src/HOL/Real/RealDef.thy
author haftmann
Tue Nov 06 08:47:25 2007 +0100 (2007-11-06)
changeset 25303 0699e20feabd
parent 25162 ad4d5365d9d8
child 25502 9200b36280c0
permissions -rw-r--r--
renamed lordered_*_* to lordered_*_add_*; further localization
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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" where
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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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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" where
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  "real_of_preal m = Abs_Real(realrel``{(m + 1, 1)})"
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instance real :: zero
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  real_zero_def: "0 == Abs_Real(realrel``{(1, 1)})" ..
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lemmas [code func del] = real_zero_def
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instance real :: one
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  real_one_def: "1 == Abs_Real(realrel``{(1 + 1, 1)})" ..
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lemmas [code func del] = real_one_def
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instance real :: plus
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  real_add_def: "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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lemmas [code func del] = real_add_def
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instance real :: minus
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  real_minus_def: "- r ==  contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
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  real_diff_def: "r - (s::real) == r + - s" ..
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lemmas [code func del] = real_minus_def real_diff_def
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instance real :: times
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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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lemmas [code func del] = real_mult_def
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instance real :: inverse
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  real_inverse_def: "inverse (R::real) == (THE S. (R = 0 & S = 0) | S * R = 1)"
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  real_divide_def: "R / (S::real) == R * inverse S" ..
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lemmas [code func del] = real_inverse_def real_divide_def
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instance real :: ord
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  real_le_def: "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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lemmas [code func del] = real_le_def real_less_def
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instance real :: abs
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  real_abs_def:  "abs (r::real) == (if r < 0 then - r else r)" ..
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instance real :: sgn
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  real_sgn_def: "sgn x == (if x=0 then 0 else if 0<x then 1 else - 1)" ..
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subsection {* Equivalence relation over positive reals *}
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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: 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: 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: add_ac)
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  finally have "(x1 + y2) + x = (x2 + y1) + x" .
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  thus ?thesis by (rule add_right_imp_eq)
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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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declare Abs_Real_inject [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 {* Addition and Subtraction *}
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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: add_assoc)
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apply (rule add_left_commute [of ab, THEN ssubst])
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apply (simp add: add_assoc [symmetric])
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apply (simp add: 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_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 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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instance real :: ab_group_add
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proof
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  fix x y z :: real
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  show "(x + y) + z = x + (y + z)"
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    by (cases x, cases y, cases z, simp add: real_add add_assoc)
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  show "x + y = y + x"
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    by (cases x, cases y, simp add: real_add add_commute)
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  show "0 + x = x"
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    by (cases x, simp add: real_add real_zero_def add_ac)
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  show "- x + x = 0"
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    by (cases x, simp add: real_minus real_add real_zero_def add_commute)
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  show "x - y = x + - y"
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    by (simp add: real_diff_def)
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qed
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subsection {* 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: add_left_commute add_assoc [symmetric])
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apply (simp add: add_assoc right_distrib [symmetric])
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apply (simp add: 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: mult_commute 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 add_ac 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 right_distrib add_ac 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 right_distrib
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                  mult_1_right mult_ac 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 right_distrib add_ac 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 "(1::preal) < 1 + 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)
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qed
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instance real :: comm_ring_1
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proof
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  fix x y z :: real
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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 (rule real_add_mult_distrib)
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  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
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qed
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subsection {* Inverse and Division *}
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lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
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by (simp add: real_zero_def 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``{(1, inverse (D) + 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) + 1, 1)})" 
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       in exI)
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apply (auto simp add: real_mult preal_mult_inverse_right ring_simps)
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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 (drule real_mult_inverse_left_ex, safe)
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apply (rule theI, assumption, rename_tac z)
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apply (subgoal_tac "(z * x) * y = z * (x * y)")
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apply (simp add: mult_commute)
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apply (rule mult_assoc)
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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 \<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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subsection{*The @{text "\<le>"} Ordering*}
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lemma real_le_refl: "w \<le> (w::real)"
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by (cases w, force simp add: real_le_def)
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text{*The arithmetic decision procedure is not set up for type preal.
