author  paulson 
Sun, 15 Feb 2004 10:46:37 +0100  
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permissions  rwrr 
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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)" 

110 
by (unfold realrel_def, blast) 

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112 
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) *) 

123 
lemmas equiv_realrel_iff = 

124 
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) 

135 
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" 

144 
apply (rule inj_on_inverseI) 

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apply (rule Rep_REAL_inverse) 

146 
done 

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148 
(** real_of_preal: the injection from preal to real **) 

149 
lemma inj_real_of_preal: "inj(real_of_preal)" 

150 
apply (rule inj_onI) 

151 
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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159 
lemma eq_Abs_REAL: 

160 
"(!!x y. z = Abs_REAL(realrel``{(x,y)}) ==> P) ==> P" 

161 
apply (rule_tac x1 = z in Rep_REAL [unfolded REAL_def, THEN quotientE]) 

162 
apply (drule_tac f = Abs_REAL in arg_cong) 

163 
apply (case_tac "x") 

164 
apply (simp add: Rep_REAL_inverse) 

165 
done 

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14329  168 
subsection{*Congruence property for addition*} 
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170 
lemma real_add_congruent2_lemma: 

171 
"[a + ba = aa + b; ab + bc = ac + bb] 

172 
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" 

173 
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]) 

176 
apply (simp add: preal_add_ac) 

177 
done 

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lemma real_add: 

180 
"Abs_REAL(realrel``{(x1,y1)}) + Abs_REAL(realrel``{(x2,y2)}) = 

181 
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]) 

184 
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) 
192 
done 

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194 
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" 

202 
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) 
205 
done 

206 

207 
lemma real_add_zero_right: "z + (0::real) = z" 

14334  208 
by (simp add: real_add_zero_left real_add_commute) 
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instance real :: plus_ac0 

211 
by (intro_classes, 

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(assumption  

213 
rule real_add_commute real_add_assoc real_add_zero_left)+) 

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14334  216 
subsection{*Additive Inverse on real*} 
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lemma real_minus_congruent: 

219 
"congruent realrel (%p. (%(x,y). realrel``{(y,x)}) p)" 

220 
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)})" 

226 
apply (unfold real_minus_def) 

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apply (rule_tac f = Abs_REAL in arg_cong) 

228 
apply (simp add: realrel_in_real [THEN Abs_REAL_inverse] 

229 
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) 
236 
done 

237 

238 

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subsection{*Congruence property for multiplication*} 
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14329  241 
lemma real_mult_congruent2_lemma: 
242 
"!!(x1::preal). [ x1 + y2 = x2 + y1 ] ==> 

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x * x1 + y * y1 + (x * y2 + x2 * y) = 
244 
x * x2 + y * y2 + (x * y1 + x1 * y)" 

245 
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]) 

247 
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) 

250 
done 

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252 
lemma real_mult_congruent2: 

253 
"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) 
256 
apply (unfold split_def) 

257 
apply (simp add: preal_mult_commute preal_add_commute) 

258 
apply (auto simp add: real_mult_congruent2_lemma) 

259 
done 

260 

261 
lemma real_mult: 

262 
"Abs_REAL((realrel``{(x1,y1)})) * Abs_REAL((realrel``{(x2,y2)})) = 

263 
Abs_REAL(realrel `` {(x1*x2+y1*y2,x1*y2+x2*y1)})" 

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apply (unfold real_mult_def) 

265 
apply (simp add: equiv_realrel [THEN UN_equiv_class2] real_mult_congruent2) 

266 
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) 
272 
done 

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274 
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) 
279 
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) 

14269  286 
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) 
293 
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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298 
prefer 2 apply (simp add: preal_self_less_add_left) 
14269  299 
apply (unfold real_zero_def real_one_def) 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

300 
apply (auto simp add: preal_add_right_cancel_iff) 
14269  301 
done 
302 

14329  303 
subsection{*existence of inverse*} 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

304 

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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  306 
apply (unfold real_zero_def) 
307 
apply (auto simp add: preal_add_commute) 

308 
done 

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:
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diff
changeset

313 
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" 
14269  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  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:
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diff
changeset

317 
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) 
14334  318 
apply (rule_tac 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

319 
x = "Abs_REAL (realrel `` { (preal_of_rat 1, 
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

