src/HOL/Real/RealDef.thy
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generic theorems about exponentials; general tidying up
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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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paulson
parents: 14348
diff changeset
   300
apply (auto simp add: preal_add_right_cancel_iff)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   301
done
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   302
14329
ff3210fe968f re-organized some hyperreal and real lemmas
paulson
parents: 14270
diff changeset
   303
subsection{*existence of inverse*}
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   304
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   305
lemma real_zero_iff: "Abs_REAL (realrel `` {(x, x)}) = 0"
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   306
apply (unfold real_zero_def)
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   307
apply (auto simp add: preal_add_commute)
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   308
done
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   309
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   310
text{*Instead of using an existential quantifier and constructing the inverse
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   311
within the proof, we could define the inverse explicitly.*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   312
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   313
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   314
apply (unfold real_zero_def real_one_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   315
apply (rule eq_Abs_REAL [of x])
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   316
apply (cut_tac x = xa and y = y in linorder_less_linear)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   317
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   318
apply (rule_tac
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   319
        x = "Abs_REAL (realrel `` { (preal_of_rat 1, 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   320
                            inverse (D) + preal_of_rat 1)}) " 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   321
       in exI)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   322
apply (rule_tac [2]
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   323
        x = "Abs_REAL (realrel `` { (inverse (D) + preal_of_rat 1,
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   324
                   preal_of_rat 1)})" 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   325
       in exI)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   326
apply (auto simp add: real_mult preal_mult_1_right
14329
ff3210fe968f re-organized some hyperreal and real lemmas
paulson
parents: 14270
diff changeset
   327
              preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   328
              preal_mult_inverse_right preal_add_ac preal_mult_ac)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   329
done
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   330
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   331
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   332
apply (unfold real_inverse_def)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   333
apply (frule real_mult_inverse_left_ex, safe)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   334
apply (rule someI2, auto)
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   335
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   336
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14335
diff changeset
   337
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14335
diff changeset
   338
subsection{*The Real Numbers form a Field*}
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14335
diff changeset
   339
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   340
instance real :: field
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   341
proof
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   342
  fix x y z :: real
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   343
  show "(x + y) + z = x + (y + z)" by (rule real_add_assoc)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   344
  show "x + y = y + x" by (rule real_add_commute)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   345
  show "0 + x = x" by simp
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   346
  show "- x + x = 0" by (rule real_add_minus_left)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   347
  show "x - y = x + (-y)" by (simp add: real_diff_def)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   348
  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   349
  show "x * y = y * x" by (rule real_mult_commute)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   350
  show "1 * x = x" by (rule real_mult_1)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   351
  show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   352
  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   353
  show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14426
diff changeset
   354
  show "x / y = x * inverse y" by (simp add: real_divide_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   355
qed
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   356
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   357
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14335
diff changeset
   358
text{*Inverse of zero!  Useful to simplify certain equations*}
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   359
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   360
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   361
apply (unfold real_inverse_def)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   362
apply (rule someI2)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   363
apply (auto simp add: zero_neq_one)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   364
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   365
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   366
instance real :: division_by_zero
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   367
proof
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   368
  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   369
qed
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   370
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   371
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   372
(*Pull negations out*)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   373
declare minus_mult_right [symmetric, simp] 
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   374
        minus_mult_left [symmetric, simp]
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   375
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   376
lemma real_mult_1_right: "z * (1::real) = z"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   377
  by (rule Ring_and_Field.mult_1_right)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   378
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   379
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   380
subsection{*The @{text "\<le>"} Ordering*}
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   381
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   382
lemma real_le_refl: "w \<le> (w::real)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   383
apply (rule eq_Abs_REAL [of w])
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   384
apply (force simp add: real_le_def)
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   385
done
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   386
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   387
text{*The arithmetic decision procedure is not set up for type preal.
