src/HOL/Real/RealDef.thy
author webertj
Wed Jul 26 19:23:04 2006 +0200 (2006-07-26)
changeset 20217 25b068a99d2b
parent 19765 dfe940911617
child 20485 3078fd2eec7b
permissions -rw-r--r--
linear arithmetic splits certain operators (e.g. min, max, abs)
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(*  Title       : Real/RealDef.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
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    Additional contributions by Jeremy Avigad
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*)
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header{*Defining the Reals from the Positive Reals*}
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theory RealDef
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imports PReal
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uses ("real_arith.ML")
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begin
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definition
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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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definition
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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 = 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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const_syntax (xsymbols)
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  Reals  ("\<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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proof -
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  have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
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        respects2 realrel"
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    by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
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  thus ?thesis
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    by (simp add: real_add_def UN_UN_split_split_eq
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                  UN_equiv_class2 [OF equiv_realrel equiv_realrel])
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qed
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lemma real_add_commute: "(z::real) + w = w + z"
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by (cases z, cases w, simp add: real_add preal_add_ac)
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lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)"
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by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc)
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lemma real_add_zero_left: "(0::real) + z = z"
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by (cases z, simp add: real_add real_zero_def preal_add_ac)
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instance real :: comm_monoid_add
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  by (intro_classes,
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      (assumption | 
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       rule real_add_commute real_add_assoc real_add_zero_left)+)
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subsection{*Additive Inverse on real*}
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lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
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proof -
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  have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
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    by (simp add: congruent_def preal_add_commute) 
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  thus ?thesis
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    by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
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qed
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lemma real_add_minus_left: "(-z) + z = (0::real)"
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by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute)
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subsection{*Congruence property for multiplication*}
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lemma real_mult_congruent2_lemma:
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     "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
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          x * x1 + y * y1 + (x * y2 + y * x2) =
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          x * x2 + y * y2 + (x * y1 + y * x1)"
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apply (simp add: preal_add_left_commute preal_add_assoc [symmetric])
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apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric])
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apply (simp add: preal_add_commute)
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done
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lemma real_mult_congruent2:
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    "(%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)
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     respects2 realrel"
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apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
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apply (simp add: preal_mult_commute preal_add_commute)
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apply (auto simp add: real_mult_congruent2_lemma)
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done
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lemma real_mult:
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      "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
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       Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
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by (simp add: real_mult_def UN_UN_split_split_eq
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         UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
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lemma real_mult_commute: "(z::real) * w = w * z"
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by (cases z, cases w, simp add: real_mult preal_add_ac preal_mult_ac)
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lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
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apply (cases z1, cases z2, cases z3)
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apply (simp add: real_mult preal_add_mult_distrib2 preal_add_ac preal_mult_ac)
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done
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lemma real_mult_1: "(1::real) * z = z"
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apply (cases z)
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apply (simp add: real_mult real_one_def preal_add_mult_distrib2
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                 preal_mult_1_right preal_mult_ac preal_add_ac)
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done
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lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
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apply (cases z1, cases z2, cases w)
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apply (simp add: real_add real_mult preal_add_mult_distrib2 
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                 preal_add_ac preal_mult_ac)
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done
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text{*one and zero are distinct*}
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lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
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proof -
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  have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1"
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    by (simp add: preal_self_less_add_left) 
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  thus ?thesis
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    by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff)
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qed
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subsection{*existence of inverse*}
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lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
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by (simp add: real_zero_def preal_add_commute)
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text{*Instead of using an existential quantifier and constructing the inverse
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within the proof, we could define the inverse explicitly.*}
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lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
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apply (simp add: real_zero_def real_one_def, cases x)
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apply (cut_tac x = xa and y = y in linorder_less_linear)
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apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
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apply (rule_tac
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        x = "Abs_Real (realrel `` { (preal_of_rat 1, 
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                            inverse (D) + preal_of_rat 1)}) " 
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       in exI)
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apply (rule_tac [2]
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        x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1,
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                   preal_of_rat 1)})" 
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       in exI)
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apply (auto simp add: real_mult preal_mult_1_right
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              preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1
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              preal_mult_inverse_right preal_add_ac preal_mult_ac)
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done
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lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
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apply (simp add: real_inverse_def)
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apply (frule real_mult_inverse_left_ex, safe)
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apply (rule someI2, auto)
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done
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subsection{*The Real Numbers form a Field*}
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instance real :: field
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proof
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  fix x y z :: real
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  show "- x + x = 0" by (rule real_add_minus_left)
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  show "x - y = x + (-y)" by (simp add: real_diff_def)
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  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
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  show "x * y = y * x" by (rule real_mult_commute)
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  show "1 * x = x" by (rule real_mult_1)
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  show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib)
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  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
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  show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
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  show "x / y = x * inverse y" by (simp add: real_divide_def)
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qed
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text{*Inverse of zero!  Useful to simplify certain equations*}
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lemma INVERSE_ZERO: "inverse 0 = (0::real)"
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by (simp add: real_inverse_def)
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instance real :: division_by_zero
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proof
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  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
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qed
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(*Pull negations out*)
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declare minus_mult_right [symmetric, simp] 
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        minus_mult_left [symmetric, simp]
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lemma real_mult_1_right: "z * (1::real) = z"
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  by (rule OrderedGroup.mult_1_right)
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paulson@14269
   317
paulson@14365
   318
subsection{*The @{text "\<le>"} Ordering*}
paulson@14269
   319
paulson@14365
   320
lemma real_le_refl: "w \<le> (w::real)"
paulson@14484
   321
by (cases w, force simp add: real_le_def)
paulson@14269
   322
paulson@14378
   323
text{*The arithmetic decision procedure is not set up for type preal.
