author  haftmann 
Fri, 20 Jul 2007 14:28:01 +0200  
changeset 23879  4776af8be741 
parent 23482  2f4be6844f7c 
child 24075  366d4d234814 
permissions  rwrr 
5588  1 
(* Title : Real/RealDef.thy 
7219  2 
ID : $Id$ 
5588  3 
Author : Jacques D. Fleuriot 
4 
Copyright : 1998 University of Cambridge 

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

5 
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 
16819  6 
Additional contributions by Jeremy Avigad 
14269  7 
*) 
8 

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

9 
header{*Defining the Reals from the Positive Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

10 

15131  11 
theory RealDef 
15140  12 
imports PReal 
16417  13 
uses ("real_arith.ML") 
15131  14 
begin 
5588  15 

19765  16 
definition 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20554
diff
changeset

17 
realrel :: "((preal * preal) * (preal * preal)) set" where 
19765  18 
"realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" 
14269  19 

14484  20 
typedef (Real) real = "UNIV//realrel" 
14269  21 
by (auto simp add: quotient_def) 
5588  22 

19765  23 
definition 
14484  24 
(** these don't use the overloaded "real" function: users don't see them **) 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
20554
diff
changeset

25 
real_of_preal :: "preal => real" where 
23288  26 
"real_of_preal m = Abs_Real(realrel``{(m + 1, 1)})" 
14484  27 

14269  28 
consts 
29 
(*overloaded constant for injecting other types into "real"*) 

30 
real :: "'a => real" 

5588  31 

23879  32 
instance real :: zero 
33 
real_zero_def: "0 == Abs_Real(realrel``{(1, 1)})" .. 

5588  34 

23879  35 
instance real :: one 
36 
real_one_def: "1 == Abs_Real(realrel``{(1 + 1, 1)})" .. 

5588  37 

23879  38 
instance real :: plus 
39 
real_add_def: "z + w == 

14484  40 
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 
23879  41 
{ Abs_Real(realrel``{(x+u, y+v)}) })" .. 
10606  42 

23879  43 
instance real :: minus 
44 
real_minus_def: " r == contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" 

45 
real_diff_def: "r  (s::real) == r +  s" .. 

14484  46 

23879  47 
instance real :: times 
14484  48 
real_mult_def: 
49 
"z * w == 

50 
contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 

23879  51 
{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" .. 
5588  52 

23879  53 
instance real :: inverse 
54 
real_inverse_def: "inverse (R::real) == (THE S. (R = 0 & S = 0)  S * R = 1)" 

55 
real_divide_def: "R / (S::real) == R * inverse S" .. 

14269  56 

23879  57 
instance real :: ord 
58 
real_le_def: "z \<le> (w::real) == 

14484  59 
\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w" 
23879  60 
real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)" .. 
5588  61 

23879  62 
instance real :: abs 
63 
real_abs_def: "abs (r::real) == (if r < 0 then  r else r)" .. 

14334  64 

65 

23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

66 
subsection {* Equivalence relation over positive reals *} 
14269  67 

14270  68 
lemma preal_trans_lemma: 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

69 
assumes "x + y1 = x1 + y" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

70 
and "x + y2 = x2 + y" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

71 
shows "x1 + y2 = x2 + (y1::preal)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

72 
proof  
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

73 
have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

74 
also have "... = (x2 + y) + x1" by (simp add: prems) 
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

75 
also have "... = x2 + (x1 + y)" by (simp add: add_ac) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

76 
also have "... = x2 + (x + y1)" by (simp add: prems) 
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

77 
also have "... = (x2 + y1) + x" by (simp add: add_ac) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

78 
finally have "(x1 + y2) + x = (x2 + y1) + x" . 
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

79 
thus ?thesis by (rule add_right_imp_eq) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

80 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

81 

14269  82 

14484  83 
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)" 
84 
by (simp add: realrel_def) 

14269  85 

86 
lemma equiv_realrel: "equiv UNIV realrel" 

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

87 
apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

88 
apply (blast dest: preal_trans_lemma) 
14269  89 
done 
90 

14497  91 
text{*Reduces equality of equivalence classes to the @{term realrel} relation: 
92 
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *} 

14269  93 
lemmas equiv_realrel_iff = 
94 
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] 

