src/HOL/Real/RealDef.thy
author paulson
Tue Feb 10 12:02:11 2004 +0100 (2004-02-10)
changeset 14378 69c4d5997669
parent 14369 c50188fe6366
child 14387 e96d5c42c4b0
permissions -rw-r--r--
generic of_nat and of_int functions, and generalization of iszero
and neg
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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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    Description : The reals
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*)
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theory RealDef = PReal:
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constdefs
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  realrel   ::  "((preal * preal) * (preal * preal)) set"
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  "realrel == {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
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typedef (REAL)  real = "UNIV//realrel"
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  by (auto simp add: quotient_def)
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instance real :: ord ..
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instance real :: zero ..
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instance real :: one ..
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instance real :: plus ..
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instance real :: times ..
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instance real :: minus ..
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instance real :: inverse ..
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consts
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   (*Overloaded constant denoting the Real subset of enclosing
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     types such as hypreal and complex*)
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   Reals :: "'a set"
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   (*overloaded constant for injecting other types into "real"*)
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   real :: "'a => real"
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defs (overloaded)
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  real_zero_def:
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  "0 == Abs_REAL(realrel``{(preal_of_rat 1, preal_of_rat 1)})"
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  real_one_def:
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  "1 == Abs_REAL(realrel``
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               {(preal_of_rat 1 + preal_of_rat 1,
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		 preal_of_rat 1)})"
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  real_minus_def:
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  "- R ==  Abs_REAL(UN (x,y):Rep_REAL(R). realrel``{(y,x)})"
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  real_diff_def:
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  "R - (S::real) == R + - S"
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  real_inverse_def:
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  "inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1)"
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  real_divide_def:
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  "R / (S::real) == R * inverse S"
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constdefs
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  (** these don't use the overloaded "real" function: users don't see them **)
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  real_of_preal :: "preal => real"
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  "real_of_preal m     ==
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           Abs_REAL(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})"
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defs (overloaded)
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  real_add_def:
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  "P+Q == Abs_REAL(\<Union>p1\<in>Rep_REAL(P). \<Union>p2\<in>Rep_REAL(Q).
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                   (%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)"
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  real_mult_def:
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  "P*Q == Abs_REAL(\<Union>p1\<in>Rep_REAL(P). \<Union>p2\<in>Rep_REAL(Q).
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                   (%(x1,y1). (%(x2,y2). realrel``{(x1*x2+y1*y2,x1*y2+x2*y1)})
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		   p2) p1)"
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  real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)"
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  real_le_def:
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  "P \<le> (Q::real) == \<exists>x1 y1 x2 y2. x1 + y2 \<le> x2 + y1 &
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                            (x1,y1) \<in> Rep_REAL(P) & (x2,y2) \<in> Rep_REAL(Q)"
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  real_abs_def:  "abs (r::real) == (if 0 \<le> r then r else -r)"
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syntax (xsymbols)
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  Reals     :: "'a set"                   ("\<real>")
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subsection{*Proving that realrel is an equivalence relation*}
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lemma preal_trans_lemma:
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  assumes "x + y1 = x1 + y"
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      and "x + y2 = x2 + y"
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  shows "x1 + y2 = x2 + (y1::preal)"
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proof -
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  have "(x1 + y2) + x = (x + y2) + x1" by (simp add: preal_add_ac) 
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  also have "... = (x2 + y) + x1"  by (simp add: prems)
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  also have "... = x2 + (x1 + y)"  by (simp add: preal_add_ac)
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  also have "... = x2 + (x + y1)"  by (simp add: prems)
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  also have "... = (x2 + y1) + x"  by (simp add: preal_add_ac)
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  finally have "(x1 + y2) + x = (x2 + y1) + x" .
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  thus ?thesis by (simp add: preal_add_right_cancel_iff) 
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qed
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lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)): realrel) = (x1 + y2 = x2 + y1)"
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by (unfold realrel_def, blast)
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lemma realrel_refl: "(x,x): realrel"
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apply (case_tac "x")
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apply (simp add: realrel_def)
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done
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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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(* (realrel `` {x} = realrel `` {y}) = ((x,y) : 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 (unfold REAL_def realrel_def quotient_def, blast)
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lemma inj_on_Abs_REAL: "inj_on Abs_REAL REAL"
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apply (rule inj_on_inverseI)
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apply (erule Abs_REAL_inverse)
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done
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declare inj_on_Abs_REAL [THEN inj_on_iff, simp]
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declare Abs_REAL_inverse [simp]
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lemmas eq_realrelD = equiv_realrel [THEN [2] eq_equiv_class]
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lemma inj_Rep_REAL: "inj Rep_REAL"
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apply (rule inj_on_inverseI)
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apply (rule Rep_REAL_inverse)