paulson@14378
   324
  This lemma is currently unused, but it could simplify the proofs of the
paulson@14378
   325
  following two lemmas.*}
paulson@14378
   326
lemma preal_eq_le_imp_le:
paulson@14378
   327
  assumes eq: "a+b = c+d" and le: "c \<le> a"
paulson@14378
   328
  shows "b \<le> (d::preal)"
paulson@14378
   329
proof -
paulson@14378
   330
  have "c+d \<le> a+d" by (simp add: prems preal_cancels)
paulson@14378
   331
  hence "a+b \<le> a+d" by (simp add: prems)
paulson@14378
   332
  thus "b \<le> d" by (simp add: preal_cancels)
paulson@14378
   333
qed
paulson@14378
   334
paulson@14378
   335
lemma real_le_lemma:
paulson@14378
   336
  assumes l: "u1 + v2 \<le> u2 + v1"
paulson@14378
   337
      and "x1 + v1 = u1 + y1"
paulson@14378
   338
      and "x2 + v2 = u2 + y2"
paulson@14378
   339
  shows "x1 + y2 \<le> x2 + (y1::preal)"
paulson@14365
   340
proof -
paulson@14378
   341
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
paulson@14378
   342
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac)
paulson@14378
   343
  also have "... \<le> (x2+y1) + (u2+v1)"
paulson@14365
   344
         by (simp add: prems preal_add_le_cancel_left)
paulson@14378
   345
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
paulson@14378
   346
qed						 
paulson@14378
   347
paulson@14378
   348
lemma real_le: 
paulson@14484
   349
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
paulson@14484
   350
      (x1 + y2 \<le> x2 + y1)"
paulson@14378
   351
apply (simp add: real_le_def) 
paulson@14387
   352
apply (auto intro: real_le_lemma)
paulson@14378
   353
done
paulson@14378
   354
paulson@14378
   355
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
nipkow@15542
   356
by (cases z, cases w, simp add: real_le)
paulson@14378
   357
paulson@14378
   358
lemma real_trans_lemma:
paulson@14378
   359
  assumes "x + v \<le> u + y"
paulson@14378
   360
      and "u + v' \<le> u' + v"
paulson@14378
   361
      and "x2 + v2 = u2 + y2"
paulson@14378
   362
  shows "x + v' \<le> u' + (y::preal)"
paulson@14378
   363
proof -
paulson@14378
   364
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac)
paulson@14378
   365
  also have "... \<le> (u+y) + (u+v')" 
paulson@14378
   366
    by (simp add: preal_add_le_cancel_right prems) 
paulson@14378
   367
  also have "... \<le> (u+y) + (u'+v)" 
paulson@14378
   368
    by (simp add: preal_add_le_cancel_left prems) 
paulson@14378
   369
  also have "... = (u'+y) + (u+v)"  by (simp add: preal_add_ac)
paulson@14378
   370
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
nipkow@15542
   371
qed
paulson@14269
   372
paulson@14365
   373
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
paulson@14484
   374
apply (cases i, cases j, cases k)
paulson@14484
   375
apply (simp add: real_le)
paulson@14378
   376
apply (blast intro: real_trans_lemma) 
paulson@14334
   377
done
paulson@14334
   378
paulson@14334
   379
(* Axiom 'order_less_le' of class 'order': *)
paulson@14334
   380
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
paulson@14365
   381
by (simp add: real_less_def)
paulson@14365
   382
paulson@14365
   383
instance real :: order
paulson@14365
   384
proof qed
paulson@14365
   385
 (assumption |
paulson@14365
   386
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
paulson@14365
   387
paulson@14378
   388
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14378
   389
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
paulson@14484
   390
apply (cases z, cases w) 
paulson@14378
   391
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels)
paulson@14334
   392
done
paulson@14334
   393
paulson@14334
   394
paulson@14334
   395
instance real :: linorder
paulson@14334
   396
  by (intro_classes, rule real_le_linear)
paulson@14334
   397
paulson@14334
   398
paulson@14378
   399
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
paulson@14484
   400
apply (cases x, cases y) 
paulson@14378
   401
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
paulson@14378
   402
                      preal_add_ac)
paulson@14378
   403
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
nipkow@15542
   404
done
paulson@14378
   405
paulson@14484
   406
lemma real_add_left_mono: 
paulson@14484
   407
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
paulson@14484
   408
proof -
paulson@14484
   409
  have "z + x - (z + y) = (z + -z) + (x - y)"
paulson@14484
   410
    by (simp add: diff_minus add_ac) 
paulson@14484
   411
  with le show ?thesis 
obua@14754
   412
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
paulson@14484
   413
qed
paulson@14334
   414
paulson@14365
   415
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
paulson@14365
   416
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14365
   417
paulson@14365
   418
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
paulson@14365
   419
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14334
