src/HOL/Real/RealDef.thy
author paulson
Sun Feb 15 10:46:37 2004 +0100 (2004-02-15)
changeset 14387 e96d5c42c4b0
parent 14378 69c4d5997669
child 14398 c5c47703f763
permissions -rw-r--r--
Polymorphic treatment of binary arithmetic using axclasses
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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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*)
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header{*Defining the Reals from the Positive Reals*}
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theory RealDef = PReal
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files ("real_arith.ML"):
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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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   319
        x = "Abs_REAL (realrel `` { (preal_of_rat 1, 
paulson@14365
   320
                            inverse (D) + preal_of_rat 1)}) " 
paulson@14334
   321
       in exI)
paulson@14334
   322
apply (rule_tac [2]
paulson@14365
   323
        x = "Abs_REAL (realrel `` { (inverse (D) + preal_of_rat 1,
paulson@14365
   324
                   preal_of_rat 1)})" 
paulson@14334
   325
       in exI)
paulson@14365
   326
apply (auto simp add: real_mult preal_mult_1_right
paulson@14329
   327
              preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1
paulson@14365
   328
              preal_mult_inverse_right preal_add_ac preal_mult_ac)
paulson@14269
   329
done
paulson@14269
   330
paulson@14365
   331
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
paulson@14269
   332
apply (unfold real_inverse_def)
paulson@14365
   333
apply (frule real_mult_inverse_left_ex, safe)
paulson@14269
   334
apply (rule someI2, auto)
paulson@14269
   335
done
paulson@14334
   336
paulson@14341
   337
paulson@14341
   338
subsection{*The Real Numbers form a Field*}
paulson@14341
   339
paulson@14334
   340
instance real :: field
paulson@14334
   341
proof
paulson@14334
   342
  fix x y z :: real
paulson@14334
   343
  show "(x + y) + z = x + (y + z)" by (rule real_add_assoc)
paulson@14334
   344
  show "x + y = y + x" by (rule real_add_commute)
paulson@14334
   345
  show "0 + x = x" by simp
paulson@14334
   346
  show "- x + x = 0" by (rule real_add_minus_left)
paulson@14334
   347
  show "x - y = x + (-y)" by (simp add: real_diff_def)
paulson@14334
   348
  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
paulson@14334
   349
  show "x * y = y * x" by (rule real_mult_commute)
paulson@14334
   350
  show "1 * x = x" by (rule real_mult_1)
paulson@14334
   351
  show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib)
paulson@14334
   352
  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
paulson@14365
   353
  show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
paulson@14334
   354
  show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: real_divide_def)
paulson@14341
   355
  assume eq: "z+x = z+y" 
paulson@14341
   356
    hence "(-z + z) + x = (-z + z) + y" by (simp only: eq real_add_assoc)
paulson@14341
   357
    thus "x = y" by (simp add: real_add_minus_left)
paulson@14334
   358
qed
paulson@14334
   359
paulson@14334
   360
paulson@14341
   361
text{*Inverse of zero!  Useful to simplify certain equations*}
paulson@14269
   362
paulson@14334
   363
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
paulson@14334
   364
apply (unfold real_inverse_def)
paulson@14334
   365
apply (rule someI2)
paulson@14334
   366
apply (auto simp add: zero_neq_one)
paulson@14269
   367
done
paulson@14334
   368
paulson@14334
   369
lemma DIVISION_BY_ZERO: "a / (0::real) = 0"
paulson@14334
   370
  by (simp add: real_divide_def INVERSE_ZERO)
paulson@14334
   371
paulson@14334
   372
instance real :: division_by_zero
paulson@14334
   373
proof
paulson@14334
   374
  fix x :: real
paulson@14334
   375
  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
paulson@14334
   376
  show "x/0 = 0" by (rule DIVISION_BY_ZERO) 
paulson@14334
   377
qed
paulson@14334
   378
paulson@14334
   379
paulson@14334
   380
(*Pull negations out*)
paulson@14334
   381
declare minus_mult_right [symmetric, simp] 
paulson@14334
   382
        minus_mult_left [symmetric, simp]
paulson@14334
   383
paulson@14334
   384
lemma real_mult_1_right: "z * (1::real) = z"
paulson@14334
   385
  by (rule Ring_and_Field.mult_1_right)
paulson@14269
   386
paulson@14269
   387
paulson@14365
   388
subsection{*The @{text "\<le>"} Ordering*}
paulson@14269
   389
paulson@14365
   390
lemma real_le_refl: "w \<le> (w::real)"
paulson@14378
   391
apply (rule eq_Abs_REAL [of w])
paulson@14365
   392
apply (force simp add: real_le_def)
paulson@14269
   393
done
paulson@14269
   394
paulson@14378
   395
text{*The arithmetic decision procedure is not set up for type preal.