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done
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(** real_of_preal: the injection from preal to real **)
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lemma inj_real_of_preal: "inj(real_of_preal)"
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apply (rule inj_onI)
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apply (unfold real_of_preal_def)
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apply (drule inj_on_Abs_REAL [THEN inj_onD])
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apply (rule realrel_in_real)+
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apply (drule eq_equiv_class)
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apply (rule equiv_realrel, blast)
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apply (simp add: realrel_def preal_add_right_cancel_iff)
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done
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lemma eq_Abs_REAL: 
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    "(!!x y. z = Abs_REAL(realrel``{(x,y)}) ==> P) ==> P"
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apply (rule_tac x1 = z in Rep_REAL [unfolded REAL_def, THEN quotientE])
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apply (drule_tac f = Abs_REAL in arg_cong)
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apply (case_tac "x")
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apply (simp add: Rep_REAL_inverse)
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done
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subsection{*Congruence property for addition*}
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lemma real_add_congruent2_lemma:
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     "[|a + ba = aa + b; ab + bc = ac + bb|]
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      ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
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apply (simp add: preal_add_assoc) 
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apply (rule preal_add_left_commute [of ab, THEN ssubst])
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apply (simp add: preal_add_assoc [symmetric])
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apply (simp add: preal_add_ac)
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done
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lemma real_add:
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  "Abs_REAL(realrel``{(x1,y1)}) + Abs_REAL(realrel``{(x2,y2)}) =
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   Abs_REAL(realrel``{(x1+x2, y1+y2)})"
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apply (simp add: real_add_def UN_UN_split_split_eq)
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apply (subst equiv_realrel [THEN UN_equiv_class2])
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apply (auto simp add: congruent2_def)
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apply (blast intro: real_add_congruent2_lemma) 
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done
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lemma real_add_commute: "(z::real) + w = w + z"
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apply (rule eq_Abs_REAL [of z])
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apply (rule eq_Abs_REAL [of w])
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apply (simp add: preal_add_ac real_add)
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done
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lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)"
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apply (rule eq_Abs_REAL [of z1])
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apply (rule eq_Abs_REAL [of z2])
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apply (rule eq_Abs_REAL [of z3])
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apply (simp add: real_add preal_add_assoc)
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done
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lemma real_add_zero_left: "(0::real) + z = z"
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apply (unfold real_of_preal_def real_zero_def)
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apply (rule eq_Abs_REAL [of z])
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apply (simp add: real_add preal_add_ac)
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done
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lemma real_add_zero_right: "z + (0::real) = z"
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by (simp add: real_add_zero_left real_add_commute)
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instance real :: plus_ac0
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  by (intro_classes,
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      (assumption | 
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       rule real_add_commute real_add_assoc real_add_zero_left)+)
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subsection{*Additive Inverse on real*}
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lemma real_minus_congruent:
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  "congruent realrel (%p. (%(x,y). realrel``{(y,x)}) p)"
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apply (unfold congruent_def, clarify)
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apply (simp add: preal_add_commute)
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done
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lemma real_minus:
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      "- (Abs_REAL(realrel``{(x,y)})) = Abs_REAL(realrel `` {(y,x)})"
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apply (unfold real_minus_def)
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apply (rule_tac f = Abs_REAL in arg_cong)
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apply (simp add: realrel_in_real [THEN Abs_REAL_inverse] 
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            UN_equiv_class [OF equiv_realrel real_minus_congruent])
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done
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lemma real_add_minus_left: "(-z) + z = (0::real)"
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apply (unfold real_zero_def)
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apply (rule eq_Abs_REAL [of z])
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apply (simp add: real_minus real_add preal_add_commute)
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done
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subsection{*Congruence property for multiplication*}
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lemma real_mult_congruent2_lemma:
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     "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
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          x * x1 + y * y1 + (x * y2 + x2 * y) =
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          x * x2 + y * y2 + (x * y1 + x1 * y)"
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apply (simp add: preal_add_left_commute preal_add_assoc [symmetric] preal_add_mult_distrib2 [symmetric])
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apply (rule preal_mult_commute [THEN subst])
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apply (rule_tac y1 = x2 in preal_mult_commute [THEN subst])
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apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric])
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apply (simp add: preal_add_commute)
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done
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lemma real_mult_congruent2:
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    "congruent2 realrel (%p1 p2.
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        (%(x1,y1). (%(x2,y2). realrel``{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)"
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apply (rule equiv_realrel [THEN congruent2_commuteI], clarify)
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apply (unfold split_def)
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apply (simp add: preal_mult_commute preal_add_commute)
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apply (auto simp add: real_mult_congruent2_lemma)
