src/HOL/Real/RealDef.thy
author huffman
Thu Jun 07 04:33:15 2007 +0200 (2007-06-07)
changeset 23289 0cf371d943b1
parent 23288 3e45b5464d2b
child 23431 25ca91279a9b
permissions -rw-r--r--
remove redundant lemmas
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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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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" where
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  "real_of_preal m = Abs_Real(realrel``{(m + 1, 1)})"
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consts
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   (*overloaded constant for injecting other types into "real"*)
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   real :: "'a => real"
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defs (overloaded)
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  real_zero_def:
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  "0 == Abs_Real(realrel``{(1, 1)})"
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  real_one_def:
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  "1 == Abs_Real(realrel``{(1 + 1, 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) == (THE 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 r < 0 then - r else r)"
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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 ring_distrib
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              preal_mult_inverse_right add_ac 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 (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)
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  also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
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  finally show ?thesis by simp
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   318
qed
paulson@14378
   319
paulson@14378
   320
lemma real_le: 
paulson@14484
   321
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
paulson@14484
   322
      (x1 + y2 \<le> x2 + y1)"
huffman@23288
   323
apply (simp add: real_le_def)
paulson@14387
   324
apply (auto intro: real_le_lemma)
paulson@14378
   325
done
paulson@14378
   326
paulson@14378
   327
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
nipkow@15542
   328
by (cases z, cases w, simp add: real_le)
paulson@14378
   329
paulson@14378
   330
lemma real_trans_lemma:
paulson@14378
   331
  assumes "x + v \<le> u + y"
paulson@14378
   332
      and "u + v' \<le> u' + v"
paulson@14378
   333
      and "x2 + v2 = u2 + y2"
paulson@14378
   334
  shows "x + v' \<le> u' + (y::preal)"
paulson@14378
   335
proof -
huffman@23288
   336
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
huffman@23288
   337
  also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
huffman@23288
   338
  also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
huffman@23288
   339
  also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)
huffman@23288
   340
  finally show ?thesis by simp
nipkow@15542
   341
qed
paulson@14269
   342
paulson@14365
   343
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
paulson@14484
   344
apply (cases i, cases j, cases k)
paulson@14484
   345
apply (simp add: real_le)
huffman@23288
   346
apply (blast intro: real_trans_lemma)
paulson@14334
   347
done
paulson@14334
   348
paulson@14334
   349
(* Axiom 'order_less_le' of class 'order': *)
paulson@14334
   350
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
paulson@14365
   351
by (simp add: real_less_def)
paulson@14365
   352
paulson@14365
   353
instance real :: order
paulson@14365
   354
proof qed
paulson@14365
   355
 (assumption |
paulson@14365
   356
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
paulson@14365
   357
paulson@14378
   358
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14378
   359
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
huffman@23288
   360
apply (cases z, cases w)
huffman@23288
   361
apply (auto simp add: real_le real_zero_def add_ac)
paulson@14334
   362
done
paulson@14334
   363
paulson@14334
   364
paulson@14334
   365
instance real :: linorder
paulson@14334
   366
  by (intro_classes, rule real_le_linear)
paulson@14334
   367
paulson@14334
   368
paulson@14378
   369
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
paulson@14484
   370
apply (cases x, cases y) 
paulson@14378
   371
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
huffman@23288
   372
                      add_ac)
huffman@23288
   373
apply (simp_all add: add_assoc [symmetric])
nipkow@15542
   374
done
paulson@14378
   375
paulson@14484
   376
lemma real_add_left_mono: 
paulson@14484
   377
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
paulson@14484
   378
proof -
paulson@14484
   379
  have "z + x - (z + y) = (z + -z) + (x - y)"