95 

96 
declare equiv_realrel_iff [simp] 

97 

14497  98 

14484  99 
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" 
100 
by (simp add: Real_def realrel_def quotient_def, blast) 

14269  101 

22958  102 
declare Abs_Real_inject [simp] 
14484  103 
declare Abs_Real_inverse [simp] 
14269  104 

105 

14484  106 
text{*Case analysis on the representation of a real number as an equivalence 
107 
class of pairs of positive reals.*} 

108 
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 

109 
"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" 

110 
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) 

111 
apply (drule arg_cong [where f=Abs_Real]) 

112 
apply (auto simp add: Rep_Real_inverse) 

14269  113 
done 
114 

115 

23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

116 
subsection {* Addition and Subtraction *} 
14269  117 

118 
lemma real_add_congruent2_lemma: 

119 
"[a + ba = aa + b; ab + bc = ac + bb] 

120 
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" 

23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

121 
apply (simp add: add_assoc) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

122 
apply (rule add_left_commute [of ab, THEN ssubst]) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

123 
apply (simp add: add_assoc [symmetric]) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

124 
apply (simp add: add_ac) 
14269  125 
done 
126 

127 
lemma real_add: 

14497  128 
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = 
129 
Abs_Real (realrel``{(x+u, y+v)})" 

130 
proof  

15169  131 
have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) 
132 
respects2 realrel" 

14497  133 
by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
134 
thus ?thesis 

135 
by (simp add: real_add_def UN_UN_split_split_eq 

14658  136 
UN_equiv_class2 [OF equiv_realrel equiv_realrel]) 
14497  137 
qed 
14269  138 

14484  139 
lemma real_minus: " Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" 
140 
proof  

15169  141 
have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" 
23288  142 
by (simp add: congruent_def add_commute) 
14484  143 
thus ?thesis 
144 
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) 

145 
qed 

14334  146 

23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

147 
instance real :: ab_group_add 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

148 
proof 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

149 
fix x y z :: real 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

150 
show "(x + y) + z = x + (y + z)" 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

151 
by (cases x, cases y, cases z, simp add: real_add add_assoc) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

152 
show "x + y = y + x" 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

153 
by (cases x, cases y, simp add: real_add add_commute) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

154 
show "0 + x = x" 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

155 
by (cases x, simp add: real_add real_zero_def add_ac) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

156 
show " x + x = 0" 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

157 
by (cases x, simp add: real_minus real_add real_zero_def add_commute) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

158 
show "x  y = x +  y" 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

159 
by (simp add: real_diff_def) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

160 
qed 
14269  161 

162 

23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

163 
subsection {* Multiplication *} 
14269  164 

14329  165 
lemma real_mult_congruent2_lemma: 
166 
"!!(x1::preal). [ x1 + y2 = x2 + y1 ] ==> 

14484  167 
x * x1 + y * y1 + (x * y2 + y * x2) = 
168 
x * x2 + y * y2 + (x * y1 + y * x1)" 

23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

169 
apply (simp add: add_left_commute add_assoc [symmetric]) 
23288  170 
apply (simp add: add_assoc right_distrib [symmetric]) 
171 
apply (simp add: add_commute) 

14269  172 
done 
173 

174 
lemma real_mult_congruent2: 

15169  175 
"(%p1 p2. 
14484  176 
(%(x1,y1). (%(x2,y2). 
15169  177 
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1) 
178 
respects2 realrel" 

14658  179 
apply (rule congruent2_commuteI [OF equiv_realrel], clarify) 
23288  180 
apply (simp add: mult_commute add_commute) 
14269  181 
apply (auto simp add: real_mult_congruent2_lemma) 
182 
done 

183 

184 
lemma real_mult: 

14484  185 
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = 
186 
Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" 

187 
by (simp add: real_mult_def UN_UN_split_split_eq 

14658  188 
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) 
14269  189 

190 
lemma real_mult_commute: "(z::real) * w = w * z" 

23288  191 
by (cases z, cases w, simp add: real_mult add_ac mult_ac) 
14269  192 

193 
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" 

14484  194 
apply (cases z1, cases z2, cases z3) 
23288  195 
apply (simp add: real_mult right_distrib add_ac mult_ac) 
14269  196 
done 
197 

198 
lemma real_mult_1: "(1::real) * z = z" 

