src/HOL/Real/RealDef.thy
author huffman
Thu, 07 Jun 2007 03:45:56 +0200
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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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   264
apply (subgoal_tac "(z * x) * y = z * (x * y)")
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   265
apply (simp add: mult_commute)
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   266
apply (rule mult_assoc)
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done
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14341
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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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   286
proof
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   287
  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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   292
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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
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
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  following two lemmas.*}
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   299
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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   301
  shows "b \<le> (d::preal)"
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   302
proof -
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diff changeset
   303
  have "c+d \<le> a+d" by (simp add: prems)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   304
  hence "a+b \<le> a+d" by (simp add: prems)
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   305
  thus "b \<le> d" by simp
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   306
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   307
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   308
lemma real_le_lemma:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   309
  assumes l: "u1 + v2 \<le> u2 + v1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   310
      and "x1 + v1 = u1 + y1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   311
      and "x2 + v2 = u2 + y2"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   312
  shows "x1 + y2 \<le> x2 + (y1::preal)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   313
proof -
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   314
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   315
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   316
  also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   317
  finally show ?thesis by simp
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   318
qed
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   319
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   320
lemma real_le: 
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   321
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   322
      (x1 + y2 \<le> x2 + y1)"
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   323
apply (simp add: real_le_def)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   324
apply (auto intro: real_le_lemma)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   325
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   326
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   327
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15229
diff changeset
   328
by (cases z, cases w, simp add: real_le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   329
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   330
lemma real_trans_lemma:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   331
  assumes "x + v \<le> u + y"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   332
      and "u + v' \<le> u' + v"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   333
      and "x2 + v2 = u2 + y2"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   334
  shows "x + v' \<le> u' + (y::preal)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   335
proof -
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   336
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   337
  also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   338
  also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   339
  also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   340
  finally show ?thesis by simp
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15229
diff changeset
   341
qed
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   342
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   343
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   344
apply (cases i, cases j, cases k)
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   345
apply (simp add: real_le)
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   346
apply (blast intro: real_trans_lemma)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   347
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   348
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   349
(* Axiom 'order_less_le' of class 'order': *)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   350
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   351
by (simp add: real_less_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   352
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   353
instance real :: order
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   354
proof qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   355
 (assumption |
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   356
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   357
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   358
(* Axiom 'linorder_linear' of class 'linorder': *)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   359
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   360
apply (cases z, cases w)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   361
apply (auto simp add: real_le real_zero_def add_ac)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   362
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   363
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   364
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   365
instance real :: linorder
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   366
  by (intro_classes, rule real_le_linear)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   367
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   368
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   369
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   370
apply (cases x, cases y) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   371
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   372
                      add_ac)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   373
apply (simp_all add: add_assoc [symmetric])
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15229
diff changeset
   374
done
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   375
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   376
lemma real_add_left_mono: 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   377
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   378
proof -
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   379
  have "z + x - (z + y) = (z + -z) + (x - y)"
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   380
    by (simp add: diff_minus add_ac) 
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   381
  with le show ?thesis 
14754
a080eeeaec14 Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents: 14738
diff changeset
   382
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   383
qed
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   384
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   385
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   386
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   387
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   388
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   389
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   390
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   391
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   392
apply (cases x, cases y)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   393
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   394
                 linorder_not_le [where 'a = preal] 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   395
                  real_zero_def real_le real_mult)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   396
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16888
diff changeset
   397
apply (auto dest!: less_add_left_Ex
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   398
     simp add: add_ac mult_ac
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   399
          right_distrib preal_self_less_add_left)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   400
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   401
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   402
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   403
apply (rule real_sum_gt_zero_less)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   404
apply (drule real_less_sum_gt_zero [of x y])
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   405
apply (drule real_mult_order, assumption)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   406
apply (simp add: right_distrib)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   407
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   408
22456
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   409
instance real :: distrib_lattice
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   410
  "inf x y \<equiv> min x y"
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   411
  "sup x y \<equiv> max x y"
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   412
  by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
6070e48ecb78 added lattice definitions
haftmann
parents: 21404
diff changeset
   413
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   414
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   415
subsection{*The Reals Form an Ordered Field*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   416
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   417
instance real :: ordered_field
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   418
proof
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   419
  fix x y z :: real
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   420
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
22962
4bb05ba38939 remove redundant lemmas
huffman
parents: 22958
diff changeset
   421
  show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
4bb05ba38939 remove redundant lemmas
huffman
parents: 22958
diff changeset
   422
  show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   423
qed
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   424
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   425
text{*The function @{term real_of_preal} requires many proofs, but it seems
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   426
to be essential for proving completeness of the reals from that of the
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   427
positive reals.*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   428
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   429