   420
paulson@14334
   421
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
paulson@14484
   422
apply (cases x, cases y)
paulson@14378
   423
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
paulson@14378
   424
                 linorder_not_le [where 'a = preal] 
paulson@14378
   425
                  real_zero_def real_le real_mult)
paulson@14365
   426
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
wenzelm@16973
   427
apply (auto dest!: less_add_left_Ex
paulson@14365
   428
     simp add: preal_add_ac preal_mult_ac 
wenzelm@16973
   429
          preal_add_mult_distrib2 preal_cancels preal_self_less_add_left)
paulson@14334
   430
done
paulson@14334
   431
paulson@14334
   432
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
paulson@14334
   433
apply (rule real_sum_gt_zero_less)
paulson@14334
   434
apply (drule real_less_sum_gt_zero [of x y])
paulson@14334
   435
apply (drule real_mult_order, assumption)
paulson@14334
   436
apply (simp add: right_distrib)
paulson@14334
   437
done
paulson@14334
   438
paulson@14365
   439
text{*lemma for proving @{term "0<(1::real)"}*}
paulson@14365
   440
lemma real_zero_le_one: "0 \<le> (1::real)"
paulson@14387
   441
by (simp add: real_zero_def real_one_def real_le 
paulson@14378
   442
                 preal_self_less_add_left order_less_imp_le)
paulson@14334
   443
paulson@14378
   444
paulson@14334
   445
subsection{*The Reals Form an Ordered Field*}
paulson@14334
   446
paulson@14334
   447
instance real :: ordered_field
paulson@14334
   448
proof
paulson@14334
   449
  fix x y z :: real
paulson@14334
   450
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
paulson@14334
   451
  show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2)
paulson@14334
   452
  show "\<bar>x\<bar> = (if x < 0 then -x else x)"
paulson@14334
   453
    by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le)
paulson@14334
   454
qed
paulson@14334
   455
paulson@14365
   456
paulson@14365
   457
paulson@14365
   458
text{*The function @{term real_of_preal} requires many proofs, but it seems
paulson@14365
   459
to be essential for proving completeness of the reals from that of the
paulson@14365
   460
positive reals.*}
paulson@14365
   461
paulson@14365
   462
lemma real_of_preal_add:
paulson@14365
   463
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
paulson@14365
   464
by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1 
paulson@14365
   465
              preal_add_ac)
paulson@14365
   466
paulson@14365
   467
lemma real_of_preal_mult:
paulson@14365
   468
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
paulson@14365
   469
by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2
paulson@14365
   470
              preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac)
paulson@14365
   471
paulson@14365
   472
paulson@14365
   473
text{*Gleason prop 9-4.4 p 127*}
paulson@14365
   474
lemma real_of_preal_trichotomy:
paulson@14365
   475
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
paulson@14484
   476
apply (simp add: real_of_preal_def real_zero_def, cases x)
paulson@14365
   477
apply (auto simp add: real_minus preal_add_ac)
paulson@14365
   478
apply (cut_tac x = x and y = y in linorder_less_linear)
paulson@14365
   479
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric])
paulson@14365
   480
done
paulson@14365
   481
paulson@14365
   482
lemma real_of_preal_leD:
paulson@14365
   483
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
paulson@14484
   484
by (simp add: real_of_preal_def real_le preal_cancels)
paulson@14365
   485
paulson@14365
   486
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
paulson@14365
   487
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
paulson@14365
   488
paulson@14365
   489
lemma real_of_preal_lessD:
paulson@14365
   490
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
paulson@14484
   491
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric] 
paulson@14484
   492
              preal_cancels) 
paulson@14484
   493
paulson@14365
   494
paulson@14365
   495
lemma real_of_preal_less_iff [simp]:
paulson@14365
   496
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
paulson@14365
   497
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
paulson@14365
   498
paulson@14365
   499
lemma real_of_preal_le_iff:
paulson@14365
   500
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
paulson@14365
   501
by (simp add: linorder_not_less [symmetric]) 
paulson@14365
   502
paulson@14365
   503
lemma real_of_preal_zero_less: "0 < real_of_preal m"
paulson@14365
   504
apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def
paulson@14365
   505
            preal_add_ac preal_cancels)
paulson@14365
   506
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
paulson@14365
   507
apply (blast intro: preal_self_less_add_left order_less_imp_le)
paulson@14365
   508
apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) 
