author | huffman |
Thu, 18 Feb 2010 14:21:44 -0800 | |
changeset 35216 | 7641e8d831d2 |
parent 35050 | 9f841f20dca6 |
child 35372 | ca158c7b1144 |
permissions | -rw-r--r-- |
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1 |
(* Title: HOL/PReal.thy |
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Author: Jacques D. Fleuriot, University of Cambridge |
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The positive reals as Dedekind sections of positive |
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rationals. Fundamentals of Abstract Analysis [Gleason- p. 121] |
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provides some of the definitions. |
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*) |
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header {* Positive real numbers *} |
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theory PReal |
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imports Rational |
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begin |
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|
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text{*Could be generalized and moved to @{text Groups}*} |
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lemma add_eq_exists: "\<exists>x. a+x = (b::rat)" |
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by (rule_tac x="b-a" in exI, simp) |
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|
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definition |
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cut :: "rat set => bool" where |
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[code del]: "cut A = ({} \<subset> A & |
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A < {r. 0 < r} & |
23 |
(\<forall>y \<in> A. ((\<forall>z. 0<z & z < y --> z \<in> A) & (\<exists>u \<in> A. y < u))))" |
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|
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lemma interval_empty_iff: |
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"{y. (x::'a::dense_linorder) < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z" |
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by (auto dest: dense) |
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|
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lemma cut_of_rat: |
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assumes q: "0 < q" shows "cut {r::rat. 0 < r & r < q}" (is "cut ?A") |
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proof - |
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from q have pos: "?A < {r. 0 < r}" by force |
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have nonempty: "{} \<subset> ?A" |
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proof |
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show "{} \<subseteq> ?A" by simp |
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show "{} \<noteq> ?A" |
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by (force simp only: q eq_commute [of "{}"] interval_empty_iff) |
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qed |
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show ?thesis |
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by (simp add: cut_def pos nonempty, |
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blast dest: dense intro: order_less_trans) |
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qed |
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44 |
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typedef preal = "{A. cut A}" |
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by (blast intro: cut_of_rat [OF zero_less_one]) |
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definition |
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preal_of_rat :: "rat => preal" where |
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"preal_of_rat q = Abs_preal {x::rat. 0 < x & x < q}" |
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definition |
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psup :: "preal set => preal" where |
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[code del]: "psup P = Abs_preal (\<Union>X \<in> P. Rep_preal X)" |
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definition |
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add_set :: "[rat set,rat set] => rat set" where |
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"add_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}" |
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definition |
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diff_set :: "[rat set,rat set] => rat set" where |
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[code del]: "diff_set A B = {w. \<exists>x. 0 < w & 0 < x & x \<notin> B & x + w \<in> A}" |
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definition |
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mult_set :: "[rat set,rat set] => rat set" where |
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"mult_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}" |
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definition |
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inverse_set :: "rat set => rat set" where |
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[code del]: "inverse_set A = {x. \<exists>y. 0 < x & x < y & inverse y \<notin> A}" |
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instantiation preal :: "{ord, plus, minus, times, inverse, one}" |
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begin |
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definition |
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preal_less_def [code del]: |
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"R < S == Rep_preal R < Rep_preal S" |
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definition |
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preal_le_def [code del]: |
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"R \<le> S == Rep_preal R \<subseteq> Rep_preal S" |
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definition |
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preal_add_def: |
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"R + S == Abs_preal (add_set (Rep_preal R) (Rep_preal S))" |
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definition |
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preal_diff_def: |
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"R - S == Abs_preal (diff_set (Rep_preal R) (Rep_preal S))" |
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definition |
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preal_mult_def: |
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"R * S == Abs_preal (mult_set (Rep_preal R) (Rep_preal S))" |
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definition |
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preal_inverse_def: |
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"inverse R == Abs_preal (inverse_set (Rep_preal R))" |
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definition "R / S = R * inverse (S\<Colon>preal)" |
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definition |
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preal_one_def: |
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"1 == preal_of_rat 1" |
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instance .. |
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end |
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text{*Reduces equality on abstractions to equality on representatives*} |
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declare Abs_preal_inject [simp] |
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declare Abs_preal_inverse [simp] |
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lemma rat_mem_preal: "0 < q ==> {r::rat. 0 < r & r < q} \<in> preal" |
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by (simp add: preal_def cut_of_rat) |
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lemma preal_nonempty: "A \<in> preal ==> \<exists>x\<in>A. 0 < x" |
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by (unfold preal_def cut_def, blast) |
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lemma preal_Ex_mem: "A \<in> preal \<Longrightarrow> \<exists>x. x \<in> A" |
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by (drule preal_nonempty, fast) |
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lemma preal_imp_psubset_positives: "A \<in> preal ==> A < {r. 0 < r}" |
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by (force simp add: preal_def cut_def) |
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lemma preal_exists_bound: "A \<in> preal ==> \<exists>x. 0 < x & x \<notin> A" |
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by (drule preal_imp_psubset_positives, auto) |
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lemma preal_exists_greater: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u" |
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by (unfold preal_def cut_def, blast) |
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lemma preal_downwards_closed: "[| A \<in> preal; y \<in> A; 0 < z; z < y |] ==> z \<in> A" |
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by (unfold preal_def cut_def, blast) |
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text{*Relaxing the final premise*} |
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lemma preal_downwards_closed': |
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"[| A \<in> preal; y \<in> A; 0 < z; z \<le> y |] ==> z \<in> A" |
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apply (simp add: order_le_less) |
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apply (blast intro: preal_downwards_closed) |
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141 |
done |
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text{*A positive fraction not in a positive real is an upper bound. |
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Gleason p. 122 - Remark (1)*} |
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lemma not_in_preal_ub: |
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assumes A: "A \<in> preal" |
148 |
and notx: "x \<notin> A" |
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149 |
and y: "y \<in> A" |
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and pos: "0 < x" |
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shows "y < x" |
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proof (cases rule: linorder_cases) |
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assume "x<y" |
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with notx show ?thesis |
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by (simp add: preal_downwards_closed [OF A y] pos) |
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156 |
next |
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assume "x=y" |
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with notx and y show ?thesis by simp |