lemma real_of_preal_add:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   430
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   431
by (simp add: real_of_preal_def real_add left_distrib add_ac)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   432
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   433
lemma real_of_preal_mult:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   434
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   435
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   436
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   437
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   438
text{*Gleason prop 9-4.4 p 127*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   439
lemma real_of_preal_trichotomy:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   440
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   441
apply (simp add: real_of_preal_def real_zero_def, cases x)
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   442
apply (auto simp add: real_minus add_ac)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   443
apply (cut_tac x = x and y = y in linorder_less_linear)
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   444
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   445
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   446
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   447
lemma real_of_preal_leD:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   448
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   449
by (simp add: real_of_preal_def real_le)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   450
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   451
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   452
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   453
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   454
lemma real_of_preal_lessD:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   455
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   456
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   457
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   458
lemma real_of_preal_less_iff [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   459
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   460
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   461
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   462
lemma real_of_preal_le_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   463
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   464
by (simp add: linorder_not_less [symmetric])
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   465
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   466
lemma real_of_preal_zero_less: "0 < real_of_preal m"
23288
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   467
apply (insert preal_self_less_add_left [of 1 m])
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   468
apply (auto simp add: real_zero_def real_of_preal_def
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   469
                      real_less_def real_le_def add_ac)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   470
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
3e45b5464d2b remove references to preal-specific theorems
huffman
parents: 23287
diff changeset
   471
apply (simp add: add_ac)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   472
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   473
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   474
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   475
by (simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   476
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   477
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
14484
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   478
proof -
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   479
  from real_of_preal_minus_less_zero
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   480
  show ?thesis by (blast dest: order_less_trans)
ef8c7c5eb01b new treatment of equivalence classes
paulson
parents: 14476
diff changeset
   481
qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   482
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   483
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   484
subsection{*Theorems About the Ordering*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   485
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   486
text{*obsolete but used a lot*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   487
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   488
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   489
by blast 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   490
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   491
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   492
by (simp add: order_le_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   493
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   494
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   495
apply (auto simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   496
apply (cut_tac x = x in real_of_preal_trichotomy)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   497
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   498
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   499
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   500
lemma real_gt_preal_preal_Ex:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   501
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   502
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   503
             intro: real_gt_zero_preal_Ex [THEN iffD1])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   504
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   505
lemma real_ge_preal_preal_Ex:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   506
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   507
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   508
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   509
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   510
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   511
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   512
            simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   513
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   514
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   515
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   516
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   517
lemma real_le_square [simp]: "(0::real) \<le> x*x"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   518
 by (rule Ring_and_Field.zero_le_square)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   519
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   520
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   521
subsection{*More Lemmas*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   522
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   523
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   524
by auto
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   525
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   526
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   527
by auto
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   528
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   529
text{*The precondition could be weakened to @{term "0\<le>x"}*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   530
lemma real_mult_less_mono:
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   531
     "[| u<v;  x<y;  (0::real) < v;  0 < x |] ==> u*x < v* y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   532
 by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   533
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   534
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   535
  by (force elim: order_less_asym
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   536
            simp add: Ring_and_Field.mult_less_cancel_right)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   537
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   538
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   539
apply (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   540
apply (blast intro: elim: order_less_asym) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   541
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   542
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   543
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
15923
01d5d0c1c078 fixed lin.arith
nipkow
parents: 15542
diff changeset
   544
by(simp add:mult_commute)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   545
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   546
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y"
22958
b3a5569a81e5 cleaned up
huffman
parents: 22456
diff changeset
   547
by (rule add_pos_pos)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   548
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   549
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y"
22958
b3a5569a81e5 cleaned up
huffman
parents: 22456
diff changeset
   550
by (rule add_nonneg_nonneg)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   551
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   552
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
22958
b3a5569a81e5 cleaned up
huffman
parents: 22456
diff changeset
   553
by (rule inverse_unique [symmetric])
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   554
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   555
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
22958
b3a5569a81e5 cleaned up
huffman
parents: 22456
diff changeset
   556
by (simp add: one_less_inverse_iff)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   557
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   558
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   559
subsection{*Embedding the Integers into the Reals*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   560
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   561
defs (overloaded)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   562
  real_of_nat_def: "real z == of_nat z"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   563
  real_of_int_def: "real z == of_int z"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   564
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   565
lemma real_eq_of_nat: "real = of_nat"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   566
  apply (rule ext)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   567
  apply (unfold real_of_nat_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   568
  apply (rule refl)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   569
  done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   570
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   571
lemma real_eq_of_int: "real = of_int"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   572
  apply (rule ext)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   573
  apply (unfold real_of_int_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   574
  apply (rule refl)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   575
  done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   576
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   577
