Section Lists_compl.
Variable A B : Set.
auto with arith. Qed.
auto with arith. Qed.
auto with arith. Qed.
auto with arith. Qed.
Inductive len : list A -> nat -> Prop :=
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| len_nil : len nil 0
| len_cons :
forall (b : A) (v : list A) (l : nat), len v l -> len (b :: v) (S l).
Hint Resolve len_nil len_cons.
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intros. discriminate.
Qed.
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simple induction v. intro. discriminate.
intros. injection. intros. absurd (l = a :: l). apply H.
exact H1.
Qed.
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intro.
rewrite <- app_nil_end; trivial with arith.
Qed. Hint Resolve app_v_nil.
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intros. inversion H. apply app_v_nil.
Qed.
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simple induction v1. simpl in |- *. trivial with arith.
intros. generalize H0. rewrite app_eq2. intros. injection H1. intro. apply H. exact H2.
Qed.
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simple induction v2. intro. rewrite H. trivial with arith.
intros. rewrite H0. auto with arith.
Qed.
Definition app_assoc_r := app_ass.
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Definition app_assoc_l := ass_app.
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simple induction v. intros. inversion H. discriminate.
intros. inversion H0. discriminate.
Qed.
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intros; inversion H; trivial with arith.
Qed.
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simple induction l. red in |- *. intro. inversion H. unfold not in |- *. intros. inversion H0.
Qed.
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simple induction l1. simpl in |- *. intros. inversion H. simpl in |- *. assumption.
intros. inversion H0. simpl in |- *. apply len_cons. apply H. exact H4.
exact H1.
Qed.
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simple induction v. auto with arith.
intros. rewrite length_eq2. apply len_cons. exact H.
Qed. Hint Resolve len_length.
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intros. rewrite H. trivial with arith.
Qed.
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simple induction v. simpl in |- *; intro. absurd (0 <> 0). unfold not in |- *; auto with arith. exact H.
simpl in |- *. intros. discriminate.
Qed.
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simple induction v1. simpl in |- *. trivial with arith.
intros. simpl in |- *. rewrite H. trivial with arith.
Qed.
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simple induction v. intro. absurd (nil <> nil :>list A). unfold not in |- *. auto with arith.
exact H.
intros. simpl in |- *. apply le_n_S. auto with arith.
Qed.

Lemma le_SO_length_v :
forall v : list A,
length v <> 0 -> 1 <= length v.