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  This lemma is currently unused, but it could simplify the proofs of the
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  following two lemmas.*}
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lemma preal_eq_le_imp_le:
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  assumes eq: "a+b = c+d" and le: "c \<le> a"
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  shows "b \<le> (d::preal)"
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proof -
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  have "c+d \<le> a+d" by (simp add: prems)
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  hence "a+b \<le> a+d" by (simp add: prems)
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  thus "b \<le> d" by simp
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qed
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lemma real_le_lemma:
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  assumes l: "u1 + v2 \<le> u2 + v1"
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      and "x1 + v1 = u1 + y1"
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      and "x2 + v2 = u2 + y2"
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  shows "x1 + y2 \<le> x2 + (y1::preal)"
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proof -
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  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
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  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
huffman@23288
   310
  also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
huffman@23288
   311
  finally show ?thesis by simp
huffman@23288
   312
qed
paulson@14378
   313
paulson@14378
   314
lemma real_le: 
paulson@14484
   315
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
paulson@14484
   316
      (x1 + y2 \<le> x2 + y1)"
huffman@23288
   317
apply (simp add: real_le_def)
paulson@14387
   318
apply (auto intro: real_le_lemma)
paulson@14378
   319
done
paulson@14378
   320
paulson@14378
   321
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
nipkow@15542
   322
by (cases z, cases w, simp add: real_le)
paulson@14378
   323
paulson@14378
   324
lemma real_trans_lemma:
paulson@14378
   325
  assumes "x + v \<le> u + y"
paulson@14378
   326
      and "u + v' \<le> u' + v"
paulson@14378
   327
      and "x2 + v2 = u2 + y2"
paulson@14378
   328
  shows "x + v' \<le> u' + (y::preal)"
paulson@14378
   329
proof -
huffman@23288
   330
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
huffman@23288
   331
  also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
huffman@23288
   332
  also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
huffman@23288
   333
  also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)
huffman@23288
   334
  finally show ?thesis by simp
nipkow@15542
   335
qed
paulson@14269
   336
paulson@14365
   337
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
paulson@14484
   338
apply (cases i, cases j, cases k)
paulson@14484
   339
apply (simp add: real_le)
huffman@23288
   340
apply (blast intro: real_trans_lemma)
paulson@14334
   341
done
paulson@14334
   342
paulson@14334
   343
(* Axiom 'order_less_le' of class 'order': *)
paulson@14334
   344
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
paulson@14365
   345
by (simp add: real_less_def)
paulson@14365
   346
paulson@14365
   347
instance real :: order
paulson@14365
   348
proof qed
paulson@14365
   349
 (assumption |
paulson@14365
   350
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
paulson@14365
   351
paulson@14378
   352
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14378
   353
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
huffman@23288
   354
apply (cases z, cases w)
huffman@23288
   355
apply (auto simp add: real_le real_zero_def add_ac)
paulson@14334
   356
done
paulson@14334
   357
paulson@14334
   358
paulson@14334
   359
instance real :: linorder
paulson@14334
   360
  by (intro_classes, rule real_le_linear)
paulson@14334
   361
paulson@14334
   362
paulson@14378
   363
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
paulson@14484
   364
apply (cases x, cases y) 
paulson@14378
   365
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
huffman@23288
   366
                      add_ac)
huffman@23288
   367
apply (simp_all add: add_assoc [symmetric])
nipkow@15542
   368
done
paulson@14378
   369
paulson@14484
   370
lemma real_add_left_mono: 
paulson@14484
   371
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
paulson@14484
   372
proof -
paulson@14484
   373
  have "z + x - (z + y) = (z + -z) + (x - y)"
paulson@14484
   374
    by (simp add: diff_minus add_ac) 
paulson@14484
   375
  with le show ?thesis 
obua@14754
   376
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
paulson@14484
   377
qed
paulson@14334
   378
paulson@14365
   379
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
paulson@14365
   380
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14365
   381
paulson@14365
   382
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
paulson@14365
   383
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14334
   384
paulson@14334
   385
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
paulson@14484
   386
apply (cases x, cases y)
paulson@14378
   387
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
paulson@14378
   388
                 linorder_not_le [where 'a = preal] 
paulson@14378
   389
                  real_zero_def real_le real_mult)
paulson@14365
   390
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
wenzelm@16973
   391
apply (auto dest!: less_add_left_Ex
huffman@23288
   392
     simp add: add_ac mult_ac
huffman@23288
   393
          right_distrib preal_self_less_add_left)
paulson@14334
   394
done
paulson@14334
   395
paulson@14334
   396
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
paulson@14334
   397
apply (rule real_sum_gt_zero_less)
paulson@14334
   398
apply (drule real_less_sum_gt_zero [of x y])
paulson@14334
   399
apply (drule real_mult_order, assumption)
paulson@14334
   400
apply (simp add: right_distrib)
paulson@14334
   401
done
paulson@14334
   402
haftmann@22456
   403
instance real :: distrib_lattice
haftmann@22456
   404
  "inf x y \<equiv> min x y"
haftmann@22456
   405
  "sup x y \<equiv> max x y"
haftmann@22456
   406
  by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
haftmann@22456
   407
paulson@14378
   408
paulson@14334
   409
subsection{*The Reals Form an Ordered Field*}
paulson@14334
   410
paulson@14334
   411
instance real :: ordered_field
paulson@14334
   412
proof
paulson@14334
   413
  fix x y z :: real
paulson@14334
   414
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
huffman@22962
   415
  show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
huffman@22962
   416
  show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
nipkow@24506
   417
  show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
nipkow@24506
   418
    by (simp only: real_sgn_def)
paulson@14334
   419
qed
paulson@14334
   420
haftmann@25303
   421
instance real :: lordered_ab_group_add ..
