author  avigad 
Wed, 13 Jul 2005 19:49:07 +0200  
changeset 16819  00d8f9300d13 
parent 16417  9bc16273c2d4 
child 16888  7cb4bcfa058e 
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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Additional contributions by Jeremy Avigad 
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*) 
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header{*Defining the Reals from the Positive Reals*} 
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15131  11 
theory RealDef 
15140  12 
imports PReal 
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uses ("real_arith.ML") 
15131  14 
begin 
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16 
constdefs 

17 
realrel :: "((preal * preal) * (preal * preal)) set" 

14269  18 
"realrel == {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" 
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14484  20 
typedef (Real) real = "UNIV//realrel" 
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by (auto simp add: quotient_def) 
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14691  23 
instance real :: "{ord, zero, one, plus, times, minus, inverse}" .. 
14269  24 

14484  25 
constdefs 
26 

27 
(** these don't use the overloaded "real" function: users don't see them **) 

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29 
real_of_preal :: "preal => real" 

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"real_of_preal m == 

31 
Abs_Real(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})" 

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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" 

37 

38 
(*overloaded constant for injecting other types into "real"*) 

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real :: "'a => real" 

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syntax (xsymbols) 
42 
Reals :: "'a set" ("\<real>") 

43 

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defs (overloaded) 
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real_zero_def: 
14484  48 
"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: 
14484  56 
" r == contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" 
57 

58 
real_add_def: 

59 
"z + w == 

60 
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 

61 
{ Abs_Real(realrel``{(x+u, y+v)}) })" 

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14269  63 
real_diff_def: 
14484  64 
"r  (s::real) == r +  s" 
65 

66 
real_mult_def: 

67 
"z * w == 

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contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 

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{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" 

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real_inverse_def: 
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"inverse (R::real) == (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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14484  77 
real_le_def: 
78 
"z \<le> (w::real) == 

79 
\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w" 

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real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)" 
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14334  83 
real_abs_def: "abs (r::real) == (if 0 \<le> r then r else r)" 
84 

85 

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14329  87 
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)) \<in> realrel) = (x1 + y2 = x2 + y1)" 
105 
by (simp add: realrel_def) 

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107 
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 
111 

14497  112 
text{*Reduces equality of equivalence classes to the @{term realrel} relation: 
113 
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *} 

14269  114 
lemmas equiv_realrel_iff = 
115 
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] 

116 

117 
declare equiv_realrel_iff [simp] 

118 

14497  119 

14484  120 
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" 
121 
by (simp add: Real_def realrel_def quotient_def, blast) 

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14484  124 
lemma inj_on_Abs_Real: "inj_on Abs_Real Real" 
14269  125 
apply (rule inj_on_inverseI) 
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apply (erule Abs_Real_inverse) 
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done 
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14484  129 
declare inj_on_Abs_Real [THEN inj_on_iff, simp] 
130 
declare Abs_Real_inverse [simp] 

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132 

14484  133 
text{*Case analysis on the representation of a real number as an equivalence 
134 
class of pairs of positive reals.*} 

135 
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 

136 
"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" 

137 
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) 

138 
apply (drule arg_cong [where f=Abs_Real]) 

139 
apply (auto simp add: Rep_Real_inverse) 

14269  140 
done 
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142 

14329  143 
subsection{*Congruence property for addition*} 
14269  144 

145 
lemma real_add_congruent2_lemma: 

146 
"[a + ba = aa + b; ab + bc = ac + bb] 

147 
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" 

148 
apply (simp add: preal_add_assoc) 

149 
apply (rule preal_add_left_commute [of ab, THEN ssubst]) 

150 
apply (simp add: preal_add_assoc [symmetric]) 

151 
apply (simp add: preal_add_ac) 

152 
done 

153 

154 
lemma real_add: 

14497  155 
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = 
156 
Abs_Real (realrel``{(x+u, y+v)})" 

157 
proof  

15169  158 
have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) 
159 
respects2 realrel" 

14497  160 
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
161 
thus ?thesis 

162 
by (simp add: real_add_def UN_UN_split_split_eq 

14658  163 
UN_equiv_class2 [OF equiv_realrel equiv_realrel]) 
14497  164 
qed 
14269  165 

