author  paulson 
Mon, 26 Jul 2004 17:34:52 +0200  
changeset 15077  89840837108e 
parent 15013  34264f5e4691 
child 15085  5693a977a767 
permissions  rwrr 
5588  1 
(* 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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13 
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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14484  17 
typedef (Real) real = "UNIV//realrel" 
14269  18 
by (auto simp add: quotient_def) 
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14691  20 
instance real :: "{ord, zero, one, plus, times, minus, inverse}" .. 
14269  21 

14484  22 
constdefs 
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24 
(** these don't use the overloaded "real" function: users don't see them **) 

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26 
real_of_preal :: "preal => real" 

27 
"real_of_preal m == 

28 
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" 

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

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real :: "'a => real" 

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syntax (xsymbols) 
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Reals :: "'a set" ("\<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 == contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" 
54 

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real_add_def: 

56 
"z + w == 

57 
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 

58 
{ Abs_Real(realrel``{(x+u, y+v)}) })" 

10606  59 

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real_diff_def: 
14484  61 
"r  (s::real) == r +  s" 
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real_mult_def: 

64 
"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" 
14269  73 

14484  74 
real_le_def: 
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"z \<le> (w::real) == 

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\<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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real_abs_def: "abs (r::real) == (if 0 \<le> r then r else r)" 
81 

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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)) \<in> realrel) = (x1 + y2 = x2 + y1)" 
102 
by (simp add: realrel_def) 

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104 
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) 
14269  107 
done 
108 

14497  109 
text{*Reduces equality of equivalence classes to the @{term realrel} relation: 
110 
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *} 

14269  111 
lemmas equiv_realrel_iff = 
112 
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] 

113 

114 
declare equiv_realrel_iff [simp] 

115 

14497  116 

14484  117 
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" 
118 
by (simp add: Real_def realrel_def quotient_def, blast) 

14269  119 

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lemma inj_on_Abs_Real: "inj_on Abs_Real Real" 
14269  122 
apply (rule inj_on_inverseI) 
14484  123 
apply (erule Abs_Real_inverse) 
14269  124 
done 
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14484  126 
declare inj_on_Abs_Real [THEN inj_on_iff, simp] 
127 
declare Abs_Real_inverse [simp] 

14269  128 

129 

14484  130 
text{*Case analysis on the representation of a real number as an equivalence 
131 
class of pairs of positive reals.*} 

132 
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 

133 
"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" 

134 
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) 

135 
apply (drule arg_cong [where f=Abs_Real]) 

136 
apply (auto simp add: Rep_Real_inverse) 

14269  137 
done 
138 

139 

14329  140 
subsection{*Congruence property for addition*} 
14269  141 

142 
lemma real_add_congruent2_lemma: 

143 
"[a + ba = aa + b; ab + bc = ac + bb] 

144 
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" 

145 
apply (simp add: preal_add_assoc) 

146 
apply (rule preal_add_left_commute [of ab, THEN ssubst]) 

147 
apply (simp add: preal_add_assoc [symmetric]) 

148 
apply (simp add: preal_add_ac) 

149 
done 

150 

151 
lemma real_add: 

14497  152 
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = 
153 
Abs_Real (realrel``{(x+u, y+v)})" 

154 
proof  

14658  155 
have "congruent2 realrel realrel 
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(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)" 
157 
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 

158 
thus ?thesis 

159 
by (simp add: real_add_def UN_UN_split_split_eq 

14658  160 
UN_equiv_class2 [OF equiv_realrel equiv_realrel]) 
14497  161 
qed 
14269  162 

163 
lemma real_add_commute: "(z::real) + w = w + z" 

14497  164 
by (cases z, cases w, simp add: real_add preal_add_ac) 
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166 
lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)" 

14497  167 
by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc) 
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169 
lemma real_add_zero_left: "(0::real) + z = z" 

14497  170 
by (cases z, simp add: real_add real_zero_def preal_add_ac) 
14269  171 