14484  199 
apply (cases z) 
23288  200 
apply (simp add: real_mult real_one_def right_distrib 
201 
mult_1_right mult_ac add_ac) 

14269  202 
done 
203 

204 
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" 

14484  205 
apply (cases z1, cases z2, cases w) 
23288  206 
apply (simp add: real_add real_mult right_distrib add_ac mult_ac) 
14269  207 
done 
208 

14329  209 
text{*one and zero are distinct*} 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

210 
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" 
14484  211 
proof  
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

212 
have "(1::preal) < 1 + 1" 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

213 
by (simp add: preal_self_less_add_left) 
14484  214 
thus ?thesis 
23288  215 
by (simp add: real_zero_def real_one_def) 
14484  216 
qed 
14269  217 

23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

218 
instance real :: comm_ring_1 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

219 
proof 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

220 
fix x y z :: real 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

221 
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

222 
show "x * y = y * x" by (rule real_mult_commute) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

223 
show "1 * x = x" by (rule real_mult_1) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

224 
show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

225 
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

226 
qed 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

227 

063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

228 
subsection {* Inverse and Division *} 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

229 

14484  230 
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" 
23288  231 
by (simp add: real_zero_def add_commute) 
14269  232 

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

233 
text{*Instead of using an existential quantifier and constructing the inverse 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

234 
within the proof, we could define the inverse explicitly.*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

235 

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

236 
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)" 
14484  237 
apply (simp add: real_zero_def real_one_def, cases x) 
14269  238 
apply (cut_tac x = xa and y = y in linorder_less_linear) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

239 
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) 
14334  240 
apply (rule_tac 
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

241 
x = "Abs_Real (realrel``{(1, inverse (D) + 1)})" 
14334  242 
in exI) 
243 
apply (rule_tac [2] 

23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

244 
x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" 
14334  245 
in exI) 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23438
diff
changeset

246 
apply (auto simp add: real_mult preal_mult_inverse_right ring_simps) 
14269  247 
done 
248 

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

249 
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)" 
14484  250 
apply (simp add: real_inverse_def) 
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

251 
apply (drule real_mult_inverse_left_ex, safe) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

252 
apply (rule theI, assumption, rename_tac z) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

253 
apply (subgoal_tac "(z * x) * y = z * (x * y)") 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

254 
apply (simp add: mult_commute) 
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

255 
apply (rule mult_assoc) 
14269  256 
done 
14334  257 

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

258 

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

259 
subsection{*The Real Numbers form a Field*} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset

260 

14334  261 
instance real :: field 
262 
proof 

263 
fix x y z :: real 

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

264 
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14426
diff
changeset

265 
show "x / y = x * inverse y" by (simp add: real_divide_def) 
14334  266 
qed 
267 

268 

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

269 
text{*Inverse of zero! Useful to simplify certain equations*} 
14269  270 

14334  271 
lemma INVERSE_ZERO: "inverse 0 = (0::real)" 
14484  272 
by (simp add: real_inverse_def) 
14334  273 

274 
instance real :: division_by_zero 

275 
proof 

276 
show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) 

277 
qed 

278 

14269  279 

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

280 
subsection{*The @{text "\<le>"} Ordering*} 
14269  281 

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

282 
lemma real_le_refl: "w \<le> (w::real)" 
14484  283 
by (cases w, force simp add: real_le_def) 
14269  284 

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

285 
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

286 
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

287 
following two lemmas.*} 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

288 
lemma preal_eq_le_imp_le: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

289 
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

290 
shows "b \<le> (d::preal)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

291 
proof  
23288  292 
have "c+d \<le> a+d" by (simp add: prems) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

293 
hence "a+b \<le> a+d" by (simp add: prems) 
23288  294 
thus "b \<le> d" by simp 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

295 
qed 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

296 

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

297 
lemma real_le_lemma: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

298 
assumes l: "u1 + v2 \<le> u2 + v1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

299 
and "x1 + v1 = u1 + y1" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

300 
and "x2 + v2 = u2 + y2" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

301 
shows "x1 + y2 \<le> x2 + (y1::preal)" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

302 
proof  
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

303 
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems) 
23288  304 
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac) 
305 
also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems) 