paulson@14484
   380
    by (simp add: diff_minus add_ac) 
paulson@14484
   381
  with le show ?thesis 
obua@14754
   382
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
paulson@14484
   383
qed
paulson@14334
   384
paulson@14365
   385
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
paulson@14365
   386
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14365
   387
paulson@14365
   388
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
paulson@14365
   389
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14334
   390
paulson@14334
   391
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
paulson@14484
   392
apply (cases x, cases y)
paulson@14378
   393
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
paulson@14378
   394
                 linorder_not_le [where 'a = preal] 
paulson@14378
   395
                  real_zero_def real_le real_mult)
paulson@14365
   396
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
wenzelm@16973
   397
apply (auto dest!: less_add_left_Ex
huffman@23288
   398
     simp add: add_ac mult_ac
huffman@23288
   399
          right_distrib preal_self_less_add_left)
paulson@14334
   400
done
paulson@14334
   401
paulson@14334
   402
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
paulson@14334
   403
apply (rule real_sum_gt_zero_less)
paulson@14334
   404
apply (drule real_less_sum_gt_zero [of x y])
paulson@14334
   405
apply (drule real_mult_order, assumption)
paulson@14334
   406
apply (simp add: right_distrib)
paulson@14334
   407
done
paulson@14334
   408
haftmann@22456
   409
instance real :: distrib_lattice
haftmann@22456
   410
  "inf x y \<equiv> min x y"
haftmann@22456
   411
  "sup x y \<equiv> max x y"
haftmann@22456
   412
  by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
haftmann@22456
   413
paulson@14378
   414
paulson@14334
   415
subsection{*The Reals Form an Ordered Field*}
paulson@14334
   416
paulson@14334
   417
instance real :: ordered_field
paulson@14334
   418
proof
paulson@14334
   419
  fix x y z :: real
paulson@14334
   420
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
huffman@22962
   421
  show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
huffman@22962
   422
  show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
paulson@14334
   423
qed
paulson@14334
   424
paulson@14365
   425
text{*The function @{term real_of_preal} requires many proofs, but it seems
paulson@14365
   426
to be essential for proving completeness of the reals from that of the
paulson@14365
   427
positive reals.*}
paulson@14365
   428
paulson@14365
   429
lemma real_of_preal_add:
paulson@14365
   430
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
huffman@23288
   431
by (simp add: real_of_preal_def real_add left_distrib add_ac)
paulson@14365
   432
paulson@14365
   433
lemma real_of_preal_mult:
paulson@14365
   434
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
huffman@23288
   435
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac)
paulson@14365
   436
paulson@14365
   437
paulson@14365
   438
text{*Gleason prop 9-4.4 p 127*}
paulson@14365
   439
lemma real_of_preal_trichotomy:
paulson@14365
   440
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
paulson@14484
   441
apply (simp add: real_of_preal_def real_zero_def, cases x)
huffman@23288
   442
apply (auto simp add: real_minus add_ac)
paulson@14365
   443
apply (cut_tac x = x and y = y in linorder_less_linear)
huffman@23288
   444
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
paulson@14365
   445
done
paulson@14365
   446
paulson@14365
   447
lemma real_of_preal_leD:
paulson@14365
   448
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
huffman@23288
   449
by (simp add: real_of_preal_def real_le)
paulson@14365
   450
paulson@14365
   451
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
paulson@14365
   452
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
paulson@14365
   453
paulson@14365
   454
lemma real_of_preal_lessD:
paulson@14365
   455
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
huffman@23288
   456
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
paulson@14365
   457
paulson@14365
   458
lemma real_of_preal_less_iff [simp]:
paulson@14365
   459
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
paulson@14365
   460
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
paulson@14365
   461
paulson@14365
   462
lemma real_of_preal_le_iff:
paulson@14365
   463
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
huffman@23288
   464
by (simp add: linorder_not_less [symmetric])
paulson@14365
   465
paulson@14365
   466
lemma real_of_preal_zero_less: "0 < real_of_preal m"
huffman@23288
   467
apply (insert preal_self_less_add_left [of 1 m])
huffman@23288
   468
apply (auto simp add: real_zero_def real_of_preal_def
huffman@23288
   469
                      real_less_def real_le_def add_ac)