paulson@14365
   509
apply (simp add: preal_add_ac) 
paulson@14365
   510
done
paulson@14365
   511
paulson@14365
   512
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
paulson@14365
   513
by (simp add: real_of_preal_zero_less)
paulson@14365
   514
paulson@14365
   515
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
paulson@14484
   516
proof -
paulson@14484
   517
  from real_of_preal_minus_less_zero
paulson@14484
   518
  show ?thesis by (blast dest: order_less_trans)
paulson@14484
   519
qed
paulson@14365
   520
paulson@14365
   521
paulson@14365
   522
subsection{*Theorems About the Ordering*}
paulson@14365
   523
paulson@14365
   524
text{*obsolete but used a lot*}
paulson@14365
   525
paulson@14365
   526
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)"
paulson@14365
   527
by blast 
paulson@14365
   528
paulson@14365
   529
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y"
paulson@14365
   530
by (simp add: order_le_less)
paulson@14365
   531
paulson@14365
   532
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
paulson@14365
   533
apply (auto simp add: real_of_preal_zero_less)
paulson@14365
   534
apply (cut_tac x = x in real_of_preal_trichotomy)
paulson@14365
   535
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
paulson@14365
   536
done
paulson@14365
   537
paulson@14365
   538
lemma real_gt_preal_preal_Ex:
paulson@14365
   539
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   540
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
paulson@14365
   541
             intro: real_gt_zero_preal_Ex [THEN iffD1])
paulson@14365
   542
paulson@14365
   543
lemma real_ge_preal_preal_Ex:
paulson@14365
   544
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   545
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
paulson@14365
   546
paulson@14365
   547
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
paulson@14365
   548
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
paulson@14365
   549
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
paulson@14365
   550
            simp add: real_of_preal_zero_less)
paulson@14365
   551
paulson@14365
   552
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
paulson@14365
   553
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
paulson@14365
   554
paulson@14334
   555
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)"
obua@14738
   556
  by (rule OrderedGroup.add_less_le_mono)
paulson@14334
   557
paulson@14334
   558
lemma real_add_le_less_mono:
paulson@14334
   559
     "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z"
obua@14738
   560
  by (rule OrderedGroup.add_le_less_mono)
paulson@14334
   561
paulson@14334
   562
lemma real_le_square [simp]: "(0::real) \<le> x*x"
paulson@14334
   563
 by (rule Ring_and_Field.zero_le_square)
paulson@14334
   564
paulson@14334
   565
paulson@14334
   566
subsection{*More Lemmas*}
paulson@14334
   567
paulson@14334
   568
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14334
   569
by auto
paulson@14334
   570
paulson@14334
   571
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   572
by auto
paulson@14334
   573
paulson@14334
   574
text{*The precondition could be weakened to @{term "0\<le>x"}*}
paulson@14334
   575
lemma real_mult_less_mono:
paulson@14334
   576
     "[| u<v;  x<y;  (0::real) < v;  0 < x |] ==> u*x < v* y"
paulson@14334
   577
 by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
paulson@14334
   578
paulson@14334
   579
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
paulson@14334
   580
  by (force elim: order_less_asym
paulson@14334
   581
            simp add: Ring_and_Field.mult_less_cancel_right)
paulson@14334
   582
paulson@14334
   583
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
paulson@14365
   584
apply (simp add: mult_le_cancel_right)
paulson@14365
   585
apply (blast intro: elim: order_less_asym) 
paulson@14365
   586
done
paulson@14334
   587
paulson@14334
   588
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
nipkow@15923
   589
by(simp add:mult_commute)
paulson@14334
   590
paulson@14334
   591
text{*Only two uses?*}
paulson@14334
   592
lemma real_mult_less_mono':
paulson@14334
   593
     "[| x < y;  r1 < r2;  (0::real) \<le> r1;  0 \<le> x|] ==> r1 * x < r2 * y"
paulson@14334
   594
 by (rule Ring_and_Field.mult_strict_mono')
paulson@14334
   595
paulson@14334
   596
text{*FIXME: delete or at least combine the next two lemmas*}
paulson@14334
   597
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
obua@14738
   598
apply (drule OrderedGroup.equals_zero_I [THEN sym])
paulson@14334
   599
apply (cut_tac x = y in real_le_square) 
paulson@14476
   600
apply (auto, drule order_antisym, auto)
paulson@14334
   601
done
paulson@14334
   602
paulson@14334
   603
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
paulson@14334
   604
apply (rule_tac y = x in real_sum_squares_cancel)
paulson@14476
   605
