H0. assumption.
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destruct IHl. left. unfold subsequence2. assert (H := s0).
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unfold subsequence2 in H. destruct H. destruct H.
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exists (false::x). split. simpl. apply eq_S. assumption.
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assumption.
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(* destructurer s puis tester les deux cas : a = (car s) ou non *)
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destruct s. left. unfold subsequence2.
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Theorem subsequence_dec : forall (l s : list nat),
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{ subsequence2 l s } + { ~ subsequence2 l s }.
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Proof.
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intros l s. induction l.
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unfold subsequence2. destruct s. simpl.
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left. exists nil. easy. right.
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unfold not. intro H. destruct H. destruct H.
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rewrite combine_nil in H0.
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symmetry in H0. apply nil_cons in H0. assumption.
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destruct IHl. left. unfold subsequence2. assert (H := s0).
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unfold subsequence2 in H. destruct H. destruct H.
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exists (false::x). split. simpl. apply eq_S. assumption.
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assumption.
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(* destructurer s puis tester les deux cas : a = (car s) ou non *)
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destruct s. left. unfold subsequence2.
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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 nat)
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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 H.
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assert ({a=n0}+{a<>n0}). apply PeanoNat.Nat.eq_dec. destruct H.
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Theorem subsequence_eq_def : forall l s, subsequence l s <-> subsequence2 l s.
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Proof.
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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 (x = nil). destruct x. 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 Type)
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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=n} + {a<>n}). apply PeanoNat.Nat.eq_dec. destruct H.
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unfold subsequence. unfold subsequence2. split.
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destruct H. destruct H. destruct H.
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assert (x0 = nil). symmetry in H. apply length_zero_iff_nil in H.
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assumption. rewrite H1 in H0. simpl in H0.
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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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