src/HOL/Real/RealDef.thy
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New theory header syntax.
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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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*)
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header{*Defining the Reals from the Positive Reals*}
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theory RealDef
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import PReal
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files ("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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   156
proof -
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   157
  have "congruent2 realrel realrel
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   158
        (\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)"
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diff changeset
   159
    by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
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   160
  thus ?thesis
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   161
    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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   164
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lemma real_add_commute: "(z::real) + w = w + z"
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   166
by (cases z, cases w, simp add: real_add preal_add_ac)
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   167
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   168
lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)"
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   169
by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc)
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   170
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   171
lemma real_add_zero_left: "(0::real) + z = z"
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   172
by (cases z, simp add: real_add real_zero_def preal_add_ac)
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   173
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   174
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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   178
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   179
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subsection{*Additive Inverse on real*}
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   181
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lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
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   183
proof -
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   184
  have "congruent realrel (\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})})"
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   185
    by (simp add: congruent_def preal_add_commute) 
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   186
  thus ?thesis
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   187
    by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
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   188
qed
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   189
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   190
lemma real_add_minus_left: "(-z) + z = (0::real)"
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parents: 14484
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   191
by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute)
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diff changeset
   192
502a7c95de73 conversion of some Real theories to Isar scripts
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diff changeset
   193
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subsection{*Congruence property for multiplication*}
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   195
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   196
lemma real_mult_congruent2_lemma:
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   197
     "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
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   198
          x * x1 + y * y1 + (x * y2 + y * x2) =
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   199
          x * x2 + y * y2 + (x * y1 + y * x1)"
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   200
apply (simp add: preal_add_left_commute preal_add_assoc [symmetric])
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diff changeset
   201
apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric])
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   202
apply (simp add: preal_add_commute)
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   203
done
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diff changeset
   204
502a7c95de73 conversion of some Real theories to Isar scripts
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   205
lemma real_mult_congruent2:
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   206
    "congruent2 realrel realrel (%p1 p2.
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        (%(x1,y1). (%(x2,y2). 
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          { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)"
14658
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diff changeset
   209
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
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   210
apply (simp add: preal_mult_commute preal_add_commute)
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   211
apply (auto simp add: real_mult_congruent2_lemma)
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diff changeset
   212
done
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diff changeset
   213
502a7c95de73 conversion of some Real theories to Isar scripts
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   214
lemma real_mult:
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parents: 14476
diff changeset
   215
      "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
ef8c7c5eb01b new treatment of equivalence classes
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parents: 14476
diff changeset
   216
       Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
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parents: 14476
diff changeset
   217
by (simp add: real_mult_def UN_UN_split_split_eq
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diff changeset
   218
         UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
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diff changeset
   219
502a7c95de73 conversion of some Real theories to Isar scripts
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   220
lemma real_mult_commute: "(z::real) * w = w * z"
14497
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parents: 14484
diff changeset
   221
by (cases z, cases w, simp add: real_mult preal_add_ac preal_mult_ac)
14269
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parents: 13487
diff changeset
   222
502a7c95de73 conversion of some Real theories to Isar scripts
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   223
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
14484
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diff changeset
   224
apply (cases z1, cases z2, cases z3)
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   225
apply (simp add: real_mult preal_add_mult_distrib2 preal_add_ac preal_mult_ac)
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   226
done
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   227
502a7c95de73 conversion of some Real theories to Isar scripts
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   228
lemma real_mult_1: "(1::real) * z = z"
14484
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   229
apply (cases z)
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   230
apply (simp add: real_mult real_one_def preal_add_mult_distrib2
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   231
                 preal_mult_1_right preal_mult_ac preal_add_ac)
14269
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   232
done
502a7c95de73 conversion of some Real theories to Isar scripts
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parents: 13487
diff changeset
   233
502a7c95de73 conversion of some Real theories to Isar scripts
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   234
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
14484
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   235
apply (cases z1, cases z2, cases w)
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parents: 14476
diff changeset
   236
apply (simp add: real_add real_mult preal_add_mult_distrib2 
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parents: 14476
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   237
                 preal_add_ac preal_mult_ac)
14269
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   238
done
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parents: 13487
diff changeset
   239
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   240
text{*one and zero are distinct*}
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   241
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
14484
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parents: 14476
diff changeset
   242
proof -
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parents: 14476
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   243
  have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1"
ef8c7c5eb01b new treatment of equivalence classes
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parents: 14476
diff changeset
   244
    by (simp add: preal_self_less_add_left) 
ef8c7c5eb01b new treatment of equivalence classes
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parents: 14476
diff changeset
   245
  thus ?thesis
ef8c7c5eb01b new treatment of equivalence classes
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parents: 14476
diff changeset
   246
    by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff)
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parents: 14476
diff changeset
   247
qed