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done
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lemma real_mult:
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   "Abs_REAL((realrel``{(x1,y1)})) * Abs_REAL((realrel``{(x2,y2)})) =
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    Abs_REAL(realrel `` {(x1*x2+y1*y2,x1*y2+x2*y1)})"
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apply (unfold real_mult_def)
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apply (simp add: equiv_realrel [THEN UN_equiv_class2] real_mult_congruent2)
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done
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lemma real_mult_commute: "(z::real) * w = w * z"
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apply (rule eq_Abs_REAL [of z])
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apply (rule eq_Abs_REAL [of w])
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apply (simp add: real_mult preal_add_ac preal_mult_ac)
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done
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lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
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apply (rule eq_Abs_REAL [of z1])
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apply (rule eq_Abs_REAL [of z2])
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apply (rule eq_Abs_REAL [of z3])
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apply (simp add: preal_add_mult_distrib2 real_mult preal_add_ac preal_mult_ac)
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done
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lemma real_mult_1: "(1::real) * z = z"
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apply (unfold real_one_def)
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apply (rule eq_Abs_REAL [of z])
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apply (simp add: real_mult preal_add_mult_distrib2 preal_mult_1_right
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                 preal_mult_ac preal_add_ac)
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done
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lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
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apply (rule eq_Abs_REAL [of z1])
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apply (rule eq_Abs_REAL [of z2])
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apply (rule eq_Abs_REAL [of w])
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apply (simp add: preal_add_mult_distrib2 real_add real_mult preal_add_ac preal_mult_ac)
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done
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text{*one and zero are distinct*}
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lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
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apply (subgoal_tac "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1")
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 prefer 2 apply (simp add: preal_self_less_add_left) 
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apply (unfold real_zero_def real_one_def)
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apply (auto simp add: preal_add_right_cancel_iff)
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done
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subsection{*existence of inverse*}
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lemma real_zero_iff: "Abs_REAL (realrel `` {(x, x)}) = 0"
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apply (unfold real_zero_def)
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apply (auto simp add: preal_add_commute)
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done
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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 (unfold real_zero_def real_one_def)
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apply (rule eq_Abs_REAL [of x])
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apply (cut_tac x = xa and y = y in linorder_less_linear)
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apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
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apply (rule_tac
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        x = "Abs_REAL (realrel `` { (preal_of_rat 1, 
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                            inverse (D) + preal_of_rat 1)}) " 
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   318
       in exI)
paulson@14334
   319
apply (rule_tac [2]
paulson@14365
   320
        x = "Abs_REAL (realrel `` { (inverse (D) + preal_of_rat 1,
paulson@14365
   321
                   preal_of_rat 1)})" 
paulson@14334
   322
       in exI)
paulson@14365
   323
apply (auto simp add: real_mult preal_mult_1_right
paulson@14329
   324
              preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1
paulson@14365
   325
              preal_mult_inverse_right preal_add_ac preal_mult_ac)
paulson@14269
   326
done
paulson@14269
   327
paulson@14365
   328
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
paulson@14269
   329
apply (unfold real_inverse_def)
paulson@14365
   330
apply (frule real_mult_inverse_left_ex, safe)
paulson@14269
   331
apply (rule someI2, auto)
paulson@14269
   332
done
paulson@14334
   333
paulson@14341
   334
paulson@14341
   335
subsection{*The Real Numbers form a Field*}
paulson@14341
   336
paulson@14334
   337
instance real :: field
paulson@14334
   338
proof
paulson@14334
   339
  fix x y z :: real
paulson@14334
   340
  show "(x + y) + z = x + (y + z)" by (rule real_add_assoc)
paulson@14334
   341
  show "x + y = y + x" by (rule real_add_commute)
paulson@14334
   342
  show "0 + x = x" by simp
paulson@14334
   343
  show "- x + x = 0" by (rule real_add_minus_left)
paulson@14334
   344
  show "x - y = x + (-y)" by (simp add: real_diff_def)
paulson@14334
   345
  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
paulson@14334
   346
  show "x * y = y * x" by (rule real_mult_commute)
paulson@14334
   347
  show "1 * x = x" by (rule real_mult_1)
paulson@14334
   348
  show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib)
paulson@14334
   349
  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
paulson@14365
   350
  show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
paulson@14334
   351
  show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: real_divide_def)
paulson@14341
   352
  assume eq: "z+x = z+y" 
paulson@14341
   353
    hence "(-z + z) + x = (-z + z) + y" by (simp only: eq real_add_assoc)
paulson@14341
   354
    thus "x = y" by (simp add: real_add_minus_left)
paulson@14334
   355
qed
paulson@14334
   356
paulson@14334
   357
paulson@14341
   358
text{*Inverse of zero!  Useful to simplify certain equations*}
paulson@14269
   359
paulson@14334
   360
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
paulson@14334
   361
apply (unfold real_inverse_def)
paulson@14334
   362
apply (rule someI2)
paulson@14334
   363
apply (auto simp add: zero_neq_one)
paulson@14269
   364
done
paulson@14334
   365
paulson@14334
   366
lemma DIVISION_BY_ZERO: "a / (0::real) = 0"
paulson@14334
   367
  by (simp add: real_divide_def INVERSE_ZERO)
paulson@14334
   368
paulson@14334
   369
instance real :: division_by_zero
paulson@14334
   370
proof
paulson@14334
   371
  fix x :: real
paulson@14334
   372
  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
paulson@14334
   373
  show "x/0 = 0" by (rule DIVISION_BY_ZERO) 
paulson@14334
   374
qed
paulson@14334
   375
paulson@14334
   376
paulson@14334
   377
(*Pull negations out*)
paulson@14334
   378
declare minus_mult_right [symmetric, simp] 
paulson@14334
   379
        minus_mult_left [symmetric, simp]
paulson@14334
   380
paulson@14334
   381
lemma real_mult_1_right: "z * (1::real) = z"
paulson@14334
   382
  by (rule Ring_and_Field.mult_1_right)
paulson@14269
   383
paulson@14269
   384
paulson@14365
   385
subsection{*The @{text "\<le>"} Ordering*}
paulson@14269
   386
paulson@14365
   387
lemma real_le_refl: "w \<le> (w::real)"
paulson@14378
   388
apply (rule eq_Abs_REAL [of w])
paulson@14365
   389
apply (force simp add: real_le_def)
paulson@14269
   390
done
paulson@14269
   391
paulson@14378
   392
text{*The arithmetic decision procedure is not set up for type preal.