14738  172 
instance real :: comm_monoid_add 
14269  173 
by (intro_classes, 
174 
(assumption  

175 
rule real_add_commute real_add_assoc real_add_zero_left)+) 

176 

177 

14334  178 
subsection{*Additive Inverse on real*} 
179 

14484  180 
lemma real_minus: " Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" 
181 
proof  

182 
have "congruent realrel (\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})})" 

183 
by (simp add: congruent_def preal_add_commute) 

184 
thus ?thesis 

185 
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) 

186 
qed 

14334  187 

188 
lemma real_add_minus_left: "(z) + z = (0::real)" 

14497  189 
by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute) 
14269  190 

191 

14329  192 
subsection{*Congruence property for multiplication*} 
14269  193 

14329  194 
lemma real_mult_congruent2_lemma: 
195 
"!!(x1::preal). [ x1 + y2 = x2 + y1 ] ==> 

14484  196 
x * x1 + y * y1 + (x * y2 + y * x2) = 
197 
x * x2 + y * y2 + (x * y1 + y * x1)" 

198 
apply (simp add: preal_add_left_commute preal_add_assoc [symmetric]) 

14269  199 
apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric]) 
200 
apply (simp add: preal_add_commute) 

201 
done 

202 

203 
lemma real_mult_congruent2: 

14658  204 
"congruent2 realrel realrel (%p1 p2. 
14484  205 
(%(x1,y1). (%(x2,y2). 
206 
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)" 

14658  207 
apply (rule congruent2_commuteI [OF equiv_realrel], clarify) 
14269  208 
apply (simp add: preal_mult_commute preal_add_commute) 
209 
apply (auto simp add: real_mult_congruent2_lemma) 

210 
done 

211 

212 
lemma real_mult: 

14484  213 
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = 
214 
Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" 

215 
by (simp add: real_mult_def UN_UN_split_split_eq 

14658  216 
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) 
14269  217 

218 
lemma real_mult_commute: "(z::real) * w = w * z" 

14497  219 
by (cases z, cases w, simp add: real_mult preal_add_ac preal_mult_ac) 
14269  220 

221 
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" 

14484  222 
apply (cases z1, cases z2, cases z3) 
223 
apply (simp add: real_mult preal_add_mult_distrib2 preal_add_ac preal_mult_ac) 

14269  224 
done 
225 

226 
lemma real_mult_1: "(1::real) * z = z" 

14484  227 
apply (cases z) 
228 
apply (simp add: real_mult real_one_def preal_add_mult_distrib2 

229 
preal_mult_1_right preal_mult_ac preal_add_ac) 

14269  230 
done 
231 

232 
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" 

14484  233 
apply (cases z1, cases z2, cases w) 
234 
apply (simp add: real_add real_mult preal_add_mult_distrib2 

235 
preal_add_ac preal_mult_ac) 

14269  236 
done 
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14329  238 
text{*one and zero are distinct*} 
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lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" 
14484  240 
proof  
241 
have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1" 

242 
by (simp add: preal_self_less_add_left) 

243 
thus ?thesis 

244 
by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff) 

245 
qed 

14269  246 

14329  247 
subsection{*existence of inverse*} 
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14484  249 
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" 
14497  250 
by (simp add: real_zero_def preal_add_commute) 
14269  251 

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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  256 
apply (simp add: real_zero_def real_one_def, cases x) 
14269  257 
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  259 
apply (rule_tac 
14484  260 
x = "Abs_Real (realrel `` { (preal_of_rat 1, 
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inverse (D) + preal_of_rat 1)}) " 
14334  262 
in exI) 
263 
apply (rule_tac [2] 

14484  264 
x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1, 
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preal_of_rat 1)})" 
14334  266 
in exI) 
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apply (auto simp add: real_mult preal_mult_1_right 
14329  268 
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  270 
done 
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lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" 
14484  273 
apply (simp add: real_inverse_def) 
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apply (frule real_mult_inverse_left_ex, safe) 
14269  275 
apply (rule someI2, auto) 
276 
done 

