huffman@36787
|
1 |
(* Title: HOL/ex/Dedekind_Real.thy
|
huffman@36787
|
2 |
Author: Jacques D. Fleuriot, University of Cambridge
|
huffman@36788
|
3 |
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
|
huffman@36787
|
4 |
|
huffman@36787
|
5 |
The positive reals as Dedekind sections of positive
|
huffman@36787
|
6 |
rationals. Fundamentals of Abstract Analysis [Gleason- p. 121]
|
huffman@36787
|
7 |
provides some of the definitions.
|
huffman@36787
|
8 |
*)
|
huffman@36787
|
9 |
|
huffman@36787
|
10 |
theory Dedekind_Real
|
wenzelm@54510
|
11 |
imports Complex_Main
|
huffman@36787
|
12 |
begin
|
huffman@36787
|
13 |
|
huffman@36787
|
14 |
section {* Positive real numbers *}
|
huffman@36787
|
15 |
|
huffman@36787
|
16 |
text{*Could be generalized and moved to @{text Groups}*}
|
huffman@36787
|
17 |
lemma add_eq_exists: "\<exists>x. a+x = (b::rat)"
|
huffman@36787
|
18 |
by (rule_tac x="b-a" in exI, simp)
|
huffman@36787
|
19 |
|
huffman@36787
|
20 |
definition
|
huffman@36787
|
21 |
cut :: "rat set => bool" where
|
haftmann@37765
|
22 |
"cut A = ({} \<subset> A &
|
huffman@36787
|
23 |
A < {r. 0 < r} &
|
huffman@36787
|
24 |
(\<forall>y \<in> A. ((\<forall>z. 0<z & z < y --> z \<in> A) & (\<exists>u \<in> A. y < u))))"
|
huffman@36787
|
25 |
|
huffman@36787
|
26 |
lemma interval_empty_iff:
|
hoelzl@54352
|
27 |
"{y. (x::'a::unbounded_dense_linorder) < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
|
huffman@36787
|
28 |
by (auto dest: dense)
|
huffman@36787
|
29 |
|
huffman@36787
|
30 |
|
huffman@36787
|
31 |
lemma cut_of_rat:
|
huffman@36787
|
32 |
assumes q: "0 < q" shows "cut {r::rat. 0 < r & r < q}" (is "cut ?A")
|
huffman@36787
|
33 |
proof -
|
huffman@36787
|
34 |
from q have pos: "?A < {r. 0 < r}" by force
|
huffman@36787
|
35 |
have nonempty: "{} \<subset> ?A"
|
huffman@36787
|
36 |
proof
|
huffman@36787
|
37 |
show "{} \<subseteq> ?A" by simp
|
huffman@36787
|
38 |
show "{} \<noteq> ?A"
|
huffman@36787
|
39 |
by (force simp only: q eq_commute [of "{}"] interval_empty_iff)
|
huffman@36787
|
40 |
qed
|
huffman@36787
|
41 |
show ?thesis
|
huffman@36787
|
42 |
by (simp add: cut_def pos nonempty,
|
huffman@36787
|
43 |
blast dest: dense intro: order_less_trans)
|
huffman@36787
|
44 |
qed
|
huffman@36787
|
45 |
|
huffman@36787
|
46 |
|
wenzelm@46567
|
47 |
definition "preal = {A. cut A}"
|
wenzelm@46567
|
48 |
|
wenzelm@50849
|
49 |
typedef preal = preal
|
wenzelm@46567
|
50 |
unfolding preal_def by (blast intro: cut_of_rat [OF zero_less_one])
|
huffman@36787
|
51 |
|
huffman@36787
|
52 |
definition
|
huffman@36787
|
53 |
psup :: "preal set => preal" where
|
haftmann@37765
|
54 |
"psup P = Abs_preal (\<Union>X \<in> P. Rep_preal X)"
|
huffman@36787
|
55 |
|
huffman@36787
|
56 |
definition
|
huffman@36787
|
57 |
add_set :: "[rat set,rat set] => rat set" where
|
huffman@36787
|
58 |
"add_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}"
|
huffman@36787
|
59 |
|
huffman@36787
|
60 |
definition
|
huffman@36787
|
61 |
diff_set :: "[rat set,rat set] => rat set" where
|
haftmann@37765
|
62 |
"diff_set A B = {w. \<exists>x. 0 < w & 0 < x & x \<notin> B & x + w \<in> A}"
|
huffman@36787
|
63 |
|
huffman@36787
|
64 |
definition
|
huffman@36787
|
65 |
mult_set :: "[rat set,rat set] => rat set" where
|
huffman@36787
|
66 |
"mult_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}"
|
huffman@36787
|
67 |
|
huffman@36787
|
68 |
definition
|
huffman@36787
|
69 |
inverse_set :: "rat set => rat set" where
|
haftmann@37765
|
70 |
"inverse_set A = {x. \<exists>y. 0 < x & x < y & inverse y \<notin> A}"
|
huffman@36787
|
71 |
|
huffman@36787
|
72 |
instantiation preal :: "{ord, plus, minus, times, inverse, one}"
|
huffman@36787
|
73 |
begin
|
huffman@36787
|
74 |
|
huffman@36787
|
75 |
definition
|
haftmann@37765
|
76 |
preal_less_def:
|
huffman@36787
|
77 |
"R < S == Rep_preal R < Rep_preal S"
|
huffman@36787
|
78 |
|
huffman@36787
|
79 |
definition
|
haftmann@37765
|
80 |
preal_le_def:
|
huffman@36787
|
81 |
"R \<le> S == Rep_preal R \<subseteq> Rep_preal S"
|
huffman@36787
|
82 |
|
huffman@36787
|
83 |
definition
|
huffman@36787
|
84 |
preal_add_def:
|
huffman@36787
|
85 |
"R + S == Abs_preal (add_set (Rep_preal R) (Rep_preal S))"
|
huffman@36787
|
86 |
|
huffman@36787
|
87 |
definition
|
huffman@36787
|
88 |
preal_diff_def:
|
huffman@36787
|
89 |
"R - S == Abs_preal (diff_set (Rep_preal R) (Rep_preal S))"
|
huffman@36787
|
90 |
|
huffman@36787
|
91 |
definition
|
huffman@36787
|
92 |
preal_mult_def:
|
huffman@36787
|
93 |
"R * S == Abs_preal (mult_set (Rep_preal R) (Rep_preal S))"
|
huffman@36787
|
94 |
|
huffman@36787
|
95 |
definition
|
huffman@36787
|
96 |
preal_inverse_def:
|
huffman@36787
|
97 |
"inverse R == Abs_preal (inverse_set (Rep_preal R))"
|
huffman@36787
|
98 |
|
huffman@36787
|
99 |
definition "R / S = R * inverse (S\<Colon>preal)"
|
huffman@36787
|
100 |
|
huffman@36787
|
101 |
definition
|
huffman@36787
|
102 |
preal_one_def:
|
huffman@36787
|
103 |
"1 == Abs_preal {x. 0 < x & x < 1}"
|
huffman@36787
|
104 |
|
huffman@36787
|
105 |
instance ..
|
huffman@36787
|
106 |
|
huffman@36787
|
107 |
end
|
huffman@36787
|
108 |
|
huffman@36787
|
109 |
|
huffman@36787
|
110 |
text{*Reduces equality on abstractions to equality on representatives*}
|
huffman@36787
|
111 |
declare Abs_preal_inject [simp]
|
huffman@36787
|
112 |
declare Abs_preal_inverse [simp]
|
huffman@36787
|
113 |
|
huffman@36787
|
114 |
lemma rat_mem_preal: "0 < q ==> {r::rat. 0 < r & r < q} \<in> preal"
|
huffman@36787
|
115 |
by (simp add: preal_def cut_of_rat)
|
huffman@36787
|
116 |
|
huffman@36787
|
117 |
lemma preal_nonempty: "A \<in> preal ==> \<exists>x\<in>A. 0 < x"
|
wenzelm@47776
|
118 |
unfolding preal_def cut_def [abs_def] by blast
|
huffman@36787
|
119 |
|
huffman@36787
|
120 |
lemma preal_Ex_mem: "A \<in> preal \<Longrightarrow> \<exists>x. x \<in> A"
|
wenzelm@46567
|
121 |
apply (drule preal_nonempty)
|
wenzelm@46567
|
122 |
apply fast
|
wenzelm@46567
|
123 |
done
|
huffman@36787
|
124 |
|
huffman@36787
|
125 |
lemma preal_imp_psubset_positives: "A \<in> preal ==> A < {r. 0 < r}"
|
wenzelm@46567
|
126 |
by (force simp add: preal_def cut_def)
|
huffman@36787
|
127 |
|
huffman@36787
|
128 |
lemma preal_exists_bound: "A \<in> preal ==> \<exists>x. 0 < x & x \<notin> A"
|
wenzelm@46567
|
129 |
apply (drule preal_imp_psubset_positives)
|
wenzelm@46567
|
130 |
apply auto
|
wenzelm@46567
|
131 |
done
|
huffman@36787
|
132 |
|
huffman@36787
|
133 |
lemma preal_exists_greater: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u"
|
wenzelm@47776
|
134 |
unfolding preal_def cut_def [abs_def] by blast
|
huffman@36787
|
135 |
|
huffman@36787
|
136 |
lemma preal_downwards_closed: "[| A \<in> preal; y \<in> A; 0 < z; z < y |] ==> z \<in> A"
|
wenzelm@47776
|
137 |
unfolding preal_def cut_def [abs_def] by blast
|
huffman@36787
|
138 |
|
huffman@36787
|
139 |
text{*Relaxing the final premise*}
|
huffman@36787
|
140 |
lemma preal_downwards_closed':
|
huffman@36787
|
141 |
"[| A \<in> preal; y \<in> A; 0 < z; z \<le> y |] ==> z \<in> A"
|
huffman@36787
|
142 |
apply (simp add: order_le_less)
|
huffman@36787
|
143 |
apply (blast intro: preal_downwards_closed)
|
huffman@36787
|
144 |
done
|
huffman@36787
|
145 |
|
huffman@36787
|
146 |
text{*A positive fraction not in a positive real is an upper bound.
|
huffman@36787
|
147 |
Gleason p. 122 - Remark (1)*}
|
huffman@36787
|
148 |
|
huffman@36787
|
149 |
lemma not_in_preal_ub:
|
huffman@36787
|
150 |
assumes A: "A \<in> preal"
|
huffman@36787
|
151 |
and notx: "x \<notin> A"
|
huffman@36787
|
152 |
and y: "y \<in> A"
|
huffman@36787
|
153 |
and pos: "0 < x"
|
huffman@36787
|
154 |
shows "y < x"
|
huffman@36787
|
155 |
proof (cases rule: linorder_cases)
|
huffman@36787
|
156 |
assume "x<y"
|
huffman@36787
|
157 |
with notx show ?thesis
|
huffman@36787
|
158 |
by (simp add: preal_downwards_closed [OF A y] pos)
|
huffman@36787
|
159 |
next
|
huffman@36787
|
160 |
assume "x=y"
|
huffman@36787
|
161 |
with notx and y show ?thesis by simp
|
huffman@36787
|
162 |
next
|
huffman@36787
|
163 |
assume "y<x"
|
huffman@36787
|
164 |
thus ?thesis .
|
huffman@36787
|
165 |
qed
|
huffman@36787
|
166 |
|
huffman@36787
|
167 |
text {* preal lemmas instantiated to @{term "Rep_preal X"} *}
|
huffman@36787
|
168 |
|
huffman@36787
|
169 |
lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X"
|
huffman@36787
|
170 |
by (rule preal_Ex_mem [OF Rep_preal])
|
huffman@36787
|
171 |
|
huffman@36787
|
172 |
lemma Rep_preal_exists_bound: "\<exists>x>0. x \<notin> Rep_preal X"
|
huffman@36787
|
173 |
by (rule preal_exists_bound [OF Rep_preal])
|
huffman@36787
|
174 |
|
huffman@36787
|
175 |
lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF Rep_preal]
|
huffman@36787
|
176 |
|
huffman@36787
|
177 |
|
huffman@36787
|
178 |
subsection{*Properties of Ordering*}
|
huffman@36787
|
179 |
|
huffman@36787
|
180 |
instance preal :: order
|
huffman@36787
|
181 |
proof
|
huffman@36787
|
182 |
fix w :: preal
|
huffman@36787
|
183 |
show "w \<le> w" by (simp add: preal_le_def)
|
huffman@36787
|
184 |
next
|
huffman@36787
|
185 |
fix i j k :: preal
|
huffman@36787
|
186 |
assume "i \<le> j" and "j \<le> k"
|
huffman@36787
|
187 |
then show "i \<le> k" by (simp add: preal_le_def)
|
huffman@36787
|
188 |
next
|
huffman@36787
|
189 |
fix z w :: preal
|
huffman@36787
|
190 |
assume "z \<le> w" and "w \<le> z"
|
huffman@36787
|
191 |
then show "z = w" by (simp add: preal_le_def Rep_preal_inject)
|
huffman@36787
|
192 |
next
|
huffman@36787
|
193 |
fix z w :: preal
|
huffman@36787
|
194 |
show "z < w \<longleftrightarrow> z \<le> w \<and> \<not> w \<le> z"
|
huffman@36787
|
195 |
by (auto simp add: preal_le_def preal_less_def Rep_preal_inject)
|
huffman@36787
|
196 |
qed
|
huffman@36787
|
197 |
|
huffman@36787
|
198 |
lemma preal_imp_pos: "[|A \<in> preal; r \<in> A|] ==> 0 < r"
|
huffman@36787
|
199 |
by (insert preal_imp_psubset_positives, blast)
|
huffman@36787
|
200 |
|
huffman@36787
|
201 |
instance preal :: linorder
|
huffman@36787
|
202 |
proof
|
huffman@36787
|
203 |
fix x y :: preal
|
huffman@36787
|
204 |
show "x <= y | y <= x"
|
huffman@36787
|
205 |
apply (auto simp add: preal_le_def)
|
huffman@36787
|
206 |
apply (rule ccontr)
|
huffman@36787
|
207 |
apply (blast dest: not_in_Rep_preal_ub intro: preal_imp_pos [OF Rep_preal]
|
huffman@36787
|
208 |
elim: order_less_asym)
|
huffman@36787
|
209 |
done
|
huffman@36787
|
210 |
qed
|
huffman@36787
|
211 |
|
huffman@36787
|
212 |
instantiation preal :: distrib_lattice
|
huffman@36787
|
213 |
begin
|
huffman@36787
|
214 |
|
huffman@36787
|
215 |
definition
|
huffman@36787
|
216 |
"(inf \<Colon> preal \<Rightarrow> preal \<Rightarrow> preal) = min"
|
huffman@36787
|
217 |
|
huffman@36787
|
218 |
definition
|
huffman@36787
|
219 |
"(sup \<Colon> preal \<Rightarrow> preal \<Rightarrow> preal) = max"
|
huffman@36787
|
220 |
|
huffman@36787
|
221 |
instance
|
huffman@36787
|
222 |
by intro_classes
|
huffman@36787
|
223 |
(auto simp add: inf_preal_def sup_preal_def min_max.sup_inf_distrib1)
|
huffman@36787
|
224 |
|
huffman@36787
|
225 |
end
|
huffman@36787
|
226 |
|
huffman@36787
|
227 |
subsection{*Properties of Addition*}
|
huffman@36787
|
228 |
|
huffman@36787
|
229 |
lemma preal_add_commute: "(x::preal) + y = y + x"
|
huffman@36787
|
230 |
apply (unfold preal_add_def add_set_def)
|
huffman@36787
|
231 |
apply (rule_tac f = Abs_preal in arg_cong)
|
huffman@36787
|
232 |
apply (force simp add: add_commute)
|
huffman@36787
|
233 |
done
|
huffman@36787
|
234 |
|
huffman@36787
|
235 |
text{*Lemmas for proving that addition of two positive reals gives
|
huffman@36787
|
236 |
a positive real*}
|
huffman@36787
|
237 |
|
huffman@36787
|
238 |
text{*Part 1 of Dedekind sections definition*}
|
huffman@36787
|
239 |
lemma add_set_not_empty:
|
huffman@36787
|
240 |
"[|A \<in> preal; B \<in> preal|] ==> {} \<subset> add_set A B"
|
huffman@36787
|
241 |
apply (drule preal_nonempty)+
|
huffman@36787
|
242 |
apply (auto simp add: add_set_def)
|
huffman@36787
|
243 |
done
|
huffman@36787
|
244 |
|
huffman@36787
|
245 |
text{*Part 2 of Dedekind sections definition. A structured version of
|
huffman@36787
|
246 |
this proof is @{text preal_not_mem_mult_set_Ex} below.*}
|
huffman@36787
|
247 |
lemma preal_not_mem_add_set_Ex:
|
huffman@36787
|
248 |
"[|A \<in> preal; B \<in> preal|] ==> \<exists>q>0. q \<notin> add_set A B"
|
huffman@36787
|
249 |
apply (insert preal_exists_bound [of A] preal_exists_bound [of B], auto)
|
huffman@36787
|
250 |
apply (rule_tac x = "x+xa" in exI)
|
huffman@36787
|
251 |
apply (simp add: add_set_def, clarify)
|
huffman@36787
|
252 |
apply (drule (3) not_in_preal_ub)+
|
huffman@36787
|
253 |
apply (force dest: add_strict_mono)
|
huffman@36787
|
254 |
done
|
huffman@36787