haftmann@25303
   422
paulson@14365
   423
text{*The function @{term real_of_preal} requires many proofs, but it seems
paulson@14365
   424
to be essential for proving completeness of the reals from that of the
paulson@14365
   425
positive reals.*}
paulson@14365
   426
paulson@14365
   427
lemma real_of_preal_add:
paulson@14365
   428
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
huffman@23288
   429
by (simp add: real_of_preal_def real_add left_distrib add_ac)
paulson@14365
   430
paulson@14365
   431
lemma real_of_preal_mult:
paulson@14365
   432
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
huffman@23288
   433
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac)
paulson@14365
   434
paulson@14365
   435
paulson@14365
   436
text{*Gleason prop 9-4.4 p 127*}
paulson@14365
   437
lemma real_of_preal_trichotomy:
paulson@14365
   438
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
paulson@14484
   439
apply (simp add: real_of_preal_def real_zero_def, cases x)
huffman@23288
   440
apply (auto simp add: real_minus add_ac)
paulson@14365
   441
apply (cut_tac x = x and y = y in linorder_less_linear)
huffman@23288
   442
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
paulson@14365
   443
done
paulson@14365
   444
paulson@14365
   445
lemma real_of_preal_leD:
paulson@14365
   446
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
huffman@23288
   447
by (simp add: real_of_preal_def real_le)
paulson@14365
   448
paulson@14365
   449
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
paulson@14365
   450
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
paulson@14365
   451
paulson@14365
   452
lemma real_of_preal_lessD:
paulson@14365
   453
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
huffman@23288
   454
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
paulson@14365
   455
paulson@14365
   456
lemma real_of_preal_less_iff [simp]:
paulson@14365
   457
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
paulson@14365
   458
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
paulson@14365
   459
paulson@14365
   460
lemma real_of_preal_le_iff:
paulson@14365
   461
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
huffman@23288
   462
by (simp add: linorder_not_less [symmetric])
paulson@14365
   463
paulson@14365
   464
lemma real_of_preal_zero_less: "0 < real_of_preal m"
huffman@23288
   465
apply (insert preal_self_less_add_left [of 1 m])
huffman@23288
   466
apply (auto simp add: real_zero_def real_of_preal_def
huffman@23288
   467
                      real_less_def real_le_def add_ac)
huffman@23288
   468
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
huffman@23288
   469
apply (simp add: add_ac)
paulson@14365
   470
done
paulson@14365
   471
paulson@14365
   472
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
paulson@14365
   473
by (simp add: real_of_preal_zero_less)
paulson@14365
   474
paulson@14365
   475
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
paulson@14484
   476
proof -
paulson@14484
   477
  from real_of_preal_minus_less_zero
paulson@14484
   478
  show ?thesis by (blast dest: order_less_trans)
paulson@14484
   479
qed
paulson@14365
   480
paulson@14365
   481
paulson@14365
   482
subsection{*Theorems About the Ordering*}
paulson@14365
   483
paulson@14365
   484
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
paulson@14365
   485
apply (auto simp add: real_of_preal_zero_less)
paulson@14365
   486
apply (cut_tac x = x in real_of_preal_trichotomy)
paulson@14365
   487
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
paulson@14365
   488
done
paulson@14365
   489
paulson@14365
   490
lemma real_gt_preal_preal_Ex:
paulson@14365
   491
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   492
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
paulson@14365
   493
             intro: real_gt_zero_preal_Ex [THEN iffD1])
paulson@14365
   494
paulson@14365
   495
lemma real_ge_preal_preal_Ex:
paulson@14365
   496
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   497
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
paulson@14365
   498
paulson@14365
   499
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
paulson@14365
   500
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
paulson@14365
   501
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
paulson@14365
   502
            simp add: real_of_preal_zero_less)
paulson@14365
   503
paulson@14365
   504
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
paulson@14365
   505
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
paulson@14365
   506
paulson@14334
   507
paulson@14334
   508
subsection{*More Lemmas*}
paulson@14334
   509
paulson@14334
   510
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14334
   511
by auto
paulson@14334
   512
paulson@14334
   513
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   514
by auto
paulson@14334
   515
paulson@14334
   516
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
paulson@14334
   517
  by (force elim: order_less_asym
paulson@14334
   518
            simp add: Ring_and_Field.mult_less_cancel_right)
paulson@14334
   519
paulson@14334
   520
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
paulson@14365
   521
apply (simp add: mult_le_cancel_right)
huffman@23289
   522
apply (blast intro: elim: order_less_asym)
paulson@14365
   523
done
paulson@14334
   524
paulson@14334
   525
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
nipkow@15923
   526
by(simp add:mult_commute)
paulson@14334
   527
paulson@14365
   528
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
huffman@23289
   529
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *)
paulson@14334
   530
paulson@14334
   531
haftmann@24198
   532
subsection {* Embedding numbers into the Reals *}
haftmann@24198
   533
haftmann@24198
   534
abbreviation
haftmann@24198
   535
  real_of_nat :: "nat \<Rightarrow> real"
haftmann@24198
   536
where
haftmann@24198
   537
  "real_of_nat \<equiv> of_nat"
haftmann@24198
   538
haftmann@24198
   539
abbreviation
haftmann@24198
   540
  real_of_int :: "int \<Rightarrow> real"
haftmann@24198
   541
where
haftmann@24198
   542
  "real_of_int \<equiv> of_int"
haftmann@24198
   543
haftmann@24198
   544
abbreviation
haftmann@24198
   545
  real_of_rat :: "rat \<Rightarrow> real"
haftmann@24198
   546
where
haftmann@24198
   547
  "real_of_rat \<equiv> of_rat"
haftmann@24198
   548
haftmann@24198
   549
consts
haftmann@24198
   550
  (*overloaded constant for injecting other types into "real"*)
haftmann@24198
   551
  real :: "'a => real"
paulson@14365
   552
paulson@14378
   553
defs (overloaded)
berghofe@24534
   554
  real_of_nat_def [code inline]: "real == real_of_nat"
berghofe@24534
   555
  real_of_int_def [code inline]: "real == real_of_int"
paulson@14365
   556
avigad@16819
   557
lemma real_eq_of_nat: "real = of_nat"
haftmann@24198
   558
  unfolding real_of_nat_def ..
