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This commit is contained in:
Thomas Baruchel
parent
444d3a6550
commit
30e9a21065
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@ -188,7 +188,90 @@ Qed.
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(**
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Decidability and equivalence of all definitions of a subsequence.
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Equivalence of all definitions of a subsequence.
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*)
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Theorem subsequence_eq_def_1 {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 subsequence_eq_def_2 {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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Theorem subsequence_eq_def_3 {X: Type} :
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forall l s : list X, subsequence3 l s -> subsequence l s.
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Proof.
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intros l s. generalize l. induction s; intro l0; intro H.
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apply subsequence_nil_r. destruct H. destruct H. destruct H.
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apply IHs in H0. destruct H0. destruct H0. destruct H0.
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exists x. exists (x1:: x2). split. simpl. rewrite H0.
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reflexivity. rewrite H. rewrite H1. reflexivity.
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Qed.
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(**
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Decidability of all definitions of a subsequence.
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*)
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Theorem subsequence2_dec {X: Type}:
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@ -224,64 +307,14 @@ 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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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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apply subsequence_eq_def_2 in s0. left. assumption.
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right. unfold not. intro I.
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apply subsequence_eq_def_3 in I.
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apply subsequence_eq_def_1 in I.
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apply n in I. contradiction I.
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Qed.
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@ -290,45 +323,13 @@ Theorem subsequence_dec {X: Type}:
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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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assert ({ subsequence3 l s } + { ~ subsequence3 l s }).
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apply subsequence3_dec. assumption. destruct H0.
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apply subsequence_eq_def_3 in s0. left. assumption.
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right. unfold not. intro I.
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apply subsequence_eq_def_1 in I.
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apply subsequence_eq_def_2 in I.
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apply n in I. contradiction I.
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Qed.
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@ -446,173 +447,7 @@ 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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apply subsequence_eq_def_3. apply 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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Theorem subsequence0_eq_def_1 {X: Type} :
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forall l s : list X, subsequence3 l s -> subsequence l s.
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Proof.
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intros l s. generalize l. induction s; intro l0; intro H.
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apply subsequence_nil_r. destruct H. destruct H. destruct H.
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apply IHs in H0. destruct H0. destruct H0. destruct H0.
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exists x. exists (x1:: x2). split. simpl. rewrite H0.
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reflexivity. rewrite H. rewrite H1. reflexivity.
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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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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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