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@ -585,15 +585,18 @@ Qed.
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Theorem subsequence_double_cons {X: Type} :
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forall (u v: list X) a, subsequence (a::u) (a::v) -> subsequence u v.
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forall (u v: list X) a, subsequence (a::u) (a::v) <-> subsequence u v.
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Proof.
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intros u v a. intro H. destruct H. destruct H. destruct H.
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intros u v a. split; intro H. destruct H. destruct H. destruct H.
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destruct x; destruct x0. symmetry in H0. apply nil_cons in H0.
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contradiction. inversion H0.
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exists l. exists x0. split. inversion H. reflexivity. reflexivity.
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apply PeanoNat.Nat.neq_succ_0 in H. contradiction.
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inversion H0. exists (x1++x::l). exists x0. split. inversion H.
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reflexivity. rewrite <- app_assoc. reflexivity.
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apply subsequence_eq_def_3. exists nil. exists u.
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split. reflexivity. apply subsequence_eq_def_2. apply subsequence_eq_def_1.
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assumption.
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Qed.
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@ -617,7 +620,7 @@ Proof.
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Qed.
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Theorem subsequence_filter {X: Type} :
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Theorem subsequence_filter_2 {X: Type} :
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forall (u v: list X) f,
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subsequence u v -> subsequence (filter f u) (filter f v).
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Proof.
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@ -634,13 +637,6 @@ Proof.
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apply subsequence_nil_r. simpl. destruct (f a).
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apply subsequence_cons_r.
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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 -> subsequence l2 l3 -> subsequence l1 l3.
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Theorem subsequence_incl {X: Type} :
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forall (u v: list X), subsequence u v -> incl v u.
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