huffman@23288
   470
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
huffman@23288
   471
apply (simp add: add_ac)
paulson@14365
   472
done
paulson@14365
   473
paulson@14365
   474
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
paulson@14365
   475
by (simp add: real_of_preal_zero_less)
paulson@14365
   476
paulson@14365
   477
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
paulson@14484
   478
proof -
paulson@14484
   479
  from real_of_preal_minus_less_zero
paulson@14484
   480
  show ?thesis by (blast dest: order_less_trans)
paulson@14484
   481
qed
paulson@14365
   482
paulson@14365
   483
paulson@14365
   484
subsection{*Theorems About the Ordering*}
paulson@14365
   485
paulson@14365
   486
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
paulson@14365
   487
apply (auto simp add: real_of_preal_zero_less)
paulson@14365
   488
apply (cut_tac x = x in real_of_preal_trichotomy)
paulson@14365
   489
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
paulson@14365
   490
done
paulson@14365
   491
paulson@14365
   492
lemma real_gt_preal_preal_Ex:
paulson@14365
   493
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   494
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
paulson@14365
   495
             intro: real_gt_zero_preal_Ex [THEN iffD1])
paulson@14365
   496
paulson@14365
   497
lemma real_ge_preal_preal_Ex:
paulson@14365
   498
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   499
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
paulson@14365
   500
paulson@14365
   501
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
paulson@14365
   502
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
paulson@14365
   503
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
paulson@14365
   504
            simp add: real_of_preal_zero_less)
paulson@14365
   505
paulson@14365
   506
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
paulson@14365
   507
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
paulson@14365
   508
paulson@14334
   509
paulson@14334
   510
subsection{*More Lemmas*}
paulson@14334
   511
paulson@14334
   512
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14334
   513
by auto
paulson@14334
   514
paulson@14334
   515
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   516
by auto
paulson@14334
   517
paulson@14334
   518
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
paulson@14334
   519
  by (force elim: order_less_asym
paulson@14334
   520
            simp add: Ring_and_Field.mult_less_cancel_right)
paulson@14334
   521
paulson@14334
   522
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
paulson@14365
   523
apply (simp add: mult_le_cancel_right)
huffman@23289
   524
apply (blast intro: elim: order_less_asym)
paulson@14365
   525
done
paulson@14334
   526
paulson@14334
   527
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
nipkow@15923
   528
by(simp add:mult_commute)
paulson@14334
   529
huffman@23289
   530
(* FIXME: redundant, but used by Integration/Integral.thy in AFP *)
paulson@14334
   531
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y"
huffman@22958
   532
by (rule add_nonneg_nonneg)
paulson@14334
   533
paulson@14365
   534
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
huffman@23289
   535
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *)
paulson@14334
   536
paulson@14334
   537
paulson@14365
   538
subsection{*Embedding the Integers into the Reals*}
paulson@14365
   539
paulson@14378
   540
defs (overloaded)
paulson@14378
   541
  real_of_nat_def: "real z == of_nat z"
paulson@14378
   542
  real_of_int_def: "real z == of_int z"
paulson@14365
   543
avigad@16819
   544
lemma real_eq_of_nat: "real = of_nat"
avigad@16819
   545
  apply (rule ext)
avigad@16819
   546
  apply (unfold real_of_nat_def)
avigad@16819
   547
  apply (rule refl)
avigad@16819
   548
  done
avigad@16819
   549
avigad@16819
   550
lemma real_eq_of_int: "real = of_int"
avigad@16819
   551
  apply (rule ext)
avigad@16819
   552
  apply (unfold real_of_int_def)
avigad@16819
   553
  apply (rule refl)
avigad@16819
   554
  done
avigad@16819
   555
paulson@14365
   556
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378
   557
by (simp add: real_of_int_def) 
paulson@14365
   558
paulson@14365
   559
lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378
   560
by (simp add: real_of_int_def) 
paulson@14334
   561
avigad@16819
   562
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
paulson@14378
   563
by (simp add: real_of_int_def) 
paulson@14365
   564
avigad@16819
   565
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
paulson@14378
   566
by (simp add: real_of_int_def) 
avigad@16819
   567
avigad@16819
   568
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
avigad@16819
   569
by (simp add: real_of_int_def) 