avigad@16819
   559
avigad@16819
   560
lemma real_eq_of_int: "real = of_int"
haftmann@24198
   561
  unfolding real_of_int_def ..
avigad@16819
   562
paulson@14365
   563
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378
   564
by (simp add: real_of_int_def) 
paulson@14365
   565
paulson@14365
   566
lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378
   567
by (simp add: real_of_int_def) 
paulson@14334
   568
avigad@16819
   569
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
paulson@14378
   570
by (simp add: real_of_int_def) 
paulson@14365
   571
avigad@16819
   572
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
paulson@14378
   573
by (simp add: real_of_int_def) 
avigad@16819
   574
avigad@16819
   575
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
avigad@16819
   576
by (simp add: real_of_int_def) 
paulson@14365
   577
avigad@16819
   578
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
paulson@14378
   579
by (simp add: real_of_int_def) 
paulson@14334
   580
avigad@16819
   581
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
avigad@16819
   582
  apply (subst real_eq_of_int)+
avigad@16819
   583
  apply (rule of_int_setsum)
avigad@16819
   584
done
avigad@16819
   585
avigad@16819
   586
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
avigad@16819
   587
    (PROD x:A. real(f x))"
avigad@16819
   588
  apply (subst real_eq_of_int)+
avigad@16819
   589
  apply (rule of_int_setprod)
avigad@16819
   590
done
paulson@14365
   591
paulson@14365
   592
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
paulson@14378
   593
by (simp add: real_of_int_def) 
paulson@14365
   594
paulson@14365
   595
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
paulson@14378
   596
by (simp add: real_of_int_def) 
paulson@14365
   597
paulson@14365
   598
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
paulson@14378
   599
by (simp add: real_of_int_def) 
paulson@14365
   600
paulson@14365
   601
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378
   602
by (simp add: real_of_int_def) 
paulson@14365
   603
avigad@16819
   604
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)"
avigad@16819
   605
by (simp add: real_of_int_def) 
avigad@16819
   606
avigad@16819
   607
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)"
avigad@16819
   608
by (simp add: real_of_int_def) 
avigad@16819
   609
avigad@16819
   610
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)"
avigad@16819
   611
by (simp add: real_of_int_def)
avigad@16819
   612
avigad@16819
   613
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)"
avigad@16819
   614
by (simp add: real_of_int_def)
avigad@16819
   615
avigad@16888
   616
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
avigad@16888
   617
by (auto simp add: abs_if)
avigad@16888
   618
avigad@16819
   619
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
avigad@16819
   620
  apply (subgoal_tac "real n + 1 = real (n + 1)")
avigad@16819
   621
  apply (simp del: real_of_int_add)
avigad@16819
   622
  apply auto
avigad@16819
   623
done
avigad@16819
   624
avigad@16819
   625
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
avigad@16819
   626
  apply (subgoal_tac "real m + 1 = real (m + 1)")
avigad@16819
   627
  apply (simp del: real_of_int_add)
avigad@16819
   628
  apply simp
avigad@16819
   629
done
avigad@16819
   630
avigad@16819
   631
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
avigad@16819
   632
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   633
proof -
avigad@16819
   634
  assume "d ~= 0"
avigad@16819
   635
  have "x = (x div d) * d + x mod d"
avigad@16819
   636
    by auto
avigad@16819
   637
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   638
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
avigad@16819
   639
  then have "real x / real d = ... / real d"
avigad@16819
   640
    by simp
avigad@16819
   641
  then show ?thesis
nipkow@23477
   642
    by (auto simp add: add_divide_distrib ring_simps prems)
avigad@16819
   643
qed
avigad@16819
   644
avigad@16819
   645
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
avigad@16819
   646
    real(n div d) = real n / real d"
avigad@16819
   647
  apply (frule real_of_int_div_aux [of d n])
avigad@16819
   648
  apply simp
avigad@16819
   649
  apply (simp add: zdvd_iff_zmod_eq_0)
avigad@16819
   650
done
avigad@16819
   651
avigad@16819
   652
lemma real_of_int_div2:
avigad@16819