166 
lemma real_add_commute: "(z::real) + w = w + z" 

14497  167 
by (cases z, cases w, simp add: real_add preal_add_ac) 
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169 
lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)" 

14497  170 
by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc) 
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172 
lemma real_add_zero_left: "(0::real) + z = z" 

14497  173 
by (cases z, simp add: real_add real_zero_def preal_add_ac) 
14269  174 

14738  175 
instance real :: comm_monoid_add 
14269  176 
by (intro_classes, 
177 
(assumption  

178 
rule real_add_commute real_add_assoc real_add_zero_left)+) 

179 

180 

14334  181 
subsection{*Additive Inverse on real*} 
182 

14484  183 
lemma real_minus: " Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" 
184 
proof  

15169  185 
have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" 
14484  186 
by (simp add: congruent_def preal_add_commute) 
187 
thus ?thesis 

188 
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) 

189 
qed 

14334  190 

191 
lemma real_add_minus_left: "(z) + z = (0::real)" 

14497  192 
by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute) 
14269  193 

194 

14329  195 
subsection{*Congruence property for multiplication*} 
14269  196 

14329  197 
lemma real_mult_congruent2_lemma: 
198 
"!!(x1::preal). [ x1 + y2 = x2 + y1 ] ==> 

14484  199 
x * x1 + y * y1 + (x * y2 + y * x2) = 
200 
x * x2 + y * y2 + (x * y1 + y * x1)" 

201 
apply (simp add: preal_add_left_commute preal_add_assoc [symmetric]) 

14269  202 
apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric]) 
203 
apply (simp add: preal_add_commute) 

204 
done 

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206 
lemma real_mult_congruent2: 

15169  207 
"(%p1 p2. 
14484  208 
(%(x1,y1). (%(x2,y2). 
15169  209 
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1) 
210 
respects2 realrel" 

14658  211 
apply (rule congruent2_commuteI [OF equiv_realrel], clarify) 
14269  212 
apply (simp add: preal_mult_commute preal_add_commute) 
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apply (auto simp add: real_mult_congruent2_lemma) 

214 
done 

215 

216 
lemma real_mult: 

14484  217 
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = 
218 
Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" 

219 
by (simp add: real_mult_def UN_UN_split_split_eq 

14658  220 
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) 
14269  221 

222 
lemma real_mult_commute: "(z::real) * w = w * z" 

14497  223 
by (cases z, cases w, simp add: real_mult preal_add_ac preal_mult_ac) 
14269  224 

225 
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" 

14484  226 
apply (cases z1, cases z2, cases z3) 
227 
apply (simp add: real_mult preal_add_mult_distrib2 preal_add_ac preal_mult_ac) 

14269  228 
done 
229 

230 
lemma real_mult_1: "(1::real) * z = z" 

14484  231 
apply (cases z) 
232 
apply (simp add: real_mult real_one_def preal_add_mult_distrib2 

233 
preal_mult_1_right preal_mult_ac preal_add_ac) 

14269  234 
done 
235 

236 
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" 

14484  237 
apply (cases z1, cases z2, cases w) 
238 
apply (simp add: real_add real_mult preal_add_mult_distrib2 

239 
preal_add_ac preal_mult_ac) 

14269  240 
done 
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14329  242 
text{*one and zero are distinct*} 
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lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" 
14484  244 
proof  
245 
have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1" 

246 
by (simp add: preal_self_less_add_left) 

247 
thus ?thesis 

248 
by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff) 

249 
qed 

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14329  251 
subsection{*existence of inverse*} 
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14484  253 
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" 
14497  254 
by (simp add: real_zero_def preal_add_commute) 
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text{*Instead of using an existential quantifier and constructing the inverse 
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within the proof, we could define the inverse explicitly.*} 
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lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" 
14484  260 
apply (simp add: real_zero_def real_one_def, cases x) 
14269  261 
apply (cut_tac x = xa and y = y in linorder_less_linear) 
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apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) 
14334  263 
apply (rule_tac 
14484  264 
x = "Abs_Real (realrel `` { (preal_of_rat 1, 
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inverse (D) + preal_of_rat 1)}) " 
14334  266 
in exI) 
267 
apply (rule_tac [2] 