320 
inverse (D) + preal_of_rat 1)}) " 
14334  321 
in exI) 
322 
apply (rule_tac [2] 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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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  325 
in exI) 
14365
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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  327 
preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

328 
preal_mult_inverse_right preal_add_ac preal_mult_ac) 
14269  329 
done 
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  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  334 
apply (rule someI2, auto) 
335 
done 

14334  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  340 
instance real :: field 
341 
proof 

342 
fix x y z :: real 

343 
show "(x + y) + z = x + (y + z)" by (rule real_add_assoc) 

344 
show "x + y = y + x" by (rule real_add_commute) 

345 
show "0 + x = x" by simp 

346 
show " x + x = 0" by (rule real_add_minus_left) 

347 
show "x  y = x + (y)" by (simp add: real_diff_def) 

348 
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) 

349 
show "x * y = y * x" by (rule real_mult_commute) 

350 
show "1 * x = x" by (rule real_mult_1) 

351 
show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) 

352 
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) 

14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

353 
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) 
14334  354 
show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: real_divide_def) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset

355 
assume eq: "z+x = z+y" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset

356 
hence "(z + z) + x = (z + z) + y" by (simp only: eq real_add_assoc) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset

357 
thus "x = y" by (simp add: real_add_minus_left) 
14334  358 
qed 
359 

360 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset

361 
text{*Inverse of zero! Useful to simplify certain equations*} 
14269  362 

14334  363 
lemma INVERSE_ZERO: "inverse 0 = (0::real)" 
364 
apply (unfold real_inverse_def) 

365 
apply (rule someI2) 

366 
apply (auto simp add: zero_neq_one) 

14269  367 
done 
14334  368 

369 
lemma DIVISION_BY_ZERO: "a / (0::real) = 0" 

370 
by (simp add: real_divide_def INVERSE_ZERO) 

371 

372 
instance real :: division_by_zero 

373 
proof 

374 
fix x :: real 

375 
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) 

376 
show "x/0 = 0" by (rule DIVISION_BY_ZERO) 

377 
qed 

378 

379 

380 
(*Pull negations out*) 

381 
declare minus_mult_right [symmetric, simp] 

382 
minus_mult_left [symmetric, simp] 

383 

384 
lemma real_mult_1_right: "z * (1::real) = z" 

385 
by (rule Ring_and_Field.mult_1_right) 

14269  386 

387 

14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

388 
subsection{*The @{text "\<le>"} Ordering*} 
14269  389 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

390 
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

391 
apply (rule eq_Abs_REAL [of w]) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

392 
apply (force simp add: real_le_def) 
14269  393 
done 
394 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

395 
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

396 
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

397 
following two lemmas.*} 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

398 
lemma preal_eq_le_imp_le: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

399 
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

400 
shows "b \<le> (d::preal)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

401 
proof  
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

402 
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

403 
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

404 
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

405 
qed 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

406 

69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

407 
lemma real_le_lemma: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

408 
assumes l: "u1 + v2 \<le> u2 + v1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

409 
and "x1 + v1 = u1 + y1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

410 
and "x2 + v2 = u2 + y2" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

411 
shows "x1 + y2 \<le> x2 + (y1::preal)" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

412 
proof  
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

413 
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

414 
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

415 
also have "... \<le> (x2+y1) + (u2+v1)" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

416 
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

417 
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

418 
qed 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

419 

69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

420 
lemma real_le: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

421 
"(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

422 
(x1 + y2 \<le> x2 + y1)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

423 
apply (simp add: real_le_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

424 
apply (auto intro: real_le_lemma) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

425 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

426 

69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

427 
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

428 
apply (rule eq_Abs_REAL [of z]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

429 
apply (rule eq_Abs_REAL [of w]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

430 
apply (simp add: real_le order_antisym) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

431 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

432 

69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

433 
lemma real_trans_lemma: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

434 
assumes "x + v \<le> u + y" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

435 
and "u + v' \<le> u' + v" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

436 
and "x2 + v2 = u2 + y2" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

437 
shows "x + v' \<le> u' + (y::preal)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

438 
proof  
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

439 
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

440 
also have "... \<le> (u+y) + (u+v')" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

441 
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

442 
also have "... \<le> (u+y) + (u'+v)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

443 
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

444 
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

445 
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

446 
qed 
14269  447 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

448 
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

449 
apply (rule eq_Abs_REAL [of i]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