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159 |
next |
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assume "y<x" |
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thus ?thesis . |
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162 |
qed |
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163 |
|
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text {* preal lemmas instantiated to @{term "Rep_preal X"} *} |
165 |
||
166 |
lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X" |
|
167 |
by (rule preal_Ex_mem [OF Rep_preal]) |
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168 |
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169 |
lemma Rep_preal_exists_bound: "\<exists>x>0. x \<notin> Rep_preal X" |
|
170 |
by (rule preal_exists_bound [OF Rep_preal]) |
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171 |
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172 |
lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF Rep_preal] |
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174 |
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176 |
subsection{*@{term preal_of_prat}: the Injection from prat to preal*} |
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177 |
||
178 |
lemma rat_less_set_mem_preal: "0 < y ==> {u::rat. 0 < u & u < y} \<in> preal" |
|
179 |
by (simp add: preal_def cut_of_rat) |
|
180 |
||
181 |
lemma rat_subset_imp_le: |
|
182 |
"[|{u::rat. 0 < u & u < x} \<subseteq> {u. 0 < u & u < y}; 0<x|] ==> x \<le> y" |
|
183 |
apply (simp add: linorder_not_less [symmetric]) |
|
184 |
apply (blast dest: dense intro: order_less_trans) |
|
185 |
done |
|
186 |
||
187 |
lemma rat_set_eq_imp_eq: |
|
188 |
"[|{u::rat. 0 < u & u < x} = {u. 0 < u & u < y}; |
|
189 |
0 < x; 0 < y|] ==> x = y" |
|
190 |
by (blast intro: rat_subset_imp_le order_antisym) |
|
191 |
||
192 |
||
193 |
||
194 |
subsection{*Properties of Ordering*} |
|
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195 |
|
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196 |
instance preal :: order |
27682 | 197 |
proof |
198 |
fix w :: preal |
|
199 |
show "w \<le> w" by (simp add: preal_le_def) |
|
200 |
next |
|
201 |
fix i j k :: preal |
|
202 |
assume "i \<le> j" and "j \<le> k" |
|
203 |
then show "i \<le> k" by (simp add: preal_le_def) |
|
204 |
next |
|
205 |
fix z w :: preal |
|
206 |
assume "z \<le> w" and "w \<le> z" |
|
207 |
then show "z = w" by (simp add: preal_le_def Rep_preal_inject) |
|
208 |
next |
|
209 |
fix z w :: preal |
|
210 |
show "z < w \<longleftrightarrow> z \<le> w \<and> \<not> w \<le> z" |
|
211 |
by (auto simp add: preal_le_def preal_less_def Rep_preal_inject) |
|
212 |
qed |
|
14335 | 213 |
|
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|
214 |
lemma preal_imp_pos: "[|A \<in> preal; r \<in> A|] ==> 0 < r" |
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215 |
by (insert preal_imp_psubset_positives, blast) |
14335 | 216 |
|
27682 | 217 |
instance preal :: linorder |
218 |
proof |
|
219 |
fix x y :: preal |
|
220 |
show "x <= y | y <= x" |
|
221 |
apply (auto simp add: preal_le_def) |
|
222 |
apply (rule ccontr) |
|
223 |
apply (blast dest: not_in_Rep_preal_ub intro: preal_imp_pos [OF Rep_preal] |
|
14365
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224 |
elim: order_less_asym) |
27682 | 225 |
done |
226 |
qed |
|
14335 | 227 |
|
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|
228 |
instantiation preal :: distrib_lattice |
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changeset
|
229 |
begin |
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|
230 |
|
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changeset
|
231 |
definition |
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|
232 |
"(inf \<Colon> preal \<Rightarrow> preal \<Rightarrow> preal) = min" |
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changeset
|
233 |
|
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parents:
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changeset
|
234 |
definition |
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parents:
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changeset
|
235 |
"(sup \<Colon> preal \<Rightarrow> preal \<Rightarrow> preal) = max" |
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|
236 |
|
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|
237 |
instance |
22483 | 238 |
by intro_classes |
239 |
(auto simp add: inf_preal_def sup_preal_def min_max.sup_inf_distrib1) |
|
14335 | 240 |
|
25571
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changeset
|
241 |
end |
14335 | 242 |
|
243 |
subsection{*Properties of Addition*} |
|
244 |
||
245 |
lemma preal_add_commute: "(x::preal) + y = y + x" |
|
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|
246 |
apply (unfold preal_add_def add_set_def) |
14335 | 247 |
apply (rule_tac f = Abs_preal in arg_cong) |
14365
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|
248 |
apply (force simp add: add_commute) |
14335 | 249 |
done |
250 |
||
14365
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changeset
|
251 |
text{*Lemmas for proving that addition of two positive reals gives |
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changeset
|
252 |
a positive real*} |
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changeset
|
253 |
|
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changeset
|
254 |
lemma empty_psubset_nonempty: "a \<in> A ==> {} \<subset> A" |
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parents:
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changeset
|
255 |
by blast |
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parents:
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changeset
|
256 |
|
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changeset
|
257 |
text{*Part 1 of Dedekind sections definition*} |
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changeset
|
258 |
lemma add_set_not_empty: |
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changeset
|
259 |
"[|A \<in> preal; B \<in> preal|] ==> {} \<subset> add_set A B" |
20495 | 260 |
apply (drule preal_nonempty)+ |
14365
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parents:
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changeset
|
261 |
apply (auto simp add: add_set_def) |
14335 | 262 |
done |
263 |
||
14365
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parents:
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changeset
|
264 |
text{*Part 2 of Dedekind sections definition. A structured version of |
3d4df8c166ae
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parents:
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diff
changeset
|
265 |
this proof is @{text preal_not_mem_mult_set_Ex} below.*} |
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changeset
|
266 |
lemma preal_not_mem_add_set_Ex: |
20495 | 267 |
"[|A \<in> preal; B \<in> preal|] ==> \<exists>q>0. q \<notin> add_set A B" |
14365
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parents:
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diff
changeset
|
268 |
apply (insert preal_exists_bound [of A] preal_exists_bound [of B], auto) |
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parents:
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diff
changeset
|
269 |
apply (rule_tac x = "x+xa" in exI) |
3d4df8c166ae
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paulson
parents:
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diff
changeset
|
270 |
apply (simp add: add_set_def, clarify) |
20495 | 271 |
apply (drule (3) not_in_preal_ub)+ |
14365
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parents:
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changeset
|
272 |
apply (force dest: add_strict_mono) |
14335 | 273 |
done |
274 |
||
14365
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parents:
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changeset
|
275 |
lemma add_set_not_rat_set: |
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diff
changeset
|
276 |
assumes A: "A \<in> preal" |
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changeset
|
277 |
and B: "B \<in> preal" |
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parents:
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diff
changeset
|
278 |
shows "add_set A B < {r. 0 < r}" |
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parents:
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changeset
|
279 |
proof |
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changeset
|
280 |
from preal_imp_pos [OF A] preal_imp_pos [OF B] |
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parents:
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diff
changeset
|
281 |
show "add_set A B \<subseteq> {r. 0 < r}" by (force simp add: add_set_def) |
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parents:
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diff
changeset
|
282 |
next |
3d4df8c166ae
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parents:
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diff
changeset
|
283 |
show "add_set A B \<noteq> {r. 0 < r}" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
284 |
by (insert preal_not_mem_add_set_Ex [OF A B], blast) |
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parents:
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diff
changeset
|
285 |
qed |
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset
|
286 |
|
14335 | 287 |
text{*Part 3 of Dedekind sections definition*} |
14365
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paulson
parents:
14335
diff
changeset
|
288 |
lemma add_set_lemma3: |
3d4df8c166ae
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paulson
parents:
14335
diff
changeset
|
289 |
"[|A \<in> preal; B \<in> preal; u \<in> add_set A B; 0 < z; z < u|] |
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paulson
parents:
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diff
changeset
|
290 |
==> z \<in> add_set A B" |
3d4df8c166ae
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paulson
parents:
14335
diff
changeset
|
291 |
proof (unfold add_set_def, clarify) |
3d4df8c166ae
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paulson
parents:
14335
diff
changeset
|
292 |
fix x::rat and y::rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset
|
293 |
assume A: "A \<in> preal" |
19765 | 294 |
and B: "B \<in> preal" |
295 |
and [simp]: "0 < z" |
|
296 |
and zless: "z < x + y" |
|
297 |
and x: "x \<in> A" |
|
298 |
and y: "y \<in> B" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
299 |
have xpos [simp]: "0<x" by (rule preal_imp_pos [OF A x]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
300 |
have ypos [simp]: "0<y" by (rule preal_imp_pos [OF B y]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
301 |
have xypos [simp]: "0 < x+y" by (simp add: pos_add_strict) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset
|
302 |
let ?f = "z/(x+y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
303 |