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   578
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   579
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   580
lemma real_of_one [simp]: "real (1::int) = (1::real)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   581
by (simp add: real_of_int_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   582
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   583
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   584
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   585
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   586
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   587
by (simp add: real_of_int_def) 
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   588
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   589
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   590
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   591
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   592
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   593
by (simp add: real_of_int_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   594
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   595
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   596
  apply (subst real_eq_of_int)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   597
  apply (rule of_int_setsum)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   598
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   599
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   600
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   601
    (PROD x:A. real(f x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   602
  apply (subst real_eq_of_int)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   603
  apply (rule of_int_setprod)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   604
done
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   605
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   606
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   607
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   608
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   609
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   610
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   611
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   612
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   613
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   614
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   615
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   616
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   617
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   618
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   619
by (simp add: real_of_int_def) 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   620
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   621
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   622
by (simp add: real_of_int_def) 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   623
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   624
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   625
by (simp add: real_of_int_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   626
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   627
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   628
by (simp add: real_of_int_def)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   629
16888
7cb4bcfa058e added list of theorem changes to NEWS
avigad
parents: 16819
diff changeset
   630
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
7cb4bcfa058e added list of theorem changes to NEWS
avigad
parents: 16819
diff changeset
   631
by (auto simp add: abs_if)
7cb4bcfa058e added list of theorem changes to NEWS
avigad
parents: 16819
diff changeset
   632
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   633
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   634
  apply (subgoal_tac "real n + 1 = real (n + 1)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   635
  apply (simp del: real_of_int_add)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   636
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   637
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   638
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   639
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   640
  apply (subgoal_tac "real m + 1 = real (m + 1)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   641
  apply (simp del: real_of_int_add)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   642
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   643
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   644
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   645
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   646
    real (x div d) + (real (x mod d)) / (real d)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   647
proof -
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   648
  assume "d ~= 0"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   649
  have "x = (x div d) * d + x mod d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   650
    by auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   651
  then have "real x = real (x div d) * real d + real(x mod d)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   652
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   653
  then have "real x / real d = ... / real d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   654
    by simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   655
  then show ?thesis
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   656
    by (auto simp add: add_divide_distrib ring_eq_simps prems)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   657
qed
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   658
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   659
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   660
    real(n div d) = real n / real d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   661
  apply (frule real_of_int_div_aux [of d n])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   662
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   663
  apply (simp add: zdvd_iff_zmod_eq_0)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   664
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   665
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   666
lemma real_of_int_div2:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   667
  "0 <= real (n::int) / real (x) - real (n div x)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   668
  apply (case_tac "x = 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   669
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   670
  apply (case_tac "0 < x")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   671
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   672
  apply (subst real_of_int_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   673
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   674
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   675
  apply (subst zero_le_divide_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   676
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   677
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   678
  apply (subst real_of_int_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   679
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   680
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   681
  apply (subst zero_le_divide_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   682
  apply auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   683
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   684
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   685
lemma real_of_int_div3:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   686
  "real (n::int) / real (x) - real (n div x) <= 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   687
  apply(case_tac "x = 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   688
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   689
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   690
  apply (subst real_of_int_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   691
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   692
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   693
  apply (subst divide_le_eq)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   694
  apply clarsimp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   695
  apply (rule conjI)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   696
  apply (rule impI)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   697
  apply (rule order_less_imp_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   698
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   699
  apply (rule impI)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   700
  apply (rule order_less_imp_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   701
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   702
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   703
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   704
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   705
  by (insert real_of_int_div2 [of n x], simp)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   706
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   707
subsection{*Embedding the Naturals into the Reals*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   708
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   709
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   710
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   711
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   712
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   713
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   714
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   715
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   716
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   717
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   718
(*Not for addsimps: often the LHS is used to represent a positive natural*)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   719
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   720
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   721
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   722
lemma real_of_nat_less_iff [iff]: 
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   723
     "(real (n::nat) < real m) = (n < m)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   724
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   725
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   726
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   727
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   728
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   729
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   730
by (simp add: real_of_nat_def zero_le_imp_of_nat)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   731
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   732
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   733
by (simp add: real_of_nat_def del: of_nat_Suc)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   734