450 
apply (rule eq_Abs_REAL [of j]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

451 
apply (rule eq_Abs_REAL [of k]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

452 
apply (simp add: real_le) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

453 
apply (blast intro: real_trans_lemma) 
14334  454 
done 
455 

456 
(* Axiom 'order_less_le' of class 'order': *) 

457 
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

458 
by (simp add: real_less_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

459 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

460 
instance real :: order 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

461 
proof qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

462 
(assumption  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

463 
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

464 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

465 
(* Axiom 'linorder_linear' of class 'linorder': *) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

466 
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

467 
apply (rule eq_Abs_REAL [of z]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

468 
apply (rule eq_Abs_REAL [of w]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

469 
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

470 
apply (cut_tac x="x+ya" and y="xa+y" in linorder_linear, auto) 
14334  471 
done 
472 

473 

474 
instance real :: linorder 

475 
by (intro_classes, rule real_le_linear) 

476 

477 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

478 
lemma real_le_eq_diff: "(x \<le> y) = (xy \<le> (0::real))" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

479 
apply (rule eq_Abs_REAL [of x]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

480 
apply (rule eq_Abs_REAL [of y]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

481 
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

482 
preal_add_ac) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

483 
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

484 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

485 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

486 
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

487 
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

488 
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

489 
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

490 
done 
14334  491 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

492 
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

493 
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

494 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

495 
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

496 
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) 
14334  497 

498 
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

499 
apply (rule eq_Abs_REAL [of x]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

500 
apply (rule eq_Abs_REAL [of y]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

501 
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

502 
linorder_not_le [where 'a = preal] 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

503 
real_zero_def real_le real_mult) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

504 
{*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

505 
apply (auto dest!: less_add_left_Ex 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

506 
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

507 
preal_add_mult_distrib2 preal_cancels preal_self_less_add_right) 
14334  508 
done 
509 

510 
lemma real_mult_less_mono2: "[ (0::real) < z; x < y ] ==> z * x < z * y" 

511 
apply (rule real_sum_gt_zero_less) 

512 
apply (drule real_less_sum_gt_zero [of x y]) 

513 
apply (drule real_mult_order, assumption) 

514 
apply (simp add: right_distrib) 

515 
done 

516 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

517 
text{*lemma for proving @{term "0<(1::real)"}*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

518 
lemma real_zero_le_one: "0 \<le> (1::real)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

519 
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

520 
preal_self_less_add_left order_less_imp_le) 
14334  521 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

522 

14334  523 
subsection{*The Reals Form an Ordered Field*} 
524 

525 
instance real :: ordered_field 

526 
proof 

527 
fix x y z :: real 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

528 
show "0 < (1::real)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

529 
by (simp add: real_less_def real_zero_le_one real_zero_not_eq_one) 
14334  530 
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) 
531 
show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) 

532 
show "\<bar>x\<bar> = (if x < 0 then x else x)" 

533 
by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) 

534 
qed 

535 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

536 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

537 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

538 
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

539 
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

540 
positive reals.*} 
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 
lemma real_of_preal_add: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

543 
"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

544 
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

545 
preal_add_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

546 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

547 
lemma real_of_preal_mult: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

548 
"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

549 
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

550 
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

551 

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 
text{*Gleason prop 94.4 p 127*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

554 
lemma real_of_preal_trichotomy: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

555 
"\<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

556 
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

557 
apply (rule eq_Abs_REAL [of x]) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

558 
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

559 
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

560 
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

561 
apply (auto simp add: preal_add_commute) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

562 
done 
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_leD: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

565 
"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

566 
apply (unfold real_of_preal_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

567 
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

568 
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

569 
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

570 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

571 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

572 
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

573 
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

574 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

575 
lemma real_of_preal_lessD: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

576 
"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

577 
apply (auto simp add: real_less_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

578 
apply (drule real_of_preal_leD) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

579 
apply (auto simp add: order_le_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

580 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

581 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

582 
lemma real_of_preal_less_iff [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

583 
"(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

584 
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

585 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

586 
lemma real_of_preal_le_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

587 
"(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

588 
by (simp add: linorder_not_less [symmetric]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

589 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

590 
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

591 
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

592 
preal_add_ac preal_cancels) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

593 
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

594 
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

595 
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

596 
apply (simp add: preal_add_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

597 
done 
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 
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