|
255 |
|
huffman@36787
|
256 |
lemma add_set_not_rat_set:
|
huffman@36787
|
257 |
assumes A: "A \<in> preal"
|
huffman@36787
|
258 |
and B: "B \<in> preal"
|
huffman@36787
|
259 |
shows "add_set A B < {r. 0 < r}"
|
huffman@36787
|
260 |
proof
|
huffman@36787
|
261 |
from preal_imp_pos [OF A] preal_imp_pos [OF B]
|
huffman@36787
|
262 |
show "add_set A B \<subseteq> {r. 0 < r}" by (force simp add: add_set_def)
|
huffman@36787
|
263 |
next
|
huffman@36787
|
264 |
show "add_set A B \<noteq> {r. 0 < r}"
|
huffman@36787
|
265 |
by (insert preal_not_mem_add_set_Ex [OF A B], blast)
|
huffman@36787
|
266 |
qed
|
huffman@36787
|
267 |
|
huffman@36787
|
268 |
text{*Part 3 of Dedekind sections definition*}
|
huffman@36787
|
269 |
lemma add_set_lemma3:
|
huffman@36787
|
270 |
"[|A \<in> preal; B \<in> preal; u \<in> add_set A B; 0 < z; z < u|]
|
huffman@36787
|
271 |
==> z \<in> add_set A B"
|
huffman@36787
|
272 |
proof (unfold add_set_def, clarify)
|
huffman@36787
|
273 |
fix x::rat and y::rat
|
huffman@36787
|
274 |
assume A: "A \<in> preal"
|
huffman@36787
|
275 |
and B: "B \<in> preal"
|
huffman@36787
|
276 |
and [simp]: "0 < z"
|
huffman@36787
|
277 |
and zless: "z < x + y"
|
huffman@36787
|
278 |
and x: "x \<in> A"
|
huffman@36787
|
279 |
and y: "y \<in> B"
|
huffman@36787
|
280 |
have xpos [simp]: "0<x" by (rule preal_imp_pos [OF A x])
|
huffman@36787
|
281 |
have ypos [simp]: "0<y" by (rule preal_imp_pos [OF B y])
|
huffman@36787
|
282 |
have xypos [simp]: "0 < x+y" by (simp add: pos_add_strict)
|
huffman@36787
|
283 |
let ?f = "z/(x+y)"
|
huffman@36787
|
284 |
have fless: "?f < 1" by (simp add: zless pos_divide_less_eq)
|
huffman@36787
|
285 |
show "\<exists>x' \<in> A. \<exists>y'\<in>B. z = x' + y'"
|
huffman@36787
|
286 |
proof (intro bexI)
|
huffman@36787
|
287 |
show "z = x*?f + y*?f"
|
webertj@50977
|
288 |
by (simp add: distrib_right [symmetric] divide_inverse mult_ac
|
huffman@36787
|
289 |
order_less_imp_not_eq2)
|
huffman@36787
|
290 |
next
|
huffman@36787
|
291 |
show "y * ?f \<in> B"
|
huffman@36787
|
292 |
proof (rule preal_downwards_closed [OF B y])
|
huffman@36787
|
293 |
show "0 < y * ?f"
|
huffman@36787
|
294 |
by (simp add: divide_inverse zero_less_mult_iff)
|
huffman@36787
|
295 |
next
|
huffman@36787
|
296 |
show "y * ?f < y"
|
huffman@36787
|
297 |
by (insert mult_strict_left_mono [OF fless ypos], simp)
|
huffman@36787
|
298 |
qed
|
huffman@36787
|
299 |
next
|
huffman@36787
|
300 |
show "x * ?f \<in> A"
|
huffman@36787
|
301 |
proof (rule preal_downwards_closed [OF A x])
|
huffman@36787
|
302 |
show "0 < x * ?f"
|
huffman@36787
|
303 |
by (simp add: divide_inverse zero_less_mult_iff)
|
huffman@36787
|
304 |
next
|
huffman@36787
|
305 |
show "x * ?f < x"
|
huffman@36787
|
306 |
by (insert mult_strict_left_mono [OF fless xpos], simp)
|
huffman@36787
|
307 |
qed
|
huffman@36787
|
308 |
qed
|
huffman@36787
|
309 |
qed
|
huffman@36787
|
310 |
|
huffman@36787
|
311 |
text{*Part 4 of Dedekind sections definition*}
|
huffman@36787
|
312 |
lemma add_set_lemma4:
|
huffman@36787
|
313 |
"[|A \<in> preal; B \<in> preal; y \<in> add_set A B|] ==> \<exists>u \<in> add_set A B. y < u"
|
huffman@36787
|
314 |
apply (auto simp add: add_set_def)
|
huffman@36787
|
315 |
apply (frule preal_exists_greater [of A], auto)
|
huffman@36787
|
316 |
apply (rule_tac x="u + y" in exI)
|
huffman@36787
|
317 |
apply (auto intro: add_strict_left_mono)
|
huffman@36787
|
318 |
done
|
huffman@36787
|
319 |
|
huffman@36787
|
320 |
lemma mem_add_set:
|
huffman@36787
|
321 |
"[|A \<in> preal; B \<in> preal|] ==> add_set A B \<in> preal"
|
huffman@36787
|
322 |
apply (simp (no_asm_simp) add: preal_def cut_def)
|
huffman@36787
|
323 |
apply (blast intro!: add_set_not_empty add_set_not_rat_set
|
huffman@36787
|
324 |
add_set_lemma3 add_set_lemma4)
|
huffman@36787
|
325 |
done
|
huffman@36787
|
326 |
|
huffman@36787
|
327 |
lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
|
huffman@36787
|
328 |
apply (simp add: preal_add_def mem_add_set Rep_preal)
|
huffman@36787
|
329 |
apply (force simp add: add_set_def add_ac)
|
huffman@36787
|
330 |
done
|
huffman@36787
|
331 |
|
huffman@36787
|
332 |
instance preal :: ab_semigroup_add
|
huffman@36787
|
333 |
proof
|
huffman@36787
|
334 |
fix a b c :: preal
|
huffman@36787
|
335 |
show "(a + b) + c = a + (b + c)" by (rule preal_add_assoc)
|
huffman@36787
|
336 |
show "a + b = b + a" by (rule preal_add_commute)
|
huffman@36787
|
337 |
qed
|
huffman@36787
|
338 |
|
huffman@36787
|
339 |
|
huffman@36787
|
340 |
subsection{*Properties of Multiplication*}
|
huffman@36787
|
341 |
|
huffman@36787
|
342 |
text{*Proofs essentially same as for addition*}
|
huffman@36787
|
343 |
|
huffman@36787
|
344 |
lemma preal_mult_commute: "(x::preal) * y = y * x"
|
huffman@36787
|
345 |
apply (unfold preal_mult_def mult_set_def)
|
huffman@36787
|
346 |
apply (rule_tac f = Abs_preal in arg_cong)
|
huffman@36787
|
347 |
apply (force simp add: mult_commute)
|
huffman@36787
|
348 |
done
|
huffman@36787
|
349 |
|
huffman@36787
|
350 |
text{*Multiplication of two positive reals gives a positive real.*}
|
huffman@36787
|
351 |
|
huffman@36787
|
352 |
text{*Lemmas for proving positive reals multiplication set in @{typ preal}*}
|
huffman@36787
|
353 |
|
huffman@36787
|
354 |
text{*Part 1 of Dedekind sections definition*}
|
huffman@36787
|
355 |
lemma mult_set_not_empty:
|
huffman@36787
|
356 |
"[|A \<in> preal; B \<in> preal|] ==> {} \<subset> mult_set A B"
|
huffman@36787
|
357 |
apply (insert preal_nonempty [of A] preal_nonempty [of B])
|
huffman@36787
|
358 |
apply (auto simp add: mult_set_def)
|
huffman@36787
|
359 |
done
|
huffman@36787
|
360 |
|
huffman@36787
|
361 |
text{*Part 2 of Dedekind sections definition*}
|
huffman@36787
|
362 |
lemma preal_not_mem_mult_set_Ex:
|
wenzelm@41789
|
363 |
assumes A: "A \<in> preal"
|
wenzelm@41789
|
364 |
and B: "B \<in> preal"
|
wenzelm@41789
|
365 |
shows "\<exists>q. 0 < q & q \<notin> mult_set A B"
|
huffman@36787
|
366 |
proof -
|
wenzelm@41789
|
367 |
from preal_exists_bound [OF A] obtain x where 1 [simp]: "0 < x" "x \<notin> A" by blast
|
wenzelm@41789
|
368 |
from preal_exists_bound [OF B] obtain y where 2 [simp]: "0 < y" "y \<notin> B" by blast
|
huffman@36787
|
369 |
show ?thesis
|
huffman@36787
|
370 |
proof (intro exI conjI)
|
huffman@36787
|
371 |
show "0 < x*y" by (simp add: mult_pos_pos)
|
huffman@36787
|
372 |
show "x * y \<notin> mult_set A B"
|
huffman@36787
|
373 |
proof -
|
wenzelm@41789
|
374 |
{
|
wenzelm@41789
|
375 |
fix u::rat and v::rat
|
wenzelm@41789
|
376 |
assume u: "u \<in> A" and v: "v \<in> B" and xy: "x*y = u*v"
|
wenzelm@41789
|
377 |
moreover from A B 1 2 u v have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+
|
wenzelm@41789
|
378 |
moreover
|
wenzelm@41789
|
379 |
from A B 1 2 u v have "0\<le>v"
|
wenzelm@41789
|
380 |
by (blast intro: preal_imp_pos [OF B] order_less_imp_le)
|
wenzelm@41789
|
381 |
moreover
|
wenzelm@41789
|
382 |
from A B 1 `u < x` `v < y` `0 \<le> v`
|
wenzelm@41789
|
383 |
have "u*v < x*y" by (blast intro: mult_strict_mono)
|
wenzelm@41789
|
384 |
ultimately have False by force
|
wenzelm@41789
|
385 |
}
|
huffman@36787
|
386 |
thus ?thesis by (auto simp add: mult_set_def)
|
huffman@36787
|
387 |
qed
|
huffman@36787
|
388 |
qed
|
huffman@36787
|
389 |
qed
|
huffman@36787
|
390 |
|
huffman@36787
|
391 |
lemma mult_set_not_rat_set:
|
huffman@36787
|
392 |
assumes A: "A \<in> preal"
|
huffman@36787
|
393 |
and B: "B \<in> preal"
|
huffman@36787
|
394 |
shows "mult_set A B < {r. 0 < r}"
|
huffman@36787
|
395 |
proof
|
huffman@36787
|
396 |
show "mult_set A B \<subseteq> {r. 0 < r}"
|
huffman@36787
|
397 |
by (force simp add: mult_set_def
|
huffman@36787
|
398 |
intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos_pos)
|
huffman@36787
|
399 |
show "mult_set A B \<noteq> {r. 0 < r}"
|
huffman@36787
|
400 |
using preal_not_mem_mult_set_Ex [OF A B] by blast
|
huffman@36787
|
401 |
qed
|
huffman@36787
|
402 |
|
huffman@36787
|
403 |
|
huffman@36787
|
404 |
|
huffman@36787
|
405 |
text{*Part 3 of Dedekind sections definition*}
|
huffman@36787
|
406 |
lemma mult_set_lemma3:
|
huffman@36787
|
407 |
"[|A \<in> preal; B \<in> preal; u \<in> mult_set A B; 0 < z; z < u|]
|
huffman@36787
|
408 |
==> z \<in> mult_set A B"
|
huffman@36787
|
409 |
proof (unfold mult_set_def, clarify)
|
huffman@36787
|
410 |
fix x::rat and y::rat
|
huffman@36787
|
411 |
assume A: "A \<in> preal"
|
huffman@36787
|
412 |
and B: "B \<in> preal"
|
huffman@36787
|
413 |
and [simp]: "0 < z"
|
huffman@36787
|
414 |
and zless: "z < x * y"
|
huffman@36787
|
415 |
and x: "x \<in> A"
|
huffman@36787
|
416 |
and y: "y \<in> B"
|
huffman@36787
|
417 |
have [simp]: "0<y" by (rule preal_imp_pos [OF B y])
|
huffman@36787
|
418 |
show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'"
|
huffman@36787
|
419 |
proof
|
huffman@36787
|
420 |
show "\<exists>y'\<in>B. z = (z/y) * y'"
|
huffman@36787
|
421 |
proof
|
huffman@36787
|
422 |
show "z = (z/y)*y"
|
huffman@36787
|
423 |
by (simp add: divide_inverse mult_commute [of y] mult_assoc
|
huffman@36787
|
424 |
order_less_imp_not_eq2)
|
huffman@36787
|
425 |
show "y \<in> B" by fact
|
huffman@36787
|
426 |
qed
|
huffman@36787
|
427 |
next
|
huffman@36787
|
428 |
show "z/y \<in> A"
|
huffman@36787
|
429 |
proof (rule preal_downwards_closed [OF A x])
|
huffman@36787
|
430 |
show "0 < z/y"
|
huffman@36787
|
431 |
by (simp add: zero_less_divide_iff)
|
huffman@36787
|
432 |
show "z/y < x" by (simp add: pos_divide_less_eq zless)
|
huffman@36787
|
433 |
qed
|
huffman@36787
|
434 |
qed
|
huffman@36787
|
435 |
qed
|
huffman@36787
|
436 |
|
huffman@36787
|
437 |
text{*Part 4 of Dedekind sections definition*}
|
huffman@36787
|
438 |
lemma mult_set_lemma4:
|
huffman@36787
|
439 |
"[|A \<in> preal; B \<in> preal; y \<in> mult_set A B|] ==> \<exists>u \<in> mult_set A B. y < u"
|
huffman@36787
|
440 |
apply (auto simp add: mult_set_def)
|
huffman@36787
|
441 |
apply (frule preal_exists_greater [of A], auto)
|
huffman@36787
|
442 |
apply (rule_tac x="u * y" in exI)
|
huffman@36787
|
443 |
apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B]
|
huffman@36787
|
444 |
mult_strict_right_mono)
|
huffman@36787
|
445 |
done
|
huffman@36787
|
446 |
|
huffman@36787
|
447 |
|
huffman@36787
|
448 |
lemma mem_mult_set:
|
huffman@36787
|
449 |
"[|A \<in> preal; B \<in> preal|] ==> mult_set A B \<in> preal"
|
huffman@36787
|
450 |
apply (simp (no_asm_simp) add: preal_def cut_def)
|
huffman@36787
|
451 |
apply (blast intro!: mult_set_not_empty mult_set_not_rat_set
|
huffman@36787
|
452 |
mult_set_lemma3 mult_set_lemma4)
|
huffman@36787
|
453 |
done
|
huffman@36787
|
454 |
|
huffman@36787
|
455 |
lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"
|
huffman@36787
|
456 |
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
|
huffman@36787
|
457 |
apply (force simp add: mult_set_def mult_ac)
|
huffman@36787
|
458 |
done
|
huffman@36787
|
459 |
|
huffman@36787
|
460 |
instance preal :: ab_semigroup_mult
|
huffman@36787
|
461 |
proof
|
huffman@36787
|
462 |
fix a b c :: preal
|
huffman@36787
|
463 |
show "(a * b) * c = a * (b * c)" by (rule preal_mult_assoc)
|
huffman@36787
|
464 |
show "a * b = b * a" by (rule preal_mult_commute)
|
huffman@36787
|
465 |
qed
|
huffman@36787
|
466 |
|
huffman@36787
|
467 |
|
huffman@36787
|
468 |
text{* Positive real 1 is the multiplicative identity element *}
|
huffman@36787
|
469 |
|
huffman@36787
|
470 |
lemma preal_mult_1: "(1::preal) * z = z"
|
huffman@36787
|
471 |
proof (induct z)
|
huffman@36787
|
472 |
fix A :: "rat set"
|
huffman@36787
|
473 |
assume A: "A \<in> preal"
|
huffman@36787
|
474 |
have "{w. \<exists>u. 0 < u \<and> u < 1 & (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A")
|
huffman@36787
|
475 |
proof
|
huffman@36787
|
476 |
show "?lhs \<subseteq> A"
|
huffman@36787
|
477 |
proof clarify
|
huffman@36787
|
478 |
fix x::rat and u::rat and v::rat
|
huffman@36787
|
479 |
assume upos: "0<u" and "u<1" and v: "v \<in> A"
|
huffman@36787
|
480 |
have vpos: "0<v" by (rule preal_imp_pos [OF A v])
|
wenzelm@41789
|
481 |
hence "u*v < 1*v" by (simp only: mult_strict_right_mono upos `u < 1` v)
|
huffman@36787
|
482 |
thus "u * v \<in> A"
|
huffman@36787
|
483 |
by (force intro: preal_downwards_closed [OF A v] mult_pos_pos
|
huffman@36787
|
484 |
upos vpos)
|
huffman@36787
|
485 |
qed
|
huffman@36787
|
486 |
next
|
huffman@36787
|
487 |
show "A \<subseteq> ?lhs"
|
huffman@36787
|
488 |
proof clarify
|
huffman@36787
|
489 |
fix x::rat
|
huffman@36787
|
490 |
assume x: "x \<in> A"
|
huffman@36787
|
491 |
have xpos: "0<x" by (rule preal_imp_pos [OF A x])
|
huffman@36787
|
492 |
from preal_exists_greater [OF A x]
|
huffman@36787
|
493 |
obtain v where v: "v \<in> A" and xlessv: "x < v" ..
