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Diffstat (limited to 'theories7/ZArith/Zbool.v')
| -rw-r--r-- | theories7/ZArith/Zbool.v | 158 |
1 files changed, 0 insertions, 158 deletions
diff --git a/theories7/ZArith/Zbool.v b/theories7/ZArith/Zbool.v deleted file mode 100644 index 258a485d..00000000 --- a/theories7/ZArith/Zbool.v +++ /dev/null @@ -1,158 +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 *) -(************************************************************************) - -(* $Id: Zbool.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ *) - -Require BinInt. -Require Zeven. -Require Zorder. -Require Zcompare. -Require ZArith_dec. -Require Zsyntax. -Require Sumbool. - -(** The decidability of equality and order relations over - type [Z] give some boolean functions with the adequate specification. *) - -Definition Z_lt_ge_bool := [x,y:Z](bool_of_sumbool (Z_lt_ge_dec x y)). -Definition Z_ge_lt_bool := [x,y:Z](bool_of_sumbool (Z_ge_lt_dec x y)). - -Definition Z_le_gt_bool := [x,y:Z](bool_of_sumbool (Z_le_gt_dec x y)). -Definition Z_gt_le_bool := [x,y:Z](bool_of_sumbool (Z_gt_le_dec x y)). - -Definition Z_eq_bool := [x,y:Z](bool_of_sumbool (Z_eq_dec x y)). -Definition Z_noteq_bool := [x,y:Z](bool_of_sumbool (Z_noteq_dec x y)). - -Definition Zeven_odd_bool := [x:Z](bool_of_sumbool (Zeven_odd_dec x)). - -(**********************************************************************) -(** Boolean comparisons of binary integers *) - -Definition Zle_bool := - [x,y:Z]Cases `x ?= y` of SUPERIEUR => false | _ => true end. -Definition Zge_bool := - [x,y:Z]Cases `x ?= y` of INFERIEUR => false | _ => true end. -Definition Zlt_bool := - [x,y:Z]Cases `x ?= y` of INFERIEUR => true | _ => false end. -Definition Zgt_bool := - [x,y:Z]Cases ` x ?= y` of SUPERIEUR => true | _ => false end. -Definition Zeq_bool := - [x,y:Z]Cases `x ?= y` of EGAL => true | _ => false end. -Definition Zneq_bool := - [x,y:Z]Cases `x ?= y` of EGAL => false | _ => true end. - -Lemma Zle_cases : (x,y:Z)if (Zle_bool x y) then `x<=y` else `x>y`. -Proof. -Intros x y; Unfold Zle_bool Zle Zgt. -Case (Zcompare x y); Auto; Discriminate. -Qed. - -Lemma Zlt_cases : (x,y:Z)if (Zlt_bool x y) then `x<y` else `x>=y`. -Proof. -Intros x y; Unfold Zlt_bool Zlt Zge. -Case (Zcompare x y); Auto; Discriminate. -Qed. - -Lemma Zge_cases : (x,y:Z)if (Zge_bool x y) then `x>=y` else `x<y`. -Proof. -Intros x y; Unfold Zge_bool Zge Zlt. -Case (Zcompare x y); Auto; Discriminate. -Qed. - -Lemma Zgt_cases : (x,y:Z)if (Zgt_bool x y) then `x>y` else `x<=y`. -Proof. -Intros x y; Unfold Zgt_bool Zgt Zle. -Case (Zcompare x y); Auto; Discriminate. -Qed. - -(** Lemmas on [Zle_bool] used in contrib/graphs *) - -Lemma Zle_bool_imp_le : (x,y:Z) (Zle_bool x y)=true -> (Zle x y). -Proof. - Unfold