apply (simp add: add_commute)
paulson@14334
   606
done
paulson@14334
   607
paulson@14334
   608
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y"
paulson@14365
   609
by (drule add_strict_mono [of concl: 0 0], assumption, simp)
paulson@14334
   610
paulson@14334
   611
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y"
paulson@14334
   612
apply (drule order_le_imp_less_or_eq)+
paulson@14334
   613
apply (auto intro: real_add_order order_less_imp_le)
paulson@14334
   614
done
paulson@14334
   615
paulson@14365
   616
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
paulson@14365
   617
apply (case_tac "x \<noteq> 0")
paulson@14365
   618
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto)
paulson@14365
   619
done
paulson@14334
   620
paulson@14365
   621
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
paulson@14365
   622
by (auto dest: less_imp_inverse_less)
paulson@14334
   623
paulson@14365
   624
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
paulson@14365
   625
proof -
paulson@14365
   626
  have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square)
paulson@14365
   627
  thus ?thesis by simp
paulson@14365
   628
qed
paulson@14365
   629
paulson@14334
   630
paulson@14365
   631
subsection{*Embedding the Integers into the Reals*}
paulson@14365
   632
paulson@14378
   633
defs (overloaded)
paulson@14378
   634
  real_of_nat_def: "real z == of_nat z"
paulson@14378
   635
  real_of_int_def: "real z == of_int z"
paulson@14365
   636
avigad@16819
   637
lemma real_eq_of_nat: "real = of_nat"
avigad@16819
   638
  apply (rule ext)
avigad@16819
   639
  apply (unfold real_of_nat_def)
avigad@16819
   640
  apply (rule refl)
avigad@16819
   641
  done
avigad@16819
   642
avigad@16819
   643
lemma real_eq_of_int: "real = of_int"
avigad@16819
   644
  apply (rule ext)
avigad@16819
   645
  apply (unfold real_of_int_def)
avigad@16819
   646
  apply (rule refl)
avigad@16819
   647
  done
avigad@16819
   648
paulson@14365
   649
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378
   650
by (simp add: real_of_int_def) 
paulson@14365
   651
paulson@14365
   652
lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378
   653
by (simp add: real_of_int_def) 
paulson@14334
   654
avigad@16819
   655
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
paulson@14378
   656
by (simp add: real_of_int_def) 
paulson@14365
   657
avigad@16819
   658
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
paulson@14378
   659
by (simp add: real_of_int_def) 
avigad@16819
   660
avigad@16819
   661
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
avigad@16819
   662
by (simp add: real_of_int_def) 
paulson@14365
   663
avigad@16819
   664
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
paulson@14378
   665
by (simp add: real_of_int_def) 
paulson@14334
   666
avigad@16819
   667
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
avigad@16819
   668
  apply (subst real_eq_of_int)+
avigad@16819
   669
  apply (rule of_int_setsum)
avigad@16819
   670
done
avigad@16819
   671
avigad@16819
   672
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
avigad@16819
   673
    (PROD x:A. real(f x))"
avigad@16819
   674
  apply (subst real_eq_of_int)+
avigad@16819
   675
  apply (rule of_int_setprod)
avigad@16819
   676
done
paulson@14365
   677
paulson@14365
   678
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
paulson@14378
   679
by (simp add: real_of_int_def) 
paulson@14365
   680
paulson@14365
   681
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
paulson@14378
   682
by (simp add: real_of_int_def) 
paulson@14365
   683
paulson@14365
   684
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
paulson@14378
   685
by (simp add: real_of_int_def) 
paulson@14365
   686
paulson@14365
   687
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378
   688
by (simp add: real_of_int_def) 
paulson@14365
   689
avigad@16819
   690
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)"
avigad@16819
   691
by (simp add: real_of_int_def) 
avigad@16819
   692
avigad@16819
   693
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)"
avigad@16819
   694
by (simp add: real_of_int_def) 
avigad@16819
   695
avigad@16819
   696
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)"
avigad@16819
   697
by (simp add: real_of_int_def)
avigad@16819
   698
avigad@16819
   699
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)"
avigad@16819
   700
by (simp add: real_of_int_def)
avigad@16819
   701
avigad@16888
   702
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
avigad@16888
   703
by (auto simp add: abs_if)
avigad@16888
   704
avigad@16819
   705
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
avigad@16819
   706
  apply (subgoal_tac "real n + 1 = real (n + 1)")
avigad@16819
   707
  apply (simp del: real_of_int_add)
avigad@16819
   708
  apply auto
avigad@16819
   709
done
avigad@16819
   710
avigad@16819