306 
finally show ?thesis by simp 

307 
qed 

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

308 

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

309 
lemma real_le: 
14484  310 
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) = 
311 
(x1 + y2 \<le> x2 + y1)" 

23288  312 
apply (simp add: real_le_def) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

313 
apply (auto intro: real_le_lemma) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

314 
done 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

315 

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

316 
lemma real_le_anti_sym: "[ z \<le> w; w \<le> z ] ==> z = (w::real)" 
15542  317 
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

318 

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

319 
lemma real_trans_lemma: 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

320 
assumes "x + v \<le> u + y" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

321 
and "u + v' \<le> u' + v" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

322 
and "x2 + v2 = u2 + y2" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

323 
shows "x + v' \<le> u' + (y::preal)" 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

324 
proof  
23288  325 
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac) 
326 
also have "... \<le> (u+y) + (u+v')" by (simp add: prems) 

327 
also have "... \<le> (u+y) + (u'+v)" by (simp add: prems) 

328 
also have "... = (u'+y) + (u+v)" by (simp add: add_ac) 

329 
finally show ?thesis by simp 

15542  330 
qed 
14269  331 

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

332 
lemma real_le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::real)" 
14484  333 
apply (cases i, cases j, cases k) 
334 
apply (simp add: real_le) 

23288  335 
apply (blast intro: real_trans_lemma) 
14334  336 
done 
337 

338 
(* Axiom 'order_less_le' of class 'order': *) 

339 
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

340 
by (simp add: real_less_def) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

341 

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

342 
instance real :: order 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

343 
proof qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

344 
(assumption  
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

345 
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

346 

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

347 
(* Axiom 'linorder_linear' of class 'linorder': *) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

348 
lemma real_le_linear: "(z::real) \<le> w  w \<le> z" 
23288  349 
apply (cases z, cases w) 
350 
apply (auto simp add: real_le real_zero_def add_ac) 

14334  351 
done 
352 

353 

354 
instance real :: linorder 

355 
by (intro_classes, rule real_le_linear) 

356 

357 

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

358 
lemma real_le_eq_diff: "(x \<le> y) = (xy \<le> (0::real))" 
14484  359 
apply (cases x, cases y) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

360 
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus 
23288  361 
add_ac) 
362 
apply (simp_all add: add_assoc [symmetric]) 

15542  363 
done 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

364 

14484  365 
lemma real_add_left_mono: 
366 
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)" 

367 
proof  

368 
have "z + x  (z + y) = (z + z) + (x  y)" 

369 
by (simp add: diff_minus add_ac) 

370 
with le show ?thesis 

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

371 
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) 
14484  372 
qed 
14334  373 

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

374 
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

375 
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

376 

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

377 
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

378 
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) 
14334  379 

380 
lemma real_mult_order: "[ 0 < x; 0 < y ] ==> (0::real) < x * y" 

14484  381 
apply (cases x, cases y) 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

382 
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

383 
linorder_not_le [where 'a = preal] 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

384 
real_zero_def real_le real_mult) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

385 
{*Reduce to the (simpler) @{text "\<le>"} relation *} 
16973  386 
apply (auto dest!: less_add_left_Ex 
23288  387 
simp add: add_ac mult_ac 
388 
right_distrib preal_self_less_add_left) 

14334  389 
done 
390 

391 
lemma real_mult_less_mono2: "[ (0::real) < z; x < y ] ==> z * x < z * y" 

392 
apply (rule real_sum_gt_zero_less) 

393 
apply (drule real_less_sum_gt_zero [of x y]) 

394 
apply (drule real_mult_order, assumption) 

395 
apply (simp add: right_distrib) 

396 
done 

397 

22456  398 
instance real :: distrib_lattice 
399 
"inf x y \<equiv> min x y" 

400 
"sup x y \<equiv> max x y" 

401 
by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1) 

402 

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

403 

14334  404 
subsection{*The Reals Form an Ordered Field*} 
405 

406 
instance real :: ordered_field 

407 
proof 

408 
fix x y z :: real 

409 
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono) 

22962  410 
show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2) 
411 
show "\<bar>x\<bar> = (if x < 0 then x else x)" by (simp only: real_abs_def) 

14334  412 
qed 
413 

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

414 
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

415 
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

416 
positive reals.*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

417 

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

418 
lemma real_of_preal_add: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

