src/HOL/Real/RealDef.thy
author avigad
Wed, 13 Jul 2005 19:49:07 +0200
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Additions to the Real (and Hyperreal) libraries: RealDef.thy: lemmas relating nats, ints, and reals reversed direction of real_of_int_mult, etc. (now they agree with nat versions) SEQ.thy and Series.thy: various additions RComplete.thy: lemmas involving floor and ceiling introduced natfloor and natceiling Log.thy: various additions
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(*  Title       : Real/RealDef.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
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    Additional contributions by Jeremy Avigad
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*)
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header{*Defining the Reals from the Positive Reals*}
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theory RealDef
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imports PReal
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uses ("real_arith.ML")
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begin
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constdefs
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  realrel   ::  "((preal * preal) * (preal * preal)) set"
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  "realrel == {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
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typedef (Real)  real = "UNIV//realrel"
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  by (auto simp add: quotient_def)
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instance real :: "{ord, zero, one, plus, times, minus, inverse}" ..
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constdefs
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  (** these don't use the overloaded "real" function: users don't see them **)
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  real_of_preal :: "preal => real"
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  "real_of_preal m     ==
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           Abs_Real(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})"
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consts
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   (*Overloaded constant denoting the Real subset of enclosing
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     types such as hypreal and complex*)
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   Reals :: "'a set"
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   (*overloaded constant for injecting other types into "real"*)
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   real :: "'a => real"
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syntax (xsymbols)
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  Reals     :: "'a set"                   ("\<real>")
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defs (overloaded)
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  real_zero_def:
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  "0 == Abs_Real(realrel``{(preal_of_rat 1, preal_of_rat 1)})"
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  real_one_def:
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  "1 == Abs_Real(realrel``
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               {(preal_of_rat 1 + preal_of_rat 1,
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		 preal_of_rat 1)})"
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  real_minus_def:
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  "- r ==  contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
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  real_add_def:
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   "z + w ==
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       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
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		 { Abs_Real(realrel``{(x+u, y+v)}) })"
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  real_diff_def:
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   "r - (s::real) == r + - s"
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  real_mult_def:
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    "z * w ==
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       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
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		 { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"
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  real_inverse_def:
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  "inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1)"
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  real_divide_def:
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  "R / (S::real) == R * inverse S"
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  real_le_def:
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   "z \<le> (w::real) == 
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    \<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w"
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  real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)"
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  real_abs_def:  "abs (r::real) == (if 0 \<le> r then r else -r)"
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subsection{*Proving that realrel is an equivalence relation*}
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lemma preal_trans_lemma:
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  assumes "x + y1 = x1 + y"
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      and "x + y2 = x2 + y"
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  shows "x1 + y2 = x2 + (y1::preal)"
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proof -
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  have "(x1 + y2) + x = (x + y2) + x1" by (simp add: preal_add_ac) 
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  also have "... = (x2 + y) + x1"  by (simp add: prems)
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  also have "... = x2 + (x1 + y)"  by (simp add: preal_add_ac)
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  also have "... = x2 + (x + y1)"  by (simp add: prems)
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  also have "... = (x2 + y1) + x"  by (simp add: preal_add_ac)
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  finally have "(x1 + y2) + x = (x2 + y1) + x" .