   711
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
avigad@16819
   712
  apply (subgoal_tac "real m + 1 = real (m + 1)")
avigad@16819
   713
  apply (simp del: real_of_int_add)
avigad@16819
   714
  apply simp
avigad@16819
   715
done
avigad@16819
   716
avigad@16819
   717
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
avigad@16819
   718
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   719
proof -
avigad@16819
   720
  assume "d ~= 0"
avigad@16819
   721
  have "x = (x div d) * d + x mod d"
avigad@16819
   722
    by auto
avigad@16819
   723
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   724
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
avigad@16819
   725
  then have "real x / real d = ... / real d"
avigad@16819
   726
    by simp
avigad@16819
   727
  then show ?thesis
avigad@16819
   728
    by (auto simp add: add_divide_distrib ring_eq_simps prems)
avigad@16819
   729
qed
avigad@16819
   730
avigad@16819
   731
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
avigad@16819
   732
    real(n div d) = real n / real d"
avigad@16819
   733
  apply (frule real_of_int_div_aux [of d n])
avigad@16819
   734
  apply simp
avigad@16819
   735
  apply (simp add: zdvd_iff_zmod_eq_0)
avigad@16819
   736
done
avigad@16819
   737
avigad@16819
   738
lemma real_of_int_div2:
avigad@16819
   739
  "0 <= real (n::int) / real (x) - real (n div x)"
avigad@16819
   740
  apply (case_tac "x = 0")
avigad@16819
   741
  apply simp
avigad@16819
   742
  apply (case_tac "0 < x")
avigad@16819
   743
  apply (simp add: compare_rls)
avigad@16819
   744
  apply (subst real_of_int_div_aux)
avigad@16819
   745
  apply simp
avigad@16819
   746
  apply simp
avigad@16819
   747
  apply (subst zero_le_divide_iff)
avigad@16819
   748
  apply auto
avigad@16819
   749
  apply (simp add: compare_rls)
avigad@16819
   750
  apply (subst real_of_int_div_aux)
avigad@16819
   751
  apply simp
avigad@16819
   752
  apply simp
avigad@16819
   753
  apply (subst zero_le_divide_iff)
avigad@16819
   754
  apply auto
avigad@16819
   755
done
avigad@16819
   756
avigad@16819
   757
lemma real_of_int_div3:
avigad@16819
   758
  "real (n::int) / real (x) - real (n div x) <= 1"
avigad@16819
   759
  apply(case_tac "x = 0")
avigad@16819
   760
  apply simp
avigad@16819
   761
  apply (simp add: compare_rls)
avigad@16819
   762
  apply (subst real_of_int_div_aux)
avigad@16819
   763
  apply assumption
avigad@16819
   764
  apply simp
avigad@16819
   765
  apply (subst divide_le_eq)
avigad@16819
   766
  apply clarsimp
avigad@16819
   767
  apply (rule conjI)
avigad@16819
   768
  apply (rule impI)
avigad@16819
   769
  apply (rule order_less_imp_le)
avigad@16819
   770
  apply simp
avigad@16819
   771
  apply (rule impI)
avigad@16819
   772
  apply (rule order_less_imp_le)
avigad@16819
   773
  apply simp
avigad@16819
   774
done
avigad@16819
   775
avigad@16819
   776
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
avigad@16819
   777
  by (insert real_of_int_div2 [of n x], simp)
paulson@14365
   778
paulson@14365
   779
subsection{*Embedding the Naturals into the Reals*}
paulson@14365
   780
paulson@14334
   781
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365
   782
by (simp add: real_of_nat_def)
paulson@14334
   783
paulson@14334
   784
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365
   785
by (simp add: real_of_nat_def)
paulson@14334
   786
paulson@14365
   787
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378
   788
by (simp add: real_of_nat_def)
paulson@14334
   789
paulson@14334
   790
(*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334
   791
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378
   792
by (simp add: real_of_nat_def)
paulson@14334
   793
paulson@14334
   794
lemma real_of_nat_less_iff [iff]: 
paulson@14334
   795
     "(real (n::nat) < real m) = (n < m)"
paulson@14365
   796
by (simp add: real_of_nat_def)
paulson@14334
   797
paulson@14334
   798
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378
   799
by (simp add: real_of_nat_def)
paulson@14334
   800
paulson@14334
   801
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
paulson@14378
   802
by (simp add: real_of_nat_def zero_le_imp_of_nat)
paulson@14334
   803
paulson@14365
   804
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378
   805
by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365
   806
paulson@14334
   807
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
paulson@14378
   808
by (simp add: real_of_nat_def)
paulson@14334
   809
avigad@16819
   810
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
avigad@16819
   811
    (SUM x:A. real(f x))"
avigad@16819
   812
  apply (subst real_eq_of_nat)+
avigad@16819
   813
  apply (rule of_nat_setsum)
avigad@16819
   814
done
avigad@16819
   815
avigad@16819
   816
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
avigad@16819
   817
    (PROD x:A. real(f x))"