paulson@14378
   393
  This lemma is currently unused, but it could simplify the proofs of the
paulson@14378
   394
  following two lemmas.*}
paulson@14378
   395
lemma preal_eq_le_imp_le:
paulson@14378
   396
  assumes eq: "a+b = c+d" and le: "c \<le> a"
paulson@14378
   397
  shows "b \<le> (d::preal)"
paulson@14378
   398
proof -
paulson@14378
   399
  have "c+d \<le> a+d" by (simp add: prems preal_cancels)
paulson@14378
   400
  hence "a+b \<le> a+d" by (simp add: prems)
paulson@14378
   401
  thus "b \<le> d" by (simp add: preal_cancels)
paulson@14378
   402
qed
paulson@14378
   403
paulson@14378
   404
lemma real_le_lemma:
paulson@14378
   405
  assumes l: "u1 + v2 \<le> u2 + v1"
paulson@14378
   406
      and "x1 + v1 = u1 + y1"
paulson@14378
   407
      and "x2 + v2 = u2 + y2"
paulson@14378
   408
  shows "x1 + y2 \<le> x2 + (y1::preal)"
paulson@14365
   409
proof -
paulson@14378
   410
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
paulson@14378
   411
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac)
paulson@14378
   412
  also have "... \<le> (x2+y1) + (u2+v1)"
paulson@14365
   413
         by (simp add: prems preal_add_le_cancel_left)
paulson@14378
   414
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
paulson@14378
   415
qed						 
paulson@14378
   416
paulson@14378
   417
lemma real_le: 
paulson@14378
   418
  "(Abs_REAL(realrel``{(x1,y1)}) \<le> Abs_REAL(realrel``{(x2,y2)})) =  
paulson@14378
   419
   (x1 + y2 \<le> x2 + y1)"
paulson@14378
   420
apply (simp add: real_le_def) 
paulson@14378
   421
apply (auto intro: real_le_lemma);
paulson@14378
   422
done
paulson@14378
   423
paulson@14378
   424
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
paulson@14378
   425
apply (rule eq_Abs_REAL [of z])
paulson@14378
   426
apply (rule eq_Abs_REAL [of w])
paulson@14378
   427
apply (simp add: real_le order_antisym) 
paulson@14378
   428
done
paulson@14378
   429
paulson@14378
   430
lemma real_trans_lemma:
paulson@14378
   431
  assumes "x + v \<le> u + y"
paulson@14378
   432
      and "u + v' \<le> u' + v"
paulson@14378
   433
      and "x2 + v2 = u2 + y2"
paulson@14378
   434
  shows "x + v' \<le> u' + (y::preal)"
paulson@14378
   435
proof -
paulson@14378
   436
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac)
paulson@14378
   437
  also have "... \<le> (u+y) + (u+v')" 
paulson@14378
   438
    by (simp add: preal_add_le_cancel_right prems) 
paulson@14378
   439
  also have "... \<le> (u+y) + (u'+v)" 
paulson@14378
   440
    by (simp add: preal_add_le_cancel_left prems) 
paulson@14378
   441
  also have "... = (u'+y) + (u+v)"  by (simp add: preal_add_ac)
paulson@14378
   442
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
paulson@14365
   443
qed						 
paulson@14269
   444
paulson@14365
   445
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
paulson@14378
   446
apply (rule eq_Abs_REAL [of i])
paulson@14378
   447
apply (rule eq_Abs_REAL [of j])
paulson@14378
   448
apply (rule eq_Abs_REAL [of k])
paulson@14378
   449
apply (simp add: real_le) 
paulson@14378
   450
apply (blast intro: real_trans_lemma) 
paulson@14334
   451
done
paulson@14334
   452
paulson@14334
   453
(* Axiom 'order_less_le' of class 'order': *)
paulson@14334
   454
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
paulson@14365
   455
by (simp add: real_less_def)
paulson@14365
   456
paulson@14365
   457
instance real :: order
paulson@14365
   458
proof qed
paulson@14365
   459
 (assumption |
paulson@14365
   460
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
paulson@14365
   461
paulson@14378
   462
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14378
   463
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
paulson@14378
   464
apply (rule eq_Abs_REAL [of z])
paulson@14378
   465
apply (rule eq_Abs_REAL [of w]) 
paulson@14378
   466
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels)
paulson@14378
   467
apply (cut_tac x="x+ya" and y="xa+y" in linorder_linear) 
paulson@14378
   468
apply (auto ); 
paulson@14334
   469
done
paulson@14334
   470
paulson@14334
   471
paulson@14334
   472
instance real :: linorder
paulson@14334
   473
  by (intro_classes, rule real_le_linear)
paulson@14334
   474
paulson@14334
   475
paulson@14378
   476
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
paulson@14378
   477
apply (rule eq_Abs_REAL [of x])
paulson@14378
   478
apply (rule eq_Abs_REAL [of y]) 
paulson@14378
   479
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
paulson@14378
   480
                      preal_add_ac)
paulson@14378
   481
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
paulson@14378
   482
done 
paulson@14378
   483
paulson@14365
   484
lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)"
paulson@14365
   485
apply (auto simp add: real_le_eq_diff [of x] real_le_eq_diff [of "z+x"])
paulson@14365
   486
apply (subgoal_tac "z + x - (z + y) = (z + -z) + (x - y)")
paulson@14365
   487
 prefer 2 apply (simp add: diff_minus add_ac, simp) 
paulson@14365
   488
done
paulson@14334
   489
paulson@14365
   490
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
paulson@14365
   491
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14365
   492
paulson@14365
   493
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
paulson@14365
   494
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14334
   495
paulson@14334
   496
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
paulson@14378
   497
apply (rule eq_Abs_REAL [of x])
paulson@14378
   498
apply (rule eq_Abs_REAL [of y])
paulson@14378
   499
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
paulson@14378
   500
                 linorder_not_le [where 'a = preal] 
paulson@14378
   501
                  real_zero_def real_le real_mult)
paulson@14365
   502
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
paulson@14378
   503
apply (auto  dest!: less_add_left_Ex 
paulson@14365