14334  277 

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subsection{*The Real Numbers form a Field*} 
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14334  281 
instance real :: field 
282 
proof 

283 
fix x y z :: real 

284 
show " x + x = 0" by (rule real_add_minus_left) 

285 
show "x  y = x + (y)" by (simp add: real_diff_def) 

286 
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) 

287 
show "x * y = y * x" by (rule real_mult_commute) 

288 
show "1 * x = x" by (rule real_mult_1) 

289 
show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) 

290 
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  293 
qed 
294 

295 

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text{*Inverse of zero! Useful to simplify certain equations*} 
14269  297 

14334  298 
lemma INVERSE_ZERO: "inverse 0 = (0::real)" 
14484  299 
by (simp add: real_inverse_def) 
14334  300 

301 
instance real :: division_by_zero 

302 
proof 

303 
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) 

304 
qed 

305 

306 

307 
(*Pull negations out*) 

308 
declare minus_mult_right [symmetric, simp] 

309 
minus_mult_left [symmetric, simp] 

310 

311 
lemma real_mult_1_right: "z * (1::real) = z" 

14738  312 
by (rule OrderedGroup.mult_1_right) 
14269  313 

314 

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subsection{*The @{text "\<le>"} Ordering*} 
14269  316 

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lemma real_le_refl: "w \<le> (w::real)" 
14484  318 
by (cases w, force simp add: real_le_def) 
14269  319 

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

320 
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:
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diff
changeset

321 
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

322 
following two lemmas.*} 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
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diff
changeset

323 
lemma preal_eq_le_imp_le: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

324 
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

325 
shows "b \<le> (d::preal)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

326 
proof  
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

327 
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

328 
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

329 
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

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

331 

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

332 
lemma real_le_lemma: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

333 
assumes l: "u1 + v2 \<le> u2 + v1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

334 
and "x1 + v1 = u1 + y1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

335 
and "x2 + v2 = u2 + y2" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

336 
shows "x1 + y2 \<le> x2 + (y1::preal)" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

337 
proof  
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

338 
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

339 
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

340 
also have "... \<le> (x2+y1) + (u2+v1)" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

341 
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

342 
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

343 
qed 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

344 

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

345 
lemma real_le: 
14484  346 
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) = 
347 
(x1 + y2 \<le> x2 + y1)" 

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

348 
apply (simp add: real_le_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

349 
apply (auto intro: real_le_lemma) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

350 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

351 

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

352 
lemma real_le_anti_sym: "[ z \<le> w; w \<le> z ] ==> z = (w::real)" 
14497  353 
by (cases z, cases w, simp add: real_le order_antisym) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

354 

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

355 
lemma real_trans_lemma: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

356 
assumes "x + v \<le> u + y" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

357 
and "u + v' \<le> u' + v" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

358 
and "x2 + v2 = u2 + y2" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

359 
shows "x + v' \<le> u' + (y::preal)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

360 
proof  
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

361 
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

362 
also have "... \<le> (u+y) + (u+v')" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

363 
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

364 
also have "... \<le> (u+y) + (u'+v)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

365 
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

366 
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

367 
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

368 
qed 
14269  369 

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

370 
lemma real_le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::real)" 
14484  371 
apply (cases i, cases j, cases k) 
372 
apply (simp add: real_le) 

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

373 
apply (blast intro: real_trans_lemma) 
14334  374 
done 
375 

376 
(* Axiom 'order_less_le' of class 'order': *) 

377 
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

378 
by (simp add: real_less_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

379 

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

380 
instance real :: order 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

381 
proof qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

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

383 
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

384 

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

385 
(* Axiom 'linorder_linear' of class 'linorder': *) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

386 
lemma real_le_linear: "(z::real) \<le> w  w \<le> z" 
14484  387 
apply (cases z, cases w) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