have fless: "?f < 1" by (simp add: zless pos_divide_less_eq) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
304 |
show "\<exists>x' \<in> A. \<exists>y'\<in>B. z = x' + y'" |
20495 | 305 |
proof (intro bexI) |
306 |
show "z = x*?f + y*?f" |
|
307 |
by (simp add: left_distrib [symmetric] divide_inverse mult_ac |
|
308 |
order_less_imp_not_eq2) |
|
309 |
next |
|
310 |
show "y * ?f \<in> B" |
|
311 |
proof (rule preal_downwards_closed [OF B y]) |
|
312 |
show "0 < y * ?f" |
|
313 |
by (simp add: divide_inverse zero_less_mult_iff) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
314 |
next |
20495 | 315 |
show "y * ?f < y" |
316 |
by (insert mult_strict_left_mono [OF fless ypos], simp) |
|
14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
317 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
318 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
319 |
show "x * ?f \<in> A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
320 |
proof (rule preal_downwards_closed [OF A x]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
321 |
show "0 < x * ?f" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
322 |
by (simp add: divide_inverse zero_less_mult_iff) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
323 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
324 |
show "x * ?f < x" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
325 |
by (insert mult_strict_left_mono [OF fless xpos], simp) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
326 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
327 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
328 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
329 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
330 |
text{*Part 4 of Dedekind sections definition*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
331 |
lemma add_set_lemma4: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
332 |
"[|A \<in> preal; B \<in> preal; y \<in> add_set A B|] ==> \<exists>u \<in> add_set A B. y < u" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
333 |
apply (auto simp add: add_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
334 |
apply (frule preal_exists_greater [of A], auto) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
335 |
apply (rule_tac x="u + y" in exI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
336 |
apply (auto intro: add_strict_left_mono) |
14335 | 337 |
done |
338 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
339 |
lemma mem_add_set: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
340 |
"[|A \<in> preal; B \<in> preal|] ==> add_set A B \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
341 |
apply (simp (no_asm_simp) add: preal_def cut_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
342 |
apply (blast intro!: add_set_not_empty add_set_not_rat_set |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
343 |
add_set_lemma3 add_set_lemma4) |
14335 | 344 |
done |
345 |
||
346 |
lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
347 |
apply (simp add: preal_add_def mem_add_set Rep_preal) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
348 |
apply (force simp add: add_set_def add_ac) |
14335 | 349 |
done |
350 |
||
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
351 |
instance preal :: ab_semigroup_add |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
352 |
proof |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
353 |
fix a b c :: preal |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
354 |
show "(a + b) + c = a + (b + c)" by (rule preal_add_assoc) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
355 |
show "a + b = b + a" by (rule preal_add_commute) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
356 |
qed |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
357 |
|
14335 | 358 |
lemma preal_add_left_commute: "x + (y + z) = y + ((x + z)::preal)" |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
359 |
by (rule add_left_commute) |
14335 | 360 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
361 |
text{* Positive Real addition is an AC operator *} |
14335 | 362 |
lemmas preal_add_ac = preal_add_assoc preal_add_commute preal_add_left_commute |
363 |
||
364 |
||
365 |
subsection{*Properties of Multiplication*} |
|
366 |
||
367 |
text{*Proofs essentially same as for addition*} |
|
368 |
||
369 |
lemma preal_mult_commute: "(x::preal) * y = y * x" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
370 |
apply (unfold preal_mult_def mult_set_def) |
14335 | 371 |
apply (rule_tac f = Abs_preal in arg_cong) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
372 |
apply (force simp add: mult_commute) |
14335 | 373 |
done |
374 |
||
15055 | 375 |
text{*Multiplication of two positive reals gives a positive real.*} |
14335 | 376 |
|
377 |
text{*Lemmas for proving positive reals multiplication set in @{typ preal}*} |
|
378 |
||
379 |
text{*Part 1 of Dedekind sections definition*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
380 |
lemma mult_set_not_empty: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
381 |
"[|A \<in> preal; B \<in> preal|] ==> {} \<subset> mult_set A B" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
382 |
apply (insert preal_nonempty [of A] preal_nonempty [of B]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
383 |
apply (auto simp add: mult_set_def) |
14335 | 384 |
done |
385 |
||
386 |
text{*Part 2 of Dedekind sections definition*} |
|
387 |
lemma preal_not_mem_mult_set_Ex: |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
388 |
assumes A: "A \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
389 |
and B: "B \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
390 |
shows "\<exists>q. 0 < q & q \<notin> mult_set A B" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
391 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
392 |
from preal_exists_bound [OF A] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
393 |
obtain x where [simp]: "0 < x" "x \<notin> A" by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
394 |
from preal_exists_bound [OF B] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
395 |
obtain y where [simp]: "0 < y" "y \<notin> B" by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
396 |
show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
397 |
proof (intro exI conjI) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
15413
diff
changeset
|
398 |
show "0 < x*y" by (simp add: mult_pos_pos) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
399 |
show "x * y \<notin> mult_set A B" |
14377 | 400 |
proof - |
401 |
{ fix u::rat and v::rat |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
402 |
assume "u \<in> A" and "v \<in> B" and "x*y = u*v" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
403 |
moreover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
404 |
with prems have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+ |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
405 |
moreover |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
406 |
with prems have "0\<le>v" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
407 |
by (blast intro: preal_imp_pos [OF B] order_less_imp_le prems) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
408 |
moreover |
14550 | 409 |
from calculation |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
410 |
have "u*v < x*y" by (blast intro: mult_strict_mono prems) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
411 |
ultimately have False by force } |
14377 | 412 |
thus ?thesis by (auto simp add: mult_set_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
413 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
414 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
415 |
qed |
14335 | 416 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
417 |
lemma mult_set_not_rat_set: |
19765 | 418 |
assumes A: "A \<in> preal" |
419 |
and B: "B \<in> preal" |
|
420 |
shows "mult_set A B < {r. 0 < r}" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
421 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
422 |
show "mult_set A B \<subseteq> {r. 0 < r}" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
423 |
by (force simp add: mult_set_def |
19765 | 424 |
intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos_pos) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
425 |
show "mult_set A B \<noteq> {r. 0 < r}" |
19765 | 426 |
using preal_not_mem_mult_set_Ex [OF A B] by blast |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
427 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
428 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
429 |
|
14335 | 430 |
|
431 |
text{*Part 3 of Dedekind sections definition*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
432 |
lemma mult_set_lemma3: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
433 |
"[|A \<in> preal; B \<in> preal; u \<in> mult_set A B; 0 < z; z < u|] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
434 |
==> z \<in> mult_set A B" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
435 |
proof (unfold mult_set_def, clarify) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
436 |
fix x::rat and y::rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
437 |
assume A: "A \<in> preal" |
19765 | 438 |
and B: "B \<in> preal" |
439 |
and [simp]: "0 < z" |
|
440 |
and zless: "z < x * y" |
|
441 |
and x: "x \<in> A" |
|
442 |
and y: "y \<in> B" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
443 |
have [simp]: "0<y" by (rule preal_imp_pos [OF B y]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
444 |
show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
445 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
446 |
show "\<exists>y'\<in>B. z = (z/y) * y'" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
447 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
448 |
show "z = (z/y)*y" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
449 |
by (simp add: divide_inverse mult_commute [of y] mult_assoc |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
450 |
order_less_imp_not_eq2) |
23389 | 451 |
show "y \<in> B" by fact |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
452 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
453 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
454 |
show "z/y \<in> A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
455 |
proof (rule preal_downwards_closed [OF A x]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
456 |
show "0 < z/y" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
457 |
by (simp add: zero_less_divide_iff) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
458 |
show "z/y < x" by (simp add: pos_divide_less_eq zless) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
459 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
460 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
461 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
462 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
463 |
text{*Part 4 of Dedekind sections definition*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