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   388
  This lemma is currently unused, but it could simplify the proofs of the
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   389
  following two lemmas.*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   390
lemma preal_eq_le_imp_le:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   391
  assumes eq: "a+b = c+d" and le: "c \<le> a"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   392
  shows "b \<le> (d::preal)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   393
proof -
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   394
  have "c+d \<le> a+d" by (simp add: prems preal_cancels)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   395
  hence "a+b \<le> a+d" by (simp add: prems)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   396
  thus "b \<le> d" by (simp add: preal_cancels)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   397
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   398
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   399
lemma real_le_lemma:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   400
  assumes l: "u1 + v2 \<le> u2 + v1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   401
      and "x1 + v1 = u1 + y1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   402
      and "x2 + v2 = u2 + y2"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   403
  shows "x1 + y2 \<le> x2 + (y1::preal)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   404
proof -
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   405
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   406
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   407
  also have "... \<le> (x2+y1) + (u2+v1)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   408
         by (simp add: prems preal_add_le_cancel_left)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   409
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   410
qed						 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   411
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   412
lemma real_le: 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   413
  "(Abs_REAL(realrel``{(x1,y1)}) \<le> Abs_REAL(realrel``{(x2,y2)})) =  
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   414
   (x1 + y2 \<le> x2 + y1)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   415
apply (simp add: real_le_def) 
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   416
apply (auto intro: real_le_lemma)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   417
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   418
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   419
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   420
apply (rule eq_Abs_REAL [of z])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   421
apply (rule eq_Abs_REAL [of w])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   422
apply (simp add: real_le order_antisym) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   423
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   424
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   425
lemma real_trans_lemma:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   426
  assumes "x + v \<le> u + y"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   427
      and "u + v' \<le> u' + v"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   428
      and "x2 + v2 = u2 + y2"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   429
  shows "x + v' \<le> u' + (y::preal)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   430
proof -
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   431
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   432
  also have "... \<le> (u+y) + (u+v')" 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   433
    by (simp add: preal_add_le_cancel_right prems) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   434
  also have "... \<le> (u+y) + (u'+v)" 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   435
    by (simp add: preal_add_le_cancel_left prems) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   436
  also have "... = (u'+y) + (u+v)"  by (simp add: preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   437
  finally show ?thesis by (simp add: preal_add_le_cancel_right)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   438
qed						 
14269
502a7c95de73 conversion of some Real theories to Isar scripts
paulson
parents: 13487
diff changeset
   439
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   440
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   441
apply (rule eq_Abs_REAL [of i])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   442
apply (rule eq_Abs_REAL [of j])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   443
apply (rule eq_Abs_REAL [of k])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   444
apply (simp add: real_le) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   445
apply (blast intro: real_trans_lemma) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   446
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   447
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   448
(* Axiom 'order_less_le' of class 'order': *)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   449
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   450
by (simp add: real_less_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   451
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   452
instance real :: order
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   453
proof qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   454
 (assumption |
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   455
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   456
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   457
(* Axiom 'linorder_linear' of class 'linorder': *)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   458
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   459
apply (rule eq_Abs_REAL [of z])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   460
apply (rule eq_Abs_REAL [of w]) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   461
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   462
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   463
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   464
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   465
instance real :: linorder
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   466
  by (intro_classes, rule real_le_linear)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   467
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   468
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   469
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   470
apply (rule eq_Abs_REAL [of x])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   471
apply (rule eq_Abs_REAL [of y]) 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   472
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   473
                      preal_add_ac)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   474
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   475
done 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   476
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   477
lemma real_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::real)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   478
apply (auto simp add: real_le_eq_diff [of x] real_le_eq_diff [of "z+x"])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   479
apply (subgoal_tac "z + x - (z + y) = (z + -z) + (x - y)")
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   480
 prefer 2 apply (simp add: diff_minus add_ac, simp) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   481
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   482
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   483
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   484
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   485
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   486
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   487
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   488
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   489
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   490
apply (rule eq_Abs_REAL [of x])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   491
apply (rule eq_Abs_REAL [of y])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   492
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   493
                 linorder_not_le [where 'a = preal] 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   494
                  real_zero_def real_le real_mult)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   495
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   496
apply (auto  dest!: less_add_left_Ex 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   497
     simp add: preal_add_ac preal_mult_ac 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   498
          preal_add_mult_distrib2 preal_cancels preal_self_less_add_right)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   499
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   500
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   501
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   502
apply (rule real_sum_gt_zero_less)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   503
apply (drule real_less_sum_gt_zero [of x y])
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   504
apply (drule real_mult_order, assumption)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   505
apply (simp add: right_distrib)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   506
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   507
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   508
text{*lemma for proving @{term "0<(1::real)"}*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   509
lemma real_zero_le_one: "0 \<le> (1::real)"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   510
by (simp add: real_zero_def real_one_def real_le 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   511
                 preal_self_less_add_left order_less_imp_le)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   512
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   513
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   514
subsection{*The Reals Form an Ordered Field*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   515
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   516
instance real :: ordered_field
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   517
proof
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   518
  fix x y z :: real
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   519
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   520
  show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   521
  show "\<bar>x\<bar> = (if x < 0 then -x else x)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   522
    by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   523