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parents: 13487
diff changeset
   248
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   249
subsection{*existence of inverse*}
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diff changeset
   250
14484
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   251
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
14497
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parents: 14484
diff changeset
   252
by (simp add: real_zero_def preal_add_commute)
14269
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parents: 13487
diff changeset
   253
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   254
text{*Instead of using an existential quantifier and constructing the inverse
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diff changeset
   255
within the proof, we could define the inverse explicitly.*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
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diff changeset
   256
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
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diff changeset
   257
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
14484
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parents: 14476
diff changeset
   258
apply (simp add: real_zero_def real_one_def, cases x)
14269
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parents: 13487
diff changeset
   259
apply (cut_tac x = xa and y = y in linorder_less_linear)
14365
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parents: 14348
diff changeset
   260
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
14334
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parents: 14329
diff changeset
   261
apply (rule_tac
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   262
        x = "Abs_Real (realrel `` { (preal_of_rat 1, 
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parents: 14348
diff changeset
   263
                            inverse (D) + preal_of_rat 1)}) " 
14334
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diff changeset
   264
       in exI)
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   265
apply (rule_tac [2]
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diff changeset
   266
        x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1,
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diff changeset
   267
                   preal_of_rat 1)})" 
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   268
       in exI)
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parents: 14348
diff changeset
   269
apply (auto simp add: real_mult preal_mult_1_right
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diff changeset
   270
              preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1
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parents: 14348
diff changeset
   271
              preal_mult_inverse_right preal_add_ac preal_mult_ac)
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   272
done
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parents: 13487
diff changeset
   273
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diff changeset
   274
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
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diff changeset
   275
apply (simp add: real_inverse_def)
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diff changeset
   276
apply (frule real_mult_inverse_left_ex, safe)
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   277
apply (rule someI2, auto)
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   278
done
14334
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parents: 14329
diff changeset
   279
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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parents: 14335
diff changeset
   280
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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parents: 14335
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   281
subsection{*The Real Numbers form a Field*}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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   282
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   283
instance real :: field
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   284
proof
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   285
  fix x y z :: real
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parents: 14329
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   286
  show "- x + x = 0" by (rule real_add_minus_left)
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parents: 14329
diff changeset
   287
  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
   288
  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
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parents: 14329
diff changeset
   289
  show "x * y = y * x" by (rule real_mult_commute)
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parents: 14329
diff changeset
   290
  show "1 * x = x" by (rule real_mult_1)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
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parents: 14329
diff changeset
   291
  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
   292
  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
14365
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paulson
parents: 14348
diff changeset
   293
  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
   294
  show "x / y = x * inverse y" by (simp add: real_divide_def)
14334
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parents: 14329
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   295
qed
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parents: 14329
diff changeset
   296
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
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parents: 14329
diff changeset
   297
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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diff changeset
   298
text{*Inverse of zero!  Useful to simplify certain equations*}
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diff changeset
   299
14334
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   300
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
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   301
by (simp add: real_inverse_def)
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   302
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
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   303
instance real :: division_by_zero
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parents: 14329
diff changeset
   304
proof
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   305
  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   306
qed
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   307
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   308
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   309
(*Pull negations out*)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   310
declare minus_mult_right [symmetric, simp] 
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   311
        minus_mult_left [symmetric, simp]
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   312
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   313
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
   314
  by (rule OrderedGroup.mult_1_right)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   315
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   316
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   317
subsection{*The @{text "\<le>"} Ordering*}
14269
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
lemma real_le_refl: "w \<le> (w::real)"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   320
by (cases w, force simp add: real_le_def)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   321
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   322
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
   323
  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
   324
  following two lemmas.*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   325
lemma preal_eq_le_imp_le:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   326
  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
   327
  shows "b \<le> (d::preal)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   328
proof -
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   329
  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
   330
  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
   331
  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
   332
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   333
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   334
lemma real_le_lemma:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   335
  assumes l: "u1 + v2 \<le> u2 + v1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   336
      and "x1 + v1 = u1 + y1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   337
      and "x2 + v2 = u2 + y2"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   338
  shows "x1 + y2 \<le> x2 + (y1::preal)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   339
proof -
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   340
  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
   341
  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
   342
  also have "... \<le> (x2+y1) + (u2+v1)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   343
         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
   344
  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
   345
qed						 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   346
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   347
lemma real_le: 
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   348
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   349
      (x1 + y2 \<le> x2 + y1)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   350
apply (simp add: real_le_def) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   351
apply (auto intro: real_le_lemma)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   352
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   353