464 |
lemma mult_set_lemma4: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
465 |
"[|A \<in> preal; B \<in> preal; y \<in> mult_set A B|] ==> \<exists>u \<in> mult_set A B. y < u" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
466 |
apply (auto simp add: mult_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
467 |
apply (frule preal_exists_greater [of A], auto) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
468 |
apply (rule_tac x="u * y" in exI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
469 |
apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
470 |
mult_strict_right_mono) |
14335 | 471 |
done |
472 |
||
473 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
474 |
lemma mem_mult_set: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
475 |
"[|A \<in> preal; B \<in> preal|] ==> mult_set A B \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
476 |
apply (simp (no_asm_simp) add: preal_def cut_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
477 |
apply (blast intro!: mult_set_not_empty mult_set_not_rat_set |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
478 |
mult_set_lemma3 mult_set_lemma4) |
14335 | 479 |
done |
480 |
||
481 |
lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
482 |
apply (simp add: preal_mult_def mem_mult_set Rep_preal) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
483 |
apply (force simp add: mult_set_def mult_ac) |
14335 | 484 |
done |
485 |
||
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
486 |
instance preal :: ab_semigroup_mult |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
487 |
proof |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
488 |
fix a b c :: preal |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
489 |
show "(a * b) * c = a * (b * c)" by (rule preal_mult_assoc) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
490 |
show "a * b = b * a" by (rule preal_mult_commute) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
491 |
qed |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
492 |
|
14335 | 493 |
lemma preal_mult_left_commute: "x * (y * z) = y * ((x * z)::preal)" |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
494 |
by (rule mult_left_commute) |
14335 | 495 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
496 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
497 |
text{* Positive Real multiplication is an AC operator *} |
14335 | 498 |
lemmas preal_mult_ac = |
499 |
preal_mult_assoc preal_mult_commute preal_mult_left_commute |
|
500 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
501 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
502 |
text{* Positive real 1 is the multiplicative identity element *} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
503 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
504 |
lemma preal_mult_1: "(1::preal) * z = z" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
505 |
unfolding preal_one_def |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
506 |
proof (induct z) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
507 |
fix A :: "rat set" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
508 |
assume A: "A \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
509 |
have "{w. \<exists>u. 0 < u \<and> u < 1 & (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
510 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
511 |
show "?lhs \<subseteq> A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
512 |
proof clarify |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
513 |
fix x::rat and u::rat and v::rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
514 |
assume upos: "0<u" and "u<1" and v: "v \<in> A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
515 |
have vpos: "0<v" by (rule preal_imp_pos [OF A v]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
516 |
hence "u*v < 1*v" by (simp only: mult_strict_right_mono prems) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
517 |
thus "u * v \<in> A" |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
15413
diff
changeset
|
518 |
by (force intro: preal_downwards_closed [OF A v] mult_pos_pos |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
15413
diff
changeset
|
519 |
upos vpos) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
520 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
521 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
522 |
show "A \<subseteq> ?lhs" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
523 |
proof clarify |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
524 |
fix x::rat |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
525 |
assume x: "x \<in> A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
526 |
have xpos: "0<x" by (rule preal_imp_pos [OF A x]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
527 |
from preal_exists_greater [OF A x] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
528 |
obtain v where v: "v \<in> A" and xlessv: "x < v" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
529 |
have vpos: "0<v" by (rule preal_imp_pos [OF A v]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
530 |
show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. x = u * v)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
531 |
proof (intro exI conjI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
532 |
show "0 < x/v" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
533 |
by (simp add: zero_less_divide_iff xpos vpos) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
534 |
show "x / v < 1" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
535 |
by (simp add: pos_divide_less_eq vpos xlessv) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
536 |
show "\<exists>v'\<in>A. x = (x / v) * v'" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
537 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
538 |
show "x = (x/v)*v" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
539 |
by (simp add: divide_inverse mult_assoc vpos |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
540 |
order_less_imp_not_eq2) |
23389 | 541 |
show "v \<in> A" by fact |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
542 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
543 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
544 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
545 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
546 |
thus "preal_of_rat 1 * Abs_preal A = Abs_preal A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
547 |
by (simp add: preal_of_rat_def preal_mult_def mult_set_def |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
548 |
rat_mem_preal A) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
549 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
550 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
551 |
instance preal :: comm_monoid_mult |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
552 |
by intro_classes (rule preal_mult_1) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
553 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
554 |
lemma preal_mult_1_right: "z * (1::preal) = z" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
555 |
by (rule mult_1_right) |
14335 | 556 |
|
557 |
||
558 |
subsection{*Distribution of Multiplication across Addition*} |
|
559 |
||
560 |
lemma mem_Rep_preal_add_iff: |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
561 |
"(z \<in> Rep_preal(R+S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x + y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
562 |
apply (simp add: preal_add_def mem_add_set Rep_preal) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
563 |
apply (simp add: add_set_def) |
14335 | 564 |
done |
565 |
||
566 |
lemma mem_Rep_preal_mult_iff: |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
567 |
"(z \<in> Rep_preal(R*S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x * y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
568 |
apply (simp add: preal_mult_def mem_mult_set Rep_preal) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
569 |
apply (simp add: mult_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
570 |
done |
14335 | 571 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
572 |
lemma distrib_subset1: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
573 |
"Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
574 |
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
575 |
apply (force simp add: right_distrib) |
14335 | 576 |
done |
577 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
578 |
lemma preal_add_mult_distrib_mean: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
579 |
assumes a: "a \<in> Rep_preal w" |
19765 | 580 |
and b: "b \<in> Rep_preal w" |
581 |
and d: "d \<in> Rep_preal x" |
|
582 |
and e: "e \<in> Rep_preal y" |
|
583 |
shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
584 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
585 |
let ?c = "(a*d + b*e)/(d+e)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
586 |
have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
587 |
by (blast intro: preal_imp_pos [OF Rep_preal] a b d e pos_add_strict)+ |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
588 |
have cpos: "0 < ?c" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
589 |
by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
590 |
show "a * d + b * e = ?c * (d + e)" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14387
diff
changeset
|
591 |
by (simp add: divide_inverse mult_assoc order_less_imp_not_eq2) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
592 |
show "?c \<in> Rep_preal w" |
20495 | 593 |
proof (cases rule: linorder_le_cases) |
594 |
assume "a \<le> b" |
|
595 |
hence "?c \<le> b" |
|
596 |
by (simp add: pos_divide_le_eq right_distrib mult_right_mono |
|
597 |
order_less_imp_le) |
|
598 |
thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos]) |
|
599 |
next |
|
600 |
assume "b \<le> a" |
|
601 |
hence "?c \<le> a" |
|
602 |
by (simp add: pos_divide_le_eq right_distrib mult_right_mono |
|
603 |
order_less_imp_le) |
|
604 |
thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal a cpos]) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
605 |
qed |
20495 | 606 |
qed |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
607 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
608 |
lemma distrib_subset2: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
609 |
"Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
610 |
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
611 |
apply (drule_tac w=w and x=x and y=y in preal_add_mult_distrib_mean, auto) |
14335 | 612 |
done |
613 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
614 |
lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)" |
15413 | 615 |
apply (rule Rep_preal_inject [THEN iffD1]) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
616 |
apply (rule equalityI [OF distrib_subset1 distrib_subset2]) |
14335 | 617 |
done |
618 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
619 |
lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