   653
  "0 <= real (n::int) / real (x) - real (n div x)"
avigad@16819
   654
  apply (case_tac "x = 0")
avigad@16819
   655
  apply simp
avigad@16819
   656
  apply (case_tac "0 < x")
avigad@16819
   657
  apply (simp add: compare_rls)
avigad@16819
   658
  apply (subst real_of_int_div_aux)
avigad@16819
   659
  apply simp
avigad@16819
   660
  apply simp
avigad@16819
   661
  apply (subst zero_le_divide_iff)
avigad@16819
   662
  apply auto
avigad@16819
   663
  apply (simp add: compare_rls)
avigad@16819
   664
  apply (subst real_of_int_div_aux)
avigad@16819
   665
  apply simp
avigad@16819
   666
  apply simp
avigad@16819
   667
  apply (subst zero_le_divide_iff)
avigad@16819
   668
  apply auto
avigad@16819
   669
done
avigad@16819
   670
avigad@16819
   671
lemma real_of_int_div3:
avigad@16819
   672
  "real (n::int) / real (x) - real (n div x) <= 1"
avigad@16819
   673
  apply(case_tac "x = 0")
avigad@16819
   674
  apply simp
avigad@16819
   675
  apply (simp add: compare_rls)
avigad@16819
   676
  apply (subst real_of_int_div_aux)
avigad@16819
   677
  apply assumption
avigad@16819
   678
  apply simp
avigad@16819
   679
  apply (subst divide_le_eq)
avigad@16819
   680
  apply clarsimp
avigad@16819
   681
  apply (rule conjI)
avigad@16819
   682
  apply (rule impI)
avigad@16819
   683
  apply (rule order_less_imp_le)
avigad@16819
   684
  apply simp
avigad@16819
   685
  apply (rule impI)
avigad@16819
   686
  apply (rule order_less_imp_le)
avigad@16819
   687
  apply simp
avigad@16819
   688
done
avigad@16819
   689
avigad@16819
   690
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
avigad@16819
   691
  by (insert real_of_int_div2 [of n x], simp)
paulson@14365
   692
paulson@14365
   693
subsection{*Embedding the Naturals into the Reals*}
paulson@14365
   694
paulson@14334
   695
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365
   696
by (simp add: real_of_nat_def)
paulson@14334
   697
paulson@14334
   698
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365
   699
by (simp add: real_of_nat_def)
paulson@14334
   700
paulson@14365
   701
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378
   702
by (simp add: real_of_nat_def)
paulson@14334
   703
paulson@14334
   704
(*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334
   705
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378
   706
by (simp add: real_of_nat_def)
paulson@14334
   707
paulson@14334
   708
lemma real_of_nat_less_iff [iff]: 
paulson@14334
   709
     "(real (n::nat) < real m) = (n < m)"
paulson@14365
   710
by (simp add: real_of_nat_def)
paulson@14334
   711
paulson@14334
   712
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378
   713
by (simp add: real_of_nat_def)
paulson@14334
   714
paulson@14334
   715
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
paulson@14378
   716
by (simp add: real_of_nat_def zero_le_imp_of_nat)
paulson@14334
   717
paulson@14365
   718
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378
   719
by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365
   720
paulson@14334
   721
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
huffman@23431
   722
by (simp add: real_of_nat_def of_nat_mult)
paulson@14334
   723
avigad@16819
   724
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
avigad@16819
   725
    (SUM x:A. real(f x))"
avigad@16819
   726
  apply (subst real_eq_of_nat)+
avigad@16819
   727
  apply (rule of_nat_setsum)
avigad@16819
   728
done
avigad@16819
   729
avigad@16819
   730
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
avigad@16819
   731
    (PROD x:A. real(f x))"
avigad@16819
   732
  apply (subst real_eq_of_nat)+
avigad@16819
   733
  apply (rule of_nat_setprod)
avigad@16819
   734
done
avigad@16819
   735
avigad@16819
   736
lemma real_of_card: "real (card A) = setsum (%x.1) A"
avigad@16819
   737
  apply (subst card_eq_setsum)
avigad@16819
   738
  apply (subst real_of_nat_setsum)
avigad@16819
   739
  apply simp
avigad@16819
   740
done
avigad@16819
   741
paulson@14334
   742
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378
   743
by (simp add: real_of_nat_def)
paulson@14334
   744
paulson@14387
   745
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
paulson@14378
   746
by (simp add: real_of_nat_def)
paulson@14334
   747
paulson@14365
   748
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
huffman@23438
   749
by (simp add: add: real_of_nat_def of_nat_diff)