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   354
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
14497
paulson
parents: 14484
diff changeset
   355
by (cases z, cases w, simp add: real_le order_antisym)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   356
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   357
lemma real_trans_lemma:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   358
  assumes "x + v \<le> u + y"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   359
      and "u + v' \<le> u' + v"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   360
      and "x2 + v2 = u2 + y2"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   361
  shows "x + v' \<le> u' + (y::preal)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   362
proof -
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   363
  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
   364
  also have "... \<le> (u+y) + (u+v')" 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   365
    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
   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_left prems) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   368
  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
   369
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   370
qed						 
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   371
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   372
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
   373
apply (cases i, cases j, cases k)
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   374
apply (simp add: real_le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   375
apply (blast intro: real_trans_lemma) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   376
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   377
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   378
(* Axiom 'order_less_le' of class 'order': *)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   379
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
   380
by (simp add: real_less_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   381
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   382
instance real :: order
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   383
proof qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   384
 (assumption |
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   385
  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
   386
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   387
(* Axiom 'linorder_linear' of class 'linorder': *)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   388
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   389
apply (cases z, cases w) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   390
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
   391
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   392
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   393
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   394
instance real :: linorder
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   395
  by (intro_classes, rule real_le_linear)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   396
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   397
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   398
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
   399
apply (cases x, cases y) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   400
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
   401
                      preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   402
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   403
done 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   404
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   405
lemma real_add_left_mono: 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   406
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   407
proof -
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   408
  have "z + x - (z + y) = (z + -z) + (x - y)"
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   409
    by (simp add: diff_minus add_ac) 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   410
  with le show ?thesis 
14754
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   411
    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
   412
qed
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   413
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   414
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
   415
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
   416
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   417
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
   418
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
   419
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   420
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   421
apply (cases x, cases y)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   422
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
   423
                 linorder_not_le [where 'a = preal] 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   424
                  real_zero_def real_le real_mult)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   425
  --{*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
   426
apply (auto  dest!: less_add_left_Ex 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   427
     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
   428
          preal_add_mult_distrib2 preal_cancels preal_self_less_add_right)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   429
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   430
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   431
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
   432
apply (rule real_sum_gt_zero_less)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   433
apply (drule real_less_sum_gt_zero [of x y])
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   434
apply (drule real_mult_order, assumption)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   435
apply (simp add: right_distrib)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   436
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   437
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   438
text{*lemma for proving @{term "0<(1::real)"}*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   439
lemma real_zero_le_one: "0 \<le> (1::real)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   440
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
   441
                 preal_self_less_add_left order_less_imp_le)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   442
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   443
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   444
subsection{*The Reals Form an Ordered Field*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   445
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   446
instance real :: ordered_field
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   447
proof
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   448
  fix x y z :: real
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   449
  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
   450
  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
   451
  show "\<bar>x\<bar> = (if x < 0 then -x else x)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   452
    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
   453
qed
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   454
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   455
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   456
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   457
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
   458
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
   459
positive reals.*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   460
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   461
lemma real_of_preal_add:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   462
     "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
   463
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
   464
              preal_add_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   465
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   466
lemma real_of_preal_mult:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   467
     "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
   468
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
   469
              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
   470
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   471
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   472
text{*Gleason prop 9-4.4 p 127*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   473
lemma real_of_preal_trichotomy:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   474
      "\<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
   475
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
   476
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
   477
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
   478
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
   479
apply (auto simp add: preal_add_commute)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   480
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   481
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   482
lemma real_of_preal_leD:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   483
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   484
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
   485
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   486
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
   487
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
   488
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   489
lemma real_of_preal_lessD:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   490
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   491
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric] 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   492
              preal_cancels) 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   493