paulson@14378
   396
  This lemma is currently unused, but it could simplify the proofs of the
paulson@14378
   397
  following two lemmas.*}
paulson@14378
   398
lemma preal_eq_le_imp_le:
paulson@14378
   399
  assumes eq: "a+b = c+d" and le: "c \<le> a"
paulson@14378
   400
  shows "b \<le> (d::preal)"
paulson@14378
   401
proof -
paulson@14378
   402
  have "c+d \<le> a+d" by (simp add: prems preal_cancels)
paulson@14378
   403
  hence "a+b \<le> a+d" by (simp add: prems)
paulson@14378
   404
  thus "b \<le> d" by (simp add: preal_cancels)
paulson@14378
   405
qed
paulson@14378
   406
paulson@14378
   407
lemma real_le_lemma:
paulson@14378
   408
  assumes l: "u1 + v2 \<le> u2 + v1"
paulson@14378
   409
      and "x1 + v1 = u1 + y1"
paulson@14378
   410
      and "x2 + v2 = u2 + y2"
paulson@14378
   411
  shows "x1 + y2 \<le> x2 + (y1::preal)"
paulson@14365
   412
proof -
paulson@14378
   413
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
paulson@14378
   414
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac)
paulson@14378
   415
  also have "... \<le> (x2+y1) + (u2+v1)"
paulson@14365
   416
         by (simp add: prems preal_add_le_cancel_left)
paulson@14378
   417
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
paulson@14378
   418
qed						 
paulson@14378
   419
paulson@14378
   420
lemma real_le: 
paulson@14378
   421
  "(Abs_REAL(realrel``{(x1,y1)}) \<le> Abs_REAL(realrel``{(x2,y2)})) =  
paulson@14378
   422
   (x1 + y2 \<le> x2 + y1)"
paulson@14378
   423
apply (simp add: real_le_def) 
paulson@14387
   424
apply (auto intro: real_le_lemma)
paulson@14378
   425
done
paulson@14378
   426
paulson@14378
   427
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
paulson@14378
   428
apply (rule eq_Abs_REAL [of z])
paulson@14378
   429
apply (rule eq_Abs_REAL [of w])
paulson@14378
   430
apply (simp add: real_le order_antisym) 
paulson@14378
   431
done
paulson@14378
   432
paulson@14378
   433
lemma real_trans_lemma:
paulson@14378
   434
  assumes "x + v \<le> u + y"
paulson@14378
   435
      and "u + v' \<le> u' + v"
paulson@14378
   436
      and "x2 + v2 = u2 + y2"
paulson@14378
   437
  shows "x + v' \<le> u' + (y::preal)"
paulson@14378
   438
proof -
paulson@14378
   439
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac)
paulson@14378
   440
  also have "... \<le> (u+y) + (u+v')" 
paulson@14378
   441
    by (simp add: preal_add_le_cancel_right prems) 
paulson@14378
   442
  also have "... \<le> (u+y) + (u'+v)" 
paulson@14378
   443
    by (simp add: preal_add_le_cancel_left prems) 
paulson@14378
   444
  also have "... = (u'+y) + (u+v)"  by (simp add: preal_add_ac)
paulson@14378
   445
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
paulson@14365
   446
qed						 
paulson@14269
   447
paulson@14365
   448
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
paulson@14378
   449
apply (rule eq_Abs_REAL [of i])
paulson@14378
   450
apply (rule eq_Abs_REAL [of j])
paulson@14378
   451
apply (rule eq_Abs_REAL [of k])
paulson@14378
   452
apply (simp add: real_le) 
paulson@14378
   453
apply (blast intro: real_trans_lemma) 
paulson@14334
   454
done
paulson@14334
   455
paulson@14334
   456
(* Axiom 'order_less_le' of class 'order': *)
paulson@14334
   457
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
paulson@14365
   458
by (simp add: real_less_def)
paulson@14365
   459
paulson@14365
   460
instance real :: order
paulson@14365
   461
proof qed
paulson@14365
   462
 (assumption |
paulson@14365
   463
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
paulson@14365
   464
paulson@14378
   465
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14378
   466
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
paulson@14378
   467
apply (rule eq_Abs_REAL [of z])
paulson@14378
   468
apply (rule eq_Abs_REAL [of w]) 
paulson@14378
   469
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels)
paulson@14387
   470
apply (cut_tac x="x+ya" and y="xa+y" in linorder_linear, auto) 
paulson@14334
   471
done
paulson@14334
   472
paulson@14334
   473
paulson@14334
   474
instance real :: linorder
paulson@14334
   475
  by (intro_classes, rule real_le_linear)
paulson@14334
   476
paulson@14334
   477
paulson@14378
   478
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
paulson@14378
   479
apply (rule eq_Abs_REAL [of x])
paulson@14378
   480
apply (rule eq_Abs_REAL [of y]) 
paulson@14378
   481
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
paulson@14378
   482
                      preal_add_ac)
