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Diffstat (limited to 'theories7/Arith/Min.v')
-rwxr-xr-x | theories7/Arith/Min.v | 84 |
1 files changed, 0 insertions, 84 deletions
diff --git a/theories7/Arith/Min.v b/theories7/Arith/Min.v deleted file mode 100755 index fd5da61a..00000000 --- a/theories7/Arith/Min.v +++ /dev/null @@ -1,84 +0,0 @@ -(************************************************************************) -(* 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. |