600 
by (simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

601 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

602 
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

603 
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

604 
apply (fast dest: order_less_trans) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

605 
done 
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 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

608 
subsection{*Theorems About the Ordering*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

609 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

610 
text{*obsolete but used a lot*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

611 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

612 
lemma real_not_refl2: "x < y ==> x \<noteq> (y::real)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

613 
by blast 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

614 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

615 
lemma real_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

616 
by (simp add: order_le_less) 
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_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

619 
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

620 
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

621 
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

622 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

623 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

624 
lemma real_gt_preal_preal_Ex: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

625 
"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

626 
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

627 
intro: real_gt_zero_preal_Ex [THEN iffD1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

628 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

629 
lemma real_ge_preal_preal_Ex: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

630 
"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

631 
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

632 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

633 
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

634 
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

635 
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

636 
simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

637 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

638 
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

639 
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

640 

14334  641 
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

642 
by (rule Ring_and_Field.add_less_le_mono) 
14334  643 

644 
lemma real_add_le_less_mono: 

645 
"!!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

646 
by (rule Ring_and_Field.add_le_less_mono) 
14334  647 

648 
lemma real_zero_less_one: "0 < (1::real)" 

649 
by (rule Ring_and_Field.zero_less_one) 

650 

651 
lemma real_le_square [simp]: "(0::real) \<le> x*x" 

652 
by (rule Ring_and_Field.zero_le_square) 

653 

654 

655 
subsection{*More Lemmas*} 

656 

657 
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" 

658 
by auto 

659 

660 
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" 

661 
by auto 

662 

663 
text{*The precondition could be weakened to @{term "0\<le>x"}*} 

664 
lemma real_mult_less_mono: 

665 
"[ u<v; x<y; (0::real) < v; 0 < x ] ==> u*x < v* y" 

666 
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) 

667 

668 
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" 

669 
by (force elim: order_less_asym 

670 
simp add: Ring_and_Field.mult_less_cancel_right) 

671 

672 
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

673 
apply (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

674 
apply (blast intro: elim: order_less_asym) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

675 
done 
14334  676 

677 
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" 

678 
by (force elim: order_less_asym 

679 
simp add: Ring_and_Field.mult_le_cancel_left) 

680 

681 
text{*Only two uses?*} 

682 
lemma real_mult_less_mono': 

683 
"[ x < y; r1 < r2; (0::real) \<le> r1; 0 \<le> x] ==> r1 * x < r2 * y" 

684 
by (rule Ring_and_Field.mult_strict_mono') 

685 

686 
text{*FIXME: delete or at least combine the next two lemmas*} 

687 
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" 

688 
apply (drule Ring_and_Field.equals_zero_I [THEN sym]) 

689 
apply (cut_tac x = y in real_le_square) 

690 
apply (auto, drule real_le_anti_sym, auto) 

691 
done 

692 

693 
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" 

694 
apply (rule_tac y = x in real_sum_squares_cancel) 

695 
apply (simp add: real_add_commute) 

696 
done 

697 

698 
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

699 
by (drule add_strict_mono [of concl: 0 0], assumption, simp) 
14334  700 

701 
lemma real_le_add_order: "[ 0 \<le> x; 0 \<le> y ] ==> (0::real) \<le> x + y" 

702 
apply (drule order_le_imp_less_or_eq)+ 

703 
apply (auto intro: real_add_order order_less_imp_le) 

704 
done 

705 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

706 
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

707 
apply (case_tac "x \<noteq> 0") 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

708 
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

709 
done 
14334  710 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

711 
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

712 
by (auto dest: less_imp_inverse_less) 
14334  713 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

714 
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

715 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

716 
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

717 
thus ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

718 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

719 

14334  720 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

721 
subsection{*Embedding the Integers into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

722 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

723 
defs (overloaded) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

724 
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

725 
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

726 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

727 
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

728 
by (simp add: real_of_int_def) 
14365
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_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

731 
by (simp add: real_of_int_def) 
14334  732 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

733 
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

734 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

735 
declare real_of_int_add [symmetric, simp] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

736 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

737 
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

738 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

739 
declare real_of_int_minus [symmetric, simp] 
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_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

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 
declare real_of_int_diff [symmetric, simp] 
14334  744 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

745 
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

746 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

747 
declare real_of_int_mult [symmetric, simp] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