419 
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" 
23288  420 
by (simp add: real_of_preal_def real_add left_distrib add_ac) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

421 

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

422 
lemma real_of_preal_mult: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

423 
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" 
23288  424 
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

425 

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

426 

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

427 
text{*Gleason prop 94.4 p 127*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

428 
lemma real_of_preal_trichotomy: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

429 
"\<exists>m. (x::real) = real_of_preal m  x = 0  x = (real_of_preal m)" 
14484  430 
apply (simp add: real_of_preal_def real_zero_def, cases x) 
23288  431 
apply (auto simp add: real_minus add_ac) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

432 
apply (cut_tac x = x and y = y in linorder_less_linear) 
23288  433 
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric]) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

434 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

435 

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

436 
lemma real_of_preal_leD: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

437 
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2" 
23288  438 
by (simp add: real_of_preal_def real_le) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

439 

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

440 
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

441 
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

442 

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

443 
lemma real_of_preal_lessD: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

444 
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2" 
23288  445 
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric]) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

446 

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

447 
lemma real_of_preal_less_iff [simp]: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

448 
"(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

449 
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

450 

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

451 
lemma real_of_preal_le_iff: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

452 
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" 
23288  453 
by (simp add: linorder_not_less [symmetric]) 
14365
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 
lemma real_of_preal_zero_less: "0 < real_of_preal m" 
23288  456 
apply (insert preal_self_less_add_left [of 1 m]) 
457 
apply (auto simp add: real_zero_def real_of_preal_def 

458 
real_less_def real_le_def add_ac) 

459 
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI) 

460 
apply (simp add: add_ac) 

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

461 
done 
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_minus_less_zero: " real_of_preal m < 0" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

464 
by (simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

465 

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

466 
lemma real_of_preal_not_minus_gt_zero: "~ 0 <  real_of_preal m" 
14484  467 
proof  
468 
from real_of_preal_minus_less_zero 

469 
show ?thesis by (blast dest: order_less_trans) 

470 
qed 

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

471 

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

474 

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

475 
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

476 
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

477 
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

478 
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

479 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

480 

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

481 
lemma real_gt_preal_preal_Ex: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

482 
"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

483 
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

484 
intro: real_gt_zero_preal_Ex [THEN iffD1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

485 

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

486 
lemma real_ge_preal_preal_Ex: 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

487 
"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

488 
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

489 

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

490 
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

491 
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

492 
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

493 
simp add: real_of_preal_zero_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

494 

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

495 
lemma real_less_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

496 
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

497 

14334  498 

499 
subsection{*More Lemmas*} 

500 

501 
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" 

502 
by auto 

503 

504 
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" 

505 
by auto 

506 

507 
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" 

508 
by (force elim: order_less_asym 

509 
simp add: Ring_and_Field.mult_less_cancel_right) 

510 

511 
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

512 
apply (simp add: mult_le_cancel_right) 
23289  513 
apply (blast intro: elim: order_less_asym) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

514 
done 
14334  515 

516 
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" 

15923  517 
by(simp add:mult_commute) 
14334  518 

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

519 
lemma real_inverse_gt_one: "[ (0::real) < x; x < 1 ] ==> 1 < inverse x" 
23289  520 
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *) 
14334  521 

522 

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

523 
subsection{*Embedding the Integers into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

524 

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

525 
defs (overloaded) 
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

526 
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

527 
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

528 

16819  529 
lemma real_eq_of_nat: "real = of_nat" 
530 
apply (rule ext) 

531 
apply (unfold real_of_nat_def) 

532 
apply (rule refl) 

533 
done 

534 

535 
lemma real_eq_of_int: "real = of_int" 

536 
apply (rule ext) 

537 
apply (unfold real_of_int_def) 

538 
apply (rule refl) 

539 
done 

540 

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

541 
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

542 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

543 

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

544 
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

545 
by (simp add: real_of_int_def) 
14334  546 

16819  547 
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

548 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

549 

16819  550 
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

551 
by (simp add: real_of_int_def) 
16819  552 

553 
lemma real_of_int_diff [simp]: "real(x  y) = real (x::int)  real y" 

554 
by (simp add: real_of_int_def) 