14484  268 
x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1, 
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preal_of_rat 1)})" 
14334  270 
in exI) 
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apply (auto simp add: real_mult preal_mult_1_right 
14329  272 
preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1 
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preal_mult_inverse_right preal_add_ac preal_mult_ac) 
14269  274 
done 
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lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" 
14484  277 
apply (simp add: real_inverse_def) 
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apply (frule real_mult_inverse_left_ex, safe) 
14269  279 
apply (rule someI2, auto) 
280 
done 

14334  281 

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subsection{*The Real Numbers form a Field*} 
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14334  285 
instance real :: field 
286 
proof 

287 
fix x y z :: real 

288 
show " x + x = 0" by (rule real_add_minus_left) 

289 
show "x  y = x + (y)" by (simp add: real_diff_def) 

290 
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) 

291 
show "x * y = y * x" by (rule real_mult_commute) 

292 
show "1 * x = x" by (rule real_mult_1) 

293 
show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) 

294 
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) 

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show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) 
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show "x / y = x * inverse y" by (simp add: real_divide_def) 
14334  297 
qed 
298 

299 

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text{*Inverse of zero! Useful to simplify certain equations*} 
14269  301 

14334  302 
lemma INVERSE_ZERO: "inverse 0 = (0::real)" 
14484  303 
by (simp add: real_inverse_def) 
14334  304 

305 
instance real :: division_by_zero 

306 
proof 

307 
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) 

308 
qed 

309 

310 

311 
(*Pull negations out*) 

312 
declare minus_mult_right [symmetric, simp] 

313 
minus_mult_left [symmetric, simp] 

314 

315 
lemma real_mult_1_right: "z * (1::real) = z" 

14738  316 
by (rule OrderedGroup.mult_1_right) 
14269  317 

318 

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subsection{*The @{text "\<le>"} Ordering*} 
14269  320 

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lemma real_le_refl: "w \<le> (w::real)" 
14484  322 
by (cases w, force simp add: real_le_def) 
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324 
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

325 
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:
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diff
changeset

326 
following two lemmas.*} 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

327 
lemma preal_eq_le_imp_le: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

328 
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

329 
shows "b \<le> (d::preal)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

330 
proof  
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

331 
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

332 
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

333 
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

334 
qed 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

335 

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

336 
lemma real_le_lemma: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

337 
assumes l: "u1 + v2 \<le> u2 + v1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

338 
and "x1 + v1 = u1 + y1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

339 
and "x2 + v2 = u2 + y2" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

340 
shows "x1 + y2 \<le> x2 + (y1::preal)" 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

341 
proof  
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

342 
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

343 
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

344 
also have "... \<le> (x2+y1) + (u2+v1)" 
14365
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paulson
parents:
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diff
changeset

345 
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

346 
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

347 
qed 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

348 

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

349 
lemma real_le: 
14484  350 
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) = 
351 
(x1 + y2 \<le> x2 + y1)" 

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

352 
apply (simp add: real_le_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

353 
apply (auto intro: real_le_lemma) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

354 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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diff
changeset

355 

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

356 
lemma real_le_anti_sym: "[ z \<le> w; w \<le> z ] ==> z = (w::real)" 
15542  357 
by (cases z, cases w, simp add: real_le) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

358 

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

359 
lemma real_trans_lemma: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

360 
assumes "x + v \<le> u + y" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

361 
and "u + v' \<le> u' + v" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

362 
and "x2 + v2 = u2 + y2" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

363 
shows "x + v' \<le> u' + (y::preal)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

364 
proof  
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

365 
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

366 
also have "... \<le> (u+y) + (u+v')" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

367 
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

368 
also have "... \<le> (u+y) + (u'+v)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

369 
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

370 
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

371 
finally show ?thesis by (simp add: preal_add_le_cancel_right) 
15542  372 
qed 
14269  373 

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

374 
lemma real_le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::real)" 
14484  375 
apply (cases i, cases j, cases k) 
376 
apply (simp add: real_le) 

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

377 
apply (blast intro: real_trans_lemma) 
14334  378 
done 
379 

380 
(* Axiom 'order_less_le' of class 'order': *) 

381 
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

382 
by (simp add: real_less_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

383 

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

384 
instance real :: order 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

385 
proof qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

386 
(assumption  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

387 
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:
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diff
changeset