|
huffman@36787
|
494 |
have vpos: "0<v" by (rule preal_imp_pos [OF A v])
|
huffman@36787
|
495 |
show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. x = u * v)"
|
huffman@36787
|
496 |
proof (intro exI conjI)
|
huffman@36787
|
497 |
show "0 < x/v"
|
huffman@36787
|
498 |
by (simp add: zero_less_divide_iff xpos vpos)
|
huffman@36787
|
499 |
show "x / v < 1"
|
huffman@36787
|
500 |
by (simp add: pos_divide_less_eq vpos xlessv)
|
huffman@36787
|
501 |
show "\<exists>v'\<in>A. x = (x / v) * v'"
|
huffman@36787
|
502 |
proof
|
huffman@36787
|
503 |
show "x = (x/v)*v"
|
huffman@36787
|
504 |
by (simp add: divide_inverse mult_assoc vpos
|
huffman@36787
|
505 |
order_less_imp_not_eq2)
|
huffman@36787
|
506 |
show "v \<in> A" by fact
|
huffman@36787
|
507 |
qed
|
huffman@36787
|
508 |
qed
|
huffman@36787
|
509 |
qed
|
huffman@36787
|
510 |
qed
|
huffman@36787
|
511 |
thus "1 * Abs_preal A = Abs_preal A"
|
huffman@36787
|
512 |
by (simp add: preal_one_def preal_mult_def mult_set_def
|
huffman@36787
|
513 |
rat_mem_preal A)
|
huffman@36787
|
514 |
qed
|
huffman@36787
|
515 |
|
huffman@36787
|
516 |
instance preal :: comm_monoid_mult
|
huffman@36787
|
517 |
by intro_classes (rule preal_mult_1)
|
huffman@36787
|
518 |
|
huffman@36787
|
519 |
|
huffman@36787
|
520 |
subsection{*Distribution of Multiplication across Addition*}
|
huffman@36787
|
521 |
|
huffman@36787
|
522 |
lemma mem_Rep_preal_add_iff:
|
huffman@36787
|
523 |
"(z \<in> Rep_preal(R+S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x + y)"
|
huffman@36787
|
524 |
apply (simp add: preal_add_def mem_add_set Rep_preal)
|
huffman@36787
|
525 |
apply (simp add: add_set_def)
|
huffman@36787
|
526 |
done
|
huffman@36787
|
527 |
|
huffman@36787
|
528 |
lemma mem_Rep_preal_mult_iff:
|
huffman@36787
|
529 |
"(z \<in> Rep_preal(R*S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x * y)"
|
huffman@36787
|
530 |
apply (simp add: preal_mult_def mem_mult_set Rep_preal)
|
huffman@36787
|
531 |
apply (simp add: mult_set_def)
|
huffman@36787
|
532 |
done
|
huffman@36787
|
533 |
|
huffman@36787
|
534 |
lemma distrib_subset1:
|
huffman@36787
|
535 |
"Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)"
|
huffman@36787
|
536 |
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
|
webertj@50977
|
537 |
apply (force simp add: distrib_left)
|
huffman@36787
|
538 |
done
|
huffman@36787
|
539 |
|
huffman@36787
|
540 |
lemma preal_add_mult_distrib_mean:
|
huffman@36787
|
541 |
assumes a: "a \<in> Rep_preal w"
|
huffman@36787
|
542 |
and b: "b \<in> Rep_preal w"
|
huffman@36787
|
543 |
and d: "d \<in> Rep_preal x"
|
huffman@36787
|
544 |
and e: "e \<in> Rep_preal y"
|
huffman@36787
|
545 |
shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)"
|
huffman@36787
|
546 |
proof
|
huffman@36787
|
547 |
let ?c = "(a*d + b*e)/(d+e)"
|
huffman@36787
|
548 |
have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e"
|
huffman@36787
|
549 |
by (blast intro: preal_imp_pos [OF Rep_preal] a b d e pos_add_strict)+
|
huffman@36787
|
550 |
have cpos: "0 < ?c"
|
huffman@36787
|
551 |
by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict)
|
huffman@36787
|
552 |
show "a * d + b * e = ?c * (d + e)"
|
huffman@36787
|
553 |
by (simp add: divide_inverse mult_assoc order_less_imp_not_eq2)
|
huffman@36787
|
554 |
show "?c \<in> Rep_preal w"
|
huffman@36787
|
555 |
proof (cases rule: linorder_le_cases)
|
huffman@36787
|
556 |
assume "a \<le> b"
|
huffman@36787
|
557 |
hence "?c \<le> b"
|
webertj@50977
|
558 |
by (simp add: pos_divide_le_eq distrib_left mult_right_mono
|
huffman@36787
|
559 |
order_less_imp_le)
|
huffman@36787
|
560 |
thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos])
|
huffman@36787
|
561 |
next
|
huffman@36787
|
562 |
assume "b \<le> a"
|
huffman@36787
|
563 |
hence "?c \<le> a"
|
webertj@50977
|
564 |
by (simp add: pos_divide_le_eq distrib_left mult_right_mono
|
huffman@36787
|
565 |
order_less_imp_le)
|
huffman@36787
|
566 |
thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal a cpos])
|
huffman@36787
|
567 |
qed
|
huffman@36787
|
568 |
qed
|
huffman@36787
|
569 |
|
huffman@36787
|
570 |
lemma distrib_subset2:
|
huffman@36787
|
571 |
"Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))"
|
huffman@36787
|
572 |
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
|
huffman@36787
|
573 |
apply (drule_tac w=w and x=x and y=y in preal_add_mult_distrib_mean, auto)
|
huffman@36787
|
574 |
done
|
huffman@36787
|
575 |
|
huffman@36787
|
576 |
lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)"
|
huffman@36787
|
577 |
apply (rule Rep_preal_inject [THEN iffD1])
|
huffman@36787
|
578 |
apply (rule equalityI [OF distrib_subset1 distrib_subset2])
|
huffman@36787
|
579 |
done
|
huffman@36787
|
580 |
|
huffman@36787
|
581 |
lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)"
|
huffman@36787
|
582 |
by (simp add: preal_mult_commute preal_add_mult_distrib2)
|
huffman@36787
|
583 |
|
huffman@36787
|
584 |
instance preal :: comm_semiring
|
huffman@36787
|
585 |
by intro_classes (rule preal_add_mult_distrib)
|
huffman@36787
|
586 |
|
huffman@36787
|
587 |
|
huffman@36787
|
588 |
subsection{*Existence of Inverse, a Positive Real*}
|
huffman@36787
|
589 |
|
huffman@36787
|
590 |
lemma mem_inv_set_ex:
|
huffman@36787
|
591 |
assumes A: "A \<in> preal" shows "\<exists>x y. 0 < x & x < y & inverse y \<notin> A"
|
huffman@36787
|
592 |
proof -
|
huffman@36787
|
593 |
from preal_exists_bound [OF A]
|
huffman@36787
|
594 |
obtain x where [simp]: "0<x" "x \<notin> A" by blast
|
huffman@36787
|
595 |
show ?thesis
|
huffman@36787
|
596 |
proof (intro exI conjI)
|
huffman@36787
|
597 |
show "0 < inverse (x+1)"
|
huffman@36787
|
598 |
by (simp add: order_less_trans [OF _ less_add_one])
|
huffman@36787
|
599 |
show "inverse(x+1) < inverse x"
|
huffman@36787
|
600 |
by (simp add: less_imp_inverse_less less_add_one)
|
huffman@36787
|
601 |
show "inverse (inverse x) \<notin> A"
|
huffman@36787
|
602 |
by (simp add: order_less_imp_not_eq2)
|
huffman@36787
|
603 |
qed
|
huffman@36787
|
604 |
qed
|
huffman@36787
|
605 |
|
huffman@36787
|
606 |
text{*Part 1 of Dedekind sections definition*}
|
huffman@36787
|
607 |
lemma inverse_set_not_empty:
|
huffman@36787
|
608 |
"A \<in> preal ==> {} \<subset> inverse_set A"
|
huffman@36787
|
609 |
apply (insert mem_inv_set_ex [of A])
|
huffman@36787
|
610 |
apply (auto simp add: inverse_set_def)
|
huffman@36787
|
611 |
done
|
huffman@36787
|
612 |
|
huffman@36787
|
613 |
text{*Part 2 of Dedekind sections definition*}
|
huffman@36787
|
614 |
|
huffman@36787
|
615 |
lemma preal_not_mem_inverse_set_Ex:
|
huffman@36787
|
616 |
assumes A: "A \<in> preal" shows "\<exists>q. 0 < q & q \<notin> inverse_set A"
|
huffman@36787
|
617 |
proof -
|
huffman@36787
|
618 |
from preal_nonempty [OF A]
|
huffman@36787
|
619 |
obtain x where x: "x \<in> A" and xpos [simp]: "0<x" ..
|
huffman@36787
|
620 |
show ?thesis
|
huffman@36787
|
621 |
proof (intro exI conjI)
|
huffman@36787
|
622 |
show "0 < inverse x" by simp
|
huffman@36787
|
623 |
show "inverse x \<notin> inverse_set A"
|
huffman@36787
|
624 |
proof -
|
huffman@36787
|
625 |
{ fix y::rat
|
huffman@36787
|
626 |
assume ygt: "inverse x < y"
|
huffman@36787
|
627 |
have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt])
|
huffman@36787
|
628 |
have iyless: "inverse y < x"
|
huffman@36787
|
629 |
by (simp add: inverse_less_imp_less [of x] ygt)
|
huffman@36787
|
630 |
have "inverse y \<in> A"
|
huffman@36787
|
631 |
by (simp add: preal_downwards_closed [OF A x] iyless)}
|
huffman@36787
|
632 |
thus ?thesis by (auto simp add: inverse_set_def)
|
huffman@36787
|
633 |
qed
|
huffman@36787
|
634 |
qed
|
huffman@36787
|
635 |
qed
|
huffman@36787
|
636 |
|
huffman@36787
|
637 |
lemma inverse_set_not_rat_set:
|
huffman@36787
|
638 |
assumes A: "A \<in> preal" shows "inverse_set A < {r. 0 < r}"
|
huffman@36787
|
639 |
proof
|
huffman@36787
|
640 |
show "inverse_set A \<subseteq> {r. 0 < r}" by (force simp add: inverse_set_def)
|
huffman@36787
|
641 |
next
|
huffman@36787
|
642 |
show "inverse_set A \<noteq> {r. 0 < r}"
|
huffman@36787
|
643 |
by (insert preal_not_mem_inverse_set_Ex [OF A], blast)
|
huffman@36787
|
644 |
qed
|
huffman@36787
|
645 |
|
huffman@36787
|
646 |
text{*Part 3 of Dedekind sections definition*}
|
huffman@36787
|
647 |
lemma inverse_set_lemma3:
|
huffman@36787
|
648 |
"[|A \<in> preal; u \<in> inverse_set A; 0 < z; z < u|]
|
huffman@36787
|
649 |
==> z \<in> inverse_set A"
|
huffman@36787
|
650 |
apply (auto simp add: inverse_set_def)
|
huffman@36787
|
651 |
apply (auto intro: order_less_trans)
|
huffman@36787
|
652 |
done
|
huffman@36787
|
653 |
|
huffman@36787
|
654 |
text{*Part 4 of Dedekind sections definition*}
|
huffman@36787
|
655 |
lemma inverse_set_lemma4:
|
huffman@36787
|
656 |
"[|A \<in> preal; y \<in> inverse_set A|] ==> \<exists>u \<in> inverse_set A. y < u"
|
huffman@36787
|
657 |
apply (auto simp add: inverse_set_def)
|
huffman@36787
|
658 |
apply (drule dense [of y])
|
huffman@36787
|
659 |
apply (blast intro: order_less_trans)
|
huffman@36787
|
660 |
done
|
huffman@36787
|
661 |
|
huffman@36787
|
662 |
|
huffman@36787
|
663 |
lemma mem_inverse_set:
|
huffman@36787
|
664 |
"A \<in> preal ==> inverse_set A \<in> preal"
|
huffman@36787
|
665 |
apply (simp (no_asm_simp) add: preal_def cut_def)
|
huffman@36787
|
666 |
apply (blast intro!: inverse_set_not_empty inverse_set_not_rat_set
|
huffman@36787
|
667 |
inverse_set_lemma3 inverse_set_lemma4)
|
huffman@36787
|
668 |
done
|
huffman@36787
|
669 |
|
huffman@36787
|
670 |
|
huffman@36787
|
671 |
subsection{*Gleason's Lemma 9-3.4, page 122*}
|
huffman@36787
|
672 |
|
huffman@36787
|
673 |
lemma Gleason9_34_exists:
|
huffman@36787
|
674 |
assumes A: "A \<in> preal"
|
huffman@36787
|
675 |
and "\<forall>x\<in>A. x + u \<in> A"
|
huffman@36787
|
676 |
and "0 \<le> z"
|
huffman@36787
|
677 |
shows "\<exists>b\<in>A. b + (of_int z) * u \<in> A"
|
huffman@36787
|
678 |
proof (cases z rule: int_cases)
|
huffman@36787
|
679 |
case (nonneg n)
|
huffman@36787
|
680 |
show ?thesis
|
wenzelm@41789
|
681 |
proof (simp add: nonneg, induct n)
|
huffman@36787
|
682 |
case 0
|
wenzelm@41789
|
683 |
from preal_nonempty [OF A]
|
wenzelm@41789
|
684 |
show ?case by force
|
wenzelm@41789
|
685 |
next
|
huffman@36787
|
686 |
case (Suc k)
|
wenzelm@41789
|
687 |
then obtain b where b: "b \<in> A" "b + of_nat k * u \<in> A" ..
|
wenzelm@41789
|
688 |
hence "b + of_int (int k)*u + u \<in> A" by (simp add: assms)
|
wenzelm@41789
|
689 |
thus ?case by (force simp add: algebra_simps b)
|
huffman@36787
|
690 |
qed
|
huffman@36787
|
691 |
next
|
huffman@36787
|
692 |
case (neg n)
|
wenzelm@41789
|
693 |
with assms show ?thesis by simp
|
huffman@36787
|
694 |
qed
|
huffman@36787
|
695 |
|
huffman@36787
|
696 |
lemma Gleason9_34_contra:
|
huffman@36787
|
697 |
assumes A: "A \<in> preal"
|
huffman@36787
|
698 |
shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A|] ==> False"
|
huffman@36787
|
699 |
proof (induct u, induct y)
|
huffman@36787
|
700 |
fix a::int and b::int
|
huffman@36787
|
701 |
fix c::int and d::int
|
huffman@36787
|
702 |
assume bpos [simp]: "0 < b"
|
huffman@36787
|
703 |
and dpos [simp]: "0 < d"
|
huffman@36787
|
704 |
and closed: "\<forall>x\<in>A. x + (Fract c d) \<in> A"
|
huffman@36787
|
705 |
and upos: "0 < Fract c d"
|
huffman@36787
|
706 |
and ypos: "0 < Fract a b"
|
huffman@36787
|
707 |
and notin: "Fract a b \<notin> A"
|
huffman@36787
|
708 |
have cpos [simp]: "0 < c"
|
huffman@36787
|
709 |
by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos)
|
huffman@36787
|
710 |
have apos [simp]: "0 < a"
|
huffman@36787
|
711 |
by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos)
|
huffman@36787
|
712 |
let ?k = "a*d"
|
huffman@36787
|
713 |
have frle: "Fract a b \<le> Fract ?k 1 * (Fract c d)"
|
huffman@36787
|
714 |
proof -
|
huffman@36787
|
715 |
have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))"
|
huffman@36787
|
716 |
by (simp add: order_less_imp_not_eq2 mult_ac)
|
huffman@36787
|
717 |
moreover
|
huffman@36787
|
718 |
have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)"
|
huffman@36787
|
719 |
by (rule mult_mono,
|
huffman@36787
|
720 |
simp_all add: int_one_le_iff_zero_less zero_less_mult_iff
|
huffman@36787
|
721 |
order_less_imp_le)
|
huffman@36787
|
722 |
ultimately
|
huffman@36787
|
723 |
show ?thesis by simp
|
huffman@36787
|
724 |
qed
|
huffman@36787
|
725 |
have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff)
|
huffman@36787
|
726 |
from Gleason9_34_exists [OF A closed k]
|
huffman@36787
|
727 |
obtain z where z: "z \<in> A"
|
huffman@36787
|
728 |
and mem: "z + of_int ?k * Fract c d \<in> A" ..