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   735
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   736
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   737
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   738
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   739
    (SUM x:A. real(f x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   740
  apply (subst real_eq_of_nat)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   741
  apply (rule of_nat_setsum)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   742
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   743
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   744
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   745
    (PROD x:A. real(f x))"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   746
  apply (subst real_eq_of_nat)+
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   747
  apply (rule of_nat_setprod)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   748
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   749
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   750
lemma real_of_card: "real (card A) = setsum (%x.1) A"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   751
  apply (subst card_eq_setsum)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   752
  apply (subst real_of_nat_setsum)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   753
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   754
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   755
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   756
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   757
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   758
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   759
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   760
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   761
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   762
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   763
by (simp add: add: real_of_nat_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   764
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   765
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   766
by (simp add: add: real_of_nat_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   767
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   768
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   769
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   770
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   771
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   772
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   773
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   774
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   775
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   776
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   777
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   778
  apply (subgoal_tac "real n + 1 = real (Suc n)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   779
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   780
  apply (auto simp add: real_of_nat_Suc)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   781
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   782
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   783
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   784
  apply (subgoal_tac "real m + 1 = real (Suc m)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   785
  apply (simp add: less_Suc_eq_le)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   786
  apply (simp add: real_of_nat_Suc)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   787
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   788
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   789
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   790
    real (x div d) + (real (x mod d)) / (real d)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   791
proof -
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   792
  assume "0 < d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   793
  have "x = (x div d) * d + x mod d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   794
    by auto
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   795
  then have "real x = real (x div d) * real d + real(x mod d)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   796
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   797
  then have "real x / real d = \<dots> / real d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   798
    by simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   799
  then show ?thesis
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   800
    by (auto simp add: add_divide_distrib ring_eq_simps prems)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   801
qed
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   802
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   803
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   804
    real(n div d) = real n / real d"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   805
  apply (frule real_of_nat_div_aux [of d n])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   806
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   807
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   808
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   809
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   810
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   811
lemma real_of_nat_div2:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   812
  "0 <= real (n::nat) / real (x) - real (n div x)"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   813
  apply(case_tac "x = 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   814
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   815
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   816
  apply (subst real_of_nat_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   817
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   818
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   819
  apply (subst zero_le_divide_iff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   820
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   821
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   822
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   823
lemma real_of_nat_div3:
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   824
  "real (n::nat) / real (x) - real (n div x) <= 1"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   825
  apply(case_tac "x = 0")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   826
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   827
  apply (simp add: compare_rls)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   828
  apply (subst real_of_nat_div_aux)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   829
  apply assumption
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   830
  apply simp
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   831
done
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   832
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   833
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   834
  by (insert real_of_nat_div2 [of n x], simp)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   835
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   836
lemma real_of_int_real_of_nat: "real (int n) = real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   837
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   838
14426
de890c247b38 fixed bugs in the setup of arithmetic procedures
paulson
parents: 14421
diff changeset
   839
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
de890c247b38 fixed bugs in the setup of arithmetic procedures
paulson
parents: 14421
diff changeset
   840
by (simp add: real_of_int_def real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   841
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   842
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   843
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   844
  apply force
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   845
  apply (simp only: real_of_int_real_of_nat)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16417
diff changeset
   846
done
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   847
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   848
subsection{*Numerals and Arithmetic*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   849
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   850
instance real :: number ..
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   851
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   852
defs (overloaded)
20485
3078fd2eec7b got rid of Numeral.bin type
haftmann
parents: 20217
diff changeset
   853
  real_number_of_def: "(number_of w :: real) == of_int w"
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   854
    --{*the type constraint is essential!*}
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   855
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   856
instance real :: number_ring
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   857
by (intro_classes, simp add: real_number_of_def) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   858
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   859
text{*Collapse applications of @{term real} to @{term number_of}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   860
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   861
by (simp add:  real_of_int_def of_int_number_of_eq)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   862
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   863
lemma real_of_nat_number_of [simp]:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   864
     "real (number_of v :: nat) =  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   865
        (if neg (number_of v :: int) then 0  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   866
         else (number_of v :: real))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   867
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   868
 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   869
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   870
use "real_arith.ML"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   871
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   872
setup real_arith_setup
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   873
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   874
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   875
lemma real_diff_mult_distrib:
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   876
  fixes a::real
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   877
  shows "a * (b - c) = a * b - a * c" 
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   878
proof -
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   879
  have "a * (b - c) = a * (b + -c)" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   880
  also have "\<dots> = (b + -c) * a" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   881
  also have "\<dots> = b*a + (-c)*a" by (rule real_add_mult_distrib)
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   882
  also have "\<dots> = a*b - a*c" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   883
  finally show ?thesis .