388 

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

389 
(* Axiom 'linorder_linear' of class 'linorder': *) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

390 
lemma real_le_linear: "(z::real) \<le> w  w \<le> z" 
14484  391 
apply (cases z, cases w) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

392 
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels) 
14334  393 
done 
394 

395 

396 
instance real :: linorder 

397 
by (intro_classes, rule real_le_linear) 

398 

399 

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

400 
lemma real_le_eq_diff: "(x \<le> y) = (xy \<le> (0::real))" 
14484  401 
apply (cases x, cases y) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

402 
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

403 
preal_add_ac) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

404 
apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) 
15542  405 
done 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

406 

14484  407 
lemma real_add_left_mono: 
408 
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)" 

409 
proof  

410 
have "z + x  (z + y) = (z + z) + (x  y)" 

411 
by (simp add: diff_minus add_ac) 

412 
with le show ?thesis 

14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset

413 
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) 
14484  414 
qed 
14334  415 

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

416 
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

417 
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

418 

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

419 
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

420 
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) 
14334  421 

422 
lemma real_mult_order: "[ 0 < x; 0 < y ] ==> (0::real) < x * y" 

14484  423 
apply (cases x, cases y) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

424 
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

425 
linorder_not_le [where 'a = preal] 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

426 
real_zero_def real_le real_mult) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

427 
{*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

428 
apply (auto dest!: less_add_left_Ex 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

429 
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

430 
preal_add_mult_distrib2 preal_cancels preal_self_less_add_right) 
14334  431 
done 
432 

433 
lemma real_mult_less_mono2: "[ (0::real) < z; x < y ] ==> z * x < z * y" 

434 
apply (rule real_sum_gt_zero_less) 

435 
apply (drule real_less_sum_gt_zero [of x y]) 

436 
apply (drule real_mult_order, assumption) 

437 
apply (simp add: right_distrib) 

438 
done 

439 

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

440 
text{*lemma for proving @{term "0<(1::real)"}*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

441 
lemma real_zero_le_one: "0 \<le> (1::real)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

442 
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

443 
preal_self_less_add_left order_less_imp_le) 
14334  444 

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

445 

14334  446 
subsection{*The Reals Form an Ordered Field*} 
447 

448 
instance real :: ordered_field 

449 
proof 

450 
fix x y z :: real 

451 
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) 

452 
show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) 

453 
show "\<bar>x\<bar> = (if x < 0 then x else x)" 

454 
by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) 

455 
qed 

456 

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

457 

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

458 

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

459 
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

460 
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

461 
positive reals.*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

462 

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

463 
lemma real_of_preal_add: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

464 
"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

465 
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

466 
preal_add_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

467 

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

468 
lemma real_of_preal_mult: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

469 
"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

470 
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

471 
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

472 

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

473 

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

474 
text{*Gleason prop 94.4 p 127*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

475 
lemma real_of_preal_trichotomy: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

476 
"\<exists>m. (x::real) = real_of_preal m  x = 0  x = (real_of_preal m)" 
14484  477 
apply (simp add: real_of_preal_def real_zero_def, cases x) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

478 
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

479 
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

480 
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

481 
apply (auto simp add: preal_add_commute) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

482 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

483 

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

484 
lemma real_of_preal_leD: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

485 
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" 
14484  486 
by (simp add: real_of_preal_def real_le preal_cancels) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

487 

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

488 
lemma real_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

489 
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

490 

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

491 
lemma real_of_preal_lessD: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

492 
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" 
14484  493 
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric] 
494 
preal_cancels) 

495 

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

496 

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

497 
lemma real_of_preal_less_iff [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

498 
"(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

499 
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

500 

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

501 
lemma real_of_preal_le_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

502 
"(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

503 
by (simp add: linorder_not_less [symmetric]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

504 

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

505 
lemma real_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

506 
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

507 
preal_add_ac preal_cancels) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

508 
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

509 
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

510 
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

511 
apply (simp add: preal_add_ac) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

512 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

513 

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

514 
lemma real_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

515 
by (simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

516 

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

517 
lemma real_of_preal_not_minus_gt_zero: "~ 0 <  real_of_preal m" 
14484  518 
proof  
519 
from real_of_preal_minus_less_zero 