|
huffman@36787
|
729 |
have less: "z + of_int ?k * Fract c d < Fract a b"
|
huffman@36787
|
730 |
by (rule not_in_preal_ub [OF A notin mem ypos])
|
huffman@36787
|
731 |
have "0<z" by (rule preal_imp_pos [OF A z])
|
huffman@36787
|
732 |
with frle and less show False by (simp add: Fract_of_int_eq)
|
huffman@36787
|
733 |
qed
|
huffman@36787
|
734 |
|
huffman@36787
|
735 |
|
huffman@36787
|
736 |
lemma Gleason9_34:
|
huffman@36787
|
737 |
assumes A: "A \<in> preal"
|
huffman@36787
|
738 |
and upos: "0 < u"
|
huffman@36787
|
739 |
shows "\<exists>r \<in> A. r + u \<notin> A"
|
huffman@36787
|
740 |
proof (rule ccontr, simp)
|
huffman@36787
|
741 |
assume closed: "\<forall>r\<in>A. r + u \<in> A"
|
huffman@36787
|
742 |
from preal_exists_bound [OF A]
|
huffman@36787
|
743 |
obtain y where y: "y \<notin> A" and ypos: "0 < y" by blast
|
huffman@36787
|
744 |
show False
|
huffman@36787
|
745 |
by (rule Gleason9_34_contra [OF A closed upos ypos y])
|
huffman@36787
|
746 |
qed
|
huffman@36787
|
747 |
|
huffman@36787
|
748 |
|
huffman@36787
|
749 |
|
huffman@36787
|
750 |
subsection{*Gleason's Lemma 9-3.6*}
|
huffman@36787
|
751 |
|
huffman@36787
|
752 |
lemma lemma_gleason9_36:
|
huffman@36787
|
753 |
assumes A: "A \<in> preal"
|
huffman@36787
|
754 |
and x: "1 < x"
|
huffman@36787
|
755 |
shows "\<exists>r \<in> A. r*x \<notin> A"
|
huffman@36787
|
756 |
proof -
|
huffman@36787
|
757 |
from preal_nonempty [OF A]
|
huffman@36787
|
758 |
obtain y where y: "y \<in> A" and ypos: "0<y" ..
|
huffman@36787
|
759 |
show ?thesis
|
huffman@36787
|
760 |
proof (rule classical)
|
huffman@36787
|
761 |
assume "~(\<exists>r\<in>A. r * x \<notin> A)"
|
huffman@36787
|
762 |
with y have ymem: "y * x \<in> A" by blast
|
huffman@36787
|
763 |
from ypos mult_strict_left_mono [OF x]
|
huffman@36787
|
764 |
have yless: "y < y*x" by simp
|
huffman@36787
|
765 |
let ?d = "y*x - y"
|
huffman@36787
|
766 |
from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto
|
huffman@36787
|
767 |
from Gleason9_34 [OF A dpos]
|
huffman@36787
|
768 |
obtain r where r: "r\<in>A" and notin: "r + ?d \<notin> A" ..
|
huffman@36787
|
769 |
have rpos: "0<r" by (rule preal_imp_pos [OF A r])
|
huffman@36787
|
770 |
with dpos have rdpos: "0 < r + ?d" by arith
|
huffman@36787
|
771 |
have "~ (r + ?d \<le> y + ?d)"
|
huffman@36787
|
772 |
proof
|
huffman@36787
|
773 |
assume le: "r + ?d \<le> y + ?d"
|
huffman@36787
|
774 |
from ymem have yd: "y + ?d \<in> A" by (simp add: eq)
|
huffman@36787
|
775 |
have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le])
|
huffman@36787
|
776 |
with notin show False by simp
|
huffman@36787
|
777 |
qed
|
huffman@36787
|
778 |
hence "y < r" by simp
|
huffman@36787
|
779 |
with ypos have dless: "?d < (r * ?d)/y"
|
huffman@36787
|
780 |
by (simp add: pos_less_divide_eq mult_commute [of ?d]
|
huffman@36787
|
781 |
mult_strict_right_mono dpos)
|
huffman@36787
|
782 |
have "r + ?d < r*x"
|
huffman@36787
|
783 |
proof -
|
huffman@36787
|
784 |
have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
|
wenzelm@54510
|
785 |
also from ypos have "... = (r/y) * (y + ?d)"
|
huffman@36787
|
786 |
by (simp only: algebra_simps divide_inverse, simp)
|
huffman@36787
|
787 |
also have "... = r*x" using ypos
|
huffman@36787
|
788 |
by simp
|
huffman@36787
|
789 |
finally show "r + ?d < r*x" .
|
huffman@36787
|
790 |
qed
|
huffman@36787
|
791 |
with r notin rdpos
|
huffman@36787
|
792 |
show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest: preal_downwards_closed [OF A])
|
huffman@36787
|
793 |
qed
|
huffman@36787
|
794 |
qed
|
huffman@36787
|
795 |
|
huffman@36787
|
796 |
subsection{*Existence of Inverse: Part 2*}
|
huffman@36787
|
797 |
|
huffman@36787
|
798 |
lemma mem_Rep_preal_inverse_iff:
|
huffman@36787
|
799 |
"(z \<in> Rep_preal(inverse R)) =
|
huffman@36787
|
800 |
(0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal R))"
|
huffman@36787
|
801 |
apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)
|
huffman@36787
|
802 |
apply (simp add: inverse_set_def)
|
huffman@36787
|
803 |
done
|
huffman@36787
|
804 |
|
huffman@36787
|
805 |
lemma Rep_preal_one:
|
huffman@36787
|
806 |
"Rep_preal 1 = {x. 0 < x \<and> x < 1}"
|
huffman@36787
|
807 |
by (simp add: preal_one_def rat_mem_preal)
|
huffman@36787
|
808 |
|
huffman@36787
|
809 |
lemma subset_inverse_mult_lemma:
|
huffman@36787
|
810 |
assumes xpos: "0 < x" and xless: "x < 1"
|
huffman@36787
|
811 |
shows "\<exists>r u y. 0 < r & r < y & inverse y \<notin> Rep_preal R &
|
huffman@36787
|
812 |
u \<in> Rep_preal R & x = r * u"
|
huffman@36787
|
813 |
proof -
|
huffman@36787
|
814 |
from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff)
|
huffman@36787
|
815 |
from lemma_gleason9_36 [OF Rep_preal this]
|
huffman@36787
|
816 |
obtain r where r: "r \<in> Rep_preal R"
|
huffman@36787
|
817 |
and notin: "r * (inverse x) \<notin> Rep_preal R" ..
|
huffman@36787
|
818 |
have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
|
huffman@36787
|
819 |
from preal_exists_greater [OF Rep_preal r]
|
huffman@36787
|
820 |
obtain u where u: "u \<in> Rep_preal R" and rless: "r < u" ..
|
huffman@36787
|
821 |
have upos: "0<u" by (rule preal_imp_pos [OF Rep_preal u])
|
huffman@36787
|
822 |
show ?thesis
|
huffman@36787
|
823 |
proof (intro exI conjI)
|
huffman@36787
|
824 |
show "0 < x/u" using xpos upos
|
huffman@36787
|
825 |
by (simp add: zero_less_divide_iff)
|
huffman@36787
|
826 |
show "x/u < x/r" using xpos upos rpos
|
huffman@36787
|
827 |
by (simp add: divide_inverse mult_less_cancel_left rless)
|
huffman@36787
|
828 |
show "inverse (x / r) \<notin> Rep_preal R" using notin
|
huffman@36787
|
829 |
by (simp add: divide_inverse mult_commute)
|
huffman@36787
|
830 |
show "u \<in> Rep_preal R" by (rule u)
|
huffman@36787
|
831 |
show "x = x / u * u" using upos
|
huffman@36787
|
832 |
by (simp add: divide_inverse mult_commute)
|
huffman@36787
|
833 |
qed
|
huffman@36787
|
834 |
qed
|
huffman@36787
|
835 |
|
huffman@36787
|
836 |
lemma subset_inverse_mult:
|
huffman@36787
|
837 |
"Rep_preal 1 \<subseteq> Rep_preal(inverse R * R)"
|
huffman@36787
|
838 |
apply (auto simp add: Bex_def Rep_preal_one mem_Rep_preal_inverse_iff
|
huffman@36787
|
839 |
mem_Rep_preal_mult_iff)
|
huffman@36787
|
840 |
apply (blast dest: subset_inverse_mult_lemma)
|
huffman@36787
|
841 |
done
|
huffman@36787
|
842 |
|
huffman@36787
|
843 |
lemma inverse_mult_subset_lemma:
|
huffman@36787
|
844 |
assumes rpos: "0 < r"
|
huffman@36787
|
845 |
and rless: "r < y"
|
huffman@36787
|
846 |
and notin: "inverse y \<notin> Rep_preal R"
|
huffman@36787
|
847 |
and q: "q \<in> Rep_preal R"
|
huffman@36787
|
848 |
shows "r*q < 1"
|
huffman@36787
|
849 |
proof -
|
huffman@36787
|
850 |
have "q < inverse y" using rpos rless
|
huffman@36787
|
851 |
by (simp add: not_in_preal_ub [OF Rep_preal notin] q)
|
huffman@36787
|
852 |
hence "r * q < r/y" using rpos
|
huffman@36787
|
853 |
by (simp add: divide_inverse mult_less_cancel_left)
|
huffman@36787
|
854 |
also have "... \<le> 1" using rpos rless
|
huffman@36787
|
855 |
by (simp add: pos_divide_le_eq)
|
huffman@36787
|
856 |
finally show ?thesis .
|
huffman@36787
|
857 |
qed
|
huffman@36787
|
858 |
|
huffman@36787
|
859 |
lemma inverse_mult_subset:
|
huffman@36787
|
860 |
"Rep_preal(inverse R * R) \<subseteq> Rep_preal 1"
|
huffman@36787
|
861 |
apply (auto simp add: Bex_def Rep_preal_one mem_Rep_preal_inverse_iff
|
huffman@36787
|
862 |
mem_Rep_preal_mult_iff)
|
huffman@36787
|
863 |
apply (simp add: zero_less_mult_iff preal_imp_pos [OF Rep_preal])
|
huffman@36787
|
864 |
apply (blast intro: inverse_mult_subset_lemma)
|
huffman@36787
|
865 |
done
|
huffman@36787
|
866 |
|
huffman@36787
|
867 |
lemma preal_mult_inverse: "inverse R * R = (1::preal)"
|
huffman@36787
|
868 |
apply (rule Rep_preal_inject [THEN iffD1])
|
huffman@36787
|
869 |
apply (rule equalityI [OF inverse_mult_subset subset_inverse_mult])
|
huffman@36787
|
870 |
done
|
huffman@36787
|
871 |
|
huffman@36787
|
872 |
lemma preal_mult_inverse_right: "R * inverse R = (1::preal)"
|
huffman@36787
|
873 |
apply (rule preal_mult_commute [THEN subst])
|
huffman@36787
|
874 |
apply (rule preal_mult_inverse)
|
huffman@36787
|
875 |
done
|
huffman@36787
|
876 |
|
huffman@36787
|
877 |
|
huffman@36787
|
878 |
text{*Theorems needing @{text Gleason9_34}*}
|
huffman@36787
|
879 |
|
huffman@36787
|
880 |
lemma Rep_preal_self_subset: "Rep_preal (R) \<subseteq> Rep_preal(R + S)"
|
huffman@36787
|
881 |
proof
|
huffman@36787
|
882 |
fix r
|
huffman@36787
|
883 |
assume r: "r \<in> Rep_preal R"
|
huffman@36787
|
884 |
have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
|
huffman@36787
|
885 |
from mem_Rep_preal_Ex
|
huffman@36787
|
886 |
obtain y where y: "y \<in> Rep_preal S" ..
|
huffman@36787
|
887 |
have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
|
huffman@36787
|
888 |
have ry: "r+y \<in> Rep_preal(R + S)" using r y
|
huffman@36787
|
889 |
by (auto simp add: mem_Rep_preal_add_iff)
|
huffman@36787
|
890 |
show "r \<in> Rep_preal(R + S)" using r ypos rpos
|
huffman@36787
|
891 |
by (simp add: preal_downwards_closed [OF Rep_preal ry])
|
huffman@36787
|
892 |
qed
|
huffman@36787
|
893 |
|
huffman@36787
|
894 |
lemma Rep_preal_sum_not_subset: "~ Rep_preal (R + S) \<subseteq> Rep_preal(R)"
|
huffman@36787
|
895 |
proof -
|
huffman@36787
|
896 |
from mem_Rep_preal_Ex
|
huffman@36787
|
897 |
obtain y where y: "y \<in> Rep_preal S" ..
|
huffman@36787
|
898 |
have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
|
huffman@36787
|
899 |
from Gleason9_34 [OF Rep_preal ypos]
|
huffman@36787
|
900 |
obtain r where r: "r \<in> Rep_preal R" and notin: "r + y \<notin> Rep_preal R" ..
|
huffman@36787
|
901 |
have "r + y \<in> Rep_preal (R + S)" using r y
|
huffman@36787
|
902 |
by (auto simp add: mem_Rep_preal_add_iff)
|
huffman@36787
|
903 |
thus ?thesis using notin by blast
|
huffman@36787
|
904 |
qed
|
huffman@36787
|
905 |
|
huffman@36787
|
906 |
lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) \<noteq> Rep_preal(R)"
|
huffman@36787
|
907 |
by (insert Rep_preal_sum_not_subset, blast)
|
huffman@36787
|
908 |
|
huffman@36787
|
909 |
text{*at last, Gleason prop. 9-3.5(iii) page 123*}
|
huffman@36787
|
910 |
lemma preal_self_less_add_left: "(R::preal) < R + S"
|
huffman@36787
|
911 |
apply (unfold preal_less_def less_le)
|
huffman@36787
|
912 |
apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])
|
huffman@36787
|
913 |
done
|
huffman@36787
|
914 |
|
huffman@36787
|
915 |
|
huffman@36787
|
916 |
subsection{*Subtraction for Positive Reals*}
|
huffman@36787
|
917 |
|
huffman@36787
|
918 |
text{*Gleason prop. 9-3.5(iv), page 123: proving @{prop "A < B ==> \<exists>D. A + D =
|
huffman@36787
|
919 |
B"}. We define the claimed @{term D} and show that it is a positive real*}
|
huffman@36787
|
920 |
|
huffman@36787
|
921 |
text{*Part 1 of Dedekind sections definition*}
|
huffman@36787
|
922 |
lemma diff_set_not_empty:
|
huffman@36787
|
923 |
"R < S ==> {} \<subset> diff_set (Rep_preal S) (Rep_preal R)"
|
huffman@36787
|
924 |
apply (auto simp add: preal_less_def diff_set_def elim!: equalityE)
|
huffman@36787
|
925 |
apply (frule_tac x1 = S in Rep_preal [THEN preal_exists_greater])
|
huffman@36787
|
926 |
apply (drule preal_imp_pos [OF Rep_preal], clarify)
|
huffman@36787
|
927 |
apply (cut_tac a=x and b=u in add_eq_exists, force)
|
huffman@36787
|
928 |
done
|
huffman@36787
|
929 |
|
huffman@36787
|
930 |
text{*Part 2 of Dedekind sections definition*}
|
huffman@36787
|
931 |
lemma diff_set_nonempty:
|
huffman@36787
|
932 |
"\<exists>q. 0 < q & q \<notin> diff_set (Rep_preal S) (Rep_preal R)"
|
huffman@36787
|
933 |
apply (cut_tac X = S in Rep_preal_exists_bound)
|
huffman@36787
|
934 |
apply (erule exE)
|
huffman@36787
|
935 |
apply (rule_tac x = x in exI, auto)
|
huffman@36787
|
936 |
apply (simp add: diff_set_def)
|
huffman@36787
|
937 |
apply (auto dest: Rep_preal [THEN preal_downwards_closed])
|
huffman@36787
|
938 |
done
|
huffman@36787
|
939 |
|
huffman@36787
|
940 |
lemma diff_set_not_rat_set:
|
huffman@36787
|
941 |
"diff_set (Rep_preal S) (Rep_preal R) < {r. 0 < r}" (is "?lhs < ?rhs")
|
huffman@36787
|
942 |
proof
|
huffman@36787
|
943 |
show "?lhs \<subseteq> ?rhs" by (auto simp add: diff_set_def)
|
huffman@36787
|
944 |
show "?lhs \<noteq> ?rhs" using diff_set_nonempty by blast
|
huffman@36787
|
945 |
qed
|
huffman@36787
|
946 |
|
huffman@36787
|
947 |
text{*Part 3 of Dedekind sections definition*}
|
huffman@36787
|
948 |
lemma diff_set_lemma3:
|
huffman@36787
|
949 |
"[|R < S; u \<in> diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|]
|
huffman@36787
|
950 |
==> z \<in> diff_set (Rep_preal S) (Rep_preal R)"
|
huffman@36787
|
951 |
apply (auto simp add: diff_set_def)
|
huffman@36787
|
952 |
apply (rule_tac x=x in exI)
|
huffman@36787
|
953 |
apply (drule Rep_preal [THEN preal_downwards_closed], auto)
|
huffman@36787
|
954 |
done
|
huffman@36787
|
955 |
|
huffman@36787
|
956 |
text{*Part 4 of Dedekind sections definition*}
|
huffman@36787
|
957 |
lemma diff_set_lemma4:
|
huffman@36787
|
958 |
"[|R < S; y \<in> diff_set (Rep_preal S) (Rep_preal R)|]
|
huffman@36787
|
959 |
==> \<exists>u \<in> diff_set (Rep_preal S) (Rep_preal R). y < u"
|
huffman@36787
|
960 |
apply (auto simp add: diff_set_def)
|
huffman@36787
|
961 |
apply (drule Rep_preal [THEN preal_exists_greater], clarify)
|
huffman@36787
|
962 |
apply (cut_tac a="x+y" and b=u in add_eq_exists, clarify)
|
huffman@36787
|
963 |
apply (rule_tac x="y+xa" in exI)
|
huffman@36787
|
964 |
apply (auto simp add: add_ac)
|
huffman@36787
|
965 |
done
|
huffman@36787
|
966 |
|
huffman@36787
|
967 |
lemma mem_diff_set:
|
huffman@36787
|
968 |
"R < S ==> diff_set (Rep_preal S) (Rep_preal R) \<in> preal"
|
wenzelm@47776
|
969 |
apply (unfold preal_def cut_def [abs_def])
|
huffman@36787
|
970 |
apply (blast intro!: diff_set_not_empty diff_set_not_rat_set
|
huffman@36787
|
971 |
diff_set_lemma3 diff_set_lemma4)
|
huffman@36787
|
972 |
done
|
huffman@36787
|
973 |
|
huffman@36787
|
974 |
lemma mem_Rep_preal_diff_iff:
|
huffman@36787
|
975 |
"R < S ==>
|
huffman@36787
|
976 |
(z \<in> Rep_preal(S-R)) =
|
huffman@36787
|
977 |
(\<exists>x. 0 < x & 0 < z & x \<notin> Rep_preal R & x + z \<in> Rep_preal S)"
|
huffman@36787
|
978 |
apply (simp add: preal_diff_def mem_diff_set Rep_preal)
|
huffman@36787
|
979 |
apply (force simp add: diff_set_def)
|
huffman@36787
|
980 |
done
|
huffman@36787
|
981 |
|
huffman@36787
|
982 |
|
huffman@36787
|
983 |
text{*proving that @{term "R + D \<le> S"}*}
|
huffman@36787
|
984 |
|
huffman@36787
|
985 |
lemma less_add_left_lemma:
|
huffman@36787
|
986 |
assumes Rless: "R < S"
|
huffman@36787
|
987 |
and a: "a \<in> Rep_preal R"
|
huffman@36787
|
988 |
and cb: "c + b \<in> Rep_preal S"
|
huffman@36787
|
989 |
and "c \<notin> Rep_preal R"
|
huffman@36787
|
990 |
and "0 < b"
|
huffman@36787
|
991 |
and "0 < c"
|
huffman@36787
|
992 |
shows "a + b \<in> Rep_preal S"
|
huffman@36787
|
993 |
proof -
|
huffman@36787
|
994 |
have "0<a" by (rule preal_imp_pos [OF Rep_preal a])
|
huffman@36787
|
995 |
moreover
|
wenzelm@41789
|
996 |
have "a < c" using assms by (blast intro: not_in_Rep_preal_ub )
|
wenzelm@41789
|
997 |
ultimately show ?thesis
|
wenzelm@41789
|
998 |
using assms by (simp add: preal_downwards_closed [OF Rep_preal cb])
|
huffman@36787
|
999 |
qed
|
huffman@36787
|
1000 |
|
huffman@36787
|
1001 |
lemma less_add_left_le1:
|
huffman@36787
|
1002 |
"R < (S::preal) ==> R + (S-R) \<le> S"
|
huffman@36787
|
1003 |
apply (auto simp add: Bex_def preal_le_def mem_Rep_preal_add_iff
|
huffman@36787
|
1004 |
mem_Rep_preal_diff_iff)
|
huffman@36787
|
1005 |
apply (blast intro: less_add_left_lemma)
|
huffman@36787
|
1006 |
done
|
huffman@36787
|
1007 |
|
huffman@36787
|
1008 |
subsection{*proving that @{term "S \<le> R + D"} --- trickier*}
|
huffman@36787
|
1009 |
|
huffman@36787
|
1010 |
lemma lemma_sum_mem_Rep_preal_ex:
|
huffman@36787
|
1011 |
"x \<in> Rep_preal S ==> \<exists>e. 0 < e & x + e \<in> Rep_preal S"
|
huffman@36787
|
1012 |
apply (drule Rep_preal [THEN preal_exists_greater], clarify)
|
huffman@36787
|
1013 |
apply (cut_tac a=x and b=u in add_eq_exists, auto)
|
huffman@36787
|
1014 |
done
|
huffman@36787
|
1015 |
|
huffman@36787
|
1016 |
lemma less_add_left_lemma2:
|
huffman@36787
|
1017 |
assumes Rless: "R < S"
|
huffman@36787
|
1018 |
and x: "x \<in> Rep_preal S"
|
huffman@36787
|
1019 |
and xnot: "x \<notin> Rep_preal R"
|
huffman@36787
|
1020 |
shows "\<exists>u v z. 0 < v & 0 < z & u \<in> Rep_preal R & z \<notin> Rep_preal R &
|
huffman@36787
|
1021 |
z + v \<in> Rep_preal S & x = u + v"
|
huffman@36787
|
1022 |
proof -
|
huffman@36787
|
1023 |
have xpos: "0<x" by (rule preal_imp_pos [OF Rep_preal x])
|
huffman@36787
|
1024 |
from lemma_sum_mem_Rep_preal_ex [OF x]
|
huffman@36787
|
1025 |
obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal S" by blast
|
huffman@36787
|
1026 |
from Gleason9_34 [OF Rep_preal epos]
|
huffman@36787
|
1027 |
obtain r where r: "r \<in> Rep_preal R" and notin: "r + e \<notin> Rep_preal R" ..