   504
     simp add: preal_add_ac preal_mult_ac 
paulson@14378
   505
          preal_add_mult_distrib2 preal_cancels preal_self_less_add_right)
paulson@14334
   506
done
paulson@14334
   507
paulson@14334
   508
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
paulson@14334
   509
apply (rule real_sum_gt_zero_less)
paulson@14334
   510
apply (drule real_less_sum_gt_zero [of x y])
paulson@14334
   511
apply (drule real_mult_order, assumption)
paulson@14334
   512
apply (simp add: right_distrib)
paulson@14334
   513
done
paulson@14334
   514
paulson@14365
   515
text{*lemma for proving @{term "0<(1::real)"}*}
paulson@14365
   516
lemma real_zero_le_one: "0 \<le> (1::real)"
paulson@14378
   517
apply (simp add: real_zero_def real_one_def real_le 
paulson@14378
   518
                 preal_self_less_add_left order_less_imp_le)
paulson@14334
   519
done
paulson@14334
   520
paulson@14378
   521
paulson@14334
   522
subsection{*The Reals Form an Ordered Field*}
paulson@14334
   523
paulson@14334
   524
instance real :: ordered_field
paulson@14334
   525
proof
paulson@14334
   526
  fix x y z :: real
paulson@14365
   527
  show "0 < (1::real)"
paulson@14365
   528
    by (simp add: real_less_def real_zero_le_one real_zero_not_eq_one)  
paulson@14334
   529
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
paulson@14334
   530
  show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2)
paulson@14334
   531
  show "\<bar>x\<bar> = (if x < 0 then -x else x)"
paulson@14334
   532
    by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le)
paulson@14334
   533
qed
paulson@14334
   534
paulson@14365
   535
paulson@14365
   536
paulson@14365
   537
text{*The function @{term real_of_preal} requires many proofs, but it seems
paulson@14365
   538
to be essential for proving completeness of the reals from that of the
paulson@14365
   539
positive reals.*}
paulson@14365
   540
paulson@14365
   541
lemma real_of_preal_add:
paulson@14365
   542
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
paulson@14365
   543
by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1 
paulson@14365
   544
              preal_add_ac)
paulson@14365
   545
paulson@14365
   546
lemma real_of_preal_mult:
paulson@14365
   547
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
paulson@14365
   548
by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2
paulson@14365
   549
              preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac)
paulson@14365
   550
paulson@14365
   551
paulson@14365
   552
text{*Gleason prop 9-4.4 p 127*}
paulson@14365
   553
lemma real_of_preal_trichotomy:
paulson@14365
   554
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
paulson@14365
   555
apply (unfold real_of_preal_def real_zero_def)
paulson@14378
   556
apply (rule eq_Abs_REAL [of x])
paulson@14365
   557
apply (auto simp add: real_minus preal_add_ac)
paulson@14365
   558
apply (cut_tac x = x and y = y in linorder_less_linear)
paulson@14365
   559
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric])
paulson@14365
   560
apply (auto simp add: preal_add_commute)
paulson@14365
   561
done
paulson@14365
   562
paulson@14365
   563
lemma real_of_preal_leD:
paulson@14365
   564
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
paulson@14365
   565
apply (unfold real_of_preal_def)
paulson@14365
   566
apply (auto simp add: real_le_def preal_add_ac)
paulson@14365
   567
apply (auto simp add: preal_add_assoc [symmetric] preal_add_right_cancel_iff)
paulson@14365
   568
apply (auto simp add: preal_add_ac preal_add_le_cancel_left)
paulson@14365
   569
done
paulson@14365
   570
paulson@14365
   571
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
paulson@14365
   572
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
paulson@14365
   573
paulson@14365
   574
lemma real_of_preal_lessD:
paulson@14365
   575
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
paulson@14365
   576
apply (auto simp add: real_less_def)
paulson@14365
   577
apply (drule real_of_preal_leD) 
paulson@14365
   578
apply (auto simp add: order_le_less) 
paulson@14365
   579
done
paulson@14365
   580
paulson@14365
   581
lemma real_of_preal_less_iff [simp]:
paulson@14365
   582
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
paulson@14365
   583
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
paulson@14365
   584
paulson@14365
   585
lemma real_of_preal_le_iff:
paulson@14365
   586
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
paulson@14365
   587
by (simp add: linorder_not_less [symmetric]) 
paulson@14365
   588
paulson@14365
   589
lemma real_of_preal_zero_less: "0 < real_of_preal m"
paulson@14365
   590
apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def
paulson@14365
   591
            preal_add_ac preal_cancels)
paulson@14365
   592
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
paulson@14365
   593
apply (blast intro: preal_self_less_add_left order_less_imp_le)
paulson@14365
   594
apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) 
paulson@14365
   595
apply (simp add: preal_add_ac) 
paulson@14365
   596
done
paulson@14365
   597
paulson@14365
   598
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
paulson@14365
   599
by (simp add: real_of_preal_zero_less)
paulson@14365
   600
paulson@14365
   601
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
paulson@14365
   602
apply (cut_tac real_of_preal_minus_less_zero)
paulson@14365
   603
apply (fast dest: order_less_trans)
paulson@14365
   604
done
paulson@14365
   605
paulson@14365
   606
paulson@14365
   607
subsection{*Theorems About the Ordering*}
paulson@14365
   608
paulson@14365
   609
text{*obsolete but used a lot*}
paulson@14365
   610
paulson@14365
   611
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)"
paulson@14365
   612
by blast 
paulson@14365
   613
paulson@14365
   614
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y"
paulson@14365