388 
apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels) 
14334  389 
done 
390 

391 

392 
instance real :: linorder 

393 
by (intro_classes, rule real_le_linear) 

394 

395 

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

396 
lemma real_le_eq_diff: "(x \<le> y) = (xy \<le> (0::real))" 
14484  397 
apply (cases x, cases y) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

398 
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

399 
preal_add_ac) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

400 
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

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

402 

14484  403 
lemma real_add_left_mono: 
404 
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)" 

405 
proof  

406 
have "z + x  (z + y) = (z + z) + (x  y)" 

407 
by (simp add: diff_minus add_ac) 

408 
with le show ?thesis 

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

409 
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) 
14484  410 
qed 
14334  411 

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

412 
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

413 
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

414 

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

415 
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

416 
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) 
14334  417 

418 
lemma real_mult_order: "[ 0 < x; 0 < y ] ==> (0::real) < x * y" 

14484  419 
apply (cases x, cases y) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

420 
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

421 
linorder_not_le [where 'a = preal] 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

422 
real_zero_def real_le real_mult) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

423 
{*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

424 
apply (auto dest!: less_add_left_Ex 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

425 
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

426 
preal_add_mult_distrib2 preal_cancels preal_self_less_add_right) 
14334  427 
done 
428 

429 
lemma real_mult_less_mono2: "[ (0::real) < z; x < y ] ==> z * x < z * y" 

430 
apply (rule real_sum_gt_zero_less) 

431 
apply (drule real_less_sum_gt_zero [of x y]) 

432 
apply (drule real_mult_order, assumption) 

433 
apply (simp add: right_distrib) 

434 
done 

435 

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

436 
text{*lemma for proving @{term "0<(1::real)"}*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

437 
lemma real_zero_le_one: "0 \<le> (1::real)" 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

438 
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

439 
preal_self_less_add_left order_less_imp_le) 
14334  440 

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

441 

14334  442 
subsection{*The Reals Form an Ordered Field*} 
443 

444 
instance real :: ordered_field 

445 
proof 

446 
fix x y z :: real 

447 
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) 

448 
show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) 

449 
show "\<bar>x\<bar> = (if x < 0 then x else x)" 

450 
by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) 

451 
qed 

452 

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

453 

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

454 

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

455 
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

456 
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

457 
positive reals.*} 
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 
lemma real_of_preal_add: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

460 
"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

461 
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

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

463 

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

464 
lemma real_of_preal_mult: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

465 
"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

466 
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

467 
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

468 

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

469 

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

470 
text{*Gleason prop 94.4 p 127*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

471 
lemma real_of_preal_trichotomy: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

472 
"\<exists>m. (x::real) = real_of_preal m  x = 0  x = (real_of_preal m)" 
14484  473 
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

474 
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

475 
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

476 
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

477 
apply (auto simp add: preal_add_commute) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

478 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

479 

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

480 
lemma real_of_preal_leD: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

481 
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" 
14484  482 
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

483 

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

484 
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

485 
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

486 

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

487 
lemma real_of_preal_lessD: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

488 
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" 
14484  489 
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric] 
490 
preal_cancels) 

491 

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

492 

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

493 
lemma real_of_preal_less_iff [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

494 
"(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

495 
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

496 

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

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

498 
"(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

499 
by (simp add: linorder_not_less [symmetric]) 
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_zero_less: "0 < real_of_preal m" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

502 
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

503 
preal_add_ac preal_cancels) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

504 
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

505 
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

506 
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

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

508 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

509 

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

510 
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

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

512 

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

513 
lemma real_of_preal_not_minus_gt_zero: "~ 0 <  real_of_preal m" 
14484  514 
proof  
515 
from real_of_preal_minus_less_zero 

516 
show ?thesis by (blast dest: order_less_trans) 

517 
qed 

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

518 

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

519 

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

520 
subsection{*Theorems About the Ordering*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

521 

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

522 
text{*obsolete but used a lot*} 
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 
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