520 
show ?thesis by (blast dest: order_less_trans) 

521 
qed 

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

522 

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

523 

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

524 
subsection{*Theorems About the Ordering*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

525 

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

526 
text{*obsolete but used a lot*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

527 

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

528 
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

529 
by blast 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

530 

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

531 
lemma real_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

532 
by (simp add: order_le_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

533 

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

534 
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

535 
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

536 
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

537 
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

538 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

539 

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

540 
lemma real_gt_preal_preal_Ex: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

541 
"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

542 
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

543 
intro: real_gt_zero_preal_Ex [THEN iffD1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

544 

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

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

546 
"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

547 
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

548 

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

549 
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

550 
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

551 
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

552 
simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

553 

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

554 
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

555 
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

556 

14334  557 
lemma real_add_less_le_mono: "[ w'<w; z'\<le>z ] ==> w' + z' < w + (z::real)" 
14738  558 
by (rule OrderedGroup.add_less_le_mono) 
14334  559 

560 
lemma real_add_le_less_mono: 

561 
"!!z z'::real. [ w'\<le>w; z'<z ] ==> w' + z' < w + z" 

14738  562 
by (rule OrderedGroup.add_le_less_mono) 
14334  563 

564 
lemma real_le_square [simp]: "(0::real) \<le> x*x" 

565 
by (rule Ring_and_Field.zero_le_square) 

566 

567 

568 
subsection{*More Lemmas*} 

569 

570 
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" 

571 
by auto 

572 

573 
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" 

574 
by auto 

575 

576 
text{*The precondition could be weakened to @{term "0\<le>x"}*} 

577 
lemma real_mult_less_mono: 

578 
"[ u<v; x<y; (0::real) < v; 0 < x ] ==> u*x < v* y" 

579 
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) 

580 

581 
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" 

582 
by (force elim: order_less_asym 

583 
simp add: Ring_and_Field.mult_less_cancel_right) 

584 

585 
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

586 
apply (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

587 
apply (blast intro: elim: order_less_asym) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

588 
done 
14334  589 

590 
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" 

15923  591 
by(simp add:mult_commute) 
14334  592 

593 
text{*Only two uses?*} 

594 
lemma real_mult_less_mono': 

595 
"[ x < y; r1 < r2; (0::real) \<le> r1; 0 \<le> x] ==> r1 * x < r2 * y" 

596 
by (rule Ring_and_Field.mult_strict_mono') 

597 

598 
text{*FIXME: delete or at least combine the next two lemmas*} 

599 
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" 

14738  600 
apply (drule OrderedGroup.equals_zero_I [THEN sym]) 
14334  601 
apply (cut_tac x = y in real_le_square) 
14476  602 
apply (auto, drule order_antisym, auto) 
14334  603 
done 
604 

605 
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" 

606 
apply (rule_tac y = x in real_sum_squares_cancel) 

14476  607 
apply (simp add: add_commute) 
14334  608 
done 
609 

610 
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

611 
by (drule add_strict_mono [of concl: 0 0], assumption, simp) 
14334  612 

613 
lemma real_le_add_order: "[ 0 \<le> x; 0 \<le> y ] ==> (0::real) \<le> x + y" 

614 
apply (drule order_le_imp_less_or_eq)+ 

615 
apply (auto intro: real_add_order order_less_imp_le) 

616 
done 

617 

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

618 
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

619 
apply (case_tac "x \<noteq> 0") 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

620 
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

621 
done 
14334  622 

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

623 
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

624 
by (auto dest: less_imp_inverse_less) 
14334  625 

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

626 
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

627 
proof  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

628 
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

629 
thus ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

630 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

631 

14334  632 

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

633 
subsection{*Embedding the Integers into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

634 

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

635 
defs (overloaded) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

636 
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

637 
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

638 

16819  639 
lemma real_eq_of_nat: "real = of_nat" 
640 
apply (rule ext) 

641 
apply (unfold real_of_nat_def) 

642 
apply (rule refl) 

643 
done 

644 

645 
lemma real_eq_of_int: "real = of_int" 

646 
apply (rule ext) 

647 
apply (unfold real_of_int_def) 

648 
apply (rule refl) 