paulson@14334
   750
nipkow@25162
   751
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
nipkow@25140
   752
by (auto simp: real_of_nat_def)
paulson@14365
   753
paulson@14365
   754
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378
   755
by (simp add: add: real_of_nat_def)
paulson@14334
   756
paulson@14365
   757
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378
   758
by (simp add: add: real_of_nat_def)
paulson@14334
   759
nipkow@25140
   760
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat))"
paulson@14378
   761
by (simp add: add: real_of_nat_def)
paulson@14334
   762
avigad@16819
   763
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
avigad@16819
   764
  apply (subgoal_tac "real n + 1 = real (Suc n)")
avigad@16819
   765
  apply simp
avigad@16819
   766
  apply (auto simp add: real_of_nat_Suc)
avigad@16819
   767
done
avigad@16819
   768
avigad@16819
   769
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
avigad@16819
   770
  apply (subgoal_tac "real m + 1 = real (Suc m)")
avigad@16819
   771
  apply (simp add: less_Suc_eq_le)
avigad@16819
   772
  apply (simp add: real_of_nat_Suc)
avigad@16819
   773
done
avigad@16819
   774
avigad@16819
   775
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
avigad@16819
   776
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   777
proof -
avigad@16819
   778
  assume "0 < d"
avigad@16819
   779
  have "x = (x div d) * d + x mod d"
avigad@16819
   780
    by auto
avigad@16819
   781
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   782
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
avigad@16819
   783
  then have "real x / real d = \<dots> / real d"
avigad@16819
   784
    by simp
avigad@16819
   785
  then show ?thesis
nipkow@23477
   786
    by (auto simp add: add_divide_distrib ring_simps prems)
avigad@16819
   787
qed
avigad@16819
   788
avigad@16819
   789
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
avigad@16819
   790
    real(n div d) = real n / real d"
avigad@16819
   791
  apply (frule real_of_nat_div_aux [of d n])
avigad@16819
   792
  apply simp
avigad@16819
   793
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
avigad@16819
   794
  apply assumption
avigad@16819
   795
done
avigad@16819
   796
avigad@16819
   797
lemma real_of_nat_div2:
avigad@16819
   798
  "0 <= real (n::nat) / real (x) - real (n div x)"
nipkow@25134
   799
apply(case_tac "x = 0")
nipkow@25134
   800
 apply (simp)
nipkow@25134
   801
apply (simp add: compare_rls)
nipkow@25134
   802
apply (subst real_of_nat_div_aux)
nipkow@25134
   803
 apply simp
nipkow@25134
   804
apply simp
nipkow@25134
   805
apply (subst zero_le_divide_iff)
nipkow@25134
   806
apply simp
avigad@16819
   807
done
avigad@16819
   808
avigad@16819
   809
lemma real_of_nat_div3:
avigad@16819
   810
  "real (n::nat) / real (x) - real (n div x) <= 1"
nipkow@25134
   811
apply(case_tac "x = 0")
nipkow@25134
   812
apply (simp)
nipkow@25134
   813
apply (simp add: compare_rls)
nipkow@25134
   814
apply (subst real_of_nat_div_aux)
nipkow@25134
   815
 apply simp
nipkow@25134
   816
apply simp
avigad@16819
   817
done
avigad@16819
   818
avigad@16819
   819
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
avigad@16819
   820
  by (insert real_of_nat_div2 [of n x], simp)
avigad@16819
   821
paulson@14365
   822
lemma real_of_int_real_of_nat: "real (int n) = real n"
paulson@14378
   823
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
paulson@14378
   824
paulson@14426
   825
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
paulson@14426
   826
by (simp add: real_of_int_def real_of_nat_def)
paulson@14334
   827
avigad@16819
   828
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
avigad@16819
   829
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
avigad@16819
   830
  apply force
avigad@16819
   831
  apply (simp only: real_of_int_real_of_nat)
avigad@16819
   832
done
paulson@14387
   833
paulson@14387
   834
subsection{*Numerals and Arithmetic*}
paulson@14387
   835
haftmann@24198
   836
instance real :: number_ring
haftmann@24198
   837
  real_number_of_def: "number_of w \<equiv> real_of_int w"
haftmann@24198
   838
  by intro_classes (simp add: real_number_of_def)
paulson@14387
   839
haftmann@24198
   840
lemma [code, code unfold]:
haftmann@24198
   841
  "number_of k = real_of_int (number_of k)"
haftmann@24198
   842
  unfolding number_of_is_id real_number_of_def ..