525 
by blast 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

526 

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

527 
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

528 
by (simp add: order_le_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

529 

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

530 
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

531 
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

532 
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

533 
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

534 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

535 

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

536 
lemma real_gt_preal_preal_Ex: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

537 
"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

538 
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

539 
intro: real_gt_zero_preal_Ex [THEN iffD1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

540 

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

541 
lemma real_ge_preal_preal_Ex: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

542 
"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

543 
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

544 

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

545 
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

546 
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

547 
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

548 
simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

549 

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

550 
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

551 
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

552 

14334  553 
lemma real_add_less_le_mono: "[ w'<w; z'\<le>z ] ==> w' + z' < w + (z::real)" 
14738  554 
by (rule OrderedGroup.add_less_le_mono) 
14334  555 

556 
lemma real_add_le_less_mono: 

557 
"!!z z'::real. [ w'\<le>w; z'<z ] ==> w' + z' < w + z" 

14738  558 
by (rule OrderedGroup.add_le_less_mono) 
14334  559 

560 
lemma real_le_square [simp]: "(0::real) \<le> x*x" 

561 
by (rule Ring_and_Field.zero_le_square) 

562 

563 

564 
subsection{*More Lemmas*} 

565 

566 
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" 

567 
by auto 

568 

569 
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" 

570 
by auto 

571 

572 
text{*The precondition could be weakened to @{term "0\<le>x"}*} 

573 
lemma real_mult_less_mono: 

574 
"[ u<v; x<y; (0::real) < v; 0 < x ] ==> u*x < v* y" 

575 
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) 

576 

577 
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" 

578 
by (force elim: order_less_asym 

579 
simp add: Ring_and_Field.mult_less_cancel_right) 

580 

581 
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

582 
apply (simp add: mult_le_cancel_right) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

583 
apply (blast intro: elim: order_less_asym) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

584 
done 
14334  585 

586 
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" 

587 
by (force elim: order_less_asym 

588 
simp add: Ring_and_Field.mult_le_cancel_left) 

589 

590 
text{*Only two uses?*} 

591 
lemma real_mult_less_mono': 

592 
"[ x < y; r1 < r2; (0::real) \<le> r1; 0 \<le> x] ==> r1 * x < r2 * y" 

593 
by (rule Ring_and_Field.mult_strict_mono') 

594 

595 
text{*FIXME: delete or at least combine the next two lemmas*} 

596 
lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" 

14738  597 
apply (drule OrderedGroup.equals_zero_I [THEN sym]) 
14334  598 
apply (cut_tac x = y in real_le_square) 
14476  599 
apply (auto, drule order_antisym, auto) 
14334  600 
done 
601 

602 
lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" 

603 
apply (rule_tac y = x in real_sum_squares_cancel) 

14476  604 
apply (simp add: add_commute) 
14334  605 
done 
606 

607 
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

608 
by (drule add_strict_mono [of concl: 0 0], assumption, simp) 
14334  609 

610 
lemma real_le_add_order: "[ 0 \<le> x; 0 \<le> y ] ==> (0::real) \<le> x + y" 

611 
apply (drule order_le_imp_less_or_eq)+ 

612 
apply (auto intro: real_add_order order_less_imp_le) 

613 
done 

614 

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

615 
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

616 
apply (case_tac "x \<noteq> 0") 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

617 
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

618 
done 
14334  619 

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

620 
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

621 
by (auto dest: less_imp_inverse_less) 
14334  622 

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

623 
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

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

625 
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

626 
thus ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

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

628 

14334  629 

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

630 
subsection{*Embedding the Integers into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

631 

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

632 
defs (overloaded) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

633 
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

634 
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

635 

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

636 
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

637 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

638 

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

639 
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

640 
by (simp add: real_of_int_def) 
14334  641 

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

642 
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

643 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

644 
declare real_of_int_add [symmetric, simp] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