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   494
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   495
lemma real_of_preal_less_iff [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   496
     "(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
   497
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
   498
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   499
lemma real_of_preal_le_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   500
     "(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
   501
by (simp add: linorder_not_less [symmetric]) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   502
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   503
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
   504
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
   505
            preal_add_ac preal_cancels)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   506
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
   507
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
   508
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
   509
apply (simp add: preal_add_ac) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   510
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   511
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   512
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
   513
by (simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   514
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   515
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
   516
proof -
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   517
  from real_of_preal_minus_less_zero
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   518
  show ?thesis by (blast dest: order_less_trans)
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   519
qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   520
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   521
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   522
subsection{*Theorems About the Ordering*}
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
text{*obsolete but used a lot*}
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
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
   527
by blast 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   528
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   529
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
   530
by (simp add: order_le_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   531
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   532
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
   533
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
   534
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
   535
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
   536
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   537
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   538
lemma real_gt_preal_preal_Ex:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   539
     "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
   540
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
   541
             intro: real_gt_zero_preal_Ex [THEN iffD1])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   542
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   543
lemma real_ge_preal_preal_Ex:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   544
     "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
   545
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
   546
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   547
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
   548
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
   549
            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
   550
            simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   551
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   552
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
   553
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
   554
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   555
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
   556
  by (rule OrderedGroup.add_less_le_mono)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   557
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   558
lemma real_add_le_less_mono:
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   559
     "!!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
   560
  by (rule OrderedGroup.add_le_less_mono)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   561
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   562
lemma real_le_square [simp]: "(0::real) \<le> x*x"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   563
 by (rule Ring_and_Field.zero_le_square)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   564
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   565
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   566
subsection{*More Lemmas*}
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
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
   569
by auto
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   570
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   571
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
   572
by auto
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   573
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   574
text{*The precondition could be weakened to @{term "0\<le>x"}*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   575
lemma real_mult_less_mono:
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   576
     "[| 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
   577
 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
   578
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   579
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
   580
  by (force elim: order_less_asym
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   581
            simp add: Ring_and_Field.mult_less_cancel_right)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   582
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   583
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
   584
apply (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   585
apply (blast intro: elim: order_less_asym) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   586
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   587
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   588
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   589
  by (force elim: order_less_asym
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   590
            simp add: Ring_and_Field.mult_le_cancel_left)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   591
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   592
text{*Only two uses?*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   593
lemma real_mult_less_mono':
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   594
     "[| 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
   595
 by (rule Ring_and_Field.mult_strict_mono')
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   596
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   597
text{*FIXME: delete or at least combine the next two lemmas*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   598
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
   599
apply (drule OrderedGroup.equals_zero_I [THEN sym])
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   600
apply (cut_tac x = y in real_le_square) 
14476
758e7acdea2f removed redundant thms
paulson
parents: 14443
diff changeset
   601
apply (auto, drule order_antisym, auto)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   602
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   603
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   604
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
   605
apply (rule_tac y = x in real_sum_squares_cancel)
14476
758e7acdea2f removed redundant thms
paulson
parents: 14443
diff changeset
   606
apply (simp add: add_commute)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   607
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   608
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   609
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
   610
by (drule add_strict_mono [of concl: 0 0], assumption, simp)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   611
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   612
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
   613
apply (drule order_le_imp_less_or_eq)+
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   614
apply (auto intro: real_add_order order_less_imp_le)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   615
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   616
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   617
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
   618
apply (case_tac "x \<noteq> 0")
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   619
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
   620
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   621
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   622
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
   623
by (auto dest: less_imp_inverse_less)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   624
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   625
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
   626
proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   627
  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
   628
  thus ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   629
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   630
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   631
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   632
subsection{*Embedding the Integers into the Reals*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   633
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   634
defs (overloaded)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   635
  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
   636
  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
   637