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  thus ?thesis by (simp add: preal_add_right_cancel_iff) 
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qed
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lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"
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by (simp add: realrel_def)
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lemma equiv_realrel: "equiv UNIV realrel"
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apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def)
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apply (blast dest: preal_trans_lemma) 
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done
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text{*Reduces equality of equivalence classes to the @{term realrel} relation:
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  @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *}
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lemmas equiv_realrel_iff = 
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       eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]
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declare equiv_realrel_iff [simp]
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lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
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by (simp add: Real_def realrel_def quotient_def, blast)
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lemma inj_on_Abs_Real: "inj_on Abs_Real Real"
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apply (rule inj_on_inverseI)
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apply (erule Abs_Real_inverse)
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done
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declare inj_on_Abs_Real [THEN inj_on_iff, simp]
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declare Abs_Real_inverse [simp]
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text{*Case analysis on the representation of a real number as an equivalence
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      class of pairs of positive reals.*}
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lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 
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     "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
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apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
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apply (drule arg_cong [where f=Abs_Real])
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apply (auto simp add: Rep_Real_inverse)
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done
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subsection{*Congruence property for addition*}
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lemma real_add_congruent2_lemma:
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     "[|a + ba = aa + b; ab + bc = ac + bb|]
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      ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
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apply (simp add: preal_add_assoc) 
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apply (rule preal_add_left_commute [of ab, THEN ssubst])
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apply (simp add: preal_add_assoc [symmetric])
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apply (simp add: preal_add_ac)
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done
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lemma real_add:
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     "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
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      Abs_Real (realrel``{(x+u, y+v)})"
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   157
proof -
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  have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
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        respects2 realrel"
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    by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
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  thus ?thesis
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    by (simp add: real_add_def UN_UN_split_split_eq
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                  UN_equiv_class2 [OF equiv_realrel equiv_realrel])
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qed
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lemma real_add_commute: "(z::real) + w = w + z"
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by (cases z, cases w, simp add: real_add preal_add_ac)
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   168
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lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)"
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by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc)
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   171
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lemma real_add_zero_left: "(0::real) + z = z"
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by (cases z, simp add: real_add real_zero_def preal_add_ac)
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   174
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instance real :: comm_monoid_add
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  by (intro_classes,
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      (assumption | 
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       rule real_add_commute real_add_assoc real_add_zero_left)+)
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subsection{*Additive Inverse on real*}
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lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
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   184
proof -
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   185
  have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
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    by (simp add: congruent_def preal_add_commute) 
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  thus ?thesis
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    by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
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   189
qed
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   190
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   191
lemma real_add_minus_left: "(-z) + z = (0::real)"
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   192
by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute)
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diff changeset
   193
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   194
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subsection{*Congruence property for multiplication*}
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   196
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   197
lemma real_mult_congruent2_lemma:
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   198
     "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
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   199
          x * x1 + y * y1 + (x * y2 + y * x2) =
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   200
          x * x2 + y * y2 + (x * y1 + y * x1)"
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   201
apply (simp add: preal_add_left_commute preal_add_assoc [symmetric])
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apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric])
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   203
apply (simp add: preal_add_commute)
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   204
done
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   205
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   206
lemma real_mult_congruent2:
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    "(%p1 p2.
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   208
        (%(x1,y1). (%(x2,y2). 
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   209
          { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
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   210
     respects2 realrel"
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   211
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
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   212
apply (simp add: preal_mult_commute preal_add_commute)
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   213
apply (auto simp add: real_mult_congruent2_lemma)
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   214
done
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diff changeset
   215
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   216
lemma real_mult:
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   217
      "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
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   218
       Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
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diff changeset
   219
by (simp add: real_mult_def UN_UN_split_split_eq
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   220
         UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
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diff changeset
   221
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   222
lemma real_mult_commute: "(z::real) * w = w * z"
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   223
by (cases z, cases w, simp add: real_mult preal_add_ac preal_mult_ac)
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   224
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   225
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
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   226
apply (cases z1, cases z2, cases z3)