paulson@14378
   483
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
paulson@14378
   484
done 
paulson@14378
   485
paulson@14365
   486
lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)"
paulson@14365
   487
apply (auto simp add: real_le_eq_diff [of x] real_le_eq_diff [of "z+x"])
paulson@14365
   488
apply (subgoal_tac "z + x - (z + y) = (z + -z) + (x - y)")
paulson@14365
   489
 prefer 2 apply (simp add: diff_minus add_ac, simp) 
paulson@14365
   490
done
paulson@14334
   491
paulson@14365
   492
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
paulson@14365
   493
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14365
   494
paulson@14365
   495
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
paulson@14365
   496
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14334
   497
paulson@14334
   498
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
paulson@14378
   499
apply (rule eq_Abs_REAL [of x])
paulson@14378
   500
apply (rule eq_Abs_REAL [of y])
paulson@14378
   501
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
paulson@14378
   502
                 linorder_not_le [where 'a = preal] 
paulson@14378
   503
                  real_zero_def real_le real_mult)
paulson@14365
   504
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
paulson@14378
   505
apply (auto  dest!: less_add_left_Ex 
paulson@14365
   506
     simp add: preal_add_ac preal_mult_ac 
paulson@14378
   507
          preal_add_mult_distrib2 preal_cancels preal_self_less_add_right)
paulson@14334
   508
done
paulson@14334
   509
paulson@14334
   510
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
paulson@14334
   511
apply (rule real_sum_gt_zero_less)
paulson@14334
   512
apply (drule real_less_sum_gt_zero [of x y])
paulson@14334
   513
apply (drule real_mult_order, assumption)
paulson@14334
   514
apply (simp add: right_distrib)
paulson@14334
   515
done
paulson@14334
   516
paulson@14365
   517
text{*lemma for proving @{term "0<(1::real)"}*}
paulson@14365
   518
lemma real_zero_le_one: "0 \<le> (1::real)"
paulson@14387
   519
by (simp add: real_zero_def real_one_def real_le 
paulson@14378
   520
                 preal_self_less_add_left order_less_imp_le)
paulson@14334
   521
paulson@14378
   522
paulson@14334
   523
subsection{*The Reals Form an Ordered Field*}
paulson@14334
   524
paulson@14334
   525
instance real :: ordered_field
paulson@14334
   526
proof
paulson@14334
   527
  fix x y z :: real
paulson@14365
   528
  show "0 < (1::real)"
paulson@14365
   529
    by (simp add: real_less_def real_zero_le_one real_zero_not_eq_one)  
paulson@14334
   530
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
paulson@14334
   531
  show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2)
paulson@14334
   532
  show "\<bar>x\<bar> = (if x < 0 then -x else x)"
paulson@14334
   533
    by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le)
paulson@14334
   534
qed
paulson@14334
   535
paulson@14365
   536
paulson@14365
   537
paulson@14365
   538
text{*The function @{term real_of_preal} requires many proofs, but it seems
paulson@14365
   539
to be essential for proving completeness of the reals from that of the
paulson@14365
   540
positive reals.*}
paulson@14365
   541
paulson@14365
   542
lemma real_of_preal_add:
paulson@14365
   543
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
paulson@14365
   544
by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1 
paulson@14365
   545
              preal_add_ac)
paulson@14365
   546
paulson@14365
   547
lemma real_of_preal_mult:
paulson@14365
   548
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
paulson@14365
   549
by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2
paulson@14365
   550
              preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac)
paulson@14365
   551
paulson@14365
   552
paulson@14365
   553
text{*Gleason prop 9-4.4 p 127*}
paulson@14365
   554
lemma real_of_preal_trichotomy:
paulson@14365
   555
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
paulson@14365
   556
apply (unfold real_of_preal_def real_zero_def)
paulson@14378
   557
apply (rule eq_Abs_REAL [of x])
paulson@14365
   558
apply (auto simp add: real_minus preal_add_ac)
paulson@14365
   559
apply (cut_tac x = x and y = y in linorder_less_linear)
paulson@14365
   560
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric])
paulson@14365
   561
apply (auto simp add: preal_add_commute)
paulson@14365
   562
done
paulson@14365
   563
paulson@14365
   564
lemma real_of_preal_leD:
paulson@14365
   565
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
paulson@14365
   566
apply (unfold real_of_preal_def)
paulson@14365
   567
apply (auto simp add: real_le_def preal_add_ac)
paulson@14365
   568
apply (auto simp add: preal_add_assoc [symmetric] preal_add_right_cancel_iff)
paulson@14365
   569
apply (auto simp add: preal_add_ac preal_add_le_cancel_left)
paulson@14365
   570
done
paulson@14365
   571
paulson@14365
   572
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
paulson@14365
   573
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
paulson@14365
   574
paulson@14365
   575
lemma real_of_preal_lessD:
paulson@14365
   576
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
paulson@14365
   577
apply (auto simp add: real_less_def)
paulson@14365
   578
apply (drule real_of_preal_leD) 
paulson@14365
   579
apply (auto simp add: order_le_less) 
paulson@14365
   580
done
paulson@14365
   581
paulson@14365
   582
lemma real_of_preal_less_iff [simp]:
paulson@14365
   583
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
paulson@14365
   584
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
paulson@14365
   585
paulson@14365
   586
lemma real_of_preal_le_iff:
paulson@14365
   587
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
paulson@14365
   588
by (simp add: linorder_not_less [symmetric]) 
paulson@14365
   589
paulson@14365
   590
lemma real_of_preal_zero_less: "0 < real_of_preal m"
paulson@14365
   591
apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def
paulson@14365
   592
            preal_add_ac preal_cancels)
paulson@14365
   593
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
paulson@14365
   594
apply (blast intro: preal_self_less_add_left order_less_imp_le)
paulson@14365
   595
apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) 
paulson@14365
   596
apply (simp add: preal_add_ac) 
paulson@14365
   597
done
paulson@14365
   598
paulson@14365
   599
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
paulson@14365
   600
by (simp add: real_of_preal_zero_less)
paulson@14365
   601
paulson@14365
   602
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
paulson@14365
   603
apply (cut_tac real_of_preal_minus_less_zero)
paulson@14365
   604
apply (fast dest: order_less_trans)
paulson@14365
   605
done
paulson@14365
   606
paulson@14365
   607
paulson@14365
   608
subsection{*Theorems About the Ordering*}
paulson@14365
   609
paulson@14365
   610
text{*obsolete but used a lot*}
paulson@14365
   611
paulson@14365
   612
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)"
paulson@14365
   613
by blast 
paulson@14365
   614
paulson@14365
   615
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y"
paulson@14365
   616
by (simp add: order_le_less)
paulson@14365
   617
paulson@14365
   618
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
paulson@14365
   619
apply (auto simp add: real_of_preal_zero_less)
paulson@14365
   620
apply (cut_tac x = x in real_of_preal_trichotomy)
paulson@14365
   621
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
paulson@14365
   622
done
paulson@14365
   623
paulson@14365
   624
lemma real_gt_preal_preal_Ex:
paulson@14365
   625
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   626
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
paulson@14365
   627
             intro: real_gt_zero_preal_Ex [THEN iffD1])
paulson@14365
   628
paulson@14365
   629
lemma real_ge_preal_preal_Ex:
paulson@14365
   630
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   631
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
paulson@14365
   632
paulson@14365
   633
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
paulson@14365
   634
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
paulson@14365
   635
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
paulson@14365
   636
            simp add: real_of_preal_zero_less)
paulson@14365
   637
paulson@14365
   638
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
paulson@14365
   639
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
paulson@14365
   640
paulson@14334
   641
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)"
paulson@14365
   642
  by (rule Ring_and_Field.add_less_le_mono)
paulson@14334
   643
paulson@14334
   644
lemma real_add_le_less_mono:
paulson@14334
   645
     "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z"
paulson@14365
   646
  by (rule Ring_and_Field.add_le_less_mono)
paulson@14334
   647
paulson@14334
   648
lemma real_zero_less_one: "0 < (1::real)"