paulson@14365
   570
avigad@16819
   571
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
paulson@14378
   572
by (simp add: real_of_int_def) 
paulson@14334
   573
avigad@16819
   574
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
avigad@16819
   575
  apply (subst real_eq_of_int)+
avigad@16819
   576
  apply (rule of_int_setsum)
avigad@16819
   577
done
avigad@16819
   578
avigad@16819
   579
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
avigad@16819
   580
    (PROD x:A. real(f x))"
avigad@16819
   581
  apply (subst real_eq_of_int)+
avigad@16819
   582
  apply (rule of_int_setprod)
avigad@16819
   583
done
paulson@14365
   584
paulson@14365
   585
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
paulson@14378
   586
by (simp add: real_of_int_def) 
paulson@14365
   587
paulson@14365
   588
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
paulson@14378
   589
by (simp add: real_of_int_def) 
paulson@14365
   590
paulson@14365
   591
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
paulson@14378
   592
by (simp add: real_of_int_def) 
paulson@14365
   593
paulson@14365
   594
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378
   595
by (simp add: real_of_int_def) 
paulson@14365
   596
avigad@16819
   597
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)"
avigad@16819
   598
by (simp add: real_of_int_def) 
avigad@16819
   599
avigad@16819
   600
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)"
avigad@16819
   601
by (simp add: real_of_int_def) 
avigad@16819
   602
avigad@16819
   603
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)"
avigad@16819
   604
by (simp add: real_of_int_def)
avigad@16819
   605
avigad@16819
   606
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)"
avigad@16819
   607
by (simp add: real_of_int_def)
avigad@16819
   608
avigad@16888
   609
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
avigad@16888
   610
by (auto simp add: abs_if)
avigad@16888
   611
avigad@16819
   612
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
avigad@16819
   613
  apply (subgoal_tac "real n + 1 = real (n + 1)")
avigad@16819
   614
  apply (simp del: real_of_int_add)
avigad@16819
   615
  apply auto
avigad@16819
   616
done
avigad@16819
   617
avigad@16819
   618
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
avigad@16819
   619
  apply (subgoal_tac "real m + 1 = real (m + 1)")
avigad@16819
   620
  apply (simp del: real_of_int_add)
avigad@16819
   621
  apply simp
avigad@16819
   622
done
avigad@16819
   623
avigad@16819
   624
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
avigad@16819
   625
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   626
proof -
avigad@16819
   627
  assume "d ~= 0"
avigad@16819
   628
  have "x = (x div d) * d + x mod d"
avigad@16819
   629
    by auto
avigad@16819
   630
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   631
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
avigad@16819
   632
  then have "real x / real d = ... / real d"
avigad@16819
   633
    by simp
avigad@16819
   634
  then show ?thesis
avigad@16819
   635
    by (auto simp add: add_divide_distrib ring_eq_simps prems)
avigad@16819
   636
qed
avigad@16819
   637
avigad@16819
   638
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
avigad@16819
   639
    real(n div d) = real n / real d"
avigad@16819
   640
  apply (frule real_of_int_div_aux [of d n])
avigad@16819
   641
  apply simp
avigad@16819
   642
  apply (simp add: zdvd_iff_zmod_eq_0)
avigad@16819
   643
done
avigad@16819
   644
avigad@16819
   645
lemma real_of_int_div2:
avigad@16819
   646
  "0 <= real (n::int) / real (x) - real (n div x)"
avigad@16819
   647
  apply (case_tac "x = 0")
avigad@16819
   648
  apply simp
avigad@16819
   649
  apply (case_tac "0 < x")
avigad@16819
   650
  apply (simp add: compare_rls)
avigad@16819
   651
  apply (subst real_of_int_div_aux)
avigad@16819
   652
  apply simp
avigad@16819
   653
  apply simp
avigad@16819
   654
  apply (subst zero_le_divide_iff)
avigad@16819
   655
  apply auto
avigad@16819
   656
  apply (simp add: compare_rls)
avigad@16819
   657
  apply (subst real_of_int_div_aux)
avigad@16819
   658
  apply simp
avigad@16819
   659
  apply simp
avigad@16819
   660
  apply (subst zero_le_divide_iff)
avigad@16819
   661
  apply auto
avigad@16819
   662
done
avigad@16819
   663
avigad@16819
   664
lemma real_of_int_div3:
avigad@16819
   665
  "real (n::int) / real (x) - real (n div x) <= 1"
avigad@16819
   666
  apply(case_tac "x = 0")
avigad@16819
   667
  apply simp
avigad@16819
   668
  apply (simp add: compare_rls)
avigad@16819
   669
  apply (subst real_of_int_div_aux)
avigad@16819
   670
  apply assumption
avigad@16819
   671
  apply simp
avigad@16819
   672
  apply (subst divide_le_eq)
avigad@16819
   673
  apply clarsimp
avigad@16819
   674
  apply (rule conjI)
avigad@16819