649 
done 

650 

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

651 
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

652 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

653 

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

654 
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

655 
by (simp add: real_of_int_def) 
14334  656 

16819  657 
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

658 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

659 

16819  660 
lemma real_of_int_minus [simp]: "real(x) = real (x::int)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

661 
by (simp add: real_of_int_def) 
16819  662 

663 
lemma real_of_int_diff [simp]: "real(x  y) = real (x::int)  real y" 

664 
by (simp add: real_of_int_def) 

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

665 

16819  666 
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

667 
by (simp add: real_of_int_def) 
14334  668 

16819  669 
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" 
670 
apply (subst real_eq_of_int)+ 

671 
apply (rule of_int_setsum) 

672 
done 

673 

674 
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 

675 
(PROD x:A. real(f x))" 

676 
apply (subst real_eq_of_int)+ 

677 
apply (rule of_int_setprod) 

678 
done 

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

679 

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

680 
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

681 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

682 

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

683 
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

684 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

685 

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

686 
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

687 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

688 

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

689 
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

690 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

691 

16819  692 
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)" 
693 
by (simp add: real_of_int_def) 

694 

695 
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)" 

696 
by (simp add: real_of_int_def) 

697 

698 
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)" 

699 
by (simp add: real_of_int_def) 

700 

701 
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)" 

702 
by (simp add: real_of_int_def) 

703 

704 
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" 

705 
apply (subgoal_tac "real n + 1 = real (n + 1)") 

706 
apply (simp del: real_of_int_add) 

707 
apply auto 

708 
done 

709 

710 
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" 

711 
apply (subgoal_tac "real m + 1 = real (m + 1)") 

712 
apply (simp del: real_of_int_add) 

713 
apply simp 

714 
done 

715 

716 
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 

717 
real (x div d) + (real (x mod d)) / (real d)" 

718 
proof  

719 
assume "d ~= 0" 

720 
have "x = (x div d) * d + x mod d" 

721 
by auto 

722 
then have "real x = real (x div d) * real d + real(x mod d)" 

723 
by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) 

724 
then have "real x / real d = ... / real d" 

725 
by simp 

726 
then show ?thesis 

727 
by (auto simp add: add_divide_distrib ring_eq_simps prems) 

728 
qed 

729 

730 
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> 

731 
real(n div d) = real n / real d" 

732 
apply (frule real_of_int_div_aux [of d n]) 

733 
apply simp 

734 
apply (simp add: zdvd_iff_zmod_eq_0) 

735 
done 

736 

737 
lemma real_of_int_div2: 

738 
"0 <= real (n::int) / real (x)  real (n div x)" 

739 
apply (case_tac "x = 0") 

740 
apply simp 

741 
apply (case_tac "0 < x") 

742 
apply (simp add: compare_rls) 

743 
apply (subst real_of_int_div_aux) 

744 
apply simp 

745 
apply simp 

746 
apply (subst zero_le_divide_iff) 

747 
apply auto 

748 
apply (simp add: compare_rls) 

749 
apply (subst real_of_int_div_aux) 

750 
apply simp 

751 
apply simp 

752 
apply (subst zero_le_divide_iff) 

753 
apply auto 

754 
done 

755 

756 
lemma real_of_int_div3: 

757 
"real (n::int) / real (x)  real (n div x) <= 1" 

758 
apply(case_tac "x = 0") 

759 
apply simp 

760 
apply (simp add: compare_rls) 

761 
apply (subst real_of_int_div_aux) 

762 
apply assumption 

763 
apply simp 

764 
apply (subst divide_le_eq) 

765 
apply clarsimp 

766 
apply (rule conjI) 

767 
apply (rule impI) 

768 
apply (rule order_less_imp_le) 

769 
apply simp 

770 
apply (rule impI) 

771 
apply (rule order_less_imp_le) 

772 
apply simp 

773 
done 

774 

775 
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 

776 
by (insert real_of_int_div2 [of n x], simp) 

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

777 

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

778 
subsection{*Embedding the Naturals into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

779 

14334  780 
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

781 
by (simp add: real_of_nat_def) 
14334  782 

783 
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

784 
by (simp add: real_of_nat_def) 
14334  785 

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

786 
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

787 
by (simp add: real_of_nat_def) 
14334  788 

789 
(*Not for addsimps: often the LHS is used to represent a positive natural*) 

