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-(************************************************************************)
-(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
-(* \VV/ **************************************************************)
-(* // * This file is distributed under the terms of the *)
-(* * GNU Lesser General Public License Version 2.1 *)
-(************************************************************************)
-Require Compare_dec.
-Require Wf_nat.
-
-Section Iter.
-Variable A : Type.
-
-Fixpoint iter (n : nat) : (A -> A) -> A -> A :=
- fun (fl : A -> A) (def : A) =>
- match n with
- | O => def
- | S m => fl (iter m fl def)
- end.
-End Iter.
-
-Theorem SSplus_lt : forall p p' : nat, p < S (S (p + p')).
- intro p; intro p'; change (S p <= S (S (p + p')));
- apply le_S; apply Gt.gt_le_S; change (p < S (p + p'));
- apply Lt.le_lt_n_Sm; apply Plus.le_plus_l.
-Qed.
-
-
-Theorem Splus_lt : forall p p' : nat, p' < S (p + p').
- intro p; intro p'; change (S p' <= S (p + p'));
- apply Gt.gt_le_S; change (p' < S (p + p')); apply Lt.le_lt_n_Sm;
- apply Plus.le_plus_r.
-Qed.
-
-Theorem le_lt_SS : forall x y, x <= y -> x < S (S y).
-intro x; intro y; intro H; change (S x <= S (S y));
- apply le_S; apply Gt.gt_le_S; change (x < S y);
- apply Lt.le_lt_n_Sm; exact H.
-Qed.
-
-Inductive max_type (m n:nat) : Set :=
- cmt : forall v, m <= v -> n <= v -> max_type m n.
-
-Definition max : forall m n:nat, max_type m n.
-intros m n; case (Compare_dec.le_gt_dec m n).
-intros h; exists n; [exact h | apply le_n].
-intros h; exists m; [apply le_n | apply Lt.lt_le_weak; exact h].
-Defined.