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

555 

16819  556 
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

557 
by (simp add: real_of_int_def) 
14334  558 

16819  559 
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" 
560 
apply (subst real_eq_of_int)+ 

561 
apply (rule of_int_setsum) 

562 
done 

563 

564 
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 

565 
(PROD x:A. real(f x))" 

566 
apply (subst real_eq_of_int)+ 

567 
apply (rule of_int_setprod) 

568 
done 

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

569 

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

570 
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

571 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

572 

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

573 
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

574 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

575 

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

576 
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

577 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

578 

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

579 
lemma real_of_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

580 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

581 

16819  582 
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)" 
583 
by (simp add: real_of_int_def) 

584 

585 
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)" 

586 
by (simp add: real_of_int_def) 

587 

588 
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)" 

589 
by (simp add: real_of_int_def) 

590 

591 
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)" 

592 
by (simp add: real_of_int_def) 

593 

16888  594 
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))" 
595 
by (auto simp add: abs_if) 

596 

16819  597 
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" 
598 
apply (subgoal_tac "real n + 1 = real (n + 1)") 

599 
apply (simp del: real_of_int_add) 

600 
apply auto 

601 
done 

602 

603 
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" 

604 
apply (subgoal_tac "real m + 1 = real (m + 1)") 

605 
apply (simp del: real_of_int_add) 

606 
apply simp 

607 
done 

608 

609 
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 

610 
real (x div d) + (real (x mod d)) / (real d)" 

611 
proof  

612 
assume "d ~= 0" 

613 
have "x = (x div d) * d + x mod d" 

614 
by auto 

615 
then have "real x = real (x div d) * real d + real(x mod d)" 

616 
by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) 

617 
then have "real x / real d = ... / real d" 

618 
by simp 

619 
then show ?thesis 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23438
diff
changeset

620 
by (auto simp add: add_divide_distrib ring_simps prems) 
16819  621 
qed 
622 

623 
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> 

624 
real(n div d) = real n / real d" 

625 
apply (frule real_of_int_div_aux [of d n]) 

626 
apply simp 

627 
apply (simp add: zdvd_iff_zmod_eq_0) 

628 
done 

629 

630 
lemma real_of_int_div2: 

631 
"0 <= real (n::int) / real (x)  real (n div x)" 

632 
apply (case_tac "x = 0") 

633 
apply simp 

634 
apply (case_tac "0 < x") 

635 
apply (simp add: compare_rls) 

636 
apply (subst real_of_int_div_aux) 

637 
apply simp 

638 
apply simp 

639 
apply (subst zero_le_divide_iff) 

640 
apply auto 

641 
apply (simp add: compare_rls) 

642 
apply (subst real_of_int_div_aux) 

643 
apply simp 

644 
apply simp 

645 
apply (subst zero_le_divide_iff) 

646 
apply auto 

647 
done 

648 

649 
lemma real_of_int_div3: 

650 
"real (n::int) / real (x)  real (n div x) <= 1" 

651 
apply(case_tac "x = 0") 

652 
apply simp 

653 
apply (simp add: compare_rls) 

654 
apply (subst real_of_int_div_aux) 

655 
apply assumption 

656 
apply simp 

657 
apply (subst divide_le_eq) 

658 
apply clarsimp 

659 
apply (rule conjI) 

660 
apply (rule impI) 

661 
apply (rule order_less_imp_le) 

662 
apply simp 

663 
apply (rule impI) 

664 
apply (rule order_less_imp_le) 

665 
apply simp 

666 
done 

667 

668 
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 

669 
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

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" 
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23289
diff
changeset

700 
by (simp add: real_of_nat_def of_nat_mult) 
14334  701 

16819  702 
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
703 
(SUM x:A. real(f x))" 

704 
apply (subst real_eq_of_nat)+ 

705 
apply (rule of_nat_setsum) 

706 
done 

707 

708 
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 

709 
(PROD x:A. real(f x))" 

710 
apply (subst real_eq_of_nat)+ 

711 
apply (rule of_nat_setprod) 

712 
done 

713 

714 
lemma real_of_card: "real (card A) = setsum (%x.1) A" 

715 
apply (subst card_eq_setsum) 

716 
apply (subst real_of_nat_setsum) 

