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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(* $Id$ *)
Require Arith.
(**************************************************************************)
(* minimum of two natural numbers *)
(**************************************************************************)
Fixpoint min [n:nat] : nat -> nat :=
[m:nat]Cases n m of
O _ => O
| (S n') O => O
| (S n') (S m') => (S (min n' m'))
end.
Lemma min_SS : (n,m:nat)((S (min n m))=(min (S n) (S m))).
Proof.
Auto with arith.
Qed.
Lemma le_min_l : (n,m:nat)(le (min n m) n).
Proof.
Induction n; Intros; Simpl; Auto with arith.
Elim m; Intros; Simpl; Auto with arith.
Qed.
Hints Resolve le_min_l : arith v62.
Lemma le_min_r : (n,m:nat)(le (min n m) m).
Proof.
Induction n; Simpl; Auto with arith.
Induction m; Simpl; Auto with arith.
Qed.
Hints Resolve le_min_r : arith v62.
(* min n m is equal to n or m *)
Lemma min_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (min n m)).
Proof.
Induction n; Simpl; Auto with arith.
Induction m; Intros; Simpl; Auto with arith.
Pattern (min n0 n1); Apply H ; Auto with arith.
Qed.
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