qed
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   524
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   525
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   526
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   527
text{*The function @{term real_of_preal} requires many proofs, but it seems
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   528
to be essential for proving completeness of the reals from that of the
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   529
positive reals.*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   530
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   531
lemma real_of_preal_add:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   532
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   533
by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   534
              preal_add_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   535
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   536
lemma real_of_preal_mult:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   537
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   538
by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   539
              preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   540
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   541
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   542
text{*Gleason prop 9-4.4 p 127*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   543
lemma real_of_preal_trichotomy:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   544
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   545
apply (unfold real_of_preal_def real_zero_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   546
apply (rule eq_Abs_REAL [of x])
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   547
apply (auto simp add: real_minus preal_add_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   548
apply (cut_tac x = x and y = y in linorder_less_linear)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   549
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   550
apply (auto simp add: preal_add_commute)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   551
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   552
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   553
lemma real_of_preal_leD:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   554
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   555
apply (unfold real_of_preal_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   556
apply (auto simp add: real_le_def preal_add_ac)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   557
apply (auto simp add: preal_add_assoc [symmetric] preal_add_right_cancel_iff)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   558
apply (auto simp add: preal_add_ac preal_add_le_cancel_left)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   559
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   560
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   561
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   562
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   563
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   564
lemma real_of_preal_lessD:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   565
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   566
apply (auto simp add: real_less_def)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   567
apply (drule real_of_preal_leD) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   568
apply (auto simp add: order_le_less) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   569
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   570
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   571
lemma real_of_preal_less_iff [simp]:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   572
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   573
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   574
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   575
lemma real_of_preal_le_iff:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   576
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   577
by (simp add: linorder_not_less [symmetric]) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   578
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   579
lemma real_of_preal_zero_less: "0 < real_of_preal m"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   580
apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   581
            preal_add_ac preal_cancels)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   582
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   583
apply (blast intro: preal_self_less_add_left order_less_imp_le)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   584
apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   585
apply (simp add: preal_add_ac) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   586
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   587
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   588
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   589
by (simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   590
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   591
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   592
apply (cut_tac real_of_preal_minus_less_zero)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   593
apply (fast dest: order_less_trans)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   594
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   595
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   596
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   597
subsection{*Theorems About the Ordering*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   598
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   599
text{*obsolete but used a lot*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   600
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   601
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   602
by blast 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   603
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   604
lemma real_le_imp_less_or_eq: "!!(x::real). x \<le> y ==> x < y | x = y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   605
by (simp add: order_le_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   606
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   607
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   608
apply (auto simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   609
apply (cut_tac x = x in real_of_preal_trichotomy)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   610
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   611
done
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   612
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   613
lemma real_gt_preal_preal_Ex:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   614
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   615
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   616
             intro: real_gt_zero_preal_Ex [THEN iffD1])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   617
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   618
lemma real_ge_preal_preal_Ex:
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   619
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   620
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   621
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   622
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   623
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   624
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   625
            simp add: real_of_preal_zero_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   626
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   627
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   628
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   629
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   630
lemma real_add_less_le_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::real)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   631
  by (rule Ring_and_Field.add_less_le_mono)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   632
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   633
lemma real_add_le_less_mono:
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   634
     "!!z z'::real. [| w'\<le>w; z'<z |] ==> w' + z' < w + z"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   635
  by (rule Ring_and_Field.add_le_less_mono)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   636
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   637
lemma real_zero_less_one: "0 < (1::real)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   638
  by (rule Ring_and_Field.zero_less_one)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   639
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   640
lemma real_le_square [simp]: "(0::real) \<le> x*x"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   641
 by (rule Ring_and_Field.zero_le_square)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   642
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   643
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   644
subsection{*More Lemmas*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   645
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   646
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   647
by auto
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   648
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   649
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   650
by auto
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   651
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   652
text{*The precondition could be weakened to @{term "0\<le>x"}*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   653
lemma real_mult_less_mono:
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   654
     "[| u<v;  x<y;  (0::real) < v;  0 < x |] ==> u*x < v* y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   655
 by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   656
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   657
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   658
  by (force elim: order_less_asym
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   659
            simp add: Ring_and_Field.mult_less_cancel_right)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   660
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   661
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   662
apply (simp add: mult_le_cancel_right)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   663
apply (blast intro: elim: order_less_asym) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   664
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   665
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   666
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   667
  by (force elim: order_less_asym
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   668