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   638
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
   639
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   640
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   641
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
   642
by (simp add: real_of_int_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   643
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   644
lemma real_of_int_add: "real (x::int) + real y = real (x + y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   645
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   646
declare real_of_int_add [symmetric, simp]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   647
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   648
lemma real_of_int_minus: "-real (x::int) = real (-x)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   649
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   650
declare real_of_int_minus [symmetric, simp]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   651
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   652
lemma real_of_int_diff: "real (x::int) - real y = real (x - y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   653
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   654
declare real_of_int_diff [symmetric, simp]
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   655
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   656
lemma real_of_int_mult: "real (x::int) * real y = real (x * y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   657
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   658
declare real_of_int_mult [symmetric, simp]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   659
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   660
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
   661
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   662
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   663
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
   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
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   666
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
   667
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   668
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   669
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
   670
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   671
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   672
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   673
subsection{*Embedding the Naturals into the Reals*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   674
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   675
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
   676
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   677
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   678
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
   679
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   680
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   681
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
   682
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   683
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   684
(*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
   685
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
   686
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   687
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   688
lemma real_of_nat_less_iff [iff]: 
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   689
     "(real (n::nat) < real m) = (n < m)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   690
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   691
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   692
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
   693
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   694
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   695
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
   696
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
   697
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   698
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
   699
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
   700
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   701
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
   702
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   703
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   704
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
   705
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   706
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   707
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
   708
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   709
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   710
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
   711
by (simp add: add: real_of_nat_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   712
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   713
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
   714
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
   715
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   716
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
   717
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   718
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   719
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
   720
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   721
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   722
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
   723
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   724
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   725
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
   726
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
   727
14426
de890c247b38 fixed bugs in the setup of arithmetic procedures
paulson
parents: 14421
diff changeset
   728
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
   729
by (simp add: real_of_int_def real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   730
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   731
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   732
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   733
subsection{*Numerals and Arithmetic*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   734
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   735
instance real :: number ..
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   736
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   737
defs (overloaded)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   738
  real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   739
    --{*the type constraint is essential!*}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   740
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   741
instance real :: number_ring
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   742
by (intro_classes, simp add: real_number_of_def) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   743
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   744
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   745
text{*Collapse applications of @{term real} to @{term number_of}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   746
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
   747
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
   748
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   749
lemma real_of_nat_number_of [simp]:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   750
     "real (number_of v :: nat) =  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   751
        (if neg (number_of v :: int) then 0  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   752
         else (number_of v :: real))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   753
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
   754
 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   755
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   756
use "real_arith.ML"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   757
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   758
setup real_arith_setup
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   759
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   760
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   761
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   762
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   763
lemma real_0_le_divide_iff:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   764
     "((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
   765
by (simp add: real_divide_def zero_le_mult_iff, auto)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   766
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   767
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
   768
by arith
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   769
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   770
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
   771
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   772
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   773
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
   774
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   775
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   776
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
   777
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   778
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   779
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
   780
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   781
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   782
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
   783
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   784
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   785
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   786
(*
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   787
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
   788
It replaces x+-y by x-y.