748 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

749 
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

750 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

751 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

752 
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

753 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

754 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

755 
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

756 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

757 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

758 
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

759 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

760 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

761 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

762 
subsection{*Embedding the Naturals into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

763 

14334  764 
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

765 
by (simp add: real_of_nat_def) 
14334  766 

767 
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

768 
by (simp add: real_of_nat_def) 
14334  769 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

770 
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

771 
by (simp add: real_of_nat_def) 
14334  772 

773 
(*Not for addsimps: often the LHS is used to represent a positive natural*) 

774 
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

775 
by (simp add: real_of_nat_def) 
14334  776 

777 
lemma real_of_nat_less_iff [iff]: 

778 
"(real (n::nat) < real m) = (n < m)" 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

779 
by (simp add: real_of_nat_def) 
14334  780 

781 
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

782 
by (simp add: real_of_nat_def) 
14334  783 

784 
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

785 
by (simp add: real_of_nat_def zero_le_imp_of_nat) 
14334  786 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

787 
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

788 
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

789 

14334  790 
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

791 
by (simp add: real_of_nat_def) 
14334  792 

793 
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

794 
by (simp add: real_of_nat_def) 
14334  795 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

796 
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

797 
by (simp add: real_of_nat_def) 
14334  798 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

799 
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

800 
by (simp add: add: real_of_nat_def) 
14334  801 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

802 
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

803 
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

804 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

805 
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

806 
by (simp add: add: real_of_nat_def) 
14334  807 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

808 
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

809 
by (simp add: add: real_of_nat_def) 
14334  810 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

811 
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

812 
by (simp add: add: real_of_nat_def) 
14334  813 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

814 
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

815 
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

816 

69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

817 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

818 
text{*Still needed for binary arithmetic*} 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

819 
lemma real_of_nat_real_of_int: "~neg z ==> real (nat z) = real z" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

820 
proof (simp add: not_neg_eq_ge_0 real_of_nat_def real_of_int_def) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

821 
assume "0 \<le> z" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

822 
hence eq: "of_nat (nat z) = z" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

823 
by (simp add: nat_0_le int_eq_of_nat[symmetric]) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

824 
have "of_nat (nat z) = of_int (of_nat (nat z))" by simp 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

825 
also have "... = of_int z" by (simp add: eq) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

826 
finally show "of_nat (nat z) = of_int z" . 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

827 
qed 
14334  828 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

829 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

830 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

831 
subsection{*Numerals and Arithmetic*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

832 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

833 
instance real :: number .. 
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 
primrec (*the type constraint is essential!*) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

836 
number_of_Pls: "number_of bin.Pls = 0" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

837 
number_of_Min: "number_of bin.Min =  (1::real)" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

838 
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

839 
(number_of w) + (number_of w)" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

840 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

841 
declare number_of_Pls [simp del] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

842 
number_of_Min [simp del] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

843 
number_of_BIT [simp del] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

844 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

845 
instance real :: number_ring 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

846 
proof 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

847 
show "Numeral0 = (0::real)" by (rule number_of_Pls) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

848 
show "1 =  (1::real)" by (rule number_of_Min) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

849 
fix w :: bin and x :: bool 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

850 
show "(number_of (w BIT x) :: real) = 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

851 
(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

852 
by (rule number_of_BIT) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

853 
qed 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

854 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

855 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

856 
text{*Collapse applications of @{term real} to @{term number_of}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

857 
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

858 
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

859 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

860 
lemma real_of_nat_number_of [simp]: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

861 
"real (number_of v :: nat) = 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

862 
(if neg (number_of v :: int) then 0 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

863 
else (number_of v :: real))" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

864 
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

865 

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 
use "real_arith.ML" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

868 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

869 
setup real_arith_setup 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

870 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

871 
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} 
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 
text{*Needed in this nonstandard form by Hyperreal/Transcendental*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

874 
lemma real_0_le_divide_iff: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

875 
"((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

876 
by (simp add: real_divide_def zero_le_mult_iff, auto) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

877 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

878 
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

879 
by arith 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

880 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

881 
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

882 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

883 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

884 
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

885 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

886 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

887 
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

888 
by auto 
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 
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

891 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

892 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

893 
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

894 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

895 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

896 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

897 
(** Simprules combining xy and 0 (needed??) **) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

898 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

899 
lemma real_0_less_diff_iff [iff]: "((0::real) < xy) = (y < x)" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