645 

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

646 
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

647 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

648 
declare real_of_int_minus [symmetric, simp] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

649 

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

650 
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

651 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

652 
declare real_of_int_diff [symmetric, simp] 
14334  653 

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

654 
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

655 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

656 
declare real_of_int_mult [symmetric, simp] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

657 

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

658 
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

659 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

660 

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

661 
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

662 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

663 

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

664 
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

665 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

666 

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

667 
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

668 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

669 

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

670 

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

671 
subsection{*Embedding the Naturals into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

672 

14334  673 
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

674 
by (simp add: real_of_nat_def) 
14334  675 

676 
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

677 
by (simp add: real_of_nat_def) 
14334  678 

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

679 
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

680 
by (simp add: real_of_nat_def) 
14334  681 

682 
(*Not for addsimps: often the LHS is used to represent a positive natural*) 

683 
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

684 
by (simp add: real_of_nat_def) 
14334  685 

686 
lemma real_of_nat_less_iff [iff]: 

687 
"(real (n::nat) < real m) = (n < m)" 

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

688 
by (simp add: real_of_nat_def) 
14334  689 

690 
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

691 
by (simp add: real_of_nat_def) 
14334  692 

693 
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

694 
by (simp add: real_of_nat_def zero_le_imp_of_nat) 
14334  695 

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

696 
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

697 
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

698 

14334  699 
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

700 
by (simp add: real_of_nat_def) 
14334  701 

702 
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

703 
by (simp add: real_of_nat_def) 
14334  704 

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

705 
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

706 
by (simp add: real_of_nat_def) 
14334  707 

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

708 
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

709 
by (simp add: add: real_of_nat_def) 
14334  710 

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

711 
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

712 
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

713 

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

714 
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

715 
by (simp add: add: real_of_nat_def) 
14334  716 

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

717 
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

718 
by (simp add: add: real_of_nat_def) 
14334  719 

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

720 
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

721 
by (simp add: add: real_of_nat_def) 
14334  722 

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

723 
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

724 
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

725 

14426  726 
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" 
727 
by (simp add: real_of_int_def real_of_nat_def) 

14334  728 

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

729 

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

730 

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

731 
subsection{*Numerals and Arithmetic*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

732 

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

733 
instance real :: number .. 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

734 

15013  735 
defs (overloaded) 
736 
real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)" 

737 
{*the type constraint is essential!*} 

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

738 

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

739 
instance real :: number_ring 
15013  740 
by (intro_classes, simp add: real_number_of_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

741 

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

742 

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

743 
text{*Collapse applications of @{term real} to @{term number_of}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

744 
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

745 
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

746 

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

747 
lemma real_of_nat_number_of [simp]: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

748 
"real (number_of v :: nat) = 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

749 
(if neg (number_of v :: int) then 0 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

750 
else (number_of v :: real))" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

751 
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

752 

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

753 

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

754 
use "real_arith.ML" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

755 

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

756 
setup real_arith_setup 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

757 

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

758 
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

759 

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

760 
text{*Needed in this nonstandard form by Hyperreal/Transcendental*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

761 
lemma real_0_le_divide_iff: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

762 
"((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

763 
by (simp add: real_divide_def zero_le_mult_iff, auto) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

764 

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

765 
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

766 
by arith 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

767 

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

768 
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

769 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

770 

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

771 
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

772 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

773 

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

774 
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

775 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

776 

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

777 
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

778 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

779 

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

780 
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

781 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

782 

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

783 

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

784 
(** Simprules combining xy and 0 (needed??) **) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

785 

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

786 
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

787 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

788 

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

789 
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

790 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

791 

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

792 
(* 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

793 
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

794 
It replaces x+y by xy. 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

795 
Addsimps [symmetric real_diff_def] 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

796 
*) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

797 

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

798 

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

799 
subsubsection{*Density of the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

800 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

801 
(*????FIXME: rename d1, d2 to x, y*) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

802 
lemma real_lbound_gt_zero: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