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apply (simp add: real_mult preal_add_mult_distrib2 preal_add_ac preal_mult_ac)
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   228
done
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   229
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   230
lemma real_mult_1: "(1::real) * z = z"
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   231
apply (cases z)
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   232
apply (simp add: real_mult real_one_def preal_add_mult_distrib2
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   233
                 preal_mult_1_right preal_mult_ac preal_add_ac)
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   234
done
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diff changeset
   235
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   236
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
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   237
apply (cases z1, cases z2, cases w)
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diff changeset
   238
apply (simp add: real_add real_mult preal_add_mult_distrib2 
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   239
                 preal_add_ac preal_mult_ac)
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   240
done
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diff changeset
   241
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   242
text{*one and zero are distinct*}
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   243
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
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diff changeset
   244
proof -
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   245
  have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1"
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diff changeset
   246
    by (simp add: preal_self_less_add_left) 
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   247
  thus ?thesis
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   248
    by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff)
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diff changeset
   249
qed
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diff changeset
   250
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   251
subsection{*existence of inverse*}
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diff changeset
   252
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   253
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
14497
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parents: 14484
diff changeset
   254
by (simp add: real_zero_def preal_add_commute)
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diff changeset
   255
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   256
text{*Instead of using an existential quantifier and constructing the inverse
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   257
within the proof, we could define the inverse explicitly.*}
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diff changeset
   258
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
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diff changeset
   259
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
14484
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parents: 14476
diff changeset
   260
apply (simp add: real_zero_def real_one_def, cases x)
14269
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diff changeset
   261
apply (cut_tac x = xa and y = y in linorder_less_linear)
14365
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paulson
parents: 14348
diff changeset
   262
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
14334
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parents: 14329
diff changeset
   263
apply (rule_tac
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diff changeset
   264
        x = "Abs_Real (realrel `` { (preal_of_rat 1, 
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diff changeset
   265
                            inverse (D) + preal_of_rat 1)}) " 
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diff changeset
   266
       in exI)
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   267
apply (rule_tac [2]
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diff changeset
   268
        x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1,
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diff changeset
   269
                   preal_of_rat 1)})" 
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   270
       in exI)
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diff changeset
   271
apply (auto simp add: real_mult preal_mult_1_right
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   272
              preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1
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diff changeset
   273
              preal_mult_inverse_right preal_add_ac preal_mult_ac)
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   274
done
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parents: 13487
diff changeset
   275
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diff changeset
   276
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
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diff changeset
   277
apply (simp add: real_inverse_def)
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parents: 14348
diff changeset
   278
apply (frule real_mult_inverse_left_ex, safe)
14269
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   279
apply (rule someI2, auto)
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diff changeset
   280
done
14334
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parents: 14329
diff changeset
   281
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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parents: 14335
diff changeset
   282
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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parents: 14335
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   283
subsection{*The Real Numbers form a Field*}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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diff changeset
   284
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   285
instance real :: field
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   286
proof
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   287
  fix x y z :: real
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parents: 14329
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   288
  show "- x + x = 0" by (rule real_add_minus_left)
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parents: 14329
diff changeset
   289
  show "x - y = x + (-y)" by (simp add: real_diff_def)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
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parents: 14329
diff changeset
   290
  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   291
  show "x * y = y * x" by (rule real_mult_commute)
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parents: 14329
diff changeset
   292
  show "1 * x = x" by (rule real_mult_1)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
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parents: 14329
diff changeset
   293
  show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   294
  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   295
  show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14426
diff changeset
   296
  show "x / y = x * inverse y" by (simp add: real_divide_def)
14334
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parents: 14329
diff changeset
   297
qed
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parents: 14329
diff changeset
   298
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
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parents: 14329
diff changeset
   299
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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   300
text{*Inverse of zero!  Useful to simplify certain equations*}
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parents: 13487
diff changeset
   301
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   302
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   303
by (simp add: real_inverse_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   304
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   305
instance real :: division_by_zero
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   306
proof
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   307
  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   308
qed
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   309
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   310
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   311
(*Pull negations out*)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   312
declare minus_mult_right [symmetric, simp] 
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   313
        minus_mult_left [symmetric, simp]
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   314
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   315
lemma real_mult_1_right: "z * (1::real) = z"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   316
  by (rule OrderedGroup.mult_1_right)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   317
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   318
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   319
subsection{*The @{text "\<le>"} Ordering*}
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   320
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   321
lemma real_le_refl: "w \<le> (w::real)"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   322
by (cases w, force simp add: real_le_def)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   323
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   324
text{*The arithmetic decision procedure is not set up for type preal.