   615
by (simp add: order_le_less)
paulson@14365
   616
paulson@14365
   617
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
paulson@14365
   618
apply (auto simp add: real_of_preal_zero_less)
paulson@14365
   619
apply (cut_tac x = x in real_of_preal_trichotomy)
paulson@14365
   620
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
paulson@14365
   621
done
paulson@14365
   622
paulson@14365
   623
lemma real_gt_preal_preal_Ex:
paulson@14365
   624
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   625
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
paulson@14365
   626
             intro: real_gt_zero_preal_Ex [THEN iffD1])
paulson@14365
   627
paulson@14365
   628
lemma real_ge_preal_preal_Ex:
paulson@14365
   629
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   630
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
paulson@14365
   631
paulson@14365
   632
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
paulson@14365
   633
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
paulson@14365
   634
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
paulson@14365
   635
            simp add: real_of_preal_zero_less)
paulson@14365
   636
paulson@14365
   637
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
paulson@14365
   638
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
paulson@14365
   639
paulson@14334
   640
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)"
paulson@14365
   641
  by (rule Ring_and_Field.add_less_le_mono)
paulson@14334
   642
paulson@14334
   643
lemma real_add_le_less_mono:
paulson@14334
   644
     "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z"
paulson@14365
   645
  by (rule Ring_and_Field.add_le_less_mono)
paulson@14334
   646
paulson@14334
   647
lemma real_zero_less_one: "0 < (1::real)"
paulson@14334
   648
  by (rule Ring_and_Field.zero_less_one)
paulson@14334
   649
paulson@14334
   650
lemma real_le_square [simp]: "(0::real) \<le> x*x"
paulson@14334
   651
 by (rule Ring_and_Field.zero_le_square)
paulson@14334
   652
paulson@14334
   653
paulson@14334
   654
subsection{*More Lemmas*}
paulson@14334
   655
paulson@14334
   656
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14334
   657
by auto
paulson@14334
   658
paulson@14334
   659
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   660
by auto
paulson@14334
   661
paulson@14334
   662
text{*The precondition could be weakened to @{term "0\<le>x"}*}
paulson@14334
   663
lemma real_mult_less_mono:
paulson@14334
   664
     "[| u<v;  x<y;  (0::real) < v;  0 < x |] ==> u*x < v* y"
paulson@14334
   665
 by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
paulson@14334
   666
paulson@14334
   667
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
paulson@14334
   668
  by (force elim: order_less_asym
paulson@14334
   669
            simp add: Ring_and_Field.mult_less_cancel_right)
paulson@14334
   670
paulson@14334
   671
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
paulson@14365
   672
apply (simp add: mult_le_cancel_right)
paulson@14365
   673
apply (blast intro: elim: order_less_asym) 
paulson@14365
   674
done
paulson@14334
   675
paulson@14334
   676
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
paulson@14334
   677
  by (force elim: order_less_asym
paulson@14334
   678
            simp add: Ring_and_Field.mult_le_cancel_left)
paulson@14334
   679
paulson@14334
   680
text{*Only two uses?*}
paulson@14334
   681
lemma real_mult_less_mono':
paulson@14334
   682
     "[| x < y;  r1 < r2;  (0::real) \<le> r1;  0 \<le> x|] ==> r1 * x < r2 * y"
paulson@14334
   683
 by (rule Ring_and_Field.mult_strict_mono')
paulson@14334
   684
paulson@14334
   685
text{*FIXME: delete or at least combine the next two lemmas*}
paulson@14334
   686
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
paulson@14334
   687
apply (drule Ring_and_Field.equals_zero_I [THEN sym])
paulson@14334
   688
apply (cut_tac x = y in real_le_square) 
paulson@14334
   689
apply (auto, drule real_le_anti_sym, auto)
paulson@14334
   690
done
paulson@14334
   691
paulson@14334
   692
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
paulson@14334
   693
apply (rule_tac y = x in real_sum_squares_cancel)
paulson@14334
   694
apply (simp add: real_add_commute)
paulson@14334
   695
done
paulson@14334
   696
paulson@14334
   697
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y"
paulson@14365
   698
by (drule add_strict_mono [of concl: 0 0], assumption, simp)
paulson@14334
   699
paulson@14334
   700
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y"
paulson@14334
   701
apply (drule order_le_imp_less_or_eq)+
paulson@14334
   702
apply (auto intro: real_add_order order_less_imp_le)
paulson@14334
   703
done
paulson@14334
   704
paulson@14365
   705
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
paulson@14365
   706
apply (case_tac "x \<noteq> 0")
paulson@14365
   707
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto)
paulson@14365
   708
done
paulson@14334
   709
paulson@14365
   710
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
paulson@14365
   711
by (auto dest: less_imp_inverse_less)
paulson@14334
   712
paulson@14365
   713
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
paulson@14365
   714
proof -
paulson@14365
   715
  have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square)
paulson@14365
   716
  thus ?thesis by simp
paulson@14365
   717
qed
paulson@14365
   718
paulson@14334
   719
paulson@14365
   720
subsection{*Embedding the Integers into the Reals*}
paulson@14365
   721
paulson@14378
   722
defs (overloaded)
paulson@14378
   723
  real_of_nat_def: "real z == of_nat z"
paulson@14378
   724
  real_of_int_def: "real z == of_int z"
paulson@14365
   725
paulson@14365
   726
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378
   727
by (simp add: real_of_int_def) 
paulson@14365
   728
paulson@14365
   729
lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378
   730
by (simp add: real_of_int_def) 
paulson@14334
   731
paulson@14365
   732
lemma real_of_int_add: "real (x::int) + real y = real (x + y)"
paulson@14378
   733
by (simp add: real_of_int_def) 
paulson@14365
   734
declare real_of_int_add [symmetric, simp]
paulson@14365
   735
paulson@14365
   736
lemma real_of_int_minus: "-real (x::int) = real (-x)"
paulson@14378
   737
by (simp add: real_of_int_def) 
paulson@14365
   738
declare real_of_int_minus [symmetric, simp]
paulson@14365
   739
paulson@14365
   740
lemma real_of_int_diff: "real (x::int) - real y = real (x - y)"
paulson@14378
   741
by (simp add: real_of_int_def) 
paulson@14365
   742
declare real_of_int_diff [symmetric, simp]
paulson@14334
   743
paulson@14365
   744
lemma real_of_int_mult: "real (x::int) * real y = real (x * y)"
paulson@14378
   745
by (simp add: real_of_int_def) 
paulson@14365
   746
declare real_of_int_mult [symmetric, simp]
paulson@14365
   747
paulson@14365
   748
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
paulson@14378
   749
by (simp add: real_of_int_def) 
paulson@14365
   750
paulson@14365
   751
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
paulson@14378
   752
by (simp add: real_of_int_def) 
paulson@14365
   753
paulson@14365
   754
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
paulson@14378
   755
by (simp add: real_of_int_def) 
paulson@14365
   756
paulson@14365
   757
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378
   758
by (simp add: real_of_int_def) 
paulson@14365
   759
paulson@14365
   760
paulson@14365
   761
subsection{*Embedding the Naturals into the Reals*}
paulson@14365
   762
paulson@14334
   763
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365
   764
by (simp add: real_of_nat_def)
paulson@14334
   765
paulson@14334
   766
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365
   767
by (simp add: real_of_nat_def)
paulson@14334
   768
paulson@14365
   769
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378
   770
by (simp add: real_of_nat_def)
paulson@14334
   771
paulson@14334
   772
(*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334
   773
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378
   774
by (simp add: real_of_nat_def)
paulson@14334
   775
paulson@14334
   776
lemma real_of_nat_less_iff [iff]: 
paulson@14334
   777
     "(real (n::nat) < real m) = (n < m)"
paulson@14365
   778
by (simp add: real_of_nat_def)
paulson@14334
   779
paulson@14334
   780
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378
   781
by (simp add: real_of_nat_def)
paulson@14334
   782
paulson@14334
   783
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
paulson@14378
   784
by (simp add: real_of_nat_def zero_le_imp_of_nat)
paulson@14334
   785
paulson@14365
   786
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378
   787
by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365
   788
paulson@14334
   789
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
paulson@14378
   790
by (simp add: real_of_nat_def)
paulson@14334
   791
paulson@14334
   792
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378
   793
by (simp add: real_of_nat_def)
paulson@14334
   794
paulson@14334
   795
lemma real_of_nat_zero_iff: "(real (n::nat) = 0) = (n = 0)"
paulson@14378
   796
by (simp add: real_of_nat_def)
paulson@14334
   797
paulson@14365
   798
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
paulson@14378
   799
by (simp add: add: real_of_nat_def) 
paulson@14334
   800
paulson@14365
   801
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
paulson@14378
   802
by (simp add: add: real_of_nat_def) 
paulson@14365
   803
paulson@14365
   804
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378
   805
by (simp add: add: real_of_nat_def)
paulson@14334
   806
paulson@14365
   807
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378
   808
by (simp add: add: real_of_nat_def)
paulson@14334
   809
paulson@14365
   810
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"
paulson@14378
   811
by (simp add: add: real_of_nat_def)
paulson@14334
   812
paulson@14365
   813
lemma real_of_int_real_of_nat: "real (int n) = real n"
paulson@14378
   814
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
paulson@14378
   815
paulson@14378
   816
paulson@14334
   817
paulson@14378
   818
text{*Still needed for binary arith*}
paulson@14365
   819
lemma real_of_nat_real_of_int: "~neg z ==> real (nat z) = real z"
paulson@14378
   820
proof (simp add: not_neg_eq_ge_0 real_of_nat_def real_of_int_def)
paulson@14378
   821
  assume "0 \<le> z"
paulson@14378
   822
  hence eq: "of_nat (nat z) = z" 
paulson@14378
   823
    by (simp add: nat_0_le int_eq_of_nat[symmetric]) 
paulson@14378
   824
  have "of_nat (nat z) = of_int (of_nat (nat z))" by simp
paulson@14378
   825
  also have "... = of_int z" by (simp add: eq)
paulson@14378
   826
  finally show "of_nat (nat z) = of_int z" .
paulson@14378
   827
qed
paulson@14334
   828
paulson@14334
   829
ML
paulson@14334
   830
{*
paulson@14334
   831
val real_abs_def = thm "real_abs_def";
paulson@14334
   832
paulson@14341
   833
val real_le_def = thm "real_le_def";
paulson@14341
   834
val real_diff_def = thm "real_diff_def";
paulson@14341
   835
val real_divide_def = thm "real_divide_def";
paulson@14341
   836
paulson@14341
   837
val realrel_iff = thm"realrel_iff";
paulson@14341
   838
val realrel_refl = thm"realrel_refl";
paulson@14341
   839
val equiv_realrel = thm"equiv_realrel";
paulson@14341
   840
val equiv_realrel_iff = thm"equiv_realrel_iff";
paulson@14341
   841
val realrel_in_real = thm"realrel_in_real";
paulson@14341
   842
val inj_on_Abs_REAL = thm"inj_on_Abs_REAL";
paulson@14341
   843
val eq_realrelD = thm"eq_realrelD";
paulson@14341
   844
val inj_Rep_REAL = thm"inj_Rep_REAL";
paulson@14341
   845
val inj_real_of_preal = thm"inj_real_of_preal";
paulson@14341
   846