paulson@14334
   649
  by (rule Ring_and_Field.zero_less_one)
paulson@14334
   650
paulson@14334
   651
lemma real_le_square [simp]: "(0::real) \<le> x*x"
paulson@14334
   652
 by (rule Ring_and_Field.zero_le_square)
paulson@14334
   653
paulson@14334
   654
paulson@14334
   655
subsection{*More Lemmas*}
paulson@14334
   656
paulson@14334
   657
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14334
   658
by auto
paulson@14334
   659
paulson@14334
   660
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   661
by auto
paulson@14334
   662
paulson@14334
   663
text{*The precondition could be weakened to @{term "0\<le>x"}*}
paulson@14334
   664
lemma real_mult_less_mono:
paulson@14334
   665
     "[| u<v;  x<y;  (0::real) < v;  0 < x |] ==> u*x < v* y"
paulson@14334
   666
 by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
paulson@14334
   667
paulson@14334
   668
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
paulson@14334
   669
  by (force elim: order_less_asym
paulson@14334
   670
            simp add: Ring_and_Field.mult_less_cancel_right)
paulson@14334
   671
paulson@14334
   672
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
paulson@14365
   673
apply (simp add: mult_le_cancel_right)
paulson@14365
   674
apply (blast intro: elim: order_less_asym) 
paulson@14365
   675
done
paulson@14334
   676
paulson@14334
   677
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
paulson@14334
   678
  by (force elim: order_less_asym
paulson@14334
   679
            simp add: Ring_and_Field.mult_le_cancel_left)
paulson@14334
   680
paulson@14334
   681
text{*Only two uses?*}
paulson@14334
   682
lemma real_mult_less_mono':
paulson@14334
   683
     "[| x < y;  r1 < r2;  (0::real) \<le> r1;  0 \<le> x|] ==> r1 * x < r2 * y"
paulson@14334
   684
 by (rule Ring_and_Field.mult_strict_mono')
paulson@14334
   685
paulson@14334
   686
text{*FIXME: delete or at least combine the next two lemmas*}
paulson@14334
   687
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
paulson@14334
   688
apply (drule Ring_and_Field.equals_zero_I [THEN sym])
paulson@14334
   689
apply (cut_tac x = y in real_le_square) 
paulson@14334
   690
apply (auto, drule real_le_anti_sym, auto)
paulson@14334
   691
done
paulson@14334
   692
paulson@14334
   693
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
paulson@14334
   694
apply (rule_tac y = x in real_sum_squares_cancel)
paulson@14334
   695
apply (simp add: real_add_commute)
paulson@14334
   696
done
paulson@14334
   697
paulson@14334
   698
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y"
paulson@14365
   699
by (drule add_strict_mono [of concl: 0 0], assumption, simp)
paulson@14334
   700
paulson@14334
   701
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y"
paulson@14334
   702
apply (drule order_le_imp_less_or_eq)+
paulson@14334
   703
apply (auto intro: real_add_order order_less_imp_le)
paulson@14334
   704
done
paulson@14334
   705
paulson@14365
   706
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
paulson@14365
   707
apply (case_tac "x \<noteq> 0")
paulson@14365
   708
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto)
paulson@14365
   709
done
paulson@14334
   710
paulson@14365
   711
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
paulson@14365
   712
by (auto dest: less_imp_inverse_less)
paulson@14334
   713
paulson@14365
   714
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
paulson@14365
   715
proof -
paulson@14365
   716
  have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square)
paulson@14365
   717
  thus ?thesis by simp
paulson@14365
   718
qed
paulson@14365
   719
paulson@14334
   720
paulson@14365
   721
subsection{*Embedding the Integers into the Reals*}
paulson@14365
   722
paulson@14378
   723
defs (overloaded)
paulson@14378
   724
  real_of_nat_def: "real z == of_nat z"
paulson@14378
   725
  real_of_int_def: "real z == of_int z"
paulson@14365
   726
paulson@14365
   727
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378
   728
by (simp add: real_of_int_def) 
paulson@14365
   729
paulson@14365
   730
lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378
   731
by (simp add: real_of_int_def) 
paulson@14334
   732
paulson@14365
   733
lemma real_of_int_add: "real (x::int) + real y = real (x + y)"
paulson@14378
   734
by (simp add: real_of_int_def) 
paulson@14365
   735
declare real_of_int_add [symmetric, simp]
paulson@14365
   736
paulson@14365
   737
lemma real_of_int_minus: "-real (x::int) = real (-x)"
paulson@14378
   738