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   325
  This lemma is currently unused, but it could simplify the proofs of the
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   326
  following two lemmas.*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   327
lemma preal_eq_le_imp_le:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   328
  assumes eq: "a+b = c+d" and le: "c \<le> a"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   329
  shows "b \<le> (d::preal)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   330
proof -
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   331
  have "c+d \<le> a+d" by (simp add: prems preal_cancels)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   332
  hence "a+b \<le> a+d" by (simp add: prems)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   333
  thus "b \<le> d" by (simp add: preal_cancels)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   334
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   335
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   336
lemma real_le_lemma:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   337
  assumes l: "u1 + v2 \<le> u2 + v1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   338
      and "x1 + v1 = u1 + y1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   339
      and "x2 + v2 = u2 + y2"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   340
  shows "x1 + y2 \<le> x2 + (y1::preal)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   341
proof -
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   342
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   343
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   344
  also have "... \<le> (x2+y1) + (u2+v1)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   345
         by (simp add: prems preal_add_le_cancel_left)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   346
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   347
qed						 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   348
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   349
lemma real_le: 
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   350
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   351
      (x1 + y2 \<le> x2 + y1)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   352
apply (simp add: real_le_def) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   353
apply (auto intro: real_le_lemma)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   354
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   355
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   356
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15229
diff changeset
   357
by (cases z, cases w, simp add: real_le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   358
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   359
lemma real_trans_lemma:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   360
  assumes "x + v \<le> u + y"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   361
      and "u + v' \<le> u' + v"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   362
      and "x2 + v2 = u2 + y2"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   363
  shows "x + v' \<le> u' + (y::preal)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   364
proof -
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   365
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   366
  also have "... \<le> (u+y) + (u+v')" 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   367
    by (simp add: preal_add_le_cancel_right prems) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   368
  also have "... \<le> (u+y) + (u'+v)" 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   369
    by (simp add: preal_add_le_cancel_left prems) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   370
  also have "... = (u'+y) + (u+v)"  by (simp add: preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   371
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15229
diff changeset
   372
qed
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   373
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   374
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   375
apply (cases i, cases j, cases k)
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   376
apply (simp add: real_le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   377
apply (blast intro: real_trans_lemma) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   378
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   379
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   380
(* Axiom 'order_less_le' of class 'order': *)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   381
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   382
by (simp add: real_less_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   383
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   384
instance real :: order
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   385
proof qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   386
 (assumption |
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   387
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   388
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   389
(* Axiom 'linorder_linear' of class 'linorder': *)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   390
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   391
apply (cases z, cases w) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   392
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   393
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   394
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   395
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   396
instance real :: linorder
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   397
  by (intro_classes, rule real_le_linear)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   398
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   399
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   400
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   401
apply (cases x, cases y) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   402
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   403
                      preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   404
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15229
diff changeset
   405
done
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   406
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   407
lemma real_add_left_mono: 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   408
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   409
proof -
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   410
  have "z + x - (z + y) = (z + -z) + (x - y)"
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   411
    by (simp add: diff_minus add_ac) 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   412
  with le show ?thesis 
14754
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   413
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   414
qed
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   415
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   416
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   417
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   418
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   419
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   420
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   421
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   422
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   423
apply (cases x, cases y)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   424
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   425
                 linorder_not_le [where 'a = preal] 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   426
                  real_zero_def real_le real_mult)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   427
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   428
apply (auto  dest!: less_add_left_Ex 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   429
     simp add: preal_add_ac preal_mult_ac 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   430
          preal_add_mult_distrib2 preal_cancels preal_self_less_add_right)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   431
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   432
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   433
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   434
apply (rule real_sum_gt_zero_less)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   435
apply (drule real_less_sum_gt_zero [of x y])
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   436
apply (drule real_mult_order, assumption)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   437
apply (simp add: right_distrib)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   438
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   439
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   440
text{*lemma for proving @{term "0<(1::real)"}*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   441
lemma real_zero_le_one: "0 \<le> (1::real)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   442
by (simp add: real_zero_def real_one_def real_le 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   443
                 preal_self_less_add_left order_less_imp_le)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   444
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   445
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   446
subsection{*The Reals Form an Ordered Field*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   447
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   448
instance real :: ordered_field
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   449
proof
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   450
  fix x y z :: real
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   451
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   452
  show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   453
  show "\<bar>x\<bar> = (if x < 0 then -x else x)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   454
    by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   455
qed
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   456
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   457
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   458
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   459
text{*The function @{term real_of_preal} requires many proofs, but it seems
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   460
to be essential for proving completeness of the reals from that of the
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   461
positive reals.*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   462
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   463
lemma real_of_preal_add:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   464
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   465
by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   466
