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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 *)
(************************************************************************)
(*i $Id: Min.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*)
Require Arith.
V7only [Import nat_scope.].
Open Local Scope nat_scope.
Implicit Variables Type m,n:nat.
(** 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.
(** Simplifications of [min] *)
Lemma min_SS : (n,m:nat)((S (min n m))=(min (S n) (S m))).
Proof.
Auto with arith.
Qed.
Lemma min_sym : (n,m:nat)(min n m)=(min m n).
Proof.
NewInduction n;NewInduction m;Simpl;Auto with arith.
Qed.
(** [min] and [le] *)
Lemma min_l : (n,m:nat)(le n m)->(min n m)=n.
Proof.
NewInduction n;NewInduction m;Simpl;Auto with arith.
Qed.
Lemma min_r : (n,m:nat)(le m n)->(min n m)=m.
Proof.
NewInduction n;NewInduction m;Simpl;Auto with arith.
Qed.
Lemma le_min_l : (n,m:nat)(le (min n m) n).
Proof.
NewInduction n; Intros; Simpl; Auto with arith.
Elim m; Intros; Simpl; Auto with arith.
Qed.
Lemma le_min_r : (n,m:nat)(le (min n m) m).
Proof.
NewInduction n; Simpl; Auto with arith.
NewInduction m; Simpl; Auto with arith.
Qed.
Hints Resolve min_l min_r le_min_l le_min_r : arith v62.
(** [min n m] is equal to [n] or [m] *)
Lemma min_dec : (n,m:nat){(min n m)=n}+{(min n m)=m}.
Proof.
NewInduction n;NewInduction m;Simpl;Auto with arith.
Elim (IHn m);Intro H;Elim H;Auto.
Qed.
Lemma min_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (min n m)).
Proof.
NewInduction n; Simpl; Auto with arith.
NewInduction m; Intros; Simpl; Auto with arith.
Pattern (min n m); Apply IHn ; Auto with arith.
Qed.
Lemma min_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (min n m)).
Proof.
NewInduction n; Simpl; Auto with arith.
NewInduction m; Intros; Simpl; Auto with arith.
Pattern (min n m); Apply IHn ; Auto with arith.
Qed.
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