620 |
by (simp add: preal_mult_commute preal_add_mult_distrib2) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
621 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
622 |
instance preal :: comm_semiring |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
623 |
by intro_classes (rule preal_add_mult_distrib) |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
624 |
|
14335 | 625 |
|
626 |
subsection{*Existence of Inverse, a Positive Real*} |
|
627 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
628 |
lemma mem_inv_set_ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
629 |
assumes A: "A \<in> preal" shows "\<exists>x y. 0 < x & x < y & inverse y \<notin> A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
630 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
631 |
from preal_exists_bound [OF A] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
632 |
obtain x where [simp]: "0<x" "x \<notin> A" by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
633 |
show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
634 |
proof (intro exI conjI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
635 |
show "0 < inverse (x+1)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
636 |
by (simp add: order_less_trans [OF _ less_add_one]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
637 |
show "inverse(x+1) < inverse x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
638 |
by (simp add: less_imp_inverse_less less_add_one) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
639 |
show "inverse (inverse x) \<notin> A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
640 |
by (simp add: order_less_imp_not_eq2) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
641 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
642 |
qed |
14335 | 643 |
|
644 |
text{*Part 1 of Dedekind sections definition*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
645 |
lemma inverse_set_not_empty: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
646 |
"A \<in> preal ==> {} \<subset> inverse_set A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
647 |
apply (insert mem_inv_set_ex [of A]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
648 |
apply (auto simp add: inverse_set_def) |
14335 | 649 |
done |
650 |
||
651 |
text{*Part 2 of Dedekind sections definition*} |
|
652 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
653 |
lemma preal_not_mem_inverse_set_Ex: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
654 |
assumes A: "A \<in> preal" shows "\<exists>q. 0 < q & q \<notin> inverse_set A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
655 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
656 |
from preal_nonempty [OF A] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
657 |
obtain x where x: "x \<in> A" and xpos [simp]: "0<x" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
658 |
show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
659 |
proof (intro exI conjI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
660 |
show "0 < inverse x" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
661 |
show "inverse x \<notin> inverse_set A" |
14377 | 662 |
proof - |
663 |
{ fix y::rat |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
664 |
assume ygt: "inverse x < y" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
665 |
have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt]) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
666 |
have iyless: "inverse y < x" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
667 |
by (simp add: inverse_less_imp_less [of x] ygt) |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
668 |
have "inverse y \<in> A" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
669 |
by (simp add: preal_downwards_closed [OF A x] iyless)} |
14377 | 670 |
thus ?thesis by (auto simp add: inverse_set_def) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
671 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
672 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
673 |
qed |
14335 | 674 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
675 |
lemma inverse_set_not_rat_set: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
676 |
assumes A: "A \<in> preal" shows "inverse_set A < {r. 0 < r}" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
677 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
678 |
show "inverse_set A \<subseteq> {r. 0 < r}" by (force simp add: inverse_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
679 |
next |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
680 |
show "inverse_set A \<noteq> {r. 0 < r}" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
681 |
by (insert preal_not_mem_inverse_set_Ex [OF A], blast) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
682 |
qed |
14335 | 683 |
|
684 |
text{*Part 3 of Dedekind sections definition*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
685 |
lemma inverse_set_lemma3: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
686 |
"[|A \<in> preal; u \<in> inverse_set A; 0 < z; z < u|] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
687 |
==> z \<in> inverse_set A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
688 |
apply (auto simp add: inverse_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
689 |
apply (auto intro: order_less_trans) |
14335 | 690 |
done |
691 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
692 |
text{*Part 4 of Dedekind sections definition*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
693 |
lemma inverse_set_lemma4: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
694 |
"[|A \<in> preal; y \<in> inverse_set A|] ==> \<exists>u \<in> inverse_set A. y < u" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
695 |
apply (auto simp add: inverse_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
696 |
apply (drule dense [of y]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
697 |
apply (blast intro: order_less_trans) |
14335 | 698 |
done |
699 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
700 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
701 |
lemma mem_inverse_set: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
702 |
"A \<in> preal ==> inverse_set A \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
703 |
apply (simp (no_asm_simp) add: preal_def cut_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
704 |
apply (blast intro!: inverse_set_not_empty inverse_set_not_rat_set |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
705 |
inverse_set_lemma3 inverse_set_lemma4) |
14335 | 706 |
done |
707 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
708 |
|
14335 | 709 |
subsection{*Gleason's Lemma 9-3.4, page 122*} |
710 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
711 |
lemma Gleason9_34_exists: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
712 |
assumes A: "A \<in> preal" |
19765 | 713 |
and "\<forall>x\<in>A. x + u \<in> A" |
714 |
and "0 \<le> z" |
|
715 |
shows "\<exists>b\<in>A. b + (of_int z) * u \<in> A" |
|
14369 | 716 |
proof (cases z rule: int_cases) |
717 |
case (nonneg n) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
718 |
show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
719 |
proof (simp add: prems, induct n) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
720 |
case 0 |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
721 |
from preal_nonempty [OF A] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
722 |
show ?case by force |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
723 |
case (Suc k) |
15013 | 724 |
from this obtain b where "b \<in> A" "b + of_nat k * u \<in> A" .. |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14377
diff
changeset
|
725 |
hence "b + of_int (int k)*u + u \<in> A" by (simp add: prems) |
29667 | 726 |
thus ?case by (force simp add: algebra_simps prems) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
727 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
728 |
next |
14369 | 729 |
case (neg n) |
730 |
with prems show ?thesis by simp |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
731 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
732 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
733 |
lemma Gleason9_34_contra: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
734 |
assumes A: "A \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
735 |
shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A|] ==> False" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
736 |
proof (induct u, induct y) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
737 |
fix a::int and b::int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
738 |
fix c::int and d::int |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
739 |
assume bpos [simp]: "0 < b" |
19765 | 740 |
and dpos [simp]: "0 < d" |
741 |
and closed: "\<forall>x\<in>A. x + (Fract c d) \<in> A" |
|
742 |
and upos: "0 < Fract c d" |
|
743 |
and ypos: "0 < Fract a b" |
|
744 |
and notin: "Fract a b \<notin> A" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
745 |
have cpos [simp]: "0 < c" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
746 |
by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
747 |
have apos [simp]: "0 < a" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
748 |
by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
749 |
let ?k = "a*d" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14377
diff
changeset
|
750 |
have frle: "Fract a b \<le> Fract ?k 1 * (Fract c d)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
751 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
752 |
have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))" |
35216 | 753 |
by (simp add: order_less_imp_not_eq2 mult_ac) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
754 |
moreover |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
755 |
have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
756 |
by (rule mult_mono, |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
757 |
simp_all add: int_one_le_iff_zero_less zero_less_mult_iff |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
758 |
order_less_imp_le) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
759 |
ultimately |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
760 |
show ?thesis by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
761 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
762 |
have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
763 |
from Gleason9_34_exists [OF A closed k] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
764 |
obtain z where z: "z \<in> A" |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14377
diff
changeset
|
765 |
and mem: "z + of_int ?k * Fract c d \<in> A" .. |
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14377
diff
changeset
|
766 |
have less: "z + of_int ?k * Fract c d < Fract a b" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
767 |
by (rule not_in_preal_ub [OF A notin mem ypos]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
768 |