790 
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

791 
by (simp add: real_of_nat_def) 
14334  792 

793 
lemma real_of_nat_less_iff [iff]: 

794 
"(real (n::nat) < real m) = (n < m)" 

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

795 
by (simp add: real_of_nat_def) 
14334  796 

797 
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

798 
by (simp add: real_of_nat_def) 
14334  799 

800 
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

801 
by (simp add: real_of_nat_def zero_le_imp_of_nat) 
14334  802 

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

803 
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

804 
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

805 

14334  806 
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

807 
by (simp add: real_of_nat_def) 
14334  808 

16819  809 
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
810 
(SUM x:A. real(f x))" 

811 
apply (subst real_eq_of_nat)+ 

812 
apply (rule of_nat_setsum) 

813 
done 

814 

815 
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 

816 
(PROD x:A. real(f x))" 

817 
apply (subst real_eq_of_nat)+ 

818 
apply (rule of_nat_setprod) 

819 
done 

820 

821 
lemma real_of_card: "real (card A) = setsum (%x.1) A" 

822 
apply (subst card_eq_setsum) 

823 
apply (subst real_of_nat_setsum) 

824 
apply simp 

825 
done 

826 

14334  827 
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

828 
by (simp add: real_of_nat_def) 
14334  829 

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

830 
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

831 
by (simp add: real_of_nat_def) 
14334  832 

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

833 
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

834 
by (simp add: add: real_of_nat_def) 
14334  835 

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

836 
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

837 
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

838 

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

839 
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

840 
by (simp add: add: real_of_nat_def) 
14334  841 

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

842 
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

843 
by (simp add: add: real_of_nat_def) 
14334  844 

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

845 
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

846 
by (simp add: add: real_of_nat_def) 
14334  847 

16819  848 
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)" 
849 
apply (subgoal_tac "real n + 1 = real (Suc n)") 

850 
apply simp 

851 
apply (auto simp add: real_of_nat_Suc) 

852 
done 

853 

854 
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)" 

855 
apply (subgoal_tac "real m + 1 = real (Suc m)") 

856 
apply (simp add: less_Suc_eq_le) 

857 
apply (simp add: real_of_nat_Suc) 

858 
done 

859 

860 
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 

861 
real (x div d) + (real (x mod d)) / (real d)" 

862 
proof  

863 
assume "0 < d" 

864 
have "x = (x div d) * d + x mod d" 

865 
by auto 

866 
then have "real x = real (x div d) * real d + real(x mod d)" 

867 
by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym]) 

868 
then have "real x / real d = \<dots> / real d" 

869 
by simp 

870 
then show ?thesis 

871 
by (auto simp add: add_divide_distrib ring_eq_simps prems) 

872 
qed 

873 

874 
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==> 

875 
real(n div d) = real n / real d" 

876 
apply (frule real_of_nat_div_aux [of d n]) 

877 
apply simp 

878 
apply (subst dvd_eq_mod_eq_0 [THEN sym]) 

879 
apply assumption 

880 
done 

881 

882 
lemma real_of_nat_div2: 

883 
"0 <= real (n::nat) / real (x)  real (n div x)" 

884 
apply(case_tac "x = 0") 

885 
apply simp 

886 
apply (simp add: compare_rls) 

887 
apply (subst real_of_nat_div_aux) 

888 
apply assumption 

889 
apply simp 

890 
apply (subst zero_le_divide_iff) 

891 
apply simp 

892 
done 

893 

894 
lemma real_of_nat_div3: 

895 
"real (n::nat) / real (x)  real (n div x) <= 1" 

896 
apply(case_tac "x = 0") 

897 
apply simp 

898 
apply (simp add: compare_rls) 

899 
apply (subst real_of_nat_div_aux) 

900 
apply assumption 

901 
apply simp 

902 
done 

903 

904 
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 

905 
by (insert real_of_nat_div2 [of n x], simp) 

906 

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

907 
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

908 
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

909 

14426  910 
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" 
911 
by (simp add: real_of_int_def real_of_nat_def) 

14334  912 

16819  913 
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x" 
914 
apply (subgoal_tac "real(int(nat x)) = real(nat x)") 