by (simp add: real_of_int_def) 
paulson@14365
   739
declare real_of_int_minus [symmetric, simp]
paulson@14365
   740
paulson@14365
   741
lemma real_of_int_diff: "real (x::int) - real y = real (x - y)"
paulson@14378
   742
by (simp add: real_of_int_def) 
paulson@14365
   743
declare real_of_int_diff [symmetric, simp]
paulson@14334
   744
paulson@14365
   745
lemma real_of_int_mult: "real (x::int) * real y = real (x * y)"
paulson@14378
   746
by (simp add: real_of_int_def) 
paulson@14365
   747
declare real_of_int_mult [symmetric, simp]
paulson@14365
   748
paulson@14365
   749
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
paulson@14378
   750
by (simp add: real_of_int_def) 
paulson@14365
   751
paulson@14365
   752
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
paulson@14378
   753
by (simp add: real_of_int_def) 
paulson@14365
   754
paulson@14365
   755
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
paulson@14378
   756
by (simp add: real_of_int_def) 
paulson@14365
   757
paulson@14365
   758
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378
   759
by (simp add: real_of_int_def) 
paulson@14365
   760
paulson@14365
   761
paulson@14365
   762
subsection{*Embedding the Naturals into the Reals*}
paulson@14365
   763
paulson@14334
   764
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365
   765
by (simp add: real_of_nat_def)
paulson@14334
   766
paulson@14334
   767
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365
   768
by (simp add: real_of_nat_def)
paulson@14334
   769
paulson@14365
   770
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378
   771
by (simp add: real_of_nat_def)
paulson@14334
   772
paulson@14334
   773
(*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334
   774
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378
   775
by (simp add: real_of_nat_def)
paulson@14334
   776
paulson@14334
   777
lemma real_of_nat_less_iff [iff]: 
paulson@14334
   778
     "(real (n::nat) < real m) = (n < m)"
paulson@14365
   779
by (simp add: real_of_nat_def)
paulson@14334
   780
paulson@14334
   781
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378
   782
by (simp add: real_of_nat_def)
paulson@14334
   783
paulson@14334
   784
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
paulson@14378
   785
by (simp add: real_of_nat_def zero_le_imp_of_nat)
paulson@14334
   786
paulson@14365
   787
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378
   788
by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365
   789
paulson@14334
   790
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
paulson@14378
   791
by (simp add: real_of_nat_def)
paulson@14334
   792
paulson@14334
   793
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378
   794
by (simp add: real_of_nat_def)
paulson@14334
   795
paulson@14387
   796
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
paulson@14378
   797
by (simp add: real_of_nat_def)
paulson@14334
   798
paulson@14365
   799
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
paulson@14378
   800
by (simp add: add: real_of_nat_def) 
paulson@14334
   801
paulson@14365
   802
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
paulson@14378
   803
by (simp add: add: real_of_nat_def) 
paulson@14365
   804
paulson@14365
   805
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378
   806
by (simp add: add: real_of_nat_def)
paulson@14334
   807
paulson@14365
   808
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378
   809
by (simp add: add: real_of_nat_def)
paulson@14334
   810
paulson@14365
   811
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"
paulson@14378
   812
by (simp add: add: real_of_nat_def)
paulson@14334
   813
paulson@14365
   814
lemma real_of_int_real_of_nat: "real (int n) = real n"
paulson@14378
   815
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
paulson@14378
   816
paulson@14378
   817
paulson@14387
   818
text{*Still needed for binary arithmetic*}
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@14387
   829
paulson@14387
   830
paulson@14387
   831
subsection{*Numerals and Arithmetic*}
paulson@14387
   832
paulson@14387
   833
instance real :: number ..
paulson@14387
   834
paulson@14387
   835
primrec (*the type constraint is essential!*)
paulson@14387
   836
  number_of_Pls: "number_of bin.Pls = 0"
paulson@14387
   837
  number_of_Min: "number_of bin.Min = - (1::real)"
paulson@14387
   838
  number_of_BIT: "number_of(w BIT x) = (if x then 1 else 0) +
paulson@14387
   839
	                               (number_of w) + (number_of w)"
