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Diffstat (limited to 'theories7/Arith/Max.v')
-rwxr-xr-x | theories7/Arith/Max.v | 87 |
1 files changed, 0 insertions, 87 deletions
diff --git a/theories7/Arith/Max.v b/theories7/Arith/Max.v deleted file mode 100755 index aea389d1..00000000 --- a/theories7/Arith/Max.v +++ /dev/null @@ -1,87 +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: Max.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. - -(** maximum of two natural numbers *) - -Fixpoint max [n:nat] : nat -> nat := -[m:nat]Cases n m of - O _ => m - | (S n') O => n - | (S n') (S m') => (S (max n' m')) - end. - -(** Simplifications of [max] *) - -Lemma max_SS : (n,m:nat)((S (max n m))=(max (S n) (S m))). -Proof. -Auto with arith. -Qed. - -Lemma max_sym : (n,m:nat)(max n m)=(max m n). -Proof. -NewInduction n;NewInduction m;Simpl;Auto with arith. -Qed. - -(** [max] and [le] *) - -Lemma max_l : (n,m:nat)(le m n)->(max n m)=n. -Proof. -NewInduction n;NewInduction m;Simpl;Auto with arith. -Qed. - -Lemma max_r : (n,m:nat)(le n m)->(max n m)=m. -Proof. -NewInduction n;NewInduction m;Simpl;Auto with arith. -Qed. - -Lemma le_max_l : (n,m:nat)(le n (max n m)). -Proof. -NewInduction n; Intros; Simpl; Auto with arith. -Elim m; Intros; Simpl; Auto with arith. -Qed. - -Lemma le_max_r : (n,m:nat)(le m (max n m)). -Proof. -NewInduction n; Simpl; Auto with arith. -NewInduction m; Simpl; Auto with arith. -Qed. -Hints Resolve max_r max_l le_max_l le_max_r: arith v62. - - -(** [max n m] is equal to [n] or [m] *) - -Lemma max_dec : (n,m:nat){(max n m)=n}+{(max n m)=m}. -Proof. -NewInduction n;NewInduction m;Simpl;Auto with arith. -Elim (IHn m);Intro H;Elim H;Auto. -Qed. - -Lemma max_case : (n,m:nat)(P:nat->Set)(P n)->(P m)->(P (max n m)). -Proof. -NewInduction n; Simpl; Auto with arith. -NewInduction m; Intros; Simpl; Auto with arith. -Pattern (max n m); Apply IHn ; Auto with arith. -Qed. - -Lemma max_case2 : (n,m:nat)(P:nat->Prop)(P n)->(P m)->(P (max n m)). -Proof. -NewInduction n; Simpl; Auto with arith. -NewInduction m; Intros; Simpl; Auto with arith. -Pattern (max n m); Apply IHn ; Auto with arith. -Qed. - - |