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Diffstat (limited to 'theories/Reals/Rdefinitions.v')
| -rw-r--r-- | theories/Reals/Rdefinitions.v | 69 |
1 files changed, 69 insertions, 0 deletions
diff --git a/theories/Reals/Rdefinitions.v b/theories/Reals/Rdefinitions.v new file mode 100644 index 00000000..33f494df --- /dev/null +++ b/theories/Reals/Rdefinitions.v @@ -0,0 +1,69 @@ +(************************************************************************) +(* 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: Rdefinitions.v,v 1.14.2.1 2004/07/16 19:31:12 herbelin Exp $ i*) + + +(*********************************************************) +(** Definitions for the axiomatization *) +(* *) +(*********************************************************) + +Require Export ZArith_base. + +Parameter R : Set. + +(* Declare Scope positive_scope with Key R *) +Delimit Scope R_scope with R. + +(* Automatically open scope R_scope for arguments of type R *) +Bind Scope R_scope with R. + +Parameter R0 : R. +Parameter R1 : R. +Parameter Rplus : R -> R -> R. +Parameter Rmult : R -> R -> R. +Parameter Ropp : R -> R. +Parameter Rinv : R -> R. +Parameter Rlt : R -> R -> Prop. +Parameter up : R -> Z. + +Infix "+" := Rplus : R_scope. +Infix "*" := Rmult : R_scope. +Notation "- x" := (Ropp x) : R_scope. +Notation "/ x" := (Rinv x) : R_scope. + +Infix "<" := Rlt : R_scope. + +(*i*******************************************************i*) + +(**********) +Definition Rgt (r1 r2:R) : Prop := (r2 < r1)%R. + +(**********) +Definition Rle (r1 r2:R) : Prop := (r1 < r2)%R \/ r1 = r2. + +(**********) +Definition Rge (r1 r2:R) : Prop := Rgt r1 r2 \/ r1 = r2. + +(**********) +Definition Rminus (r1 r2:R) : R := (r1 + - r2)%R. + +(**********) +Definition Rdiv (r1 r2:R) : R := (r1 * / r2)%R. + +Infix "-" := Rminus : R_scope. +Infix "/" := Rdiv : R_scope. + +Infix "<=" := Rle : R_scope. +Infix ">=" := Rge : R_scope. +Infix ">" := Rgt : R_scope. + +Notation "x <= y <= z" := ((x <= y)%R /\ (y <= z)%R) : R_scope. +Notation "x <= y < z" := ((x <= y)%R /\ (y < z)%R) : R_scope. +Notation "x < y < z" := ((x < y)%R /\ (y < z)%R) : R_scope. +Notation "x < y <= z" := ((x < y)%R /\ (y <= z)%R) : R_scope.
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