Zle_bool Zle. Intros x y. Unfold not. - Case (Zcompare x y); Intros; Discriminate. -Qed. - -Lemma Zle_imp_le_bool : (x,y:Z) (Zle x y) -> (Zle_bool x y)=true. -Proof. - Unfold Zle Zle_bool. Intros x y. Case (Zcompare x y); Trivial. Intro. Elim (H (refl_equal ? ?)). -Qed. - -Lemma Zle_bool_refl : (x:Z) (Zle_bool x x)=true. -Proof. - Intro. Apply Zle_imp_le_bool. Apply Zle_refl. Reflexivity. -Qed. - -Lemma Zle_bool_antisym : (x,y:Z) (Zle_bool x y)=true -> (Zle_bool y x)=true -> x=y. -Proof. - Intros. Apply Zle_antisym. Apply Zle_bool_imp_le. Assumption. - Apply Zle_bool_imp_le. Assumption. -Qed. - -Lemma Zle_bool_trans : (x,y,z:Z) (Zle_bool x y)=true -> (Zle_bool y z)=true -> (Zle_bool x z)=true. -Proof. - Intros x y z; Intros. Apply Zle_imp_le_bool. Apply Zle_trans with m:=y. Apply Zle_bool_imp_le. Assumption. - Apply Zle_bool_imp_le. Assumption. -Qed. - -Definition Zle_bool_total : (x,y:Z) {(Zle_bool x y)=true}+{(Zle_bool y x)=true}. -Proof. - Intros x y; Intros. Unfold Zle_bool. Cut (Zcompare x y)=SUPERIEUR<->(Zcompare y x)=INFERIEUR. - Case (Zcompare x y). Left . Reflexivity. - Left . Reflexivity. - Right . Rewrite (proj1 ? ? H (refl_equal ? ?)). Reflexivity. - Apply Zcompare_ANTISYM. -Defined. - -Lemma Zle_bool_plus_mono : (x,y,z,t:Z) (Zle_bool x y)=true -> (Zle_bool z t)=true -> - (Zle_bool (Zplus x z) (Zplus y t))=true. -Proof. - Intros. Apply Zle_imp_le_bool. Apply Zle_plus_plus. Apply Zle_bool_imp_le. Assumption. - Apply Zle_bool_imp_le. Assumption. -Qed. - -Lemma Zone_pos : (Zle_bool `1` `0`)=false. -Proof. - Reflexivity. -Qed. - -Lemma Zone_min_pos : (x:Z) (Zle_bool x `0`)=false -> (Zle_bool `1` x)=true. -Proof. - Intros x; Intros. Apply Zle_imp_le_bool. Change (Zle (Zs ZERO) x). Apply Zgt_le_S. Generalize H. - Unfold Zle_bool Zgt. Case (Zcompare x ZERO). Intro H0. Discriminate H0. - Intro H0. Discriminate H0. - Reflexivity. -Qed. - - - Lemma Zle_is_le_bool : (x,y:Z) (Zle x y) <-> (Zle_bool x y)=true. - Proof. - Intros. Split. Intro. Apply Zle_imp_le_bool. Assumption. - Intro. Apply Zle_bool_imp_le. Assumption. - Qed. - - Lemma Zge_is_le_bool : (x,y:Z) (Zge x y) <-> (Zle_bool y x)=true. - Proof. - Intros. Split. Intro. Apply Zle_imp_le_bool. Apply Zge_le. Assumption. - Intro. Apply Zle_ge. Apply Zle_bool_imp_le. Assumption. - Qed. - - Lemma Zlt_is_le_bool : (x,y:Z) (Zlt x y) <-> (Zle_bool x `y-1`)=true. - Proof. - Intros x y. Split. Intro. Apply Zle_imp_le_bool. Apply Zlt_n_Sm_le. Rewrite (Zs_pred y) in H. - Assumption. - Intro. Rewrite (Zs_pred y). Apply Zle_lt_n_Sm. Apply Zle_bool_imp_le. Assumption. - Qed. - - Lemma Zgt_is_le_bool : (x,y:Z) (Zgt x y) <-> (Zle_bool y `x-1`)=true. - Proof. - Intros x y. Apply iff_trans with `y < x`. Split. Exact (Zgt_lt x y). - Exact (Zlt_gt y x). - Exact (Zlt_is_le_bool y x). - Qed. - |