   675
  apply (rule impI)
avigad@16819
   676
  apply (rule order_less_imp_le)
avigad@16819
   677
  apply simp
avigad@16819
   678
  apply (rule impI)
avigad@16819
   679
  apply (rule order_less_imp_le)
avigad@16819
   680
  apply simp
avigad@16819
   681
done
avigad@16819
   682
avigad@16819
   683
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
avigad@16819
   684
  by (insert real_of_int_div2 [of n x], simp)
paulson@14365
   685
paulson@14365
   686
subsection{*Embedding the Naturals into the Reals*}
paulson@14365
   687
paulson@14334
   688
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365
   689
by (simp add: real_of_nat_def)
paulson@14334
   690
paulson@14334
   691
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365
   692
by (simp add: real_of_nat_def)
paulson@14334
   693
paulson@14365
   694
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378
   695
by (simp add: real_of_nat_def)
paulson@14334
   696
paulson@14334
   697
(*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334
   698
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378
   699
by (simp add: real_of_nat_def)
paulson@14334
   700
paulson@14334
   701
lemma real_of_nat_less_iff [iff]: 
paulson@14334
   702
     "(real (n::nat) < real m) = (n < m)"
paulson@14365
   703
by (simp add: real_of_nat_def)
paulson@14334
   704
paulson@14334
   705
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378
   706
by (simp add: real_of_nat_def)
paulson@14334
   707
paulson@14334
   708
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
paulson@14378
   709
by (simp add: real_of_nat_def zero_le_imp_of_nat)
paulson@14334
   710
paulson@14365
   711
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378
   712
by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365
   713
paulson@14334
   714
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
paulson@14378
   715
by (simp add: real_of_nat_def)
paulson@14334
   716
avigad@16819
   717
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
avigad@16819
   718
    (SUM x:A. real(f x))"
avigad@16819
   719
  apply (subst real_eq_of_nat)+
avigad@16819
   720
  apply (rule of_nat_setsum)
avigad@16819
   721
done
avigad@16819
   722
avigad@16819
   723
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
avigad@16819
   724
    (PROD x:A. real(f x))"
avigad@16819
   725
  apply (subst real_eq_of_nat)+
avigad@16819
   726
  apply (rule of_nat_setprod)
avigad@16819
   727
done
avigad@16819
   728
avigad@16819
   729
lemma real_of_card: "real (card A) = setsum (%x.1) A"
avigad@16819
   730
  apply (subst card_eq_setsum)
avigad@16819
   731
  apply (subst real_of_nat_setsum)
avigad@16819
   732
  apply simp
avigad@16819
   733
done
avigad@16819
   734
paulson@14334
   735
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378
   736
by (simp add: real_of_nat_def)
paulson@14334
   737
paulson@14387
   738
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
paulson@14378
   739
by (simp add: real_of_nat_def)
paulson@14334
   740
paulson@14365
   741
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
paulson@14378
   742
by (simp add: add: real_of_nat_def) 
paulson@14334
   743
paulson@14365
   744
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
paulson@14378
   745
by (simp add: add: real_of_nat_def) 
paulson@14365
   746
paulson@14365
   747
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378
   748
by (simp add: add: real_of_nat_def)
paulson@14334
   749
paulson@14365
   750
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378
   751
by (simp add: add: real_of_nat_def)
paulson@14334
   752
paulson@14365
   753
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"
paulson@14378
   754
by (simp add: add: real_of_nat_def)
paulson@14334
   755
avigad@16819
   756
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
avigad@16819
   757
  apply (subgoal_tac "real n + 1 = real (Suc n)")
avigad@16819
   758
  apply simp
avigad@16819
   759
  apply (auto simp add: real_of_nat_Suc)
avigad@16819
   760
done
avigad@16819
   761
avigad@16819
   762
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
avigad@16819
   763
  apply (subgoal_tac "real m + 1 = real (Suc m)")
avigad@16819
   764
  apply (simp add: less_Suc_eq_le)
avigad@16819
   765
  apply (simp add: real_of_nat_Suc)
avigad@16819
   766
done
avigad@16819
   767
avigad@16819
   768
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
avigad@16819
   769
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   770
proof -
avigad@16819
   771
  assume "0 < d"
avigad@16819
   772
  have "x = (x div d) * d + x mod d"
avigad@16819
   773
    by auto
avigad@16819
   774
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   775
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
avigad@16819
   776
  then have "real x / real d = \<dots> / real d"
avigad@16819
   777