717 
apply simp 

718 
done 

719 

14334  720 
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

721 
by (simp add: real_of_nat_def) 
14334  722 

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

723 
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

724 
by (simp add: real_of_nat_def) 
14334  725 

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

726 
lemma real_of_nat_diff: "n \<le> m ==> real (m  n) = real (m::nat)  real n" 
23438
dd824e86fa8a
remove simp attribute from of_nat_diff, for backward compatibility with zdiff_int
huffman
parents:
23431
diff
changeset

727 
by (simp add: add: real_of_nat_def of_nat_diff) 
14334  728 

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

729 
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

730 
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

731 

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

732 
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

733 
by (simp add: add: real_of_nat_def) 
14334  734 

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

735 
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

736 
by (simp add: add: real_of_nat_def) 
14334  737 

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

738 
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

739 
by (simp add: add: real_of_nat_def) 
14334  740 

16819  741 
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)" 
742 
apply (subgoal_tac "real n + 1 = real (Suc n)") 

743 
apply simp 

744 
apply (auto simp add: real_of_nat_Suc) 

745 
done 

746 

747 
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)" 

748 
apply (subgoal_tac "real m + 1 = real (Suc m)") 

749 
apply (simp add: less_Suc_eq_le) 

750 
apply (simp add: real_of_nat_Suc) 

751 
done 

752 

753 
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 

754 
real (x div d) + (real (x mod d)) / (real d)" 

755 
proof  

756 
assume "0 < d" 

757 
have "x = (x div d) * d + x mod d" 

758 
by auto 

759 
then have "real x = real (x div d) * real d + real(x mod d)" 

760 
by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym]) 

761 
then have "real x / real d = \<dots> / real d" 

762 
by simp 

763 
then show ?thesis 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23438
diff
changeset

764 
by (auto simp add: add_divide_distrib ring_simps prems) 
16819  765 
qed 
766 

767 
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==> 

768 
real(n div d) = real n / real d" 

769 
apply (frule real_of_nat_div_aux [of d n]) 

770 
apply simp 

771 
apply (subst dvd_eq_mod_eq_0 [THEN sym]) 

772 
apply assumption 

773 
done 

774 

775 
lemma real_of_nat_div2: 

776 
"0 <= real (n::nat) / real (x)  real (n div x)" 

777 
apply(case_tac "x = 0") 

778 
apply simp 

779 
apply (simp add: compare_rls) 

780 
apply (subst real_of_nat_div_aux) 

781 
apply assumption 

782 
apply simp 

783 
apply (subst zero_le_divide_iff) 

784 
apply simp 

785 
done 

786 

787 
lemma real_of_nat_div3: 

788 
"real (n::nat) / real (x)  real (n div x) <= 1" 

789 
apply(case_tac "x = 0") 

790 
apply simp 

791 
apply (simp add: compare_rls) 

792 
apply (subst real_of_nat_div_aux) 

793 
apply assumption 

794 
apply simp 

795 
done 

796 

797 
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 

798 
by (insert real_of_nat_div2 [of n x], simp) 

799 

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

800 
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

801 
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

802 

14426  803 
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" 
804 
by (simp add: real_of_int_def real_of_nat_def) 

14334  805 

16819  806 
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x" 
807 
apply (subgoal_tac "real(int(nat x)) = real(nat x)") 

808 
apply force 

809 
apply (simp only: real_of_int_real_of_nat) 

810 
done 

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

811 

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

812 
subsection{*Numerals and Arithmetic*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

813 

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

814 
instance real :: number .. 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

815 

15013  816 
defs (overloaded) 
20485  817 
real_number_of_def: "(number_of w :: real) == of_int w" 
15013  818 
{*the type constraint is essential!*} 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

819 

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

820 
instance real :: number_ring 
15013  821 
by (intro_classes, simp add: real_number_of_def) 
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 
text{*Collapse applications of @{term real} to @{term number_of}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

824 
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

825 
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

826 

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

827 
lemma real_of_nat_number_of [simp]: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

828 
"real (number_of v :: nat) = 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

829 
(if neg (number_of v :: int) then 0 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

830 
else (number_of v :: real))" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

831 
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

832 

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

833 

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

834 
use "real_arith.ML" 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

835 

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

836 
setup real_arith_setup 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