            simp add: Ring_and_Field.mult_le_cancel_left)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   669
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   670
text{*Only two uses?*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   671
lemma real_mult_less_mono':
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   672
     "[| x < y;  r1 < r2;  (0::real) \<le> r1;  0 \<le> x|] ==> r1 * x < r2 * y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   673
 by (rule Ring_and_Field.mult_strict_mono')
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   674
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   675
text{*FIXME: delete or at least combine the next two lemmas*}
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   676
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   677
apply (drule Ring_and_Field.equals_zero_I [THEN sym])
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   678
apply (cut_tac x = y in real_le_square) 
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   679
apply (auto, drule real_le_anti_sym, auto)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   680
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   681
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   682
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   683
apply (rule_tac y = x in real_sum_squares_cancel)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   684
apply (simp add: real_add_commute)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   685
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   686
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   687
lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   688
by (drule add_strict_mono [of concl: 0 0], assumption, simp)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   689
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   690
lemma real_le_add_order: "[| 0 \<le> x; 0 \<le> y |] ==> (0::real) \<le> x + y"
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   691
apply (drule order_le_imp_less_or_eq)+
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   692
apply (auto intro: real_add_order order_less_imp_le)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   693
done
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   694
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   695
lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   696
apply (case_tac "x \<noteq> 0")
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   697
apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   698
done
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   699
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   700
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   701
by (auto dest: less_imp_inverse_less)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   702
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   703
lemma real_mult_self_sum_ge_zero: "(0::real) \<le> x*x + y*y"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   704
proof -
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   705
  have "0 + 0 \<le> x*x + y*y" by (blast intro: add_mono zero_le_square)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   706
  thus ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   707
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   708
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   709
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   710
subsection{*Embedding the Integers into the Reals*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   711
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   712
defs (overloaded)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   713
  real_of_nat_def: "real z == of_nat z"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   714
  real_of_int_def: "real z == of_int z"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   715
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   716
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   717
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   718
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   719
lemma real_of_one [simp]: "real (1::int) = (1::real)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   720
by (simp add: real_of_int_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   721
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   722
lemma real_of_int_add: "real (x::int) + real y = real (x + y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   723
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   724
declare real_of_int_add [symmetric, simp]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   725
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   726
lemma real_of_int_minus: "-real (x::int) = real (-x)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   727
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   728
declare real_of_int_minus [symmetric, simp]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   729
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   730
lemma real_of_int_diff: "real (x::int) - real y = real (x - y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   731
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   732
declare real_of_int_diff [symmetric, simp]
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   733
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   734
lemma real_of_int_mult: "real (x::int) * real y = real (x * y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   735
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   736
declare real_of_int_mult [symmetric, simp]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   737
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   738
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   739
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   740
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   741
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   742
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   743
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   744
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   745
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   746
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   747
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   748
by (simp add: real_of_int_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   749
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   750
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   751
subsection{*Embedding the Naturals into the Reals*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   752
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   753
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   754
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   755
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   756
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   757
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   758
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   759
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   760
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   761
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   762
(*Not for addsimps: often the LHS is used to represent a positive natural*)
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   763
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   764
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   765
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   766
lemma real_of_nat_less_iff [iff]: 
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   767
     "(real (n::nat) < real m) = (n < m)"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   768
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   769
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   770
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   771
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   772
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   773
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   774
by (simp add: real_of_nat_def zero_le_imp_of_nat)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   775
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   776
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   777
by (simp add: real_of_nat_def del: of_nat_Suc)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   778
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   779
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   780
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   781
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   782
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   783
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   784
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   785
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   786
by (simp add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   787
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   788
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   789
by (simp add: add: real_of_nat_def) 
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   790
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   791
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   792
by (simp add: add: real_of_nat_def) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   793
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   794
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   795
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   796
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   797
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   798
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   799
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   800
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat)) = (0 \<le> n)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   801
by (simp add: add: real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   802
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14348
diff changeset
   803
lemma real_of_int_real_of_nat: "real (int n) = real n"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   804
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14369
diff changeset
   805
14426
de890c247b38 fixed bugs in the setup of arithmetic procedures
paulson
parents: 14421
diff changeset
   806
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
de890c247b38 fixed bugs in the setup of arithmetic procedures
paulson
parents: 14421
diff changeset
   807
by (simp add: real_of_int_def real_of_nat_def)
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   808
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   809
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   810
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   811
subsection{*Numerals and Arithmetic*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   812
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   813
instance real :: number ..