              preal_add_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   467
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   468
lemma real_of_preal_mult:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   469
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   470
by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   471
              preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   472
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   473
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   474
text{*Gleason prop 9-4.4 p 127*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   475
lemma real_of_preal_trichotomy:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   476
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   477
apply (simp add: real_of_preal_def real_zero_def, cases x)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   478
apply (auto simp add: real_minus preal_add_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   479
apply (cut_tac x = x and y = y in linorder_less_linear)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   480
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   481
apply (auto simp add: preal_add_commute)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   482
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   483
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   484
lemma real_of_preal_leD:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   485
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   486
by (simp add: real_of_preal_def real_le preal_cancels)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   487
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   488
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   489
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   490
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   491
lemma real_of_preal_lessD:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   492
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   493
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric] 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   494
              preal_cancels) 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   495
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   496
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   497
lemma real_of_preal_less_iff [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   498
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   499
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   500
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   501
lemma real_of_preal_le_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   502
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   503
by (simp add: linorder_not_less [symmetric]) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   504
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   505
lemma real_of_preal_zero_less: "0 < real_of_preal m"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   506
apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   507
            preal_add_ac preal_cancels)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   508
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   509
apply (blast intro: preal_self_less_add_left order_less_imp_le)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   510
apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   511
apply (simp add: preal_add_ac) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   512
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   513
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   514
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   515
by (simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   516
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   517
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   518
proof -
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   519
  from real_of_preal_minus_less_zero
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   520
  show ?thesis by (blast dest: order_less_trans)
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   521
qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   522
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   523
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   524
subsection{*Theorems About the Ordering*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   525
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   526
text{*obsolete but used a lot*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   527
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   528
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   529
by blast 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   530
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   531
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   532
by (simp add: order_le_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   533
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   534
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   535
apply (auto simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   536
apply (cut_tac x = x in real_of_preal_trichotomy)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   537
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   538
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   539
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   540
lemma real_gt_preal_preal_Ex:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   541
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   542
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   543
             intro: real_gt_zero_preal_Ex [THEN iffD1])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   544
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   545
lemma real_ge_preal_preal_Ex:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   546
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   547
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   548
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   549
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   550
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   551
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   552
            simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   553
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   554
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   555
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   556
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   557
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   558
  by (rule OrderedGroup.add_less_le_mono)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   559
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   560
lemma real_add_le_less_mono:
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   561
     "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   562
  by (rule OrderedGroup.add_le_less_mono)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   563
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   564
lemma real_le_square [simp]: "(0::real) \<le> x*x"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   565
 by (rule Ring_and_Field.zero_le_square)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   566
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   567
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   568
subsection{*More Lemmas*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   569
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   570
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   571
by auto
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   572
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   573
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   574
by auto
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   575
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   576
text{*The precondition could be weakened to @{term "0\<le>x"}*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   577
lemma real_mult_less_mono:
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   578
     "[| u<v;  x<y;  (0::real) < v;  0 < x |] ==> u*x < v* y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   579
 by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   580
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   581
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   582
  by (force elim: order_less_asym
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   583
            simp add: Ring_and_Field.mult_less_cancel_right)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   584
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   585
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   586
apply (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   587
apply (blast intro: elim: order_less_asym) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   588
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   589
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   590
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
15923
01d5d0c1c078 fixed lin.arith
nipkow
parents: 15542
diff changeset
   591
by(simp add:mult_commute)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   592
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   593
text{*Only two uses?*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   594
lemma real_mult_less_mono':
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   595
     "[| x < y;  r1 < r2;  (0::real) \<le> r1;  0 \<le> x|] ==> r1 * x < r2 * y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   596
 by (rule Ring_and_Field.mult_strict_mono')
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   597
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   598
text{*FIXME: delete or at least combine the next two lemmas*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   599
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   600
apply (drule OrderedGroup.equals_zero_I [THEN sym])
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   601
apply (cut_tac x = y in real_le_square) 
14476
758e7acdea2f removed redundant thms
paulson
parents: 14443
diff changeset
   602
apply (auto, drule order_antisym, auto)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   603
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   604
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   605
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   606
apply (rule_tac y = x in real_sum_squares_cancel)
14476
758e7acdea2f removed redundant thms
paulson
parents: 14443
diff changeset
   607
apply (simp add: add_commute)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   608
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   609
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   610
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   611
by (drule add_strict_mono [of concl: 0 0], assumption, simp)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   612
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   613