837 

19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
16973
diff
changeset

838 

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

839 
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

840 

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

841 
text{*Needed in this nonstandard form by Hyperreal/Transcendental*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

842 
lemma real_0_le_divide_iff: 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

843 
"((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

844 
by (simp add: real_divide_def zero_le_mult_iff, auto) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

845 

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

846 
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

847 
by arith 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

848 

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

849 
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

850 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

851 

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

852 
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

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

854 

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

855 
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

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

857 

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

858 
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

859 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

860 

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

861 
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

862 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

863 

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

864 

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

865 
(* 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

866 
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

867 
It replaces x+y by xy. 
15086  868 
declare real_diff_def [symmetric, simp] 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

869 
*) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

870 

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

871 

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

872 
subsubsection{*Density of the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

873 

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

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

875 
"[ (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

876 
apply (rule_tac x = " (min d1 d2) /2" in exI) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

877 
apply (simp add: min_def) 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

878 
done 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

879 

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

880 

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

881 
text{*Similar results are proved in @{text Ring_and_Field}*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

882 
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

883 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

884 

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

885 
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

886 
by auto 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

887 

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

888 

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

889 
subsection{*Absolute Value Function for the Reals*} 
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

890 

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

891 
lemma abs_minus_add_cancel: "abs(x + (y)) = abs (y + ((x::real)))" 
15003  892 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

893 

23289  894 
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

895 
lemma abs_le_interval_iff: "(abs x \<le> r) = (r\<le>x & x\<le>(r::real))" 
14738  896 
by (force simp add: OrderedGroup.abs_le_iff) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

897 

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

898 
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" 
15003  899 
by (simp add: abs_if) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

900 

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

901 
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" 
22958  902 
by (rule abs_of_nonneg [OF real_of_nat_ge_zero]) 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

903 

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

904 
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x" 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19765
diff
changeset

905 
by simp 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

906 

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

907 
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (l + m)) \<le> abs(x + l) + abs(y + m)" 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19765
diff
changeset

908 
by simp 
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset

909 

23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

910 
subsection{*Code generation using Isabelle's rats*} 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

911 

9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

912 
types_code 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

913 
real ("Rat.rat") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

914 
attach (term_of) {* 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

915 
fun term_of_real x = 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

916 
let 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

917 
val rT = HOLogic.realT 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

918 
val (p, q) = Rat.quotient_of_rat x 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

919 
in if q = 1 then HOLogic.mk_number rT p 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

920 
else Const("HOL.divide",[rT,rT] > rT) $ 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

921 
(HOLogic.mk_number rT p) $ (HOLogic.mk_number rT q) 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

922 
end; 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

923 
*} 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

924 
attach (test) {* 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

925 
fun gen_real i = 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

926 
let val p = random_range 0 i; val q = random_range 0 i; 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

927 
val r = if q=0 then Rat.rat_of_int i else Rat.rat_of_quotient(p,q) 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

928 
in if one_of [true,false] then r else Rat.neg r end; 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

929 
*} 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

930 

9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

931 
consts_code 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

932 
"0 :: real" ("Rat.zero") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

933 
"1 :: real" ("Rat.one") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

934 
"uminus :: real \<Rightarrow> real" ("Rat.neg") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

935 
"op + :: real \<Rightarrow> real \<Rightarrow> real" ("Rat.add") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

936 
"op * :: real \<Rightarrow> real \<Rightarrow> real" ("Rat.mult") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

937 
"inverse :: real \<Rightarrow> real" ("Rat.inv") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

938 
"op \<le> :: real \<Rightarrow> real \<Rightarrow> bool" ("Rat.le") 
23879  939 
"op < :: real \<Rightarrow> real \<Rightarrow> bool" ("Rat.lt") 
23031
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

940 
"op = :: real \<Rightarrow> real \<Rightarrow> bool" ("curry Rat.eq") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

941 
"real :: int \<Rightarrow> real" ("Rat.rat'_of'_int") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

942 
"real :: nat \<Rightarrow> real" ("(Rat.rat'_of'_int o {*int*})") 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

943 

9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

944 
lemma [code, code unfold]: 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

945 
"number_of k = real (number_of k :: int)" 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

946 
by simp 
9da9585c816e
added code generation based on Isabelle's rat type.
nipkow
parents:
22970
diff
changeset

947 

5588  948 
end 