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   884
qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   885
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 16973
diff changeset
   886
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   887
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   888
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   889
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   890
lemma real_0_le_divide_iff:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   891
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   892
by (simp add: real_divide_def zero_le_mult_iff, auto)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   893
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   894
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   895
by arith
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   896
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   897
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   898
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   899
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   900
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   901
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   902
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   903
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   904
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   905
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   906
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   907
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   908
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15077
diff changeset
   909
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   910
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   911
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   912
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   913
(*
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   914
FIXME: we should have this, as for type int, but many proofs would break.
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   915
It replaces x+-y by x-y.
15086
e6a2a98d5ef5 removal of more iff-rules from RealDef.thy
paulson
parents: 15085
diff changeset
   916
declare real_diff_def [symmetric, simp]
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   917
*)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   918
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   919
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   920
subsubsection{*Density of the Reals*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   921
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   922
lemma real_lbound_gt_zero:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   923
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   924
apply (rule_tac x = " (min d1 d2) /2" in exI)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   925
apply (simp add: min_def)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   926
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   927
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   928
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   929
text{*Similar results are proved in @{text Ring_and_Field}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   930
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   931
  by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   932
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   933
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   934
  by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   935
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   936
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   937
subsection{*Absolute Value Function for the Reals*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   938
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   939
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14754
diff changeset
   940
by (simp add: abs_if)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   941
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   942
lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   943
by (force simp add: Ring_and_Field.abs_less_iff)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   944
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   945
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   946
by (force simp add: OrderedGroup.abs_le_iff)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   947
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   948
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14754
diff changeset
   949
by (simp add: abs_if)
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   950
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   951
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
22958
b3a5569a81e5 cleaned up
huffman
parents: 22456
diff changeset
   952
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   953
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   954
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   955
by simp
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   956
 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   957
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   958
by simp
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   959
23031
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   960
subsection{*Code generation using Isabelle's rats*}
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   961
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   962
types_code
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   963
  real ("Rat.rat")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   964
attach (term_of) {*
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   965
fun term_of_real x =
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   966
 let 
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   967
  val rT = HOLogic.realT
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   968
  val (p, q) = Rat.quotient_of_rat x
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   969
 in if q = 1 then HOLogic.mk_number rT p
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   970
    else Const("HOL.divide",[rT,rT] ---> rT) $
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   971
           (HOLogic.mk_number rT p) $ (HOLogic.mk_number rT q)
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   972
end;
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   973
*}
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   974
attach (test) {*
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   975
fun gen_real i =
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   976
let val p = random_range 0 i; val q = random_range 0 i;
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   977
    val r = if q=0 then Rat.rat_of_int i else Rat.rat_of_quotient(p,q)
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   978
in if one_of [true,false] then r else Rat.neg r end;
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   979
*}
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   980
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   981
consts_code
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   982
  "0 :: real" ("Rat.zero")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   983
  "1 :: real" ("Rat.one")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   984
  "uminus :: real \<Rightarrow> real" ("Rat.neg")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   985
  "op + :: real \<Rightarrow> real \<Rightarrow> real" ("Rat.add")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   986
  "op * :: real \<Rightarrow> real \<Rightarrow> real" ("Rat.mult")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   987
  "inverse :: real \<Rightarrow> real" ("Rat.inv")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   988
  "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool" ("Rat.le")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   989
  "op < :: real \<Rightarrow> real \<Rightarrow> bool" ("(Rat.ord (_, _) = LESS)")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   990
  "op = :: real \<Rightarrow> real \<Rightarrow> bool" ("curry Rat.eq")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   991
  "real :: int \<Rightarrow> real" ("Rat.rat'_of'_int")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   992
  "real :: nat \<Rightarrow> real" ("(Rat.rat'_of'_int o {*int*})")
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   993
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   994
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   995
lemma [code, code unfold]:
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   996
  "number_of k = real (number_of k :: int)"
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   997
  by simp
9da9585c816e added code generation based on Isabelle's rat type.
nipkow
parents: 22970
diff changeset
   998
5588
a3ab526bb891 Revised version with Abelian group simprocs
paulson
parents:
diff changeset
   999
end