have "0<z" by (rule preal_imp_pos [OF A z]) |
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14377
diff
changeset
|
769 |
with frle and less show False by (simp add: Fract_of_int_eq) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
770 |
qed |
14335 | 771 |
|
772 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
773 |
lemma Gleason9_34: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
774 |
assumes A: "A \<in> preal" |
19765 | 775 |
and upos: "0 < u" |
776 |
shows "\<exists>r \<in> A. r + u \<notin> A" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
777 |
proof (rule ccontr, simp) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
778 |
assume closed: "\<forall>r\<in>A. r + u \<in> A" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
779 |
from preal_exists_bound [OF A] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
780 |
obtain y where y: "y \<notin> A" and ypos: "0 < y" by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
781 |
show False |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
782 |
by (rule Gleason9_34_contra [OF A closed upos ypos y]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
783 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
784 |
|
14335 | 785 |
|
786 |
||
787 |
subsection{*Gleason's Lemma 9-3.6*} |
|
788 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
789 |
lemma lemma_gleason9_36: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
790 |
assumes A: "A \<in> preal" |
19765 | 791 |
and x: "1 < x" |
792 |
shows "\<exists>r \<in> A. r*x \<notin> A" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
793 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
794 |
from preal_nonempty [OF A] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
795 |
obtain y where y: "y \<in> A" and ypos: "0<y" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
796 |
show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
797 |
proof (rule classical) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
798 |
assume "~(\<exists>r\<in>A. r * x \<notin> A)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
799 |
with y have ymem: "y * x \<in> A" by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
800 |
from ypos mult_strict_left_mono [OF x] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
801 |
have yless: "y < y*x" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
802 |
let ?d = "y*x - y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
803 |
from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
804 |
from Gleason9_34 [OF A dpos] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
805 |
obtain r where r: "r\<in>A" and notin: "r + ?d \<notin> A" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
806 |
have rpos: "0<r" by (rule preal_imp_pos [OF A r]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
807 |
with dpos have rdpos: "0 < r + ?d" by arith |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
808 |
have "~ (r + ?d \<le> y + ?d)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
809 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
810 |
assume le: "r + ?d \<le> y + ?d" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
811 |
from ymem have yd: "y + ?d \<in> A" by (simp add: eq) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
812 |
have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
813 |
with notin show False by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
814 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
815 |
hence "y < r" by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
816 |
with ypos have dless: "?d < (r * ?d)/y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
817 |
by (simp add: pos_less_divide_eq mult_commute [of ?d] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
818 |
mult_strict_right_mono dpos) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
819 |
have "r + ?d < r*x" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
820 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
821 |
have "r + ?d < r + (r * ?d)/y" by (simp add: dless) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
822 |
also with ypos have "... = (r/y) * (y + ?d)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32115
diff
changeset
|
823 |
by (simp only: algebra_simps divide_inverse, simp) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
824 |
also have "... = r*x" using ypos |
35216 | 825 |
by simp |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
826 |
finally show "r + ?d < r*x" . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
827 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
828 |
with r notin rdpos |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
829 |
show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest: preal_downwards_closed [OF A]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
830 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
831 |
qed |
14335 | 832 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
833 |
subsection{*Existence of Inverse: Part 2*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
834 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
835 |
lemma mem_Rep_preal_inverse_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
836 |
"(z \<in> Rep_preal(inverse R)) = |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
837 |
(0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal R))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
838 |
apply (simp add: preal_inverse_def mem_inverse_set Rep_preal) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
839 |
apply (simp add: inverse_set_def) |
14335 | 840 |
done |
841 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
842 |
lemma Rep_preal_of_rat: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
843 |
"0 < q ==> Rep_preal (preal_of_rat q) = {x. 0 < x \<and> x < q}" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
844 |
by (simp add: preal_of_rat_def rat_mem_preal) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
845 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
846 |
lemma subset_inverse_mult_lemma: |
19765 | 847 |
assumes xpos: "0 < x" and xless: "x < 1" |
848 |
shows "\<exists>r u y. 0 < r & r < y & inverse y \<notin> Rep_preal R & |
|
849 |
u \<in> Rep_preal R & x = r * u" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
850 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
851 |
from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
852 |
from lemma_gleason9_36 [OF Rep_preal this] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
853 |
obtain r where r: "r \<in> Rep_preal R" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
854 |
and notin: "r * (inverse x) \<notin> Rep_preal R" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
855 |
have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
856 |
from preal_exists_greater [OF Rep_preal r] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
857 |
obtain u where u: "u \<in> Rep_preal R" and rless: "r < u" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
858 |
have upos: "0<u" by (rule preal_imp_pos [OF Rep_preal u]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
859 |
show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
860 |
proof (intro exI conjI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
861 |
show "0 < x/u" using xpos upos |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
862 |
by (simp add: zero_less_divide_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
863 |
show "x/u < x/r" using xpos upos rpos |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14387
diff
changeset
|
864 |
by (simp add: divide_inverse mult_less_cancel_left rless) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
865 |
show "inverse (x / r) \<notin> Rep_preal R" using notin |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14387
diff
changeset
|
866 |
by (simp add: divide_inverse mult_commute) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
867 |
show "u \<in> Rep_preal R" by (rule u) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
868 |
show "x = x / u * u" using upos |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14387
diff
changeset
|
869 |
by (simp add: divide_inverse mult_commute) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
870 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
871 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
872 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
873 |
lemma subset_inverse_mult: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
874 |
"Rep_preal(preal_of_rat 1) \<subseteq> Rep_preal(inverse R * R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
875 |
apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
876 |
mem_Rep_preal_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
877 |
apply (blast dest: subset_inverse_mult_lemma) |
14335 | 878 |
done |
879 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
880 |
lemma inverse_mult_subset_lemma: |
19765 | 881 |
assumes rpos: "0 < r" |
882 |
and rless: "r < y" |
|
883 |
and notin: "inverse y \<notin> Rep_preal R" |
|
884 |
and q: "q \<in> Rep_preal R" |
|
885 |
shows "r*q < 1" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
886 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
887 |
have "q < inverse y" using rpos rless |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
888 |
by (simp add: not_in_preal_ub [OF Rep_preal notin] q) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
889 |
hence "r * q < r/y" using rpos |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14387
diff
changeset
|
890 |
by (simp add: divide_inverse mult_less_cancel_left) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
891 |
also have "... \<le> 1" using rpos rless |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
892 |
by (simp add: pos_divide_le_eq) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
893 |
finally show ?thesis . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
894 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
895 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
896 |
lemma inverse_mult_subset: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
897 |
"Rep_preal(inverse R * R) \<subseteq> Rep_preal(preal_of_rat 1)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
898 |
apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
899 |
mem_Rep_preal_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
900 |
apply (simp add: zero_less_mult_iff preal_imp_pos [OF Rep_preal]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
901 |
apply (blast intro: inverse_mult_subset_lemma) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
902 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
903 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
904 |
lemma preal_mult_inverse: "inverse R * R = (1::preal)" |
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
905 |
unfolding preal_one_def |
15413 | 906 |
apply (rule Rep_preal_inject [THEN iffD1]) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
907 |
apply (rule equalityI [OF inverse_mult_subset subset_inverse_mult]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