915 
apply force 

916 
apply (simp only: real_of_int_real_of_nat) 

917 
done 

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

918 

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

919 
subsection{*Numerals and Arithmetic*} 
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 
instance real :: number .. 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

922 

15013  923 
defs (overloaded) 
924 
real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)" 

925 
{*the type constraint is essential!*} 

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

926 

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

927 
instance real :: number_ring 
15013  928 
by (intro_classes, simp add: real_number_of_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

929 

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

930 

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

931 
text{*Collapse applications of @{term real} to @{term number_of}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

932 
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

933 
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

934 

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

935 
lemma real_of_nat_number_of [simp]: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

936 
"real (number_of v :: nat) = 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

937 
(if neg (number_of v :: int) then 0 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

938 
else (number_of v :: real))" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

939 
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

940 

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

941 

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

942 
use "real_arith.ML" 
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 
setup real_arith_setup 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

945 

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

946 
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

947 

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

948 
text{*Needed in this nonstandard form by Hyperreal/Transcendental*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

949 
lemma real_0_le_divide_iff: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

950 
"((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

951 
by (simp add: real_divide_def zero_le_mult_iff, auto) 
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 real_add_minus_iff [simp]: "(x +  a = (0::real)) = (x=a)" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

954 
by arith 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

955 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

956 
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = x)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

957 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

958 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

959 
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < x)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

960 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

961 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

962 
lemma real_0_less_add_iff: "((0::real) < x+y) = (x < y)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

963 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

964 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

965 
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> x)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

966 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

967 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15077
diff
changeset

968 
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (x \<le> y)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

969 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

970 

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

971 

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

972 
(* 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

973 
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

974 
It replaces x+y by xy. 
15086  975 
declare real_diff_def [symmetric, simp] 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

976 
*) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

977 

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

978 

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

979 
subsubsection{*Density of the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

980 

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

981 
lemma real_lbound_gt_zero: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

982 
"[ (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

983 
apply (rule_tac x = " (min d1 d2) /2" in exI) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

984 
apply (simp add: min_def) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

985 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

986 

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

987 

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

988 
text{*Similar results are proved in @{text Ring_and_Field}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

989 
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

990 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

991 

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

992 
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

993 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

994 

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

995 

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

996 
subsection{*Absolute Value Function for the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

997 

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

998 
lemma abs_minus_add_cancel: "abs(x + (y)) = abs (y + ((x::real)))" 
15003  999 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1000 

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

1001 
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

1002 
by (force simp add: Ring_and_Field.abs_less_iff) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1003 

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

1004 
lemma abs_le_interval_iff: "(abs x \<le> r) = (r\<le>x & x\<le>(r::real))" 
14738  1005 
by (force simp add: OrderedGroup.abs_le_iff) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1006 

14484  1007 
(*FIXME: used only once, in SEQ.ML*) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1008 
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" 
15003  1009 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1010 

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

1011 
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" 
15229  1012 
by (simp add: real_of_nat_ge_zero) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1013 

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

1014 
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

1015 
apply (simp add: linorder_not_less) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1016 
apply (auto intro: abs_ge_self [THEN order_trans]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1017 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1018 

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

1019 
text{*Used only in Hyperreal/Lim.ML*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1020 
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

1021 
apply (simp add: real_add_assoc) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1022 
apply (rule_tac a1 = y in add_left_commute [THEN ssubst]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1023 
apply (rule real_add_assoc [THEN subst]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1024 
apply (rule abs_triangle_ineq) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1025 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1026 

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

1027 

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

1028 

14334  1029 
ML 
1030 
{* 

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

1031 
val real_lbound_gt_zero = thm"real_lbound_gt_zero"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1032 
val real_less_half_sum = thm"real_less_half_sum"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1033 
val real_gt_half_sum = thm"real_gt_half_sum"; 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset

1034 

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

1035 
val abs_interval_iff = thm"abs_interval_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1036 
val abs_le_interval_iff = thm"abs_le_interval_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

1037 
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

1038 
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

1039 
val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq"; 
14334  1040 
*} 
10752
c4f1bf2acf4c
tidying, and separation of HOLHyperreal from HOLReal
paulson
parents:
10648
diff
changeset

1041 

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

1042 

5588  1043 
end 