15086
e6a2a98d5ef5 removal of more iff-rules from RealDef.thy
paulson
parents: 15085
diff changeset
   789
declare real_diff_def [symmetric, simp]
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   790
*)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   791
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   792
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   793
subsubsection{*Density of the Reals*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   794
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   795
lemma real_lbound_gt_zero:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   796
     "[| (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
   797
apply (rule_tac x = " (min d1 d2) /2" in exI)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   798
apply (simp add: min_def)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   799
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   800
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   801
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   802
text{*Similar results are proved in @{text Ring_and_Field}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   803
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
   804
  by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   805
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   806
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
   807
  by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   808
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   809
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   810
subsection{*Absolute Value Function for the Reals*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   811
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   812
text{*FIXME: these should go!*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   813
lemma abs_eqI1: "(0::real)\<le>x ==> abs x = x"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14754
diff changeset
   814
by (simp add: abs_if)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   815
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   816
lemma abs_eqI2: "(0::real) < x ==> abs x = x"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14754
diff changeset
   817
by (simp add: abs_if)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   818
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   819
lemma abs_minus_eqI2: "x < (0::real) ==> abs x = -x"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14754
diff changeset
   820
by (simp add: abs_if linorder_not_less [symmetric])
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   821
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   822
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
   823
by (simp add: abs_if)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   824
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   825
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
   826
by (force simp add: Ring_and_Field.abs_less_iff)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   827
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   828
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
   829
by (force simp add: OrderedGroup.abs_le_iff)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   830
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   831
(*FIXME: used only once, in SEQ.ML*)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   832
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
   833
by (simp add: abs_if)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   834
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   835
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   836
by (auto intro: abs_eqI1 simp add: real_of_nat_ge_zero)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   837
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   838
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
   839
apply (simp add: linorder_not_less)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   840
apply (auto intro: abs_ge_self [THEN order_trans])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   841
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   842
 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   843
text{*Used only in Hyperreal/Lim.ML*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   844
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
   845
apply (simp add: real_add_assoc)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   846
apply (rule_tac a1 = y in add_left_commute [THEN ssubst])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   847
apply (rule real_add_assoc [THEN subst])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   848
apply (rule abs_triangle_ineq)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   849
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   850
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   851
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   852
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   853
ML
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   854
{*
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   855
val real_lbound_gt_zero = thm"real_lbound_gt_zero";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   856
val real_less_half_sum = thm"real_less_half_sum";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   857
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
   858
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   859
val abs_eqI1 = thm"abs_eqI1";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   860
val abs_eqI2 = thm"abs_eqI2";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   861
val abs_minus_eqI2 = thm"abs_minus_eqI2";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   862
val abs_ge_zero = thm"abs_ge_zero";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   863
val abs_idempotent = thm"abs_idempotent";
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   864
val abs_eq_0 = thm"abs_eq_0";
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   865
val abs_ge_self = thm"abs_ge_self";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   866
val abs_ge_minus_self = thm"abs_ge_minus_self";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   867
val abs_mult = thm"abs_mult";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   868
val abs_inverse = thm"abs_inverse";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   869
val abs_triangle_ineq = thm"abs_triangle_ineq";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   870
val abs_minus_cancel = thm"abs_minus_cancel";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   871
val abs_minus_add_cancel = thm"abs_minus_add_cancel";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   872
val abs_interval_iff = thm"abs_interval_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   873
val abs_le_interval_iff = thm"abs_le_interval_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   874
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
   875
val abs_le_zero_iff = thm"abs_le_zero_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   876
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
   877
val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq";
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   878
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   879
val abs_mult_less = thm"abs_mult_less";
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   880
*}
10752
c4f1bf2acf4c tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents: 10648
diff changeset
   881
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   882
5588
a3ab526bb891 Revised version with Abelian group simprocs
paulson
parents:
diff changeset
   883
end