|
huffman@36787
|
1028 |
with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub)
|
huffman@36787
|
1029 |
from add_eq_exists [of r x]
|
huffman@36787
|
1030 |
obtain y where eq: "x = r+y" by auto
|
huffman@36787
|
1031 |
show ?thesis
|
huffman@36787
|
1032 |
proof (intro exI conjI)
|
huffman@36787
|
1033 |
show "r \<in> Rep_preal R" by (rule r)
|
huffman@36787
|
1034 |
show "r + e \<notin> Rep_preal R" by (rule notin)
|
huffman@36787
|
1035 |
show "r + e + y \<in> Rep_preal S" using xe eq by (simp add: add_ac)
|
huffman@36787
|
1036 |
show "x = r + y" by (simp add: eq)
|
huffman@36787
|
1037 |
show "0 < r + e" using epos preal_imp_pos [OF Rep_preal r]
|
huffman@36787
|
1038 |
by simp
|
huffman@36787
|
1039 |
show "0 < y" using rless eq by arith
|
huffman@36787
|
1040 |
qed
|
huffman@36787
|
1041 |
qed
|
huffman@36787
|
1042 |
|
huffman@36787
|
1043 |
lemma less_add_left_le2: "R < (S::preal) ==> S \<le> R + (S-R)"
|
huffman@36787
|
1044 |
apply (auto simp add: preal_le_def)
|
huffman@36787
|
1045 |
apply (case_tac "x \<in> Rep_preal R")
|
huffman@36787
|
1046 |
apply (cut_tac Rep_preal_self_subset [of R], force)
|
huffman@36787
|
1047 |
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_diff_iff)
|
huffman@36787
|
1048 |
apply (blast dest: less_add_left_lemma2)
|
huffman@36787
|
1049 |
done
|
huffman@36787
|
1050 |
|
huffman@36787
|
1051 |
lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S"
|
huffman@36787
|
1052 |
by (blast intro: antisym [OF less_add_left_le1 less_add_left_le2])
|
huffman@36787
|
1053 |
|
huffman@36787
|
1054 |
lemma less_add_left_Ex: "R < (S::preal) ==> \<exists>D. R + D = S"
|
huffman@36787
|
1055 |
by (fast dest: less_add_left)
|
huffman@36787
|
1056 |
|
huffman@36787
|
1057 |
lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T"
|
huffman@36787
|
1058 |
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc)
|
huffman@36787
|
1059 |
apply (rule_tac y1 = D in preal_add_commute [THEN subst])
|
huffman@36787
|
1060 |
apply (auto intro: preal_self_less_add_left simp add: preal_add_assoc [symmetric])
|
huffman@36787
|
1061 |
done
|
huffman@36787
|
1062 |
|
huffman@36787
|
1063 |
lemma preal_add_less2_mono2: "R < (S::preal) ==> T + R < T + S"
|
huffman@36787
|
1064 |
by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of T])
|
huffman@36787
|
1065 |
|
huffman@36787
|
1066 |
lemma preal_add_right_less_cancel: "R + T < S + T ==> R < (S::preal)"
|
huffman@36787
|
1067 |
apply (insert linorder_less_linear [of R S], auto)
|
huffman@36787
|
1068 |
apply (drule_tac R = S and T = T in preal_add_less2_mono1)
|
huffman@36787
|
1069 |
apply (blast dest: order_less_trans)
|
huffman@36787
|
1070 |
done
|
huffman@36787
|
1071 |
|
huffman@36787
|
1072 |
lemma preal_add_left_less_cancel: "T + R < T + S ==> R < (S::preal)"
|
huffman@36787
|
1073 |
by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of T])
|
huffman@36787
|
1074 |
|
huffman@36787
|
1075 |
lemma preal_add_less_cancel_left: "(T + (R::preal) < T + S) = (R < S)"
|
huffman@36787
|
1076 |
by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)
|
huffman@36787
|
1077 |
|
huffman@36787
|
1078 |
lemma preal_add_le_cancel_left: "(T + (R::preal) \<le> T + S) = (R \<le> S)"
|
huffman@36787
|
1079 |
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_left)
|
huffman@36787
|
1080 |
|
huffman@36787
|
1081 |
lemma preal_add_right_cancel: "(R::preal) + T = S + T ==> R = S"
|
huffman@36787
|
1082 |
apply (insert linorder_less_linear [of R S], safe)
|
huffman@36787
|
1083 |
apply (drule_tac [!] T = T in preal_add_less2_mono1, auto)
|
huffman@36787
|
1084 |
done
|
huffman@36787
|
1085 |
|
huffman@36787
|
1086 |
lemma preal_add_left_cancel: "C + A = C + B ==> A = (B::preal)"
|
huffman@36787
|
1087 |
by (auto intro: preal_add_right_cancel simp add: preal_add_commute)
|
huffman@36787
|
1088 |
|
huffman@36787
|
1089 |
instance preal :: linordered_cancel_ab_semigroup_add
|
huffman@36787
|
1090 |
proof
|
huffman@36787
|
1091 |
fix a b c :: preal
|
huffman@36787
|
1092 |
show "a + b = a + c \<Longrightarrow> b = c" by (rule preal_add_left_cancel)
|
huffman@36787
|
1093 |
show "a \<le> b \<Longrightarrow> c + a \<le> c + b" by (simp only: preal_add_le_cancel_left)
|
huffman@36787
|
1094 |
qed
|
huffman@36787
|
1095 |
|
huffman@36787
|
1096 |
|
huffman@36787
|
1097 |
subsection{*Completeness of type @{typ preal}*}
|
huffman@36787
|
1098 |
|
huffman@36787
|
1099 |
text{*Prove that supremum is a cut*}
|
huffman@36787
|
1100 |
|
huffman@36787
|
1101 |
text{*Part 1 of Dedekind sections definition*}
|
huffman@36787
|
1102 |
|
huffman@36787
|
1103 |
lemma preal_sup_set_not_empty:
|
huffman@36787
|
1104 |
"P \<noteq> {} ==> {} \<subset> (\<Union>X \<in> P. Rep_preal(X))"
|
huffman@36787
|
1105 |
apply auto
|
huffman@36787
|
1106 |
apply (cut_tac X = x in mem_Rep_preal_Ex, auto)
|
huffman@36787
|
1107 |
done
|
huffman@36787
|
1108 |
|
huffman@36787
|
1109 |
|
huffman@36787
|
1110 |
text{*Part 2 of Dedekind sections definition*}
|
huffman@36787
|
1111 |
|
huffman@36787
|
1112 |
lemma preal_sup_not_exists:
|
huffman@36787
|
1113 |
"\<forall>X \<in> P. X \<le> Y ==> \<exists>q. 0 < q & q \<notin> (\<Union>X \<in> P. Rep_preal(X))"
|
huffman@36787
|
1114 |
apply (cut_tac X = Y in Rep_preal_exists_bound)
|
huffman@36787
|
1115 |
apply (auto simp add: preal_le_def)
|
huffman@36787
|
1116 |
done
|
huffman@36787
|
1117 |
|
huffman@36787
|
1118 |
lemma preal_sup_set_not_rat_set:
|
huffman@36787
|
1119 |
"\<forall>X \<in> P. X \<le> Y ==> (\<Union>X \<in> P. Rep_preal(X)) < {r. 0 < r}"
|
huffman@36787
|
1120 |
apply (drule preal_sup_not_exists)
|
huffman@36787
|
1121 |
apply (blast intro: preal_imp_pos [OF Rep_preal])
|
huffman@36787
|
1122 |
done
|
huffman@36787
|
1123 |
|
huffman@36787
|
1124 |
text{*Part 3 of Dedekind sections definition*}
|
huffman@36787
|
1125 |
lemma preal_sup_set_lemma3:
|
huffman@36787
|
1126 |
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u|]
|
huffman@36787
|
1127 |
==> z \<in> (\<Union>X \<in> P. Rep_preal(X))"
|
huffman@36787
|
1128 |
by (auto elim: Rep_preal [THEN preal_downwards_closed])
|
huffman@36787
|
1129 |
|
huffman@36787
|
1130 |
text{*Part 4 of Dedekind sections definition*}
|
huffman@36787
|
1131 |
lemma preal_sup_set_lemma4:
|
huffman@36787
|
1132 |
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; y \<in> (\<Union>X \<in> P. Rep_preal(X)) |]
|
huffman@36787
|
1133 |
==> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u"
|
huffman@36787
|
1134 |
by (blast dest: Rep_preal [THEN preal_exists_greater])
|
huffman@36787
|
1135 |
|
huffman@36787
|
1136 |
lemma preal_sup:
|
huffman@36787
|
1137 |
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y|] ==> (\<Union>X \<in> P. Rep_preal(X)) \<in> preal"
|
wenzelm@47776
|
1138 |
apply (unfold preal_def cut_def [abs_def])
|
huffman@36787
|
1139 |
apply (blast intro!: preal_sup_set_not_empty preal_sup_set_not_rat_set
|
huffman@36787
|
1140 |
preal_sup_set_lemma3 preal_sup_set_lemma4)
|
huffman@36787
|
1141 |
done
|
huffman@36787
|
1142 |
|
huffman@36787
|
1143 |
lemma preal_psup_le:
|
huffman@36787
|
1144 |
"[| \<forall>X \<in> P. X \<le> Y; x \<in> P |] ==> x \<le> psup P"
|
huffman@36787
|
1145 |
apply (simp (no_asm_simp) add: preal_le_def)
|
huffman@36787
|
1146 |
apply (subgoal_tac "P \<noteq> {}")
|
huffman@36787
|
1147 |
apply (auto simp add: psup_def preal_sup)
|
huffman@36787
|
1148 |
done
|
huffman@36787
|
1149 |
|
huffman@36787
|
1150 |
lemma psup_le_ub: "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y"
|
huffman@36787
|
1151 |
apply (simp (no_asm_simp) add: preal_le_def)
|
huffman@36787
|
1152 |
apply (simp add: psup_def preal_sup)
|
huffman@36787
|
1153 |
apply (auto simp add: preal_le_def)
|
huffman@36787
|
1154 |
done
|
huffman@36787
|
1155 |
|
huffman@36787
|
1156 |
text{*Supremum property*}
|
huffman@36787
|
1157 |
lemma preal_complete:
|
huffman@36787
|
1158 |
"[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> (\<exists>X \<in> P. Z < X) = (Z < psup P)"
|
huffman@36787
|
1159 |
apply (simp add: preal_less_def psup_def preal_sup)
|
huffman@36787
|
1160 |
apply (auto simp add: preal_le_def)
|
huffman@36787
|
1161 |
apply (rename_tac U)
|
huffman@36787
|
1162 |
apply (cut_tac x = U and y = Z in linorder_less_linear)
|
huffman@36787
|
1163 |
apply (auto simp add: preal_less_def)
|
huffman@36787
|
1164 |
done
|
huffman@36787
|
1165 |
|
huffman@36787
|
1166 |
section {*Defining the Reals from the Positive Reals*}
|
huffman@36787
|
1167 |
|
huffman@36787
|
1168 |
definition
|
huffman@36787
|
1169 |
realrel :: "((preal * preal) * (preal * preal)) set" where
|
haftmann@37765
|
1170 |
"realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
|
huffman@36787
|
1171 |
|
wenzelm@46567
|
1172 |
definition "Real = UNIV//realrel"
|
wenzelm@46567
|
1173 |
|
wenzelm@50849
|
1174 |
typedef real = Real
|
wenzelm@46567
|
1175 |
morphisms Rep_Real Abs_Real
|
wenzelm@46567
|
1176 |
unfolding Real_def by (auto simp add: quotient_def)
|
huffman@36787
|
1177 |
|
huffman@36787
|
1178 |
definition
|
huffman@36787
|
1179 |
(** these don't use the overloaded "real" function: users don't see them **)
|
huffman@36787
|
1180 |
real_of_preal :: "preal => real" where
|
haftmann@37765
|
1181 |
"real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})"
|
huffman@36787
|
1182 |
|
huffman@36787
|
1183 |
instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}"
|
huffman@36787
|
1184 |
begin
|
huffman@36787
|
1185 |
|
huffman@36787
|
1186 |
definition
|
haftmann@37765
|
1187 |
real_zero_def: "0 = Abs_Real(realrel``{(1, 1)})"
|
huffman@36787
|
1188 |
|
huffman@36787
|
1189 |
definition
|
haftmann@37765
|
1190 |
real_one_def: "1 = Abs_Real(realrel``{(1 + 1, 1)})"
|
huffman@36787
|
1191 |
|
huffman@36787
|
1192 |
definition
|
haftmann@37765
|
1193 |
real_add_def: "z + w =
|
haftmann@40091
|
1194 |
the_elem (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
|
huffman@36787
|
1195 |
{ Abs_Real(realrel``{(x+u, y+v)}) })"
|
huffman@36787
|
1196 |
|
huffman@36787
|
1197 |
definition
|
haftmann@40091
|
1198 |
real_minus_def: "- r = the_elem (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
|
huffman@36787
|
1199 |
|
huffman@36787
|
1200 |
definition
|
haftmann@37765
|
1201 |
real_diff_def: "r - (s::real) = r + - s"
|
huffman@36787
|
1202 |
|
huffman@36787
|
1203 |
definition
|
haftmann@37765
|
1204 |
real_mult_def:
|
huffman@36787
|
1205 |
"z * w =
|
haftmann@40091
|
1206 |
the_elem (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
|
huffman@36787
|
1207 |
{ Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"
|
huffman@36787
|
1208 |
|
huffman@36787
|
1209 |
definition
|
haftmann@37765
|
1210 |
real_inverse_def: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)"
|
huffman@36787
|
1211 |
|
huffman@36787
|
1212 |
definition
|
haftmann@37765
|
1213 |
real_divide_def: "R / (S::real) = R * inverse S"
|
huffman@36787
|
1214 |
|
huffman@36787
|
1215 |
definition
|
haftmann@37765
|
1216 |
real_le_def: "z \<le> (w::real) \<longleftrightarrow>
|
huffman@36787
|
1217 |
(\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w)"
|
huffman@36787
|
1218 |
|
huffman@36787
|
1219 |
definition
|
haftmann@37765
|
1220 |
real_less_def: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
|
huffman@36787
|
1221 |
|
huffman@36787
|
1222 |
definition
|
huffman@36787
|
1223 |
real_abs_def: "abs (r::real) = (if r < 0 then - r else r)"
|
huffman@36787
|
1224 |
|
huffman@36787
|
1225 |
definition
|
huffman@36787
|
1226 |
real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)"
|
huffman@36787
|
1227 |
|
huffman@36787
|
1228 |
instance ..