val eq_Abs_REAL = thm"eq_Abs_REAL";
paulson@14341
   847
val real_minus_congruent = thm"real_minus_congruent";
paulson@14341
   848
val real_minus = thm"real_minus";
paulson@14341
   849
val real_add = thm"real_add";
paulson@14341
   850
val real_add_commute = thm"real_add_commute";
paulson@14341
   851
val real_add_assoc = thm"real_add_assoc";
paulson@14341
   852
val real_add_zero_left = thm"real_add_zero_left";
paulson@14341
   853
val real_add_zero_right = thm"real_add_zero_right";
paulson@14341
   854
paulson@14334
   855
val real_mult = thm"real_mult";
paulson@14334
   856
val real_mult_commute = thm"real_mult_commute";
paulson@14334
   857
val real_mult_assoc = thm"real_mult_assoc";
paulson@14334
   858
val real_mult_1 = thm"real_mult_1";
paulson@14334
   859
val real_mult_1_right = thm"real_mult_1_right";
paulson@14334
   860
val preal_le_linear = thm"preal_le_linear";
paulson@14365
   861
val real_mult_inverse_left = thm"real_mult_inverse_left";
paulson@14334
   862
val real_not_refl2 = thm"real_not_refl2";
paulson@14334
   863
val real_of_preal_add = thm"real_of_preal_add";
paulson@14334
   864
val real_of_preal_mult = thm"real_of_preal_mult";
paulson@14334
   865
val real_of_preal_trichotomy = thm"real_of_preal_trichotomy";
paulson@14334
   866
val real_of_preal_minus_less_zero = thm"real_of_preal_minus_less_zero";
paulson@14334
   867
val real_of_preal_not_minus_gt_zero = thm"real_of_preal_not_minus_gt_zero";
paulson@14334
   868
val real_of_preal_zero_less = thm"real_of_preal_zero_less";
paulson@14334
   869
val real_le_imp_less_or_eq = thm"real_le_imp_less_or_eq";
paulson@14334
   870
val real_le_refl = thm"real_le_refl";
paulson@14334
   871
val real_le_linear = thm"real_le_linear";
paulson@14334
   872
val real_le_trans = thm"real_le_trans";
paulson@14334
   873
val real_le_anti_sym = thm"real_le_anti_sym";
paulson@14334
   874
val real_less_le = thm"real_less_le";
paulson@14334
   875
val real_less_sum_gt_zero = thm"real_less_sum_gt_zero";
paulson@14334
   876
val real_gt_zero_preal_Ex = thm "real_gt_zero_preal_Ex";
paulson@14334
   877
val real_gt_preal_preal_Ex = thm "real_gt_preal_preal_Ex";
paulson@14334
   878
val real_ge_preal_preal_Ex = thm "real_ge_preal_preal_Ex";
paulson@14334
   879
val real_less_all_preal = thm "real_less_all_preal";
paulson@14334
   880
val real_less_all_real2 = thm "real_less_all_real2";
paulson@14334
   881
val real_of_preal_le_iff = thm "real_of_preal_le_iff";
paulson@14334
   882
val real_mult_order = thm "real_mult_order";
paulson@14334
   883
val real_zero_less_one = thm "real_zero_less_one";
paulson@14334
   884
val real_add_less_le_mono = thm "real_add_less_le_mono";
paulson@14334
   885
val real_add_le_less_mono = thm "real_add_le_less_mono";
paulson@14334
   886
val real_add_order = thm "real_add_order";
paulson@14334
   887
val real_le_add_order = thm "real_le_add_order";
paulson@14334
   888
val real_le_square = thm "real_le_square";
paulson@14334
   889
val real_mult_less_mono2 = thm "real_mult_less_mono2";
paulson@14334
   890
paulson@14334
   891
val real_mult_less_iff1 = thm "real_mult_less_iff1";
paulson@14334
   892
val real_mult_le_cancel_iff1 = thm "real_mult_le_cancel_iff1";
paulson@14334
   893
val real_mult_le_cancel_iff2 = thm "real_mult_le_cancel_iff2";
paulson@14334
   894
val real_mult_less_mono = thm "real_mult_less_mono";
paulson@14334
   895
val real_mult_less_mono' = thm "real_mult_less_mono'";
paulson@14334
   896
val real_sum_squares_cancel = thm "real_sum_squares_cancel";
paulson@14334
   897
val real_sum_squares_cancel2 = thm "real_sum_squares_cancel2";
paulson@14334
   898
paulson@14334
   899
val real_mult_left_cancel = thm"real_mult_left_cancel";
paulson@14334
   900
val real_mult_right_cancel = thm"real_mult_right_cancel";
paulson@14365
   901
val real_inverse_unique = thm "real_inverse_unique";
paulson@14365
   902
val real_inverse_gt_one = thm "real_inverse_gt_one";
paulson@14365
   903
paulson@14365
   904
val real_of_int_zero = thm"real_of_int_zero";
paulson@14365
   905
val real_of_one = thm"real_of_one";
paulson@14365
   906
val real_of_int_add = thm"real_of_int_add";
paulson@14365
   907
val real_of_int_minus = thm"real_of_int_minus";
paulson@14365
   908
val real_of_int_diff = thm"real_of_int_diff";
paulson@14365
   909
val real_of_int_mult = thm"real_of_int_mult";
paulson@14365
   910
val real_of_int_real_of_nat = thm"real_of_int_real_of_nat";
paulson@14365
   911
val real_of_int_inject = thm"real_of_int_inject";
paulson@14365
   912
val real_of_int_less_iff = thm"real_of_int_less_iff";
paulson@14365
   913
val real_of_int_le_iff = thm"real_of_int_le_iff";
paulson@14334
   914
val real_of_nat_zero = thm "real_of_nat_zero";
paulson@14334
   915
val real_of_nat_one = thm "real_of_nat_one";
paulson@14334
   916
val real_of_nat_add = thm "real_of_nat_add";
paulson@14334
   917
val real_of_nat_Suc = thm "real_of_nat_Suc";
paulson@14334
   918
val real_of_nat_less_iff = thm "real_of_nat_less_iff";
paulson@14334
   919
val real_of_nat_le_iff = thm "real_of_nat_le_iff";
paulson@14334
   920
val real_of_nat_ge_zero = thm "real_of_nat_ge_zero";
paulson@14365
   921
val real_of_nat_Suc_gt_zero = thm "real_of_nat_Suc_gt_zero";
paulson@14334
   922
val real_of_nat_mult = thm "real_of_nat_mult";
paulson@14334
   923
val real_of_nat_inject = thm "real_of_nat_inject";
paulson@14334
   924
val real_of_nat_diff = thm "real_of_nat_diff";
paulson@14334
   925
val real_of_nat_zero_iff = thm "real_of_nat_zero_iff";
paulson@14334
   926
val real_of_nat_gt_zero_cancel_iff = thm "real_of_nat_gt_zero_cancel_iff";
paulson@14334
   927
val real_of_nat_le_zero_cancel_iff = thm "real_of_nat_le_zero_cancel_iff";
paulson@14334
   928
val not_real_of_nat_less_zero = thm "not_real_of_nat_less_zero";
paulson@14334
   929
val real_of_nat_ge_zero_cancel_iff = thm "real_of_nat_ge_zero_cancel_iff";
paulson@14334
   930
*}
paulson@10752
   931
paulson@5588
   932
end