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   614
apply (drule order_le_imp_less_or_eq)+
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   615
apply (auto intro: real_add_order order_less_imp_le)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   616
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   617
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   618
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   619
apply (case_tac "x \<noteq> 0")
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   620
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   621
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   622
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   623
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   624
by (auto dest: less_imp_inverse_less)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   625
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   626
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   627
proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   628
  have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   629
  thus ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   630
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   631
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   632
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   633
subsection{*Embedding the Integers into the Reals*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   634
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   635
defs (overloaded)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   636
  real_of_nat_def: "real z == of_nat z"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   637
  real_of_int_def: "real z == of_int z"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   638
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   639
lemma real_eq_of_nat: "real = of_nat"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   640
  apply (rule ext)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   641
  apply (unfold real_of_nat_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   642
  apply (rule refl)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   643
  done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   644
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   645
lemma real_eq_of_int: "real = of_int"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   646
  apply (rule ext)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   647
  apply (unfold real_of_int_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   648
  apply (rule refl)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   649
  done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   650
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   651
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   652
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   653
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   654
lemma real_of_one [simp]: "real (1::int) = (1::real)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   655
by (simp add: real_of_int_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   656
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   657
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   658
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   659
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   660
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   661
by (simp add: real_of_int_def) 
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   662
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   663
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   664
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   665
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   666
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   667
by (simp add: real_of_int_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   668
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   669
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   670
  apply (subst real_eq_of_int)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   671
  apply (rule of_int_setsum)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   672
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   673
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   674
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   675
    (PROD x:A. real(f x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   676
  apply (subst real_eq_of_int)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   677
  apply (rule of_int_setprod)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   678
done
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   679
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   680
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   681
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   682
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   683
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   684
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   685
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   686
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   687
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   688
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   689
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   690
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   691
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   692
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   693
by (simp add: real_of_int_def) 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   694
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   695
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   696
by (simp add: real_of_int_def) 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   697
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   698
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   699
by (simp add: real_of_int_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   700
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   701
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   702
by (simp add: real_of_int_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   703
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   704
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   705
  apply (subgoal_tac "real n + 1 = real (n + 1)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   706
  apply (simp del: real_of_int_add)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   707
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   708
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   709
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   710
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   711
  apply (subgoal_tac "real m + 1 = real (m + 1)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   712
  apply (simp del: real_of_int_add)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   713
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   714
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   715
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   716
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   717
    real (x div d) + (real (x mod d)) / (real d)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   718
proof -
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   719
  assume "d ~= 0"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   720
  have "x = (x div d) * d + x mod d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   721
    by auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   722
  then have "real x = real (x div d) * real d + real(x mod d)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   723
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   724
  then have "real x / real d = ... / real d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   725
    by simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   726
  then show ?thesis
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   727
    by (auto simp add: add_divide_distrib ring_eq_simps prems)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   728
qed
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   729
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   730
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   731
    real(n div d) = real n / real d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   732
  apply (frule real_of_int_div_aux [of d n])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   733
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   734
  apply (simp add: zdvd_iff_zmod_eq_0)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   735
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   736
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   737
lemma real_of_int_div2:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   738
  "0 <= real (n::int) / real (x) - real (n div x)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   739
  apply (case_tac "x = 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   740
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   741
  apply (case_tac "0 < x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   742
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   743
  apply (subst real_of_int_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   744
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   745
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   746
  apply (subst zero_le_divide_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   747
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   748
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   749
  apply (subst real_of_int_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   750
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   751
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   752
  apply (subst zero_le_divide_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   753
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   754
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   755
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   756
lemma real_of_int_div3:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   757
  "real (n::int) / real (x) - real (n div x) <= 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   758
  apply(case_tac "x = 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   759
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   760
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   761
  apply (subst real_of_int_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   762
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   763
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   764
  apply (subst divide_le_eq)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   765
  apply clarsimp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   766
  apply (rule conjI)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   767
  apply (rule impI)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   768