|
huffman@36787
|
1229 |
|
huffman@36787
|
1230 |
end
|
huffman@36787
|
1231 |
|
huffman@36787
|
1232 |
subsection {* Equivalence relation over positive reals *}
|
huffman@36787
|
1233 |
|
huffman@36787
|
1234 |
lemma preal_trans_lemma:
|
huffman@36787
|
1235 |
assumes "x + y1 = x1 + y"
|
wenzelm@41789
|
1236 |
and "x + y2 = x2 + y"
|
huffman@36787
|
1237 |
shows "x1 + y2 = x2 + (y1::preal)"
|
huffman@36787
|
1238 |
proof -
|
huffman@36787
|
1239 |
have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac)
|
wenzelm@41789
|
1240 |
also have "... = (x2 + y) + x1" by (simp add: assms)
|
huffman@36787
|
1241 |
also have "... = x2 + (x1 + y)" by (simp add: add_ac)
|
wenzelm@41789
|
1242 |
also have "... = x2 + (x + y1)" by (simp add: assms)
|
huffman@36787
|
1243 |
also have "... = (x2 + y1) + x" by (simp add: add_ac)
|
huffman@36787
|
1244 |
finally have "(x1 + y2) + x = (x2 + y1) + x" .
|
huffman@36787
|
1245 |
thus ?thesis by (rule add_right_imp_eq)
|
huffman@36787
|
1246 |
qed
|
huffman@36787
|
1247 |
|
huffman@36787
|
1248 |
|
huffman@36787
|
1249 |
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"
|
huffman@36787
|
1250 |
by (simp add: realrel_def)
|
huffman@36787
|
1251 |
|
huffman@36787
|
1252 |
lemma equiv_realrel: "equiv UNIV realrel"
|
huffman@36787
|
1253 |
apply (auto simp add: equiv_def refl_on_def sym_def trans_def realrel_def)
|
huffman@36787
|
1254 |
apply (blast dest: preal_trans_lemma)
|
huffman@36787
|
1255 |
done
|
huffman@36787
|
1256 |
|
huffman@36787
|
1257 |
text{*Reduces equality of equivalence classes to the @{term realrel} relation:
|
huffman@36787
|
1258 |
@{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *}
|
huffman@36787
|
1259 |
lemmas equiv_realrel_iff =
|
huffman@36787
|
1260 |
eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]
|
huffman@36787
|
1261 |
|
huffman@36787
|
1262 |
declare equiv_realrel_iff [simp]
|
huffman@36787
|
1263 |
|
huffman@36787
|
1264 |
|
huffman@36787
|
1265 |
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
|
huffman@36787
|
1266 |
by (simp add: Real_def realrel_def quotient_def, blast)
|
huffman@36787
|
1267 |
|
huffman@36787
|
1268 |
declare Abs_Real_inject [simp]
|
huffman@36787
|
1269 |
declare Abs_Real_inverse [simp]
|
huffman@36787
|
1270 |
|
huffman@36787
|
1271 |
|
huffman@36787
|
1272 |
text{*Case analysis on the representation of a real number as an equivalence
|
huffman@36787
|
1273 |
class of pairs of positive reals.*}
|
huffman@36787
|
1274 |
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]:
|
huffman@36787
|
1275 |
"(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
|
huffman@36787
|
1276 |
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
|
huffman@36787
|
1277 |
apply (drule arg_cong [where f=Abs_Real])
|
huffman@36787
|
1278 |
apply (auto simp add: Rep_Real_inverse)
|
huffman@36787
|
1279 |
done
|
huffman@36787
|
1280 |
|
huffman@36787
|
1281 |
|
huffman@36787
|
1282 |
subsection {* Addition and Subtraction *}
|
huffman@36787
|
1283 |
|
huffman@36787
|
1284 |
lemma real_add_congruent2_lemma:
|
huffman@36787
|
1285 |
"[|a + ba = aa + b; ab + bc = ac + bb|]
|
huffman@36787
|
1286 |
==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
|
huffman@36787
|
1287 |
apply (simp add: add_assoc)
|
huffman@36787
|
1288 |
apply (rule add_left_commute [of ab, THEN ssubst])
|
huffman@36787
|
1289 |
apply (simp add: add_assoc [symmetric])
|
huffman@36787
|
1290 |
apply (simp add: add_ac)
|
huffman@36787
|
1291 |
done
|
huffman@36787
|
1292 |
|
huffman@36787
|
1293 |
lemma real_add:
|
huffman@36787
|
1294 |
"Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
|
huffman@36787
|
1295 |
Abs_Real (realrel``{(x+u, y+v)})"
|
huffman@36787
|
1296 |
proof -
|
huffman@36787
|
1297 |
have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
|
huffman@36787
|
1298 |
respects2 realrel"
|
haftmann@41070
|
1299 |
by (auto simp add: congruent2_def, blast intro: real_add_congruent2_lemma)
|
huffman@36787
|
1300 |
thus ?thesis
|
huffman@36787
|
1301 |
by (simp add: real_add_def UN_UN_split_split_eq
|
huffman@36787
|
1302 |
UN_equiv_class2 [OF equiv_realrel equiv_realrel])
|
huffman@36787
|
1303 |
qed
|
huffman@36787
|
1304 |
|
huffman@36787
|
1305 |
lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
|
huffman@36787
|
1306 |
proof -
|
huffman@36787
|
1307 |
have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
|
haftmann@41070
|
1308 |
by (auto simp add: congruent_def add_commute)
|
huffman@36787
|
1309 |
thus ?thesis
|
huffman@36787
|
1310 |
by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
|
huffman@36787
|
1311 |
qed
|
huffman@36787
|
1312 |
|
huffman@36787
|
1313 |
instance real :: ab_group_add
|
huffman@36787
|
1314 |
proof
|
huffman@36787
|
1315 |
fix x y z :: real
|
huffman@36787
|
1316 |
show "(x + y) + z = x + (y + z)"
|
huffman@36787
|
1317 |
by (cases x, cases y, cases z, simp add: real_add add_assoc)
|
huffman@36787
|
1318 |
show "x + y = y + x"
|
huffman@36787
|
1319 |
by (cases x, cases y, simp add: real_add add_commute)
|
huffman@36787
|
1320 |
show "0 + x = x"
|
huffman@36787
|
1321 |
by (cases x, simp add: real_add real_zero_def add_ac)
|
huffman@36787
|
1322 |
show "- x + x = 0"
|
huffman@36787
|
1323 |
by (cases x, simp add: real_minus real_add real_zero_def add_commute)
|
huffman@36787
|
1324 |
show "x - y = x + - y"
|
huffman@36787
|
1325 |
by (simp add: real_diff_def)
|
huffman@36787
|
1326 |
qed
|
huffman@36787
|
1327 |
|
huffman@36787
|
1328 |
|
huffman@36787
|
1329 |
subsection {* Multiplication *}
|
huffman@36787
|
1330 |
|
huffman@36787
|
1331 |
lemma real_mult_congruent2_lemma:
|
huffman@36787
|
1332 |
"!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
|
huffman@36787
|
1333 |
x * x1 + y * y1 + (x * y2 + y * x2) =
|
huffman@36787
|
1334 |
x * x2 + y * y2 + (x * y1 + y * x1)"
|
huffman@36787
|
1335 |
apply (simp add: add_left_commute add_assoc [symmetric])
|
webertj@50977
|
1336 |
apply (simp add: add_assoc distrib_left [symmetric])
|
huffman@36787
|
1337 |
apply (simp add: add_commute)
|
huffman@36787
|
1338 |
done
|
huffman@36787
|
1339 |
|
huffman@36787
|
1340 |
lemma real_mult_congruent2:
|
huffman@36787
|
1341 |
"(%p1 p2.
|
huffman@36787
|
1342 |
(%(x1,y1). (%(x2,y2).
|
huffman@36787
|
1343 |
{ Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
|
huffman@36787
|
1344 |
respects2 realrel"
|
huffman@36787
|
1345 |
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
|
huffman@36787
|
1346 |
apply (simp add: mult_commute add_commute)
|
huffman@36787
|
1347 |
apply (auto simp add: real_mult_congruent2_lemma)
|
huffman@36787
|
1348 |
done
|
huffman@36787
|
1349 |
|
huffman@36787
|
1350 |
lemma real_mult:
|
huffman@36787
|
1351 |
"Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
|
huffman@36787
|
1352 |
Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
|
huffman@36787
|
1353 |
by (simp add: real_mult_def UN_UN_split_split_eq
|
huffman@36787
|
1354 |
UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
|
huffman@36787
|
1355 |
|
huffman@36787
|
1356 |
lemma real_mult_commute: "(z::real) * w = w * z"
|
huffman@36787
|
1357 |
by (cases z, cases w, simp add: real_mult add_ac mult_ac)
|
huffman@36787
|
1358 |
|
huffman@36787
|
1359 |
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
|
huffman@36787
|
1360 |
apply (cases z1, cases z2, cases z3)
|
huffman@36787
|
1361 |
apply (simp add: real_mult algebra_simps)
|
huffman@36787
|
1362 |
done
|
huffman@36787
|
1363 |
|
huffman@36787
|
1364 |
lemma real_mult_1: "(1::real) * z = z"
|
huffman@36787
|
1365 |
apply (cases z)
|
huffman@36787
|
1366 |
apply (simp add: real_mult real_one_def algebra_simps)
|
huffman@36787
|
1367 |
done
|
huffman@36787
|
1368 |
|
huffman@36787
|
1369 |
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
|
huffman@36787
|
1370 |
apply (cases z1, cases z2, cases w)
|
huffman@36787
|
1371 |
apply (simp add: real_add real_mult algebra_simps)
|
huffman@36787
|
1372 |
done
|
huffman@36787
|
1373 |
|
huffman@36787
|
1374 |
text{*one and zero are distinct*}
|
huffman@36787
|
1375 |
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
|
huffman@36787
|
1376 |
proof -
|
huffman@36787
|
1377 |
have "(1::preal) < 1 + 1"
|
huffman@36787
|
1378 |
by (simp add: preal_self_less_add_left)
|
huffman@36787
|
1379 |
thus ?thesis
|
huffman@36787
|
1380 |
by (simp add: real_zero_def real_one_def)
|
huffman@36787
|
1381 |
qed
|
huffman@36787
|
1382 |
|
huffman@36787
|
1383 |
instance real :: comm_ring_1
|
huffman@36787
|
1384 |
proof
|
huffman@36787
|
1385 |
fix x y z :: real
|
huffman@36787
|
1386 |
show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
|
huffman@36787
|
1387 |
show "x * y = y * x" by (rule real_mult_commute)
|
huffman@36787
|
1388 |
show "1 * x = x" by (rule real_mult_1)
|
huffman@36787
|
1389 |
show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib)
|
huffman@36787
|
1390 |
show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
|
huffman@36787
|
1391 |
qed
|
huffman@36787
|
1392 |
|
huffman@36787
|
1393 |
subsection {* Inverse and Division *}
|
huffman@36787
|
1394 |
|
huffman@36787
|
1395 |
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
|
huffman@36787
|
1396 |
by (simp add: real_zero_def add_commute)
|
huffman@36787
|
1397 |
|
huffman@36787
|
1398 |
text{*Instead of using an existential quantifier and constructing the inverse
|
huffman@36787
|
1399 |
within the proof, we could define the inverse explicitly.*}
|
huffman@36787
|
1400 |
|
huffman@36787
|
1401 |
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
|
huffman@36787
|
1402 |
apply (simp add: real_zero_def real_one_def, cases x)
|
huffman@36787
|
1403 |
apply (cut_tac x = xa and y = y in linorder_less_linear)
|
huffman@36787
|
1404 |
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
|
huffman@36787
|
1405 |
apply (rule_tac
|
huffman@36787
|
1406 |
x = "Abs_Real (realrel``{(1, inverse (D) + 1)})"
|
huffman@36787
|
1407 |
in exI)
|
huffman@36787
|
1408 |
apply (rule_tac [2]
|
huffman@36787
|
1409 |
x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})"
|
huffman@36787
|
1410 |
in exI)
|
huffman@36787
|
1411 |
apply (auto simp add: real_mult preal_mult_inverse_right algebra_simps)
|
huffman@36787
|
1412 |
done
|
huffman@36787
|
1413 |
|
huffman@36787
|
1414 |
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
|
huffman@36787
|
1415 |
apply (simp add: real_inverse_def)
|
huffman@36787
|
1416 |
apply (drule real_mult_inverse_left_ex, safe)
|
huffman@36787
|
1417 |
apply (rule theI, assumption, rename_tac z)
|
huffman@36787
|
1418 |
apply (subgoal_tac "(z * x) * y = z * (x * y)")
|
huffman@36787
|
1419 |
apply (simp add: mult_commute)
|
huffman@36787
|
1420 |
apply (rule mult_assoc)
|
huffman@36787
|
1421 |
done
|
huffman@36787
|
1422 |
|
huffman@36787
|
1423 |
|
huffman@36787
|
1424 |
subsection{*The Real Numbers form a Field*}
|
huffman@36787
|
1425 |
|
huffman@36787
|
1426 |
instance real :: field_inverse_zero
|
huffman@36787
|
1427 |
proof
|
huffman@36787
|
1428 |
fix x y z :: real
|
huffman@36787
|
1429 |
show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
|
huffman@36787
|
1430 |
show "x / y = x * inverse y" by (simp add: real_divide_def)
|
huffman@36787
|
1431 |
show "inverse 0 = (0::real)" by (simp add: real_inverse_def)
|
huffman@36787
|
1432 |
qed
|
huffman@36787
|
1433 |
|
huffman@36787
|
1434 |
|
huffman@36787
|
1435 |
subsection{*The @{text "\<le>"} Ordering*}
|
huffman@36787
|
1436 |
|
huffman@36787
|
1437 |
lemma real_le_refl: "w \<le> (w::real)"
|
huffman@36787
|
1438 |
by (cases w, force simp add: real_le_def)
|
huffman@36787
|
1439 |
|
huffman@36787
|
1440 |
text{*The arithmetic decision procedure is not set up for type preal.