avigad@16819
   818
  apply (subst real_eq_of_nat)+
avigad@16819
   819
  apply (rule of_nat_setprod)
avigad@16819
   820
done
avigad@16819
   821
avigad@16819
   822
lemma real_of_card: "real (card A) = setsum (%x.1) A"
avigad@16819
   823
  apply (subst card_eq_setsum)
avigad@16819
   824
  apply (subst real_of_nat_setsum)
avigad@16819
   825
  apply simp
avigad@16819
   826
done
avigad@16819
   827
paulson@14334
   828
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378
   829
by (simp add: real_of_nat_def)
paulson@14334
   830
paulson@14387
   831
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
paulson@14378
   832
by (simp add: real_of_nat_def)
paulson@14334
   833
paulson@14365
   834
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
paulson@14378
   835
by (simp add: add: real_of_nat_def) 
paulson@14334
   836
paulson@14365
   837
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
paulson@14378
   838
by (simp add: add: real_of_nat_def) 
paulson@14365
   839
paulson@14365
   840
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378
   841
by (simp add: add: real_of_nat_def)
paulson@14334
   842
paulson@14365
   843
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378
   844
by (simp add: add: real_of_nat_def)
paulson@14334
   845
paulson@14365
   846
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"
paulson@14378
   847
by (simp add: add: real_of_nat_def)
paulson@14334
   848
avigad@16819
   849
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
avigad@16819
   850
  apply (subgoal_tac "real n + 1 = real (Suc n)")
avigad@16819
   851
  apply simp
avigad@16819
   852
  apply (auto simp add: real_of_nat_Suc)
avigad@16819
   853
done
avigad@16819
   854
avigad@16819
   855
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
avigad@16819
   856
  apply (subgoal_tac "real m + 1 = real (Suc m)")
avigad@16819
   857
  apply (simp add: less_Suc_eq_le)
avigad@16819
   858
  apply (simp add: real_of_nat_Suc)
avigad@16819
   859
done
avigad@16819
   860
avigad@16819
   861
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
avigad@16819
   862
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   863
proof -
avigad@16819
   864
  assume "0 < d"
avigad@16819
   865
  have "x = (x div d) * d + x mod d"
avigad@16819
   866
    by auto
avigad@16819
   867
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   868
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
avigad@16819
   869
  then have "real x / real d = \<dots> / real d"
avigad@16819
   870
    by simp
avigad@16819
   871
  then show ?thesis
avigad@16819
   872
    by (auto simp add: add_divide_distrib ring_eq_simps prems)
avigad@16819
   873
qed
avigad@16819
   874
avigad@16819
   875
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
avigad@16819
   876
    real(n div d) = real n / real d"
avigad@16819
   877
  apply (frule real_of_nat_div_aux [of d n])
avigad@16819
   878
  apply simp
avigad@16819
   879
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
avigad@16819
   880
  apply assumption
avigad@16819
   881
done
avigad@16819
   882
avigad@16819
   883
lemma real_of_nat_div2:
avigad@16819
   884
  "0 <= real (n::nat) / real (x) - real (n div x)"
avigad@16819
   885
  apply(case_tac "x = 0")
avigad@16819
   886
  apply simp
avigad@16819
   887
  apply (simp add: compare_rls)
avigad@16819
   888
  apply (subst real_of_nat_div_aux)
avigad@16819
   889
  apply assumption
avigad@16819
   890
  apply simp
avigad@16819
   891
  apply (subst zero_le_divide_iff)
avigad@16819
   892
  apply simp
avigad@16819
   893
done
avigad@16819
   894
avigad@16819
   895
lemma real_of_nat_div3:
avigad@16819
   896
  "real (n::nat) / real (x) - real (n div x) <= 1"
avigad@16819
   897
  apply(case_tac "x = 0")
avigad@16819
   898
  apply simp
avigad@16819
   899
  apply (simp add: compare_rls)
avigad@16819
   900
  apply (subst real_of_nat_div_aux)
avigad@16819
   901
  apply assumption
avigad@16819
   902
  apply simp
avigad@16819
   903
done
avigad@16819
   904
avigad@16819
   905
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
avigad@16819
   906
  by (insert real_of_nat_div2 [of n x], simp)
avigad@16819
   907
paulson@14365
   908
lemma real_of_int_real_of_nat: "real (int n) = real n"
paulson@14378
   909
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
paulson@14378
   910
paulson@14426
   911
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
paulson@14426
   912
by (simp add: real_of_int_def real_of_nat_def)
paulson@14334
   913
avigad@16819
   914
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
avigad@16819
   915
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
avigad@16819
   916
  apply force
avigad@16819
   917
  apply (simp only: real_of_int_real_of_nat)
avigad@16819
   918
done
paulson@14387
   919
paulson@14387
   920
subsection{*Numerals and Arithmetic*}
paulson@14387
   921
paulson@14387
   922
instance real :: number ..