908 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
909 |
|
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
910 |
lemma preal_mult_inverse_right: "R * inverse R = (1::preal)" |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
911 |
apply (rule preal_mult_commute [THEN subst]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
912 |
apply (rule preal_mult_inverse) |
14335 | 913 |
done |
914 |
||
915 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
916 |
text{*Theorems needing @{text Gleason9_34}*} |
14335 | 917 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
918 |
lemma Rep_preal_self_subset: "Rep_preal (R) \<subseteq> Rep_preal(R + S)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
919 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
920 |
fix r |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
921 |
assume r: "r \<in> Rep_preal R" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
922 |
have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
923 |
from mem_Rep_preal_Ex |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
924 |
obtain y where y: "y \<in> Rep_preal S" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
925 |
have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
926 |
have ry: "r+y \<in> Rep_preal(R + S)" using r y |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
927 |
by (auto simp add: mem_Rep_preal_add_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
928 |
show "r \<in> Rep_preal(R + S)" using r ypos rpos |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
929 |
by (simp add: preal_downwards_closed [OF Rep_preal ry]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
930 |
qed |
14335 | 931 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
932 |
lemma Rep_preal_sum_not_subset: "~ Rep_preal (R + S) \<subseteq> Rep_preal(R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
933 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
934 |
from mem_Rep_preal_Ex |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
935 |
obtain y where y: "y \<in> Rep_preal S" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
936 |
have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
937 |
from Gleason9_34 [OF Rep_preal ypos] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
938 |
obtain r where r: "r \<in> Rep_preal R" and notin: "r + y \<notin> Rep_preal R" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
939 |
have "r + y \<in> Rep_preal (R + S)" using r y |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
940 |
by (auto simp add: mem_Rep_preal_add_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
941 |
thus ?thesis using notin by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
942 |
qed |
14335 | 943 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
944 |
lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) \<noteq> Rep_preal(R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
945 |
by (insert Rep_preal_sum_not_subset, blast) |
14335 | 946 |
|
947 |
text{*at last, Gleason prop. 9-3.5(iii) page 123*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
948 |
lemma preal_self_less_add_left: "(R::preal) < R + S" |
26806 | 949 |
apply (unfold preal_less_def less_le) |
14335 | 950 |
apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym]) |
951 |
done |
|
952 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
953 |
lemma preal_self_less_add_right: "(R::preal) < S + R" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
954 |
by (simp add: preal_add_commute preal_self_less_add_left) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
955 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
956 |
lemma preal_not_eq_self: "x \<noteq> x + (y::preal)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
957 |
by (insert preal_self_less_add_left [of x y], auto) |
14335 | 958 |
|
959 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
960 |
subsection{*Subtraction for Positive Reals*} |
14335 | 961 |
|
22710 | 962 |
text{*Gleason prop. 9-3.5(iv), page 123: proving @{prop "A < B ==> \<exists>D. A + D = |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
963 |
B"}. We define the claimed @{term D} and show that it is a positive real*} |
14335 | 964 |
|
965 |
text{*Part 1 of Dedekind sections definition*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
966 |
lemma diff_set_not_empty: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
967 |
"R < S ==> {} \<subset> diff_set (Rep_preal S) (Rep_preal R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
968 |
apply (auto simp add: preal_less_def diff_set_def elim!: equalityE) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
969 |
apply (frule_tac x1 = S in Rep_preal [THEN preal_exists_greater]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
970 |
apply (drule preal_imp_pos [OF Rep_preal], clarify) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
971 |
apply (cut_tac a=x and b=u in add_eq_exists, force) |
14335 | 972 |
done |
973 |
||
974 |
text{*Part 2 of Dedekind sections definition*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
975 |
lemma diff_set_nonempty: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
976 |
"\<exists>q. 0 < q & q \<notin> diff_set (Rep_preal S) (Rep_preal R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
977 |
apply (cut_tac X = S in Rep_preal_exists_bound) |
14335 | 978 |
apply (erule exE) |
979 |
apply (rule_tac x = x in exI, auto) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
980 |
apply (simp add: diff_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
981 |
apply (auto dest: Rep_preal [THEN preal_downwards_closed]) |
14335 | 982 |
done |
983 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
984 |
lemma diff_set_not_rat_set: |
19765 | 985 |
"diff_set (Rep_preal S) (Rep_preal R) < {r. 0 < r}" (is "?lhs < ?rhs") |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
986 |
proof |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
987 |
show "?lhs \<subseteq> ?rhs" by (auto simp add: diff_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
988 |
show "?lhs \<noteq> ?rhs" using diff_set_nonempty by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
989 |
qed |
14335 | 990 |
|
991 |
text{*Part 3 of Dedekind sections definition*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
992 |
lemma diff_set_lemma3: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
993 |
"[|R < S; u \<in> diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
994 |
==> z \<in> diff_set (Rep_preal S) (Rep_preal R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
995 |
apply (auto simp add: diff_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
996 |
apply (rule_tac x=x in exI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
997 |
apply (drule Rep_preal [THEN preal_downwards_closed], auto) |
14335 | 998 |
done |
999 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1000 |
text{*Part 4 of Dedekind sections definition*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1001 |
lemma diff_set_lemma4: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1002 |
"[|R < S; y \<in> diff_set (Rep_preal S) (Rep_preal R)|] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1003 |
==> \<exists>u \<in> diff_set (Rep_preal S) (Rep_preal R). y < u" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1004 |
apply (auto simp add: diff_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1005 |
apply (drule Rep_preal [THEN preal_exists_greater], clarify) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1006 |
apply (cut_tac a="x+y" and b=u in add_eq_exists, clarify) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1007 |
apply (rule_tac x="y+xa" in exI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1008 |
apply (auto simp add: add_ac) |
14335 | 1009 |
done |
1010 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1011 |
lemma mem_diff_set: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1012 |
"R < S ==> diff_set (Rep_preal S) (Rep_preal R) \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1013 |
apply (unfold preal_def cut_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1014 |
apply (blast intro!: diff_set_not_empty diff_set_not_rat_set |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1015 |
diff_set_lemma3 diff_set_lemma4) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1016 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1017 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1018 |
lemma mem_Rep_preal_diff_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1019 |
"R < S ==> |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1020 |
(z \<in> Rep_preal(S-R)) = |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1021 |
(\<exists>x. 0 < x & 0 < z & x \<notin> Rep_preal R & x + z \<in> Rep_preal S)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1022 |
apply (simp add: preal_diff_def mem_diff_set Rep_preal) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1023 |
apply (force simp add: diff_set_def) |
14335 | 1024 |
done |
1025 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1026 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1027 |
text{*proving that @{term "R + D \<le> S"}*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1028 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1029 |
lemma less_add_left_lemma: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1030 |
assumes Rless: "R < S" |
19765 | 1031 |
and a: "a \<in> Rep_preal R" |
1032 |
and cb: "c + b \<in> Rep_preal S" |
|
1033 |
and "c \<notin> Rep_preal R" |
|
1034 |
and "0 < b" |
|
1035 |
and "0 < c" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1036 |
shows "a + b \<in> Rep_preal S" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1037 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1038 |
have "0<a" by (rule preal_imp_pos [OF Rep_preal a]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1039 |
moreover |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1040 |
have "a < c" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1041 |
by (blast intro: not_in_Rep_preal_ub ) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1042 |
ultimately show ?thesis using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1043 |
by (simp add: preal_downwards_closed [OF Rep_preal cb]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1044 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1045 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1046 |
lemma less_add_left_le1: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1047 |
"R < (S::preal) ==> R + (S-R) \<le> S" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1048 |
apply (auto simp add: Bex_def preal_le_def mem_Rep_preal_add_iff |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1049 |
mem_Rep_preal_diff_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1050 |
apply (blast intro: less_add_left_lemma) |
14335 | 1051 |
done |
1052 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1053 |
subsection{*proving that @{term "S \<le> R + D"} --- trickier*} |