|
huffman@36787
|
1441 |
This lemma is currently unused, but it could simplify the proofs of the
|
huffman@36787
|
1442 |
following two lemmas.*}
|
huffman@36787
|
1443 |
lemma preal_eq_le_imp_le:
|
huffman@36787
|
1444 |
assumes eq: "a+b = c+d" and le: "c \<le> a"
|
huffman@36787
|
1445 |
shows "b \<le> (d::preal)"
|
huffman@36787
|
1446 |
proof -
|
wenzelm@41789
|
1447 |
have "c+d \<le> a+d" by (simp add: le)
|
wenzelm@41789
|
1448 |
hence "a+b \<le> a+d" by (simp add: eq)
|
huffman@36787
|
1449 |
thus "b \<le> d" by simp
|
huffman@36787
|
1450 |
qed
|
huffman@36787
|
1451 |
|
huffman@36787
|
1452 |
lemma real_le_lemma:
|
huffman@36787
|
1453 |
assumes l: "u1 + v2 \<le> u2 + v1"
|
wenzelm@41789
|
1454 |
and "x1 + v1 = u1 + y1"
|
wenzelm@41789
|
1455 |
and "x2 + v2 = u2 + y2"
|
huffman@36787
|
1456 |
shows "x1 + y2 \<le> x2 + (y1::preal)"
|
huffman@36787
|
1457 |
proof -
|
wenzelm@41789
|
1458 |
have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: assms)
|
huffman@36787
|
1459 |
hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
|
wenzelm@41789
|
1460 |
also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: assms)
|
huffman@36787
|
1461 |
finally show ?thesis by simp
|
huffman@36787
|
1462 |
qed
|
huffman@36787
|
1463 |
|
huffman@36787
|
1464 |
lemma real_le:
|
huffman@36787
|
1465 |
"(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =
|
huffman@36787
|
1466 |
(x1 + y2 \<le> x2 + y1)"
|
huffman@36787
|
1467 |
apply (simp add: real_le_def)
|
huffman@36787
|
1468 |
apply (auto intro: real_le_lemma)
|
huffman@36787
|
1469 |
done
|
huffman@36787
|
1470 |
|
huffman@36787
|
1471 |
lemma real_le_antisym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
|
huffman@36787
|
1472 |
by (cases z, cases w, simp add: real_le)
|
huffman@36787
|
1473 |
|
huffman@36787
|
1474 |
lemma real_trans_lemma:
|
huffman@36787
|
1475 |
assumes "x + v \<le> u + y"
|
wenzelm@41789
|
1476 |
and "u + v' \<le> u' + v"
|
wenzelm@41789
|
1477 |
and "x2 + v2 = u2 + y2"
|
huffman@36787
|
1478 |
shows "x + v' \<le> u' + (y::preal)"
|
huffman@36787
|
1479 |
proof -
|
huffman@36787
|
1480 |
have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
|
wenzelm@41789
|
1481 |
also have "... \<le> (u+y) + (u+v')" by (simp add: assms)
|
wenzelm@41789
|
1482 |
also have "... \<le> (u+y) + (u'+v)" by (simp add: assms)
|
huffman@36787
|
1483 |
also have "... = (u'+y) + (u+v)" by (simp add: add_ac)
|
huffman@36787
|
1484 |
finally show ?thesis by simp
|
huffman@36787
|
1485 |
qed
|
huffman@36787
|
1486 |
|
huffman@36787
|
1487 |
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
|
huffman@36787
|
1488 |
apply (cases i, cases j, cases k)
|
huffman@36787
|
1489 |
apply (simp add: real_le)
|
huffman@36787
|
1490 |
apply (blast intro: real_trans_lemma)
|
huffman@36787
|
1491 |
done
|
huffman@36787
|
1492 |
|
huffman@36787
|
1493 |
instance real :: order
|
huffman@36787
|
1494 |
proof
|
huffman@36787
|
1495 |
fix u v :: real
|
huffman@36787
|
1496 |
show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u"
|
huffman@36787
|
1497 |
by (auto simp add: real_less_def intro: real_le_antisym)
|
huffman@36787
|
1498 |
qed (assumption | rule real_le_refl real_le_trans real_le_antisym)+
|
huffman@36787
|
1499 |
|
huffman@36787
|
1500 |
(* Axiom 'linorder_linear' of class 'linorder': *)
|
huffman@36787
|
1501 |
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
|
huffman@36787
|
1502 |
apply (cases z, cases w)
|
huffman@36787
|
1503 |
apply (auto simp add: real_le real_zero_def add_ac)
|
huffman@36787
|
1504 |
done
|
huffman@36787
|
1505 |
|
huffman@36787
|
1506 |
instance real :: linorder
|
huffman@36787
|
1507 |
by (intro_classes, rule real_le_linear)
|
huffman@36787
|
1508 |
|
huffman@36787
|
1509 |
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
|
huffman@36787
|
1510 |
apply (cases x, cases y)
|
huffman@36787
|
1511 |
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
|
huffman@36787
|
1512 |
add_ac)
|
huffman@36787
|
1513 |
apply (simp_all add: add_assoc [symmetric])
|
huffman@36787
|
1514 |
done
|
huffman@36787
|
1515 |
|
huffman@36787
|
1516 |
lemma real_add_left_mono:
|
huffman@36787
|
1517 |
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
|
huffman@36787
|
1518 |
proof -
|
huffman@36787
|
1519 |
have "z + x - (z + y) = (z + -z) + (x - y)"
|
huffman@36787
|
1520 |
by (simp add: algebra_simps)
|
huffman@36787
|
1521 |
with le show ?thesis
|
haftmann@55682
|
1522 |
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"])
|
huffman@36787
|
1523 |
qed
|
huffman@36787
|
1524 |
|
huffman@36787
|
1525 |
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
|
haftmann@55682
|
1526 |
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S])
|
huffman@36787
|
1527 |
|
huffman@36787
|
1528 |
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
|
haftmann@55682
|
1529 |
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S])
|
huffman@36787
|
1530 |
|
huffman@36787
|
1531 |
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
|
huffman@36787
|
1532 |
apply (cases x, cases y)
|
huffman@36787
|
1533 |
apply (simp add: linorder_not_le [where 'a = real, symmetric]
|
huffman@36787
|
1534 |
linorder_not_le [where 'a = preal]
|
huffman@36787
|
1535 |
real_zero_def real_le real_mult)
|
huffman@36787
|
1536 |
--{*Reduce to the (simpler) @{text "\<le>"} relation *}
|
huffman@36787
|
1537 |
apply (auto dest!: less_add_left_Ex
|
huffman@36787
|
1538 |
simp add: algebra_simps preal_self_less_add_left)
|
huffman@36787
|
1539 |
done
|
huffman@36787
|
1540 |
|
huffman@36787
|
1541 |
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
|
huffman@36787
|
1542 |
apply (rule real_sum_gt_zero_less)
|
huffman@36787
|
1543 |
apply (drule real_less_sum_gt_zero [of x y])
|
huffman@36787
|
1544 |
apply (drule real_mult_order, assumption)
|
haftmann@55682
|
1545 |
apply (simp add: algebra_simps)
|
huffman@36787
|
1546 |
done
|
huffman@36787
|
1547 |
|
huffman@36787
|
1548 |
instantiation real :: distrib_lattice
|
huffman@36787
|
1549 |
begin
|
huffman@36787
|
1550 |
|
huffman@36787
|
1551 |
definition
|
huffman@36787
|
1552 |
"(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min"
|
huffman@36787
|
1553 |
|
huffman@36787
|
1554 |
definition
|
huffman@36787
|
1555 |
"(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max"
|
huffman@36787
|
1556 |
|
huffman@36787
|
1557 |
instance
|
huffman@36787
|
1558 |
by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
|
huffman@36787
|
1559 |
|
huffman@36787
|
1560 |
end
|
huffman@36787
|
1561 |
|
huffman@36787
|
1562 |
|
huffman@36787
|
1563 |
subsection{*The Reals Form an Ordered Field*}
|
huffman@36787
|
1564 |
|
huffman@36787
|
1565 |
instance real :: linordered_field_inverse_zero
|
huffman@36787
|
1566 |
proof
|
huffman@36787
|
1567 |
fix x y z :: real
|
huffman@36787
|
1568 |
show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
|
huffman@36787
|
1569 |
show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
|
huffman@36787
|
1570 |
show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
|
huffman@36787
|
1571 |
show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
|
huffman@36787
|
1572 |
by (simp only: real_sgn_def)
|
huffman@36787
|
1573 |
qed
|
huffman@36787
|
1574 |
|
huffman@36787
|
1575 |
text{*The function @{term real_of_preal} requires many proofs, but it seems
|
huffman@36787
|
1576 |
to be essential for proving completeness of the reals from that of the
|
huffman@36787
|
1577 |
positive reals.*}
|
huffman@36787
|
1578 |
|
huffman@36787
|
1579 |
lemma real_of_preal_add:
|
huffman@36787
|
1580 |
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
|
huffman@36787
|
1581 |
by (simp add: real_of_preal_def real_add algebra_simps)
|
huffman@36787
|
1582 |
|
huffman@36787
|
1583 |
lemma real_of_preal_mult:
|
huffman@36787
|
1584 |
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
|
huffman@36787
|
1585 |
by (simp add: real_of_preal_def real_mult algebra_simps)
|
huffman@36787
|
1586 |
|
huffman@36787
|
1587 |
|
huffman@36787
|
1588 |
text{*Gleason prop 9-4.4 p 127*}
|
huffman@36787
|
1589 |
lemma real_of_preal_trichotomy:
|
huffman@36787
|
1590 |
"\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
|
huffman@36787
|
1591 |
apply (simp add: real_of_preal_def real_zero_def, cases x)
|
huffman@36787
|
1592 |
apply (auto simp add: real_minus add_ac)
|
huffman@36787
|
1593 |
apply (cut_tac x = x and y = y in linorder_less_linear)
|
huffman@36787
|
1594 |
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
|
huffman@36787
|
1595 |
done
|
huffman@36787
|
1596 |
|
huffman@36787
|
1597 |
lemma real_of_preal_leD:
|
huffman@36787
|
1598 |
"real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
|
huffman@36787
|
1599 |
by (simp add: real_of_preal_def real_le)
|
huffman@36787
|
1600 |
|
huffman@36787
|
1601 |
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
|
huffman@36787
|
1602 |
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
|
huffman@36787
|
1603 |
|
huffman@36787
|
1604 |
lemma real_of_preal_lessD:
|
huffman@36787
|
1605 |
"real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
|
huffman@36787
|
1606 |
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
|
huffman@36787
|
1607 |
|
huffman@36787
|
1608 |
lemma real_of_preal_less_iff [simp]:
|
huffman@36787
|
1609 |
"(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
|
huffman@36787
|
1610 |
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
|
huffman@36787
|
1611 |
|
huffman@36787
|
1612 |
lemma real_of_preal_le_iff:
|
huffman@36787
|
1613 |
"(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
|
huffman@36787
|
1614 |
by (simp add: linorder_not_less [symmetric])
|
huffman@36787
|
1615 |
|
huffman@36787
|
1616 |
lemma real_of_preal_zero_less: "0 < real_of_preal m"
|
huffman@36787
|
1617 |
apply (insert preal_self_less_add_left [of 1 m])
|
huffman@36787
|
1618 |
apply (auto simp add: real_zero_def real_of_preal_def
|
huffman@36787
|
1619 |
real_less_def real_le_def add_ac)
|
huffman@36787
|
1620 |
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
|
huffman@36787
|
1621 |
apply (simp add: add_ac)
|
huffman@36787
|
1622 |
done
|
huffman@36787
|
1623 |
|
huffman@36787
|
1624 |
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
|
huffman@36787
|
1625 |
by (simp add: real_of_preal_zero_less)
|
huffman@36787
|
1626 |
|
huffman@36787
|
1627 |
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
|
huffman@36787
|
1628 |
proof -
|
huffman@36787
|
1629 |
from real_of_preal_minus_less_zero
|
huffman@36787
|
1630 |
show ?thesis by (blast dest: order_less_trans)
|
huffman@36787
|
1631 |
qed
|
huffman@36787
|
1632 |
|
huffman@36787
|
1633 |
|
huffman@36787
|
1634 |
subsection{*Theorems About the Ordering*}
|
huffman@36787
|
1635 |
|
huffman@36787
|
1636 |
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
|
huffman@36787
|
1637 |
apply (auto simp add: real_of_preal_zero_less)
|
huffman@36787
|
1638 |
apply (cut_tac x = x in real_of_preal_trichotomy)
|
huffman@36787
|
1639 |
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
|
huffman@36787
|
1640 |
done
|
huffman@36787
|
1641 |
|
huffman@36787
|
1642 |
lemma real_gt_preal_preal_Ex:
|
huffman@36787
|
1643 |
"real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
|
huffman@36787
|
1644 |
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
|
huffman@36787
|
1645 |
intro: real_gt_zero_preal_Ex [THEN iffD1])
|
huffman@36787
|
1646 |
|
huffman@36787
|
1647 |
lemma real_ge_preal_preal_Ex:
|
huffman@36787
|
1648 |
"real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
|
huffman@36787
|
1649 |
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
|
huffman@36787
|
1650 |
|
huffman@36787
|
1651 |
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
|
huffman@36787
|
1652 |
by (auto elim: order_le_imp_less_or_eq [THEN disjE]
|
huffman@36787
|
1653 |
intro: real_of_preal_zero_less [THEN [2] order_less_trans]
|
huffman@36787
|
1654 |
simp add: real_of_preal_zero_less)
|
huffman@36787
|
1655 |
|
huffman@36787
|
1656 |
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
|
huffman@36787
|
1657 |
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
|
huffman@36787
|
1658 |
|
huffman@36787
|
1659 |
subsection {* Completeness of Positive Reals *}
|
huffman@36787
|
1660 |
|
huffman@36787
|
1661 |
text {*
|
huffman@36787
|
1662 |
Supremum property for the set of positive reals
|
huffman@36787
|
1663 |
|
huffman@36787
|
1664 |
Let @{text "P"} be a non-empty set of positive reals, with an upper
|
huffman@36787
|
1665 |
bound @{text "y"}. Then @{text "P"} has a least upper bound
|
huffman@36787
|
1666 |
(written @{text "S"}).
|
huffman@36787
|
1667 |
|
huffman@36787
|
1668 |
FIXME: Can the premise be weakened to @{text "\<forall>x \<in> P. x\<le> y"}?
|
huffman@36787
|
1669 |
*}
|
huffman@36787
|
1670 |
|
huffman@36787
|
1671 |
lemma posreal_complete:
|
huffman@36787
|
1672 |
assumes positive_P: "\<forall>x \<in> P. (0::real) < x"
|
huffman@36787
|
1673 |
and not_empty_P: "\<exists>x. x \<in> P"
|
huffman@36787
|
1674 |
and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y"
|
huffman@36787
|
1675 |
shows "\<exists>S. \<forall>y. (\<exists>x \<in> P. y < x) = (y < S)"
|
huffman@36787
|
1676 |
proof (rule exI, rule allI)
|
huffman@36787
|
1677 |
fix y
|
huffman@36787
|
1678 |
let ?pP = "{w. real_of_preal w \<in> P}"
|
huffman@36787
|
1679 |
|
huffman@36787
|
1680 |
show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))"
|
huffman@36787
|
1681 |
proof (cases "0 < y")
|
huffman@36787
|
1682 |
assume neg_y: "\<not> 0 < y"
|
huffman@36787
|
1683 |
show ?thesis
|
huffman@36787
|
1684 |
proof
|
huffman@36787
|
1685 |
assume "\<exists>x\<in>P. y < x"
|
huffman@36787
|
1686 |
have "\<forall>x. y < real_of_preal x"
|
huffman@36787
|
1687 |
using neg_y by (rule real_less_all_real2)
|
huffman@36787
|
1688 |
thus "y < real_of_preal (psup ?pP)" ..
|
huffman@36787
|
1689 |
next
|
huffman@36787
|
1690 |
assume "y < real_of_preal (psup ?pP)"
|
huffman@36787
|
1691 |
obtain "x" where x_in_P: "x \<in> P" using not_empty_P ..
|
huffman@36787
|
1692 |
hence "0 < x" using positive_P by simp
|
huffman@36787
|
1693 |
hence "y < x" using neg_y by simp
|
huffman@36787
|
1694 |
thus "\<exists>x \<in> P. y < x" using x_in_P ..
|
huffman@36787
|
1695 |
qed
|
huffman@36787
|
1696 |
next
|
huffman@36787
|
1697 |
assume pos_y: "0 < y"
|
huffman@36787
|
1698 |
|
huffman@36787
|
1699 |
then obtain py where y_is_py: "y = real_of_preal py"
|
huffman@36787
|
1700 |
by (auto simp add: real_gt_zero_preal_Ex)
|
huffman@36787
|
1701 |
|
huffman@36787
|
1702 |
obtain a where "a \<in> P" using not_empty_P ..
|
huffman@36787
|
1703 |
with positive_P have a_pos: "0 < a" ..
|
huffman@36787
|
1704 |
then obtain pa where "a = real_of_preal pa"
|
huffman@36787
|
1705 |
by (auto simp add: real_gt_zero_preal_Ex)
|
huffman@36787
|
1706 |
hence "pa \<in> ?pP" using `a \<in> P` by auto
|
huffman@36787
|
1707 |
hence pP_not_empty: "?pP \<noteq> {}" by auto
|
huffman@36787
|
1708 |
|
huffman@36787
|
1709 |
obtain sup where sup: "\<forall>x \<in> P. x < sup"
|
huffman@36787
|
1710 |
using upper_bound_Ex ..
|
huffman@36787
|
1711 |
from this and `a \<in> P` have "a < sup" ..
|
huffman@36787
|
1712 |
hence "0 < sup" using a_pos by arith
|
huffman@36787
|
1713 |
then obtain possup where "sup = real_of_preal possup"
|
huffman@36787
|
1714 |
by (auto simp add: real_gt_zero_preal_Ex)
|
huffman@36787
|
1715 |
hence "\<forall>X \<in> ?pP. X \<le> possup"
|
huffman@36787
|
1716 |
using sup by (auto simp add: real_of_preal_lessI)
|
huffman@36787
|
1717 |
with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)"
|
huffman@36787
|
1718 |
by (rule preal_complete)
|
huffman@36787
|
1719 |
|
huffman@36787
|
1720 |
show ?thesis
|
huffman@36787
|
1721 |
proof
|
huffman@36787
|
1722 |
assume "\<exists>x \<in> P. y < x"
|
huffman@36787
|
1723 |
then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" ..