paulson@14387
   840
paulson@14387
   841
declare number_of_Pls [simp del]
paulson@14387
   842
        number_of_Min [simp del]
paulson@14387
   843
        number_of_BIT [simp del]
paulson@14387
   844
paulson@14387
   845
instance real :: number_ring
paulson@14387
   846
proof
paulson@14387
   847
  show "Numeral0 = (0::real)" by (rule number_of_Pls)
paulson@14387
   848
  show "-1 = - (1::real)" by (rule number_of_Min)
paulson@14387
   849
  fix w :: bin and x :: bool
paulson@14387
   850
  show "(number_of (w BIT x) :: real) =
paulson@14387
   851
        (if x then 1 else 0) + number_of w + number_of w"
paulson@14387
   852
    by (rule number_of_BIT)
paulson@14387
   853
qed
paulson@14387
   854
paulson@14387
   855
paulson@14387
   856
text{*Collapse applications of @{term real} to @{term number_of}*}
paulson@14387
   857
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
paulson@14387
   858
by (simp add:  real_of_int_def of_int_number_of_eq)
paulson@14387
   859
paulson@14387
   860
lemma real_of_nat_number_of [simp]:
paulson@14387
   861
     "real (number_of v :: nat) =  
paulson@14387
   862
        (if neg (number_of v :: int) then 0  
paulson@14387
   863
         else (number_of v :: real))"
paulson@14387
   864
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
paulson@14387
   865
 
paulson@14387
   866
paulson@14387
   867
use "real_arith.ML"
paulson@14387
   868
paulson@14387
   869
setup real_arith_setup
paulson@14387
   870
paulson@14387
   871
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
paulson@14387
   872
paulson@14387
   873
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
paulson@14387
   874
lemma real_0_le_divide_iff:
paulson@14387
   875
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
paulson@14387
   876
by (simp add: real_divide_def zero_le_mult_iff, auto)
paulson@14387
   877
paulson@14387
   878
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
paulson@14387
   879
by arith
paulson@14387
   880
paulson@14387
   881
lemma real_add_eq_0_iff [iff]: "(x+y = (0::real)) = (y = -x)"
paulson@14387
   882
by auto
paulson@14387
   883
paulson@14387
   884
lemma real_add_less_0_iff [iff]: "(x+y < (0::real)) = (y < -x)"
paulson@14387
   885
by auto
paulson@14387
   886
paulson@14387
   887
lemma real_0_less_add_iff [iff]: "((0::real) < x+y) = (-x < y)"
paulson@14387
   888
by auto
paulson@14387
   889
paulson@14387
   890
lemma real_add_le_0_iff [iff]: "(x+y \<le> (0::real)) = (y \<le> -x)"
paulson@14387
   891
by auto
paulson@14387
   892
paulson@14387
   893
lemma real_0_le_add_iff [iff]: "((0::real) \<le> x+y) = (-x \<le> y)"
paulson@14387
   894
by auto
paulson@14387
   895
paulson@14387
   896
paulson@14387
   897
(** Simprules combining x-y and 0 (needed??) **)
paulson@14387
   898
paulson@14387
   899
lemma real_0_less_diff_iff [iff]: "((0::real) < x-y) = (y < x)"
paulson@14387
   900
by auto
paulson@14387
   901
paulson@14387
   902
lemma real_0_le_diff_iff [iff]: "((0::real) \<le> x-y) = (y \<le> x)"
paulson@14387
   903
by auto
paulson@14387
   904
paulson@14387
   905
(*
paulson@14387
   906
FIXME: we should have this, as for type int, but many proofs would break.
paulson@14387
   907
It replaces x+-y by x-y.
paulson@14387
   908
Addsimps [symmetric real_diff_def]
paulson@14387
   909
*)
paulson@14387
   910
paulson@14387
   911
paulson@14387
   912
subsubsection{*Density of the Reals*}
paulson@14387
   913
paulson@14387
   914
lemma real_lbound_gt_zero:
paulson@14387
   915
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
paulson@14387
   916
apply (rule_tac x = " (min d1 d2) /2" in exI)
paulson@14387
   917
apply (simp add: min_def)
paulson@14387
   918
done
paulson@14387
   919
paulson@14387
   920
paulson@14387
   921
text{*Similar results are proved in @{text Ring_and_Field}*}
paulson@14387
   922
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
paulson@14387
   923
  by auto
paulson@14387
   924
paulson@14387
   925
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
paulson@14387
   926
  by auto
paulson@14387
   927
paulson@14387
   928
lemma real_dense: "x < y ==> \<exists>r::real. x < r & r < y"
paulson@14387
   929
  by (rule Ring_and_Field.dense)
paulson@14387
   930
paulson@14387
   931
paulson@14387
   932
subsection{*Absolute Value Function for the Reals*}
paulson@14387
   933
paulson@14387
   934
text{*FIXME: these should go!*}
paulson@14387
   935
lemma abs_eqI1: "(0::real)\<le>x ==> abs x = x"
paulson@14387
   936
by (unfold real_abs_def, simp)