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   814
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   815
primrec (*the type constraint is essential!*)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   816
  number_of_Pls: "number_of bin.Pls = 0"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   817
  number_of_Min: "number_of bin.Min = - (1::real)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   818
  number_of_BIT: "number_of(w BIT x) = (if x then 1 else 0) +
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   819
	                               (number_of w) + (number_of w)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   820
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   821
declare number_of_Pls [simp del]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   822
        number_of_Min [simp del]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   823
        number_of_BIT [simp del]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   824
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   825
instance real :: number_ring
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   826
proof
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   827
  show "Numeral0 = (0::real)" by (rule number_of_Pls)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   828
  show "-1 = - (1::real)" by (rule number_of_Min)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   829
  fix w :: bin and x :: bool
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   830
  show "(number_of (w BIT x) :: real) =
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   831
        (if x then 1 else 0) + number_of w + number_of w"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   832
    by (rule number_of_BIT)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   833
qed
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   834
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   835
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   836
text{*Collapse applications of @{term real} to @{term number_of}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   837
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   838
by (simp add:  real_of_int_def of_int_number_of_eq)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   839
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   840
lemma real_of_nat_number_of [simp]:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   841
     "real (number_of v :: nat) =  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   842
        (if neg (number_of v :: int) then 0  
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   843
         else (number_of v :: real))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   844
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   845
 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   846
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   847
use "real_arith.ML"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   848
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   849
setup real_arith_setup
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   850
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   851
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   852
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   853
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   854
lemma real_0_le_divide_iff:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   855
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   856
by (simp add: real_divide_def zero_le_mult_iff, auto)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   857
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   858
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   859
by arith
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   860
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   861
lemma real_add_eq_0_iff [iff]: "(x+y = (0::real)) = (y = -x)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   862
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   863
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   864
lemma real_add_less_0_iff [iff]: "(x+y < (0::real)) = (y < -x)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   865
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   866
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   867
lemma real_0_less_add_iff [iff]: "((0::real) < x+y) = (-x < y)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   868
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   869
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   870
lemma real_add_le_0_iff [iff]: "(x+y \<le> (0::real)) = (y \<le> -x)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   871
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   872
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   873
lemma real_0_le_add_iff [iff]: "((0::real) \<le> x+y) = (-x \<le> y)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   874
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   875
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   876
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   877
(** Simprules combining x-y and 0 (needed??) **)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   878
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   879
lemma real_0_less_diff_iff [iff]: "((0::real) < x-y) = (y < x)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   880
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   881
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   882
lemma real_0_le_diff_iff [iff]: "((0::real) \<le> x-y) = (y \<le> x)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   883
by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   884
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   885
(*
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   886
FIXME: we should have this, as for type int, but many proofs would break.
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   887
It replaces x+-y by x-y.