paulson@14387
   843
paulson@14387
   844
paulson@14387
   845
text{*Collapse applications of @{term real} to @{term number_of}*}
paulson@14387
   846
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
paulson@14387
   847
by (simp add:  real_of_int_def of_int_number_of_eq)
paulson@14387
   848
paulson@14387
   849
lemma real_of_nat_number_of [simp]:
paulson@14387
   850
     "real (number_of v :: nat) =  
paulson@14387
   851
        (if neg (number_of v :: int) then 0  
paulson@14387
   852
         else (number_of v :: real))"
paulson@14387
   853
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
paulson@14387
   854
 
paulson@14387
   855
paulson@14387
   856
use "real_arith.ML"
wenzelm@24075
   857
declaration {* K real_arith_setup *}
paulson@14387
   858
kleing@19023
   859
paulson@14387
   860
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
paulson@14387
   861
paulson@14387
   862
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
paulson@14387
   863
lemma real_0_le_divide_iff:
paulson@14387
   864
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
paulson@14387
   865
by (simp add: real_divide_def zero_le_mult_iff, auto)
paulson@14387
   866
paulson@14387
   867
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
paulson@14387
   868
by arith
paulson@14387
   869
paulson@15085
   870
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
paulson@14387
   871
by auto
paulson@14387
   872
paulson@15085
   873
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
paulson@14387
   874
by auto
paulson@14387
   875
paulson@15085
   876
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
paulson@14387
   877
by auto
paulson@14387
   878
paulson@15085
   879
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
paulson@14387
   880
by auto
paulson@14387
   881
paulson@15085
   882
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
paulson@14387
   883
by auto
paulson@14387
   884
paulson@14387
   885
paulson@14387
   886
(*
paulson@14387
   887
FIXME: we should have this, as for type int, but many proofs would break.
paulson@14387
   888
It replaces x+-y by x-y.
paulson@15086
   889
declare real_diff_def [symmetric, simp]
paulson@14387
   890
*)
paulson@14387
   891
paulson@14387
   892
paulson@14387
   893
subsubsection{*Density of the Reals*}
paulson@14387
   894
paulson@14387
   895
lemma real_lbound_gt_zero:
paulson@14387
   896
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
paulson@14387
   897
apply (rule_tac x = " (min d1 d2) /2" in exI)
paulson@14387
   898
apply (simp add: min_def)
paulson@14387
   899
done
paulson@14387
   900
paulson@14387
   901
paulson@14387
   902
text{*Similar results are proved in @{text Ring_and_Field}*}
paulson@14387
   903
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
paulson@14387
   904
  by auto
paulson@14387
   905
paulson@14387
   906
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
paulson@14387
   907
  by auto
paulson@14387
   908
paulson@14387
   909
paulson@14387
   910
subsection{*Absolute Value Function for the Reals*}
paulson@14387
   911
paulson@14387
   912
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
paulson@15003
   913
by (simp add: abs_if)
paulson@14387
   914
huffman@23289
   915
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
paulson@14387
   916
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
obua@14738
   917
by (force simp add: OrderedGroup.abs_le_iff)
paulson@14387
   918
paulson@14387
   919
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
paulson@15003
   920
by (simp add: abs_if)
paulson@14387
   921
paulson@14387
   922
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
huffman@22958
   923
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
paulson@14387
   924
paulson@14387
   925
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
webertj@20217
   926
by simp
paulson@14387
   927
 
paulson@14387
   928
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
webertj@20217
   929
by simp
paulson@14387
   930
berghofe@24534
   931
berghofe@24534
   932
subsection {* Implementation of rational real numbers as pairs of integers *}
berghofe@24534
   933
berghofe@24534
   934
definition
haftmann@24623
   935
  Ratreal :: "int \<times> int \<Rightarrow> real"
berghofe@24534
   936
where
haftmann@24623
   937
  "Ratreal = INum"
berghofe@24534
   938
haftmann@24623
   939
code_datatype Ratreal
berghofe@24534
   940
haftmann@24623
   941
lemma Ratreal_simp:
haftmann@24623
   942
  "Ratreal (k, l) = real_of_int k / real_of_int l"
haftmann@24623
   943
  unfolding Ratreal_def INum_def by simp
berghofe@24534
   944
haftmann@24623
   945
lemma Ratreal_zero [simp]: "Ratreal 0\<^sub>N = 0"
haftmann@24623
   946
  by (simp add: Ratreal_simp)
berghofe@24534
   947
haftmann@24623
   948
lemma Ratreal_lit [simp]: "Ratreal i\<^sub>N = real_of_int i"
haftmann@24623
   949
  by (simp add: Ratreal_simp)
berghofe@24534
   950
berghofe@24534
   951
lemma zero_real_code [code, code unfold]:
haftmann@24623
   952
  "0 = Ratreal 0\<^sub>N" by simp
berghofe@24534
   953
berghofe@24534
   954
lemma one_real_code [code, code unfold]:
haftmann@24623
   955
  "1 = Ratreal 1\<^sub>N" by simp
berghofe@24534
   956
berghofe@24534
   957
instance real :: eq ..