paulson@14387
   923
paulson@15013
   924
defs (overloaded)
paulson@15013
   925
  real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)"
paulson@15013
   926
    --{*the type constraint is essential!*}
paulson@14387
   927
paulson@14387
   928
instance real :: number_ring
paulson@15013
   929
by (intro_classes, simp add: real_number_of_def) 
paulson@14387
   930
paulson@14387
   931
text{*Collapse applications of @{term real} to @{term number_of}*}
paulson@14387
   932
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
paulson@14387
   933
by (simp add:  real_of_int_def of_int_number_of_eq)
paulson@14387
   934
paulson@14387
   935
lemma real_of_nat_number_of [simp]:
paulson@14387
   936
     "real (number_of v :: nat) =  
paulson@14387
   937
        (if neg (number_of v :: int) then 0  
paulson@14387
   938
         else (number_of v :: real))"
paulson@14387
   939
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
paulson@14387
   940
 
paulson@14387
   941
paulson@14387
   942
use "real_arith.ML"
paulson@14387
   943
paulson@14387
   944
setup real_arith_setup
paulson@14387
   945
kleing@19023
   946
kleing@19023
   947
lemma real_diff_mult_distrib:
kleing@19023
   948
  fixes a::real
kleing@19023
   949
  shows "a * (b - c) = a * b - a * c" 
kleing@19023
   950
proof -
kleing@19023
   951
  have "a * (b - c) = a * (b + -c)" by simp
kleing@19023
   952
  also have "\<dots> = (b + -c) * a" by simp
kleing@19023
   953
  also have "\<dots> = b*a + (-c)*a" by (rule real_add_mult_distrib)
kleing@19023
   954
  also have "\<dots> = a*b - a*c" by simp
kleing@19023
   955
  finally show ?thesis .
kleing@19023
   956
qed
kleing@19023
   957
kleing@19023
   958
paulson@14387
   959
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
paulson@14387
   960
paulson@14387
   961
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
paulson@14387
   962
lemma real_0_le_divide_iff:
paulson@14387
   963
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
paulson@14387
   964
by (simp add: real_divide_def zero_le_mult_iff, auto)
paulson@14387
   965
paulson@14387
   966
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
paulson@14387
   967
by arith
paulson@14387
   968
paulson@15085
   969
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
paulson@14387
   970
by auto
paulson@14387
   971
paulson@15085
   972
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
paulson@14387
   973
by auto
paulson@14387
   974
paulson@15085
   975
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
paulson@14387
   976
by auto
paulson@14387
   977
paulson@15085
   978
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
paulson@14387
   979
by auto
paulson@14387
   980
paulson@15085
   981
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
paulson@14387
   982
by auto
paulson@14387
   983
paulson@14387
   984
paulson@14387
   985
(*
paulson@14387
   986
FIXME: we should have this, as for type int, but many proofs would break.
paulson@14387
   987
It replaces x+-y by x-y.
paulson@15086
   988
declare real_diff_def [symmetric, simp]
paulson@14387
   989
*)
paulson@14387
   990
paulson@14387
   991
paulson@14387
   992
subsubsection{*Density of the Reals*}
paulson@14387
   993
paulson@14387
   994
lemma real_lbound_gt_zero:
paulson@14387
   995
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
paulson@14387
   996
apply (rule_tac x = " (min d1 d2) /2" in exI)
paulson@14387
   997
apply (simp add: min_def)
paulson@14387
   998
done
paulson@14387
   999
paulson@14387
  1000
paulson@14387
  1001
text{*Similar results are proved in @{text Ring_and_Field}*}
paulson@14387
  1002
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
paulson@14387
  1003
  by auto
paulson@14387
  1004
paulson@14387
  1005
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
paulson@14387
  1006
  by auto
paulson@14387
  1007
paulson@14387
  1008
paulson@14387
  1009
subsection{*Absolute Value Function for the Reals*}
paulson@14387
  1010
paulson@14387
  1011
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
paulson@15003
  1012
by (simp add: abs_if)
paulson@14387
  1013
paulson@14387
  1014
lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))"
paulson@14387
  1015
by (force simp add: Ring_and_Field.abs_less_iff)
paulson@14387
  1016
paulson@14387
  1017
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
obua@14738
  1018
by (force simp add: OrderedGroup.abs_le_iff)
paulson@14387
  1019
paulson@14387
  1020
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
paulson@15003
  1021
by (simp add: abs_if)
paulson@14387
  1022
paulson@14387
  1023
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
paulson@15229
  1024
by (simp add: real_of_nat_ge_zero)
paulson@14387
  1025
paulson@14387
  1026
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
webertj@20217
  1027
by simp
paulson@14387
  1028
 
paulson@14387
  1029
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
webertj@20217
  1030
by simp
paulson@14387
  1031
paulson@5588
  1032
end