  apply (rule order_less_imp_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   769
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   770
  apply (rule impI)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   771
  apply (rule order_less_imp_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   772
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   773
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   774
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   775
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   776
  by (insert real_of_int_div2 [of n x], simp)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   777
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   778
subsection{*Embedding the Naturals into the Reals*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   779
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   780
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   781
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   782
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   783
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   784
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   785
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   786
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   787
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   788
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   789
(*Not for addsimps: often the LHS is used to represent a positive natural*)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   790
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   791
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   792
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   793
lemma real_of_nat_less_iff [iff]: 
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   794
     "(real (n::nat) < real m) = (n < m)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   795
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   796
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   797
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   798
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   799
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   800
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   801
by (simp add: real_of_nat_def zero_le_imp_of_nat)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   802
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   803
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   804
by (simp add: real_of_nat_def del: of_nat_Suc)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   805
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   806
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   807
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   808
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   809
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   810
    (SUM x:A. real(f x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   811
  apply (subst real_eq_of_nat)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   812
  apply (rule of_nat_setsum)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   813
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   814
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   815
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   816
    (PROD x:A. real(f x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   817
  apply (subst real_eq_of_nat)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   818
  apply (rule of_nat_setprod)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   819
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   820
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   821
lemma real_of_card: "real (card A) = setsum (%x.1) A"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   822
  apply (subst card_eq_setsum)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   823
  apply (subst real_of_nat_setsum)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   824
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   825
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   826
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   827
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   828
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   829
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   830
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   831
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   832
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   833
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   834
by (simp add: add: real_of_nat_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   835
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   836
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   837
by (simp add: add: real_of_nat_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   838
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   839
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   840
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   841
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   842
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   843
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   844
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   845
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   846
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   847
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   848
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   849
  apply (subgoal_tac "real n + 1 = real (Suc n)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   850
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   851
  apply (auto simp add: real_of_nat_Suc)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   852
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   853
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   854
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   855
  apply (subgoal_tac "real m + 1 = real (Suc m)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   856
  apply (simp add: less_Suc_eq_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   857
  apply (simp add: real_of_nat_Suc)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   858
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   859
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   860
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   861
    real (x div d) + (real (x mod d)) / (real d)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   862
proof -
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   863
  assume "0 < d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   864
  have "x = (x div d) * d + x mod d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   865
    by auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   866
  then have "real x = real (x div d) * real d + real(x mod d)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   867
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   868
  then have "real x / real d = \<dots> / real d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   869
    by simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   870
  then show ?thesis
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   871
    by (auto simp add: add_divide_distrib ring_eq_simps prems)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   872
qed
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   873
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   874
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   875
    real(n div d) = real n / real d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   876
  apply (frule real_of_nat_div_aux [of d n])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   877
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   878
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   879
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   880
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   881
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   882
lemma real_of_nat_div2:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   883
  "0 <= real (n::nat) / real (x) - real (n div x)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   884
  apply(case_tac "x = 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   885
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   886
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   887
  apply (subst real_of_nat_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   888
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   889
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   890
  apply (subst zero_le_divide_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   891
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   892
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   893
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   894
lemma real_of_nat_div3:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   895
  "real (n::nat) / real (x) - real (n div x) <= 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   896
  apply(case_tac "x = 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   897
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   898
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   899
  apply (subst real_of_nat_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   900
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   901
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   902
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   903
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   904
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   905
  by (insert real_of_nat_div2 [of n x], simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   906
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   907
lemma real_of_int_real_of_nat: "real (int n) = real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   908
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   909
14426
de890c247b38 fixed bugs in the setup of arithmetic procedures
paulson
parents: 14421
diff changeset
   910
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
de890c247b38 fixed bugs in the setup of arithmetic procedures
paulson
parents: 14421
diff changeset
   911
by (simp add: real_of_int_def real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   912
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   913
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   914
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   915
  apply force
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   916
  apply (simp only: real_of_int_real_of_nat)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   917
done
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   918
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   919
subsection{*Numerals and Arithmetic*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   920
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   921
instance real :: number ..