berghofe@24534
   958
haftmann@24623
   959
lemma real_eq_code [code]: "Ratreal x = Ratreal y \<longleftrightarrow> normNum x = normNum y"
haftmann@24623
   960
  unfolding Ratreal_def INum_normNum_iff ..
berghofe@24534
   961
haftmann@24623
   962
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> normNum x \<le>\<^sub>N normNum y"
berghofe@24534
   963
proof -
haftmann@24623
   964
  have "normNum x \<le>\<^sub>N normNum y \<longleftrightarrow> Ratreal (normNum x) \<le> Ratreal (normNum y)" 
haftmann@24623
   965
    by (simp add: Ratreal_def del: normNum)
haftmann@24623
   966
  also have "\<dots> = (Ratreal x \<le> Ratreal y)" by (simp add: Ratreal_def)
berghofe@24534
   967
  finally show ?thesis by simp
berghofe@24534
   968
qed
berghofe@24534
   969
haftmann@24623
   970
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> normNum x <\<^sub>N normNum y"
berghofe@24534
   971
proof -
haftmann@24623
   972
  have "normNum x <\<^sub>N normNum y \<longleftrightarrow> Ratreal (normNum x) < Ratreal (normNum y)" 
haftmann@24623
   973
    by (simp add: Ratreal_def del: normNum)
haftmann@24623
   974
  also have "\<dots> = (Ratreal x < Ratreal y)" by (simp add: Ratreal_def)
berghofe@24534
   975
  finally show ?thesis by simp
berghofe@24534
   976
qed
berghofe@24534
   977
haftmann@24623
   978
lemma real_add_code [code]: "Ratreal x + Ratreal y = Ratreal (x +\<^sub>N y)"
haftmann@24623
   979
  unfolding Ratreal_def by simp
berghofe@24534
   980
haftmann@24623
   981
lemma real_mul_code [code]: "Ratreal x * Ratreal y = Ratreal (x *\<^sub>N y)"
haftmann@24623
   982
  unfolding Ratreal_def by simp
berghofe@24534
   983
haftmann@24623
   984
lemma real_neg_code [code]: "- Ratreal x = Ratreal (~\<^sub>N x)"
haftmann@24623
   985
  unfolding Ratreal_def by simp
berghofe@24534
   986
haftmann@24623
   987
lemma real_sub_code [code]: "Ratreal x - Ratreal y = Ratreal (x -\<^sub>N y)"
haftmann@24623
   988
  unfolding Ratreal_def by simp
berghofe@24534
   989
haftmann@24623
   990
lemma real_inv_code [code]: "inverse (Ratreal x) = Ratreal (Ninv x)"
haftmann@24623
   991
  unfolding Ratreal_def Ninv real_divide_def by simp
berghofe@24534
   992
haftmann@24623
   993
lemma real_div_code [code]: "Ratreal x / Ratreal y = Ratreal (x \<div>\<^sub>N y)"
haftmann@24623
   994
  unfolding Ratreal_def by simp
berghofe@24534
   995
haftmann@24623
   996
text {* Setup for SML code generator *}
nipkow@23031
   997
nipkow@23031
   998
types_code
berghofe@24534
   999
  real ("(int */ int)")
nipkow@23031
  1000
attach (term_of) {*
berghofe@24534
  1001
fun term_of_real (p, q) =
haftmann@24623
  1002
  let
haftmann@24623
  1003
    val rT = HOLogic.realT
berghofe@24534
  1004
  in
berghofe@24534
  1005
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
haftmann@24623
  1006
    else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $
berghofe@24534
  1007
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24534
  1008
  end;
nipkow@23031
  1009
*}
nipkow@23031
  1010
attach (test) {*
nipkow@23031
  1011
fun gen_real i =
berghofe@24534
  1012
  let
berghofe@24534
  1013
    val p = random_range 0 i;
berghofe@24534
  1014
    val q = random_range 1 (i + 1);
berghofe@24534
  1015
    val g = Integer.gcd p q;
wenzelm@24630
  1016
    val p' = p div g;
wenzelm@24630
  1017
    val q' = q div g;
berghofe@24534
  1018
  in
berghofe@24534
  1019
    (if one_of [true, false] then p' else ~ p',
berghofe@24534
  1020
     if p' = 0 then 0 else q')
berghofe@24534
  1021
  end;
nipkow@23031
  1022
*}
nipkow@23031
  1023
nipkow@23031
  1024
consts_code
haftmann@24623
  1025
  Ratreal ("(_)")
berghofe@24534
  1026
berghofe@24534
  1027
consts_code
berghofe@24534
  1028
  "of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int")
berghofe@24534
  1029
attach {*
berghofe@24534
  1030
fun real_of_int 0 = (0, 0)
berghofe@24534
  1031
  | real_of_int i = (i, 1);
berghofe@24534
  1032
*}
berghofe@24534
  1033
berghofe@24534
  1034
declare real_of_int_of_nat_eq [symmetric, code]
nipkow@23031
  1035
paulson@5588
  1036
end