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Diffstat (limited to 'theories7/Arith/Minus.v')
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diff --git a/theories7/Arith/Minus.v b/theories7/Arith/Minus.v deleted file mode 100755 index 709d5f0b..00000000 --- a/theories7/Arith/Minus.v +++ /dev/null @@ -1,120 +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: Minus.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*) - -(** Subtraction (difference between two natural numbers) *) - -Require Lt. -Require Le. - -V7only [Import nat_scope.]. -Open Local Scope nat_scope. - -Implicit Variables Type m,n,p:nat. - -(** 0 is right neutral *) - -Lemma minus_n_O : (n:nat)(n=(minus n O)). -Proof. -NewInduction n; Simpl; Auto with arith. -Qed. -Hints Resolve minus_n_O : arith v62. - -(** Permutation with successor *) - -Lemma minus_Sn_m : (n,m:nat)(le m n)->((S (minus n m))=(minus (S n) m)). -Proof. -Intros n m Le; Pattern m n; Apply le_elim_rel; Simpl; Auto with arith. -Qed. -Hints Resolve minus_Sn_m : arith v62. - -Theorem pred_of_minus : (x:nat)(pred x)=(minus x (S O)). -Intro x; NewInduction x; Simpl; Auto with arith. -Qed. - -(** Diagonal *) - -Lemma minus_n_n : (n:nat)(O=(minus n n)). -Proof. -NewInduction n; Simpl; Auto with arith. -Qed. -Hints Resolve minus_n_n : arith v62. - -(** Simplification *) - -Lemma minus_plus_simpl : - (n,m,p:nat)((minus n m)=(minus (plus p n) (plus p m))). -Proof. - NewInduction p; Simpl; Auto with arith. -Qed. -Hints Resolve minus_plus_simpl : arith v62. - -(** Relation with plus *) - -Lemma plus_minus : (n,m,p:nat)(n=(plus m p))->(p=(minus n m)). -Proof. -Intros n m p; Pattern m n; Apply nat_double_ind; Simpl; Intros. -Replace (minus n0 O) with n0; Auto with arith. -Absurd O=(S (plus n0 p)); Auto with arith. -Auto with arith. -Qed. -Hints Immediate plus_minus : arith v62. - -Lemma minus_plus : (n,m:nat)(minus (plus n m) n)=m. -Symmetry; Auto with arith. -Qed. -Hints Resolve minus_plus : arith v62. - -Lemma le_plus_minus : (n,m:nat)(le n m)->(m=(plus n (minus m n))). -Proof. -Intros n m Le; Pattern n m; Apply le_elim_rel; Simpl; Auto with arith. -Qed. -Hints Resolve le_plus_minus : arith v62. - -Lemma le_plus_minus_r : (n,m:nat)(le n m)->(plus n (minus m n))=m. -Proof. -Symmetry; Auto with arith. -Qed. -Hints Resolve le_plus_minus_r : arith v62. - -(** Relation with order *) - -Theorem le_minus: (i,h:nat) (le (minus i h) i). -Proof. -Intros i h;Pattern i h; Apply nat_double_ind; [ - Auto -| Auto -| Intros m n H; Simpl; Apply le_trans with m:=m; Auto ]. -Qed. - -Lemma lt_minus : (n,m:nat)(le m n)->(lt O m)->(lt (minus n m) n). -Proof. -Intros n m Le; Pattern m n; Apply le_elim_rel; Simpl; Auto with arith. -Intros; Absurd (lt O O); Auto with arith. -Intros p q lepq Hp gtp. -Elim (le_lt_or_eq O p); Auto with arith. -Auto with arith. -NewInduction 1; Elim minus_n_O; Auto with arith. -Qed. -Hints Resolve lt_minus : arith v62. - -Lemma lt_O_minus_lt : (n,m:nat)(lt O (minus n m))->(lt m n). -Proof. -Intros n m; Pattern n m; Apply nat_double_ind; Simpl; Auto with arith. -Intros; Absurd (lt O O); Trivial with arith. -Qed. -Hints Immediate lt_O_minus_lt : arith v62. - -Theorem inj_minus_aux: (x,y:nat) ~(le y x) -> (minus x y) = O. -Intros y x; Pattern y x ; Apply nat_double_ind; [ - Simpl; Trivial with arith -| Intros n H; Absurd (le O (S n)); [ Assumption | Apply le_O_n] -| Simpl; Intros n m H1 H2; Apply H1; - Unfold not ; Intros H3; Apply H2; Apply le_n_S; Assumption]. -Qed. |