    by simp
avigad@16819
   778
  then show ?thesis
avigad@16819
   779
    by (auto simp add: add_divide_distrib ring_eq_simps prems)
avigad@16819
   780
qed
avigad@16819
   781
avigad@16819
   782
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
avigad@16819
   783
    real(n div d) = real n / real d"
avigad@16819
   784
  apply (frule real_of_nat_div_aux [of d n])
avigad@16819
   785
  apply simp
avigad@16819
   786
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
avigad@16819
   787
  apply assumption
avigad@16819
   788
done
avigad@16819
   789
avigad@16819
   790
lemma real_of_nat_div2:
avigad@16819
   791
  "0 <= real (n::nat) / real (x) - real (n div x)"
avigad@16819
   792
  apply(case_tac "x = 0")
avigad@16819
   793
  apply simp
avigad@16819
   794
  apply (simp add: compare_rls)
avigad@16819
   795
  apply (subst real_of_nat_div_aux)
avigad@16819
   796
  apply assumption
avigad@16819
   797
  apply simp
avigad@16819
   798
  apply (subst zero_le_divide_iff)
avigad@16819
   799
  apply simp
avigad@16819
   800
done
avigad@16819
   801
avigad@16819
   802
lemma real_of_nat_div3:
avigad@16819
   803
  "real (n::nat) / real (x) - real (n div x) <= 1"
avigad@16819
   804
  apply(case_tac "x = 0")
avigad@16819
   805
  apply simp
avigad@16819
   806
  apply (simp add: compare_rls)
avigad@16819
   807
  apply (subst real_of_nat_div_aux)
avigad@16819
   808
  apply assumption
avigad@16819
   809
  apply simp
avigad@16819
   810
done
avigad@16819
   811
avigad@16819
   812
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
avigad@16819
   813
  by (insert real_of_nat_div2 [of n x], simp)
avigad@16819
   814
paulson@14365
   815
lemma real_of_int_real_of_nat: "real (int n) = real n"
paulson@14378
   816
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
paulson@14378
   817
paulson@14426
   818
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
paulson@14426
   819
by (simp add: real_of_int_def real_of_nat_def)
paulson@14334
   820
avigad@16819
   821
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
avigad@16819
   822
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
avigad@16819
   823
  apply force
avigad@16819
   824
  apply (simp only: real_of_int_real_of_nat)
avigad@16819
   825
done
paulson@14387
   826
paulson@14387
   827
subsection{*Numerals and Arithmetic*}
paulson@14387
   828
paulson@14387
   829
instance real :: number ..
paulson@14387
   830
paulson@15013
   831
defs (overloaded)
haftmann@20485
   832
  real_number_of_def: "(number_of w :: real) == of_int w"
paulson@15013
   833
    --{*the type constraint is essential!*}
paulson@14387
   834
paulson@14387
   835
instance real :: number_ring
paulson@15013
   836
by (intro_classes, simp add: real_number_of_def) 
paulson@14387
   837
paulson@14387
   838
text{*Collapse applications of @{term real} to @{term number_of}*}
paulson@14387
   839
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
paulson@14387
   840
by (simp add:  real_of_int_def of_int_number_of_eq)
paulson@14387
   841
paulson@14387
   842
lemma real_of_nat_number_of [simp]:
paulson@14387
   843
     "real (number_of v :: nat) =  
paulson@14387
   844
        (if neg (number_of v :: int) then 0  
paulson@14387
   845
         else (number_of v :: real))"
paulson@14387
   846
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
paulson@14387
   847
 
paulson@14387
   848
paulson@14387
   849
use "real_arith.ML"
paulson@14387
   850
paulson@14387
   851
setup real_arith_setup
paulson@14387
   852
kleing@19023
   853
paulson@14387
   854
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
paulson@14387
   855
paulson@14387
   856
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
paulson@14387
   857
lemma real_0_le_divide_iff:
paulson@14387
   858
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
paulson@14387
   859
by (simp add: real_divide_def zero_le_mult_iff, auto)
paulson@14387
   860
paulson@14387
   861
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
paulson@14387
   862
by arith
paulson@14387
   863
paulson@15085
   864
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
paulson@14387
   865
by auto
paulson@14387
   866
paulson@15085
   867
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
paulson@14387
   868
by auto
paulson@14387
   869
paulson@15085
   870
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
paulson@14387
   871
by auto
paulson@14387
   872
paulson@15085
   873
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
paulson@14387
   874
by auto
paulson@14387
   875
paulson@15085
   876
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
paulson@14387
   877
by auto
paulson@14387
   878
paulson@14387
   879
paulson@14387
   880
(*
paulson@14387
   881
FIXME: we should have this, as for type int, but many proofs would break.
paulson@14387
   882
It replaces x+-y by x-y.