|
huffman@36787
|
1724 |
hence "0 < x" using pos_y by arith
|
huffman@36787
|
1725 |
then obtain px where x_is_px: "x = real_of_preal px"
|
huffman@36787
|
1726 |
by (auto simp add: real_gt_zero_preal_Ex)
|
huffman@36787
|
1727 |
|
huffman@36787
|
1728 |
have py_less_X: "\<exists>X \<in> ?pP. py < X"
|
huffman@36787
|
1729 |
proof
|
huffman@36787
|
1730 |
show "py < px" using y_is_py and x_is_px and y_less_x
|
huffman@36787
|
1731 |
by (simp add: real_of_preal_lessI)
|
huffman@36787
|
1732 |
show "px \<in> ?pP" using x_in_P and x_is_px by simp
|
huffman@36787
|
1733 |
qed
|
huffman@36787
|
1734 |
|
huffman@36787
|
1735 |
have "(\<exists>X \<in> ?pP. py < X) ==> (py < psup ?pP)"
|
huffman@36787
|
1736 |
using psup by simp
|
huffman@36787
|
1737 |
hence "py < psup ?pP" using py_less_X by simp
|
huffman@36787
|
1738 |
thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})"
|
huffman@36787
|
1739 |
using y_is_py and pos_y by (simp add: real_of_preal_lessI)
|
huffman@36787
|
1740 |
next
|
huffman@36787
|
1741 |
assume y_less_psup: "y < real_of_preal (psup ?pP)"
|
huffman@36787
|
1742 |
|
huffman@36787
|
1743 |
hence "py < psup ?pP" using y_is_py
|
huffman@36787
|
1744 |
by (simp add: real_of_preal_lessI)
|
huffman@36787
|
1745 |
then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP"
|
huffman@36787
|
1746 |
using psup by auto
|
huffman@36787
|
1747 |
then obtain x where x_is_X: "x = real_of_preal X"
|
huffman@36787
|
1748 |
by (simp add: real_gt_zero_preal_Ex)
|
huffman@36787
|
1749 |
hence "y < x" using py_less_X and y_is_py
|
huffman@36787
|
1750 |
by (simp add: real_of_preal_lessI)
|
huffman@36787
|
1751 |
|
huffman@36787
|
1752 |
moreover have "x \<in> P" using x_is_X and X_in_pP by simp
|
huffman@36787
|
1753 |
|
huffman@36787
|
1754 |
ultimately show "\<exists> x \<in> P. y < x" ..
|
huffman@36787
|
1755 |
qed
|
huffman@36787
|
1756 |
qed
|
huffman@36787
|
1757 |
qed
|
huffman@36787
|
1758 |
|
huffman@36787
|
1759 |
text {*
|
hoelzl@55715
|
1760 |
\medskip Completeness
|
huffman@36787
|
1761 |
*}
|
huffman@36787
|
1762 |
|
huffman@36787
|
1763 |
lemma reals_complete:
|
hoelzl@55715
|
1764 |
fixes S :: "real set"
|
huffman@36787
|
1765 |
assumes notempty_S: "\<exists>X. X \<in> S"
|
hoelzl@55715
|
1766 |
and exists_Ub: "bdd_above S"
|
hoelzl@55715
|
1767 |
shows "\<exists>x. (\<forall>s\<in>S. s \<le> x) \<and> (\<forall>y. (\<forall>s\<in>S. s \<le> y) \<longrightarrow> x \<le> y)"
|
huffman@36787
|
1768 |
proof -
|
huffman@36787
|
1769 |
obtain X where X_in_S: "X \<in> S" using notempty_S ..
|
hoelzl@55715
|
1770 |
obtain Y where Y_isUb: "\<forall>s\<in>S. s \<le> Y"
|
hoelzl@55715
|
1771 |
using exists_Ub by (auto simp: bdd_above_def)
|
huffman@36787
|
1772 |
let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}"
|
huffman@36787
|
1773 |
|
huffman@36787
|
1774 |
{
|
huffman@36787
|
1775 |
fix x
|
hoelzl@55715
|
1776 |
assume S_le_x: "\<forall>s\<in>S. s \<le> x"
|
huffman@36787
|
1777 |
{
|
huffman@36787
|
1778 |
fix s
|
huffman@36787
|
1779 |
assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}"
|
huffman@36787
|
1780 |
hence "\<exists> x \<in> S. s = x + -X + 1" ..
|
wenzelm@54510
|
1781 |
then obtain x1 where x1: "x1 \<in> S" "s = x1 + (-X) + 1" ..
|
wenzelm@54510
|
1782 |
then have "x1 \<le> x" using S_le_x by simp
|
wenzelm@54510
|
1783 |
with x1 have "s \<le> x + - X + 1" by arith
|
huffman@36787
|
1784 |
}
|
hoelzl@55715
|
1785 |
then have "\<forall>s\<in>?SHIFT. s \<le> x + (-X) + 1"
|
hoelzl@55715
|
1786 |
by auto
|
huffman@36787
|
1787 |
} note S_Ub_is_SHIFT_Ub = this
|
huffman@36787
|
1788 |
|
hoelzl@55715
|
1789 |
have *: "\<forall>s\<in>?SHIFT. s \<le> Y + (-X) + 1" using Y_isUb by (rule S_Ub_is_SHIFT_Ub)
|
hoelzl@55715
|
1790 |
have "\<forall>s\<in>?SHIFT. s < Y + (-X) + 2"
|
hoelzl@55715
|
1791 |
proof
|
hoelzl@55715
|
1792 |
fix s assume "s\<in>?SHIFT"
|
hoelzl@55715
|
1793 |
with * have "s \<le> Y + (-X) + 1" by simp
|
hoelzl@55715
|
1794 |
also have "\<dots> < Y + (-X) + 2" by simp
|
hoelzl@55715
|
1795 |
finally show "s < Y + (-X) + 2" .
|
hoelzl@55715
|
1796 |
qed
|
huffman@36787
|
1797 |
moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto
|
huffman@36787
|
1798 |
moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT"
|
huffman@36787
|
1799 |
using X_in_S and Y_isUb by auto
|
hoelzl@55715
|
1800 |
ultimately obtain t where t_is_Lub: "\<forall>y. (\<exists>x\<in>?SHIFT. y < x) = (y < t)"
|
hoelzl@55715
|
1801 |
using posreal_complete [of ?SHIFT] unfolding bdd_above_def by blast
|
huffman@36787
|
1802 |
|
huffman@36787
|
1803 |
show ?thesis
|
huffman@36787
|
1804 |
proof
|
hoelzl@55715
|
1805 |
show "(\<forall>s\<in>S. s \<le> (t + X + (-1))) \<and> (\<forall>y. (\<forall>s\<in>S. s \<le> y) \<longrightarrow> (t + X + (-1)) \<le> y)"
|
hoelzl@55715
|
1806 |
proof safe
|
hoelzl@55715
|
1807 |
fix x
|
hoelzl@55715
|
1808 |
assume "\<forall>s\<in>S. s \<le> x"
|
hoelzl@55715
|
1809 |
hence "\<forall>s\<in>?SHIFT. s \<le> x + (-X) + 1"
|
hoelzl@55715
|
1810 |
using S_Ub_is_SHIFT_Ub by simp
|
hoelzl@55715
|
1811 |
then have "\<not> x + (-X) + 1 < t"
|
hoelzl@55715
|
1812 |
by (subst t_is_Lub[rule_format, symmetric]) (simp add: not_less)
|
hoelzl@55715
|
1813 |
thus "t + X + -1 \<le> x" by arith
|
huffman@36787
|
1814 |
next
|
hoelzl@55715
|
1815 |
fix y
|
hoelzl@55715
|
1816 |
assume y_in_S: "y \<in> S"
|
hoelzl@55715
|
1817 |
obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty ..
|
hoelzl@55715
|
1818 |
hence "\<exists> x \<in> S. u = x + - X + 1" by simp
|
hoelzl@55715
|
1819 |
then obtain "x" where x_and_u: "u = x + - X + 1" ..
|
hoelzl@55715
|
1820 |
have u_le_t: "u \<le> t"
|
hoelzl@55715
|
1821 |
proof (rule dense_le)
|
hoelzl@55715
|
1822 |
fix x assume "x < u" then have "x < t"
|
hoelzl@55715
|
1823 |
using u_in_shift t_is_Lub by auto
|
hoelzl@55715
|
1824 |
then show "x \<le> t" by simp
|
hoelzl@55715
|
1825 |
qed
|
huffman@36787
|
1826 |
|
hoelzl@55715
|
1827 |
show "y \<le> t + X + -1"
|
hoelzl@55715
|
1828 |
proof cases
|
hoelzl@55715
|
1829 |
assume "y \<le> x"
|
hoelzl@55715
|
1830 |
moreover have "x = u + X + - 1" using x_and_u by arith
|
hoelzl@55715
|
1831 |
moreover have "u + X + - 1 \<le> t + X + -1" using u_le_t by arith
|
hoelzl@55715
|
1832 |
ultimately show "y \<le> t + X + -1" by arith
|
hoelzl@55715
|
1833 |
next
|
hoelzl@55715
|
1834 |
assume "~(y \<le> x)"
|
hoelzl@55715
|
1835 |
hence x_less_y: "x < y" by arith
|
huffman@36787
|
1836 |
|
hoelzl@55715
|
1837 |
have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp
|
hoelzl@55715
|
1838 |
hence "0 < x + (-X) + 1" by simp
|
hoelzl@55715
|
1839 |
hence "0 < y + (-X) + 1" using x_less_y by arith
|
hoelzl@55715
|
1840 |
hence *: "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp
|
hoelzl@55715
|
1841 |
have "y + (-X) + 1 \<le> t"
|
hoelzl@55715
|
1842 |
proof (rule dense_le)
|
hoelzl@55715
|
1843 |
fix x assume "x < y + (-X) + 1" then have "x < t"
|
hoelzl@55715
|
1844 |
using * t_is_Lub by auto
|
hoelzl@55715
|
1845 |
then show "x \<le> t" by simp
|
hoelzl@55715
|
1846 |
qed
|
hoelzl@55715
|
1847 |
thus ?thesis by simp
|
huffman@36787
|
1848 |
qed
|
huffman@36787
|
1849 |
qed
|
huffman@36787
|
1850 |
qed
|
huffman@36787
|
1851 |
qed
|
huffman@36787
|
1852 |
|
huffman@36787
|
1853 |
subsection {* The Archimedean Property of the Reals *}
|
huffman@36787
|
1854 |
|
huffman@36787
|
1855 |
theorem reals_Archimedean:
|
huffman@36787
|
1856 |
fixes x :: real
|
huffman@36787
|
1857 |
assumes x_pos: "0 < x"
|
huffman@36787
|
1858 |
shows "\<exists>n. inverse (of_nat (Suc n)) < x"
|
huffman@36787
|
1859 |
proof (rule ccontr)
|
huffman@36787
|
1860 |
assume contr: "\<not> ?thesis"
|
huffman@36787
|
1861 |
have "\<forall>n. x * of_nat (Suc n) <= 1"
|
huffman@36787
|
1862 |
proof
|
huffman@36787
|
1863 |
fix n
|
huffman@36787
|
1864 |
from contr have "x \<le> inverse (of_nat (Suc n))"
|
huffman@36787
|
1865 |
by (simp add: linorder_not_less)
|
huffman@36787
|
1866 |
hence "x \<le> (1 / (of_nat (Suc n)))"
|
huffman@36787
|
1867 |
by (simp add: inverse_eq_divide)
|
huffman@36787
|
1868 |
moreover have "(0::real) \<le> of_nat (Suc n)"
|
huffman@36787
|
1869 |
by (rule of_nat_0_le_iff)
|
huffman@36787
|
1870 |
ultimately have "x * of_nat (Suc n) \<le> (1 / of_nat (Suc n)) * of_nat (Suc n)"
|
huffman@36787
|
1871 |
by (rule mult_right_mono)
|
huffman@36787
|
1872 |
thus "x * of_nat (Suc n) \<le> 1" by (simp del: of_nat_Suc)
|
huffman@36787
|
1873 |
qed
|
hoelzl@55715
|
1874 |
hence 2: "bdd_above {z. \<exists>n. z = x * (of_nat (Suc n))}"
|
hoelzl@55715
|
1875 |
by (auto intro!: bdd_aboveI[of _ 1])
|
hoelzl@55715
|
1876 |
have 1: "\<exists>X. X \<in> {z. \<exists>n. z = x* (of_nat (Suc n))}" by auto
|
hoelzl@55715
|
1877 |
obtain t where
|
hoelzl@55715
|
1878 |
upper: "\<And>z. z \<in> {z. \<exists>n. z = x * of_nat (Suc n)} \<Longrightarrow> z \<le> t" and
|
hoelzl@55715
|
1879 |
least: "\<And>y. (\<And>a. a \<in> {z. \<exists>n. z = x * of_nat (Suc n)} \<Longrightarrow> a \<le> y) \<Longrightarrow> t \<le> y"
|
hoelzl@55715
|
1880 |
using reals_complete[OF 1 2] by auto
|
huffman@36787
|
1881 |
|
hoelzl@55715
|
1882 |
|
hoelzl@55715
|
1883 |
have "t \<le> t + - x"
|
hoelzl@55715
|
1884 |
proof (rule least)
|
hoelzl@55715
|
1885 |
fix a assume a: "a \<in> {z. \<exists>n. z = x * (of_nat (Suc n))}"
|
hoelzl@55715
|
1886 |
have "\<forall>n::nat. x * of_nat n \<le> t + - x"
|
hoelzl@55715
|
1887 |
proof
|
hoelzl@55715
|
1888 |
fix n
|
hoelzl@55715
|
1889 |
have "x * of_nat (Suc n) \<le> t"
|
hoelzl@55715
|
1890 |
by (simp add: upper)
|
hoelzl@55715
|
1891 |
hence "x * (of_nat n) + x \<le> t"
|
hoelzl@55715
|
1892 |
by (simp add: distrib_left)
|
hoelzl@55715
|
1893 |
thus "x * (of_nat n) \<le> t + - x" by arith
|
hoelzl@55715
|
1894 |
qed hence "\<forall>m. x * of_nat (Suc m) \<le> t + - x" by (simp del: of_nat_Suc)
|
hoelzl@55715
|
1895 |
with a show "a \<le> t + - x"
|
hoelzl@55715
|
1896 |
by auto
|
huffman@36787
|
1897 |
qed
|
huffman@36787
|
1898 |
thus False using x_pos by arith
|
huffman@36787
|
1899 |
qed
|
huffman@36787
|
1900 |
|
huffman@36787
|
1901 |
text {*
|
haftmann@37363
|
1902 |
There must be other proofs, e.g. @{text Suc} of the largest
|
huffman@36787
|
1903 |
integer in the cut representing @{text "x"}.
|
huffman@36787
|
1904 |
*}
|
huffman@36787
|
1905 |
|
huffman@36787
|
1906 |
lemma reals_Archimedean2: "\<exists>n. (x::real) < of_nat (n::nat)"
|
huffman@36787
|
1907 |
proof cases
|
huffman@36787
|
1908 |
assume "x \<le> 0"
|
huffman@36787
|
1909 |
hence "x < of_nat (1::nat)" by simp
|
huffman@36787
|
1910 |
thus ?thesis ..
|
huffman@36787
|
1911 |
next
|
huffman@36787
|
1912 |
assume "\<not> x \<le> 0"
|
huffman@36787
|
1913 |
hence x_greater_zero: "0 < x" by simp
|
huffman@36787
|
1914 |
hence "0 < inverse x" by simp
|
huffman@36787
|
1915 |
then obtain n where "inverse (of_nat (Suc n)) < inverse x"
|
huffman@36787
|
1916 |
using reals_Archimedean by blast
|
huffman@36787
|
1917 |
hence "inverse (of_nat (Suc n)) * x < inverse x * x"
|
huffman@36787
|
1918 |
using x_greater_zero by (rule mult_strict_right_mono)
|
huffman@36787
|
1919 |
hence "inverse (of_nat (Suc n)) * x < 1"
|
huffman@36787
|
1920 |
using x_greater_zero by simp
|
huffman@36787
|
1921 |
hence "of_nat (Suc n) * (inverse (of_nat (Suc n)) * x) < of_nat (Suc n) * 1"
|
huffman@36787
|
1922 |
by (rule mult_strict_left_mono) (simp del: of_nat_Suc)
|
huffman@36787
|
1923 |
hence "x < of_nat (Suc n)"
|
huffman@36787
|
1924 |
by (simp add: algebra_simps del: of_nat_Suc)
|
huffman@36787
|
1925 |
thus "\<exists>(n::nat). x < of_nat n" ..
|
huffman@36787
|
1926 |
qed
|
huffman@36787
|
1927 |
|
huffman@36787
|
1928 |
instance real :: archimedean_field
|
huffman@36787
|
1929 |
proof
|
huffman@36787
|
1930 |
fix r :: real
|
huffman@36787
|
1931 |
obtain n :: nat where "r < of_nat n"
|
huffman@36787
|
1932 |
using reals_Archimedean2 ..
|
huffman@36787
|
1933 |
then have "r \<le> of_int (int n)"
|
huffman@36787
|
1934 |
by simp
|
huffman@36787
|
1935 |
then show "\<exists>z. r \<le> of_int z" ..
|
huffman@36787
|
1936 |
qed
|
huffman@36787
|
1937 |
|
huffman@36787
|
1938 |
end
|