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   922
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   923
defs (overloaded)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   924
  real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   925
    --{*the type constraint is essential!*}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   926
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   927
instance real :: number_ring
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   928
by (intro_classes, simp add: real_number_of_def) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   929
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   930
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   931
text{*Collapse applications of @{term real} to @{term number_of}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   932
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   933
by (simp add:  real_of_int_def of_int_number_of_eq)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   934
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   935
lemma real_of_nat_number_of [simp]:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   936
     "real (number_of v :: nat) =  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   937
        (if neg (number_of v :: int) then 0  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   938
         else (number_of v :: real))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   939
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   940
 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   941
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   942
use "real_arith.ML"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   943
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   944
setup real_arith_setup
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   945
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   946
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   947
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   948
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   949
lemma real_0_le_divide_iff:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   950
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   951
by (simp add: real_divide_def zero_le_mult_iff, auto)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   952
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   953
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   954
by arith
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   955
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   956
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   957
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   958
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   959
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   960
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   961
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   962
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   963
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   964
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   965
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   966
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   967
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   968
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   969
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   970
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   971
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   972
(*
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   973
FIXME: we should have this, as for type int, but many proofs would break.
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   974
It replaces x+-y by x-y.
15086
e6a2a98d5ef5 removal of more iff-rules from RealDef.thy
paulson
parents: 15085
diff changeset
   975
declare real_diff_def [symmetric, simp]
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   976
*)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   977
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   978
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   979
subsubsection{*Density of the Reals*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   980
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   981
lemma real_lbound_gt_zero:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   982
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   983
apply (rule_tac x = " (min d1 d2) /2" in exI)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   984
apply (simp add: min_def)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   985
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   986
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   987
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   988
text{*Similar results are proved in @{text Ring_and_Field}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   989
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   990
  by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   991
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   992
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   993
  by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   994
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   995
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   996
subsection{*Absolute Value Function for the Reals*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   997
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   998
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14754
diff changeset
   999
by (simp add: abs_if)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1000
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1001
lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1002
by (force simp add: Ring_and_Field.abs_less_iff)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1003
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1004
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
  1005
by (force simp add: OrderedGroup.abs_le_iff)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1006
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
  1007
(*FIXME: used only once, in SEQ.ML*)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1008
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14754
diff changeset
  1009
by (simp add: abs_if)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1010
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1011
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15169
diff changeset
  1012
by (simp add: real_of_nat_ge_zero)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1013
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1014
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1015
apply (simp add: linorder_not_less)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1016
apply (auto intro: abs_ge_self [THEN order_trans])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1017
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1018
 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1019
text{*Used only in Hyperreal/Lim.ML*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1020
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1021
apply (simp add: real_add_assoc)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1022
apply (rule_tac a1 = y in add_left_commute [THEN ssubst])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1023
apply (rule real_add_assoc [THEN subst])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1024
apply (rule abs_triangle_ineq)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1025
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1026
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1027
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1028
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
  1029
ML
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
  1030
{*
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1031
val real_lbound_gt_zero = thm"real_lbound_gt_zero";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1032
val real_less_half_sum = thm"real_less_half_sum";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1033
val real_gt_half_sum = thm"real_gt_half_sum";
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14335
diff changeset
  1034
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1035
val abs_interval_iff = thm"abs_interval_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1036
val abs_le_interval_iff = thm"abs_le_interval_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1037
val abs_add_one_gt_zero = thm"abs_add_one_gt_zero";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1038
val abs_add_one_not_less_self = thm"abs_add_one_not_less_self";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1039
val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq";
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
  1040
*}
10752
c4f1bf2acf4c tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents: 10648
diff changeset
  1041
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1042
5588
a3ab526bb891 Revised version with Abelian group simprocs
paulson
parents:
diff changeset
  1043
end