paulson@15086
   883
declare real_diff_def [symmetric, simp]
paulson@14387
   884
*)
paulson@14387
   885
paulson@14387
   886
paulson@14387
   887
subsubsection{*Density of the Reals*}
paulson@14387
   888
paulson@14387
   889
lemma real_lbound_gt_zero:
paulson@14387
   890
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
paulson@14387
   891
apply (rule_tac x = " (min d1 d2) /2" in exI)
paulson@14387
   892
apply (simp add: min_def)
paulson@14387
   893
done
paulson@14387
   894
paulson@14387
   895
paulson@14387
   896
text{*Similar results are proved in @{text Ring_and_Field}*}
paulson@14387
   897
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
paulson@14387
   898
  by auto
paulson@14387
   899
paulson@14387
   900
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
paulson@14387
   901
  by auto
paulson@14387
   902
paulson@14387
   903
paulson@14387
   904
subsection{*Absolute Value Function for the Reals*}
paulson@14387
   905
paulson@14387
   906
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
paulson@15003
   907
by (simp add: abs_if)
paulson@14387
   908
huffman@23289
   909
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
paulson@14387
   910
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
obua@14738
   911
by (force simp add: OrderedGroup.abs_le_iff)
paulson@14387
   912
paulson@14387
   913
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
paulson@15003
   914
by (simp add: abs_if)
paulson@14387
   915
paulson@14387
   916
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
huffman@22958
   917
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
paulson@14387
   918
paulson@14387
   919
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
webertj@20217
   920
by simp
paulson@14387
   921
 
paulson@14387
   922
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
webertj@20217
   923
by simp
paulson@14387
   924
nipkow@23031
   925
subsection{*Code generation using Isabelle's rats*}
nipkow@23031
   926
nipkow@23031
   927
types_code
nipkow@23031
   928
  real ("Rat.rat")
nipkow@23031
   929
attach (term_of) {*
nipkow@23031
   930
fun term_of_real x =
nipkow@23031
   931
 let 
nipkow@23031
   932
  val rT = HOLogic.realT
nipkow@23031
   933
  val (p, q) = Rat.quotient_of_rat x
nipkow@23031
   934
 in if q = 1 then HOLogic.mk_number rT p
nipkow@23031
   935
    else Const("HOL.divide",[rT,rT] ---> rT) $
nipkow@23031
   936
           (HOLogic.mk_number rT p) $ (HOLogic.mk_number rT q)
nipkow@23031
   937
end;
nipkow@23031
   938
*}
nipkow@23031
   939
attach (test) {*
nipkow@23031
   940
fun gen_real i =
nipkow@23031
   941
let val p = random_range 0 i; val q = random_range 0 i;
nipkow@23031
   942
    val r = if q=0 then Rat.rat_of_int i else Rat.rat_of_quotient(p,q)
nipkow@23031
   943
in if one_of [true,false] then r else Rat.neg r end;
nipkow@23031
   944
*}
nipkow@23031
   945
nipkow@23031
   946
consts_code
nipkow@23031
   947
  "0 :: real" ("Rat.zero")
nipkow@23031
   948
  "1 :: real" ("Rat.one")
nipkow@23031
   949
  "uminus :: real \<Rightarrow> real" ("Rat.neg")
nipkow@23031
   950
  "op + :: real \<Rightarrow> real \<Rightarrow> real" ("Rat.add")
nipkow@23031
   951
  "op * :: real \<Rightarrow> real \<Rightarrow> real" ("Rat.mult")
nipkow@23031
   952
  "inverse :: real \<Rightarrow> real" ("Rat.inv")
nipkow@23031
   953
  "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool" ("Rat.le")
nipkow@23031
   954
  "op < :: real \<Rightarrow> real \<Rightarrow> bool" ("(Rat.ord (_, _) = LESS)")
nipkow@23031
   955
  "op = :: real \<Rightarrow> real \<Rightarrow> bool" ("curry Rat.eq")
nipkow@23031
   956
  "real :: int \<Rightarrow> real" ("Rat.rat'_of'_int")
nipkow@23031
   957
  "real :: nat \<Rightarrow> real" ("(Rat.rat'_of'_int o {*int*})")
nipkow@23031
   958
nipkow@23031
   959
nipkow@23031
   960
lemma [code, code unfold]:
nipkow@23031
   961
  "number_of k = real (number_of k :: int)"
nipkow@23031
   962
  by simp
nipkow@23031
   963
paulson@5588
   964
end