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   888
Addsimps [symmetric real_diff_def]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   889
*)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   890
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   891
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   892
subsubsection{*Density of the Reals*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   893
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   894
lemma real_lbound_gt_zero:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   895
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   896
apply (rule_tac x = " (min d1 d2) /2" in exI)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   897
apply (simp add: min_def)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   898
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   899
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   900
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   901
text{*Similar results are proved in @{text Ring_and_Field}*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   902
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   903
  by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   904
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   905
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   906
  by auto
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   907
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   908
lemma real_dense: "x < y ==> \<exists>r::real. x < r & r < y"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   909
  by (rule Ring_and_Field.dense)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   910
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   911
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   912
subsection{*Absolute Value Function for the Reals*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   913
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   914
text{*FIXME: these should go!*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   915
lemma abs_eqI1: "(0::real)\<le>x ==> abs x = x"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   916
by (unfold real_abs_def, simp)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   917
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   918
lemma abs_eqI2: "(0::real) < x ==> abs x = x"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   919
by (unfold real_abs_def, simp)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   920
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   921
lemma abs_minus_eqI2: "x < (0::real) ==> abs x = -x"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   922
by (simp add: real_abs_def linorder_not_less [symmetric])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   923
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   924
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   925
by (unfold real_abs_def, simp)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   926
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   927
lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   928
by (force simp add: Ring_and_Field.abs_less_iff)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   929
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   930
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   931
by (force simp add: Ring_and_Field.abs_le_iff)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   932
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   933
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   934
by (unfold real_abs_def, auto)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   935
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   936
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   937
by (auto intro: abs_eqI1 simp add: real_of_nat_ge_zero)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   938
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   939
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   940
apply (simp add: linorder_not_less)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   941
apply (auto intro: abs_ge_self [THEN order_trans])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   942
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   943
 
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   944
text{*Used only in Hyperreal/Lim.ML*}
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   945
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   946
apply (simp add: real_add_assoc)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   947
apply (rule_tac a1 = y in add_left_commute [THEN ssubst])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   948
apply (rule real_add_assoc [THEN subst])
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   949
apply (rule abs_triangle_ineq)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   950
done
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   951
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   952
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   953
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   954
ML
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   955
{*
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   956
val real_0_le_divide_iff = thm"real_0_le_divide_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   957
val real_add_minus_iff = thm"real_add_minus_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   958
val real_add_eq_0_iff = thm"real_add_eq_0_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   959
val real_add_less_0_iff = thm"real_add_less_0_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   960
val real_0_less_add_iff = thm"real_0_less_add_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   961
val real_add_le_0_iff = thm"real_add_le_0_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   962
val real_0_le_add_iff = thm"real_0_le_add_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   963
val real_0_less_diff_iff = thm"real_0_less_diff_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   964
val real_0_le_diff_iff = thm"real_0_le_diff_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   965
val real_lbound_gt_zero = thm"real_lbound_gt_zero";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   966
val real_less_half_sum = thm"real_less_half_sum";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   967
val real_gt_half_sum = thm"real_gt_half_sum";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   968
val real_dense = thm"real_dense";
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14335
diff changeset
   969
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   970
val abs_eqI1 = thm"abs_eqI1";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   971
val abs_eqI2 = thm"abs_eqI2";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   972
val abs_minus_eqI2 = thm"abs_minus_eqI2";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   973
val abs_ge_zero = thm"abs_ge_zero";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   974
val abs_idempotent = thm"abs_idempotent";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   975
val abs_zero_iff = thm"abs_zero_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   976
val abs_ge_self = thm"abs_ge_self";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   977
val abs_ge_minus_self = thm"abs_ge_minus_self";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   978
val abs_mult = thm"abs_mult";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   979
val abs_inverse = thm"abs_inverse";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   980
val abs_triangle_ineq = thm"abs_triangle_ineq";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   981
val abs_minus_cancel = thm"abs_minus_cancel";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   982
val abs_minus_add_cancel = thm"abs_minus_add_cancel";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   983
val abs_interval_iff = thm"abs_interval_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   984
val abs_le_interval_iff = thm"abs_le_interval_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   985
val abs_add_one_gt_zero = thm"abs_add_one_gt_zero";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   986
val abs_le_zero_iff = thm"abs_le_zero_iff";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   987
val abs_real_of_nat_cancel = thm"abs_real_of_nat_cancel";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   988
val abs_add_one_not_less_self = thm"abs_add_one_not_less_self";
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   989
val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq";
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   990
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   991
val abs_mult_less = thm"abs_mult_less";
14334
6137d24eef79 tweaking of lemmas in RealDef, RealOrd
paulson
parents: 14329
diff changeset
   992
*}
10752
c4f1bf2acf4c tidying, and separation of HOL-Hyperreal from HOL-Real
paulson
parents: 10648
diff changeset
   993
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   994
5588
a3ab526bb891 Revised version with Abelian group simprocs
paulson
parents:
diff changeset
   995
end