paulson@14387
   937
paulson@14387
   938
lemma abs_eqI2: "(0::real) < x ==> abs x = x"
paulson@14387
   939
by (unfold real_abs_def, simp)
paulson@14387
   940
paulson@14387
   941
lemma abs_minus_eqI2: "x < (0::real) ==> abs x = -x"
paulson@14387
   942
by (simp add: real_abs_def linorder_not_less [symmetric])
paulson@14387
   943
paulson@14387
   944
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
paulson@14387
   945
by (unfold real_abs_def, simp)
paulson@14387
   946
paulson@14387
   947
lemma abs_minus_one [simp]: "abs (-1) = (1::real)"
paulson@14387
   948
by (unfold real_abs_def, simp)
paulson@14387
   949
paulson@14387
   950
lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))"
paulson@14387
   951
by (force simp add: Ring_and_Field.abs_less_iff)
paulson@14387
   952
paulson@14387
   953
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
paulson@14387
   954
by (force simp add: Ring_and_Field.abs_le_iff)
paulson@14387
   955
paulson@14387
   956
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
paulson@14387
   957
by (unfold real_abs_def, auto)
paulson@14387
   958
paulson@14387
   959
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
paulson@14387
   960
by (auto intro: abs_eqI1 simp add: real_of_nat_ge_zero)
paulson@14387
   961
paulson@14387
   962
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
paulson@14387
   963
apply (simp add: linorder_not_less)
paulson@14387
   964
apply (auto intro: abs_ge_self [THEN order_trans])
paulson@14387
   965
done
paulson@14387
   966
 
paulson@14387
   967
text{*Used only in Hyperreal/Lim.ML*}
paulson@14387
   968
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
paulson@14387
   969
apply (simp add: real_add_assoc)
paulson@14387
   970
apply (rule_tac a1 = y in add_left_commute [THEN ssubst])
paulson@14387
   971
apply (rule real_add_assoc [THEN subst])
paulson@14387
   972
apply (rule abs_triangle_ineq)
paulson@14387
   973
done
paulson@14387
   974
paulson@14387
   975
paulson@14387
   976
paulson@14334
   977
ML
paulson@14334
   978
{*
paulson@14387
   979
val real_0_le_divide_iff = thm"real_0_le_divide_iff";
paulson@14387
   980
val real_add_minus_iff = thm"real_add_minus_iff";
paulson@14387
   981
val real_add_eq_0_iff = thm"real_add_eq_0_iff";
paulson@14387
   982
val real_add_less_0_iff = thm"real_add_less_0_iff";
paulson@14387
   983
val real_0_less_add_iff = thm"real_0_less_add_iff";
paulson@14387
   984
val real_add_le_0_iff = thm"real_add_le_0_iff";
paulson@14387
   985
val real_0_le_add_iff = thm"real_0_le_add_iff";
paulson@14387
   986
val real_0_less_diff_iff = thm"real_0_less_diff_iff";
paulson@14387
   987
val real_0_le_diff_iff = thm"real_0_le_diff_iff";
paulson@14387
   988
val real_lbound_gt_zero = thm"real_lbound_gt_zero";
paulson@14387
   989
val real_less_half_sum = thm"real_less_half_sum";
paulson@14387
   990
val real_gt_half_sum = thm"real_gt_half_sum";
paulson@14387
   991
val real_dense = thm"real_dense";
paulson@14341
   992
paulson@14387
   993
val abs_eqI1 = thm"abs_eqI1";
paulson@14387
   994
val abs_eqI2 = thm"abs_eqI2";
paulson@14387
   995
val abs_minus_eqI2 = thm"abs_minus_eqI2";
paulson@14387
   996
val abs_ge_zero = thm"abs_ge_zero";
paulson@14387
   997
val abs_idempotent = thm"abs_idempotent";
paulson@14387
   998
val abs_zero_iff = thm"abs_zero_iff";
paulson@14387
   999
val abs_ge_self = thm"abs_ge_self";
paulson@14387
  1000
val abs_ge_minus_self = thm"abs_ge_minus_self";
paulson@14387
  1001
val abs_mult = thm"abs_mult";
paulson@14387
  1002
val abs_inverse = thm"abs_inverse";
paulson@14387
  1003
val abs_triangle_ineq = thm"abs_triangle_ineq";
paulson@14387
  1004
val abs_minus_cancel = thm"abs_minus_cancel";
paulson@14387
  1005
val abs_minus_add_cancel = thm"abs_minus_add_cancel";
paulson@14387
  1006
val abs_minus_one = thm"abs_minus_one";
paulson@14387
  1007
val abs_interval_iff = thm"abs_interval_iff";
paulson@14387
  1008
val abs_le_interval_iff = thm"abs_le_interval_iff";
paulson@14387
  1009
val abs_add_one_gt_zero = thm"abs_add_one_gt_zero";
paulson@14387
  1010
val abs_le_zero_iff = thm"abs_le_zero_iff";
paulson@14387
  1011
val abs_real_of_nat_cancel = thm"abs_real_of_nat_cancel";
paulson@14387
  1012
val abs_add_one_not_less_self = thm"abs_add_one_not_less_self";
paulson@14387
  1013
val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq";
paulson@14334
  1014
paulson@14387
  1015
val abs_mult_less = thm"abs_mult_less";
paulson@14334
  1016
*}
paulson@10752
  1017
paulson@14387
  1018
paulson@5588
  1019
end