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author | herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2007-07-13 14:55:03 +0000 |
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committer | herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2007-07-13 14:55:03 +0000 |
commit | 246909f2c587ed798bc42b65fb90c7b77dfa52b7 (patch) | |
tree | 0ea9ba42784b09b9478b32d4dafc01c3b81124d1 /theories/Num/LtProps.v | |
parent | a2b33be15a16de033506da6a4e8b407eaf951054 (diff) |
Répertoire Numbers poursuit l'objectif entamé en syntaxe V7 dans le
répertoire Num. Suppression de ce dernier de l'archive courante.
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@9995 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Num/LtProps.v')
-rw-r--r-- | theories/Num/LtProps.v | 82 |
1 files changed, 0 insertions, 82 deletions
diff --git a/theories/Num/LtProps.v b/theories/Num/LtProps.v deleted file mode 100644 index 79f0f3303..000000000 --- a/theories/Num/LtProps.v +++ /dev/null @@ -1,82 +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 *) -(************************************************************************) - -Require Export Axioms. -Require Export AddProps. -Require Export NeqProps. - -(** This file contains basic properties of the less than relation *) - - -Lemma lt_anti_sym : (x,y:N)x<y->~(y<x). -Red; Intros x y lt1 lt2; Apply (lt_anti_refl x); EAuto with num. -Qed. -Hints Resolve lt_anti_refl : num. - -Lemma eq_not_lt : (x,y:N)(x=y)->~(x<y). -Red; Intros x y eq1 lt1; Apply (lt_anti_refl x); EAuto with num. -Qed. -Hints Resolve eq_not_lt : num. - -Lemma lt_0_1 : (zero<one). -EAuto with num. -Qed. -Hints Resolve lt_0_1 : num. - - -Lemma eq_lt_x_Sy : (x,y:N)(x=y)->(x<(S y)). -EAuto with num. -Qed. -Hints Resolve eq_lt_x_Sy : num. - -Lemma lt_lt_x_Sy : (x,y:N)(x<y)->(x<(S y)). -EAuto with num. -Qed. -Hints Immediate lt_lt_x_Sy : num. - -Lemma lt_Sx_y_lt : (x,y:N)((S x)<y)->(x<y). -EAuto with num. -Qed. -Hints Immediate lt_Sx_y_lt : num. - -(** Relating [<] and [=] *) - -Lemma lt_neq : (x,y:N)(x<y)->(x<>y). -Red; Intros x y lt1 eq1; Apply (lt_anti_refl x); EAuto with num. -Qed. -Hints Immediate lt_neq : num. - -Lemma lt_neq_sym : (x,y:N)(y<x)->(x<>y). -Intros x y lt1 ; Apply neq_sym; Auto with num. -Qed. -Hints Immediate lt_neq_sym : num. - -(** Application to inequalities properties *) - -Lemma neq_x_Sx : (x:N)x<>(S x). -Auto with num. -Qed. -Hints Resolve neq_x_Sx : num. - -Lemma neq_0_1 : zero<>one. -Auto with num. -Qed. -Hints Resolve neq_0_1 : num. - -(** Relating [<] and [+] *) - -Lemma lt_add_compat_r : (x,y,z:N)(x<y)->((z+x)<(z+y)). -Intros x y z H; Apply lt_eq_compat with (x+z) (y+z); Auto with num. -Qed. -Hints Resolve lt_add_compat_r : num. - -Lemma lt_add_compat : (x1,x2,y1,y2:N)(x1<x2)->(y1<y2)->((x1+y1)<(x2+y2)). -Intros; Apply lt_trans with (x1+y2); Auto with num. -Qed. -Hints Immediate lt_add_compat : num. - |