803 
"[ (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

804 
apply (rule_tac x = " (min d1 d2) /2" in exI) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

805 
apply (simp add: min_def) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

806 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

807 

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

808 

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

809 
text{*Similar results are proved in @{text Ring_and_Field}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

810 
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

811 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

812 

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

813 
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

814 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

815 

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

816 

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

817 
subsection{*Absolute Value Function for the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

818 

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

819 
text{*FIXME: these should go!*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

820 
lemma abs_eqI1: "(0::real)\<le>x ==> abs x = x" 
15003  821 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

822 

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

823 
lemma abs_eqI2: "(0::real) < x ==> abs x = x" 
15003  824 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

825 

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

826 
lemma abs_minus_eqI2: "x < (0::real) ==> abs x = x" 
15003  827 
by (simp add: abs_if linorder_not_less [symmetric]) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

828 

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

829 
lemma abs_minus_add_cancel: "abs(x + (y)) = abs (y + ((x::real)))" 
15003  830 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

831 

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

832 
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

833 
by (force simp add: Ring_and_Field.abs_less_iff) 
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 
lemma abs_le_interval_iff: "(abs x \<le> r) = (r\<le>x & x\<le>(r::real))" 
14738  836 
by (force simp add: OrderedGroup.abs_le_iff) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

837 

14484  838 
(*FIXME: used only once, in SEQ.ML*) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

839 
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" 
15003  840 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

841 

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

842 
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

843 
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

844 

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

845 
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

846 
apply (simp add: linorder_not_less) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

847 
apply (auto intro: abs_ge_self [THEN order_trans]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

848 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

849 

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

850 
text{*Used only in Hyperreal/Lim.ML*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

851 
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

852 
apply (simp add: real_add_assoc) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

853 
apply (rule_tac a1 = y in add_left_commute [THEN ssubst]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

854 
apply (rule real_add_assoc [THEN subst]) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

855 
apply (rule abs_triangle_ineq) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

856 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

857 

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

858 

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

859 

14334  860 
ML 
861 
{* 

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

862 
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

863 
val real_add_minus_iff = thm"real_add_minus_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

864 
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

865 
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

866 
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

867 
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

868 
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

869 
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

870 
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

871 
val real_lbound_gt_zero = thm"real_lbound_gt_zero"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

872 
val real_less_half_sum = thm"real_less_half_sum"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

873 
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

874 

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

875 
val abs_eqI1 = thm"abs_eqI1"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

876 
val abs_eqI2 = thm"abs_eqI2"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

877 
val abs_minus_eqI2 = thm"abs_minus_eqI2"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

878 
val abs_ge_zero = thm"abs_ge_zero"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

879 
val abs_idempotent = thm"abs_idempotent"; 
14738  880 
val abs_eq_0 = thm"abs_eq_0"; 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

881 
val abs_ge_self = thm"abs_ge_self"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

882 
val abs_ge_minus_self = thm"abs_ge_minus_self"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

883 
val abs_mult = thm"abs_mult"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

884 
val abs_inverse = thm"abs_inverse"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

885 
val abs_triangle_ineq = thm"abs_triangle_ineq"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

886 
val abs_minus_cancel = thm"abs_minus_cancel"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

887 
val abs_minus_add_cancel = thm"abs_minus_add_cancel"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

888 
val abs_interval_iff = thm"abs_interval_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

889 
val abs_le_interval_iff = thm"abs_le_interval_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

890 
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

891 
val abs_le_zero_iff = thm"abs_le_zero_iff"; 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

892 
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

893 
val abs_sum_triangle_ineq = thm"abs_sum_triangle_ineq"; 
14334  894 

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

895 
val abs_mult_less = thm"abs_mult_less"; 
14334  896 
*} 
10752
c4f1bf2acf4c
tidying, and separation of HOLHyperreal from HOLReal
paulson
parents:
10648
diff
changeset

897 

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

898 

5588  899 
end 