900 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

901 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

902 
lemma real_0_le_diff_iff [iff]: "((0::real) \<le> xy) = (y \<le> x)" 
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 
(* 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

906 
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

907 
It replaces x+y by xy. 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

908 
Addsimps [symmetric real_diff_def] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

909 
*) 
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 
subsubsection{*Density of 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 
lemma real_lbound_gt_zero: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

915 
"[ (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

916 
apply (rule_tac x = " (min d1 d2) /2" in exI) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

917 
apply (simp add: min_def) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

918 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

919 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

920 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

921 
text{*Similar results are proved in @{text Ring_and_Field}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

922 
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

923 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

924 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

925 
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

926 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

927 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

928 
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

929 
by (rule Ring_and_Field.dense) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

930 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

931 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

932 
subsection{*Absolute Value Function for the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

933 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

934 
text{*FIXME: these should go!*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

935 
lemma abs_eqI1: "(0::real)\<le>x ==> abs x = x" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

936 
by (unfold real_abs_def, simp) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

937 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

938 
lemma abs_eqI2: "(0::real) < x ==> abs x = x" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

939 
by (unfold real_abs_def, simp) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

940 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

941 
lemma abs_minus_eqI2: "x < (0::real) ==> abs x = x" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

942 
by (simp add: real_abs_def linorder_not_less [symmetric]) 
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 
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

945 
by (unfold real_abs_def, simp) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

946 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

947 
lemma abs_minus_one [simp]: "abs (1) = (1::real)" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

948 
by (unfold real_abs_def, simp) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

949 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

950 
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

951 
by (force simp add: Ring_and_Field.abs_less_iff) 
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 
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

954 
by (force simp add: Ring_and_Field.abs_le_iff) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

955 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

956 
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

957 
by (unfold real_abs_def, auto) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

958 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

959 
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

960 
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

961 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

962 
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

963 
apply (simp add: linorder_not_less) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

964 
apply (auto intro: abs_ge_self [THEN order_trans]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

965 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

966 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

967 
text{*Used only in Hyperreal/Lim.ML*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

968 
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

969 
apply (simp add: real_add_assoc) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

970 
apply (rule_tac a1 = y in add_left_commute [THEN ssubst]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

971 
apply (rule real_add_assoc [THEN subst]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

972 
apply (rule abs_triangle_ineq) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

973 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

974 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

975 

e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

976 

14334  977 
ML 
978 
{* 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

979 
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

980 
val real_add_minus_iff = thm"real_add_minus_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

981 
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

982 
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

983 
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

984 
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

985 
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

986 
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

987 
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

988 
val real_lbound_gt_zero = thm"real_lbound_gt_zero"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

989 
val real_less_half_sum = thm"real_less_half_sum"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

990 
val real_gt_half_sum = thm"real_gt_half_sum"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

991 
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

992 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

993 
val abs_eqI1 = thm"abs_eqI1"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

994 
val abs_eqI2 = thm"abs_eqI2"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

995 
val abs_minus_eqI2 = thm"abs_minus_eqI2"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

996 
val abs_ge_zero = thm"abs_ge_zero"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

997 
val abs_idempotent = thm"abs_idempotent"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

998 
val abs_zero_iff = thm"abs_zero_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

999 
val abs_ge_self = thm"abs_ge_self"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1000 
val abs_ge_minus_self = thm"abs_ge_minus_self"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1001 
val abs_mult = thm"abs_mult"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1002 
val abs_inverse = thm"abs_inverse"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1003 
val abs_triangle_ineq = thm"abs_triangle_ineq"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1004 
val abs_minus_cancel = thm"abs_minus_cancel"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1005 
val abs_minus_add_cancel = thm"abs_minus_add_cancel"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1006 
val abs_minus_one = thm"abs_minus_one"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1007 
val abs_interval_iff = thm"abs_interval_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1008 
val abs_le_interval_iff = thm"abs_le_interval_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1009 
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

1010 
val abs_le_zero_iff = thm"abs_le_zero_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1011 
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

1012 
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

1013 
val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq"; 
14334  1014 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1015 
val abs_mult_less = thm"abs_mult_less"; 
14334  1016 
*} 
10752
c4f1bf2acf4c
tidying, and separation of HOLHyperreal from HOLReal
paulson
parents:
10648
diff
changeset

1017 

14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1018 

5588  1019 
end 