14335 | 1054 |
|
1055 |
lemma lemma_sum_mem_Rep_preal_ex: |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1056 |
"x \<in> Rep_preal S ==> \<exists>e. 0 < e & x + e \<in> Rep_preal S" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1057 |
apply (drule Rep_preal [THEN preal_exists_greater], clarify) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1058 |
apply (cut_tac a=x and b=u in add_eq_exists, auto) |
14335 | 1059 |
done |
1060 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1061 |
lemma less_add_left_lemma2: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1062 |
assumes Rless: "R < S" |
19765 | 1063 |
and x: "x \<in> Rep_preal S" |
1064 |
and xnot: "x \<notin> Rep_preal R" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1065 |
shows "\<exists>u v z. 0 < v & 0 < z & u \<in> Rep_preal R & z \<notin> Rep_preal R & |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1066 |
z + v \<in> Rep_preal S & x = u + v" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1067 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1068 |
have xpos: "0<x" by (rule preal_imp_pos [OF Rep_preal x]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1069 |
from lemma_sum_mem_Rep_preal_ex [OF x] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1070 |
obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal S" by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1071 |
from Gleason9_34 [OF Rep_preal epos] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1072 |
obtain r where r: "r \<in> Rep_preal R" and notin: "r + e \<notin> Rep_preal R" .. |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1073 |
with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1074 |
from add_eq_exists [of r x] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1075 |
obtain y where eq: "x = r+y" by auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1076 |
show ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1077 |
proof (intro exI conjI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1078 |
show "r \<in> Rep_preal R" by (rule r) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1079 |
show "r + e \<notin> Rep_preal R" by (rule notin) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1080 |
show "r + e + y \<in> Rep_preal S" using xe eq by (simp add: add_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1081 |
show "x = r + y" by (simp add: eq) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1082 |
show "0 < r + e" using epos preal_imp_pos [OF Rep_preal r] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1083 |
by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1084 |
show "0 < y" using rless eq by arith |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1085 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1086 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1087 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1088 |
lemma less_add_left_le2: "R < (S::preal) ==> S \<le> R + (S-R)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1089 |
apply (auto simp add: preal_le_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1090 |
apply (case_tac "x \<in> Rep_preal R") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1091 |
apply (cut_tac Rep_preal_self_subset [of R], force) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1092 |
apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_diff_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1093 |
apply (blast dest: less_add_left_lemma2) |
14335 | 1094 |
done |
1095 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1096 |
lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S" |
27682 | 1097 |
by (blast intro: antisym [OF less_add_left_le1 less_add_left_le2]) |
14335 | 1098 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1099 |
lemma less_add_left_Ex: "R < (S::preal) ==> \<exists>D. R + D = S" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1100 |
by (fast dest: less_add_left) |
14335 | 1101 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1102 |
lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1103 |
apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc) |
14335 | 1104 |
apply (rule_tac y1 = D in preal_add_commute [THEN subst]) |
1105 |
apply (auto intro: preal_self_less_add_left simp add: preal_add_assoc [symmetric]) |
|
1106 |
done |
|
1107 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1108 |
lemma preal_add_less2_mono2: "R < (S::preal) ==> T + R < T + S" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1109 |
by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of T]) |
14335 | 1110 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1111 |
lemma preal_add_right_less_cancel: "R + T < S + T ==> R < (S::preal)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1112 |
apply (insert linorder_less_linear [of R S], auto) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1113 |
apply (drule_tac R = S and T = T in preal_add_less2_mono1) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1114 |
apply (blast dest: order_less_trans) |
14335 | 1115 |
done |
1116 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1117 |
lemma preal_add_left_less_cancel: "T + R < T + S ==> R < (S::preal)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1118 |
by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of T]) |
14335 | 1119 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1120 |
lemma preal_add_less_cancel_right: "((R::preal) + T < S + T) = (R < S)" |
14335 | 1121 |
by (blast intro: preal_add_less2_mono1 preal_add_right_less_cancel) |
1122 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1123 |
lemma preal_add_less_cancel_left: "(T + (R::preal) < T + S) = (R < S)" |
14335 | 1124 |
by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel) |
1125 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1126 |
lemma preal_add_le_cancel_right: "((R::preal) + T \<le> S + T) = (R \<le> S)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1127 |
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_right) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1128 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1129 |
lemma preal_add_le_cancel_left: "(T + (R::preal) \<le> T + S) = (R \<le> S)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1130 |
by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_left) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1131 |
|
14335 | 1132 |
lemma preal_add_less_mono: |
1133 |
"[| x1 < y1; x2 < y2 |] ==> x1 + x2 < y1 + (y2::preal)" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1134 |
apply (auto dest!: less_add_left_Ex simp add: preal_add_ac) |
14335 | 1135 |
apply (rule preal_add_assoc [THEN subst]) |
1136 |
apply (rule preal_self_less_add_right) |
|
1137 |
done |
|
1138 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1139 |
lemma preal_add_right_cancel: "(R::preal) + T = S + T ==> R = S" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1140 |
apply (insert linorder_less_linear [of R S], safe) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1141 |
apply (drule_tac [!] T = T in preal_add_less2_mono1, auto) |
14335 | 1142 |
done |
1143 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1144 |
lemma preal_add_left_cancel: "C + A = C + B ==> A = (B::preal)" |
14335 | 1145 |
by (auto intro: preal_add_right_cancel simp add: preal_add_commute) |
1146 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1147 |
lemma preal_add_left_cancel_iff: "(C + A = C + B) = ((A::preal) = B)" |
14335 | 1148 |
by (fast intro: preal_add_left_cancel) |
1149 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1150 |
lemma preal_add_right_cancel_iff: "(A + C = B + C) = ((A::preal) = B)" |
14335 | 1151 |
by (fast intro: preal_add_right_cancel) |
1152 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1153 |
lemmas preal_cancels = |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1154 |
preal_add_less_cancel_right preal_add_less_cancel_left |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1155 |
preal_add_le_cancel_right preal_add_le_cancel_left |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1156 |
preal_add_left_cancel_iff preal_add_right_cancel_iff |
14335 | 1157 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
32960
diff
changeset
|
1158 |
instance preal :: linordered_cancel_ab_semigroup_add |
23285
c95a4f6b3881
instance preal :: ordered_cancel_ab_semigroup_add
huffman
parents:
22710
diff
changeset
|
1159 |
proof |
c95a4f6b3881
instance preal :: ordered_cancel_ab_semigroup_add
huffman
parents:
22710
diff
changeset
|
1160 |
fix a b c :: preal |
c95a4f6b3881
instance preal :: ordered_cancel_ab_semigroup_add
huffman
parents:
22710
diff
changeset
|
1161 |
show "a + b = a + c \<Longrightarrow> b = c" by (rule preal_add_left_cancel) |
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23285
diff
changeset
|
1162 |
show "a \<le> b \<Longrightarrow> c + a \<le> c + b" by (simp only: preal_add_le_cancel_left) |
23285
c95a4f6b3881
instance preal :: ordered_cancel_ab_semigroup_add
huffman
parents:
22710
diff
changeset
|
1163 |
qed |
c95a4f6b3881
instance preal :: ordered_cancel_ab_semigroup_add
huffman
parents:
22710
diff
changeset
|
1164 |
|
14335 | 1165 |
|
1166 |
subsection{*Completeness of type @{typ preal}*} |
|
1167 |
||
1168 |
text{*Prove that supremum is a cut*} |
|
1169 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1170 |
text{*Part 1 of Dedekind sections definition*} |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1171 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1172 |
lemma preal_sup_set_not_empty: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1173 |
"P \<noteq> {} ==> {} \<subset> (\<Union>X \<in> P. Rep_preal(X))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1174 |
apply auto |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1175 |
apply (cut_tac X = x in mem_Rep_preal_Ex, auto) |
14335 | 1176 |
done |
1177 |
||
1178 |
||
1179 |
text{*Part 2 of Dedekind sections definition*} |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1180 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1181 |
lemma preal_sup_not_exists: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1182 |
"\<forall>X \<in> P. X \<le> Y ==> \<exists>q. 0 < q & q \<notin> (\<Union>X \<in> P. Rep_preal(X))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1183 |
apply (cut_tac X = Y in Rep_preal_exists_bound) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1184 |
apply (auto simp add: preal_le_def) |
14335 | 1185 |
done |
1186 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1187 |
lemma preal_sup_set_not_rat_set: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1188 |
"\<forall>X \<in> P. X \<le> Y ==> (\<Union>X \<in> P. Rep_preal(X)) < {r. 0 < r}" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1189 |
apply (drule preal_sup_not_exists) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1190 |
apply (blast intro: preal_imp_pos [OF Rep_preal]) |
14335 | 1191 |
done |
1192 |
||
1193 |
text{*Part 3 of Dedekind sections definition*} |
|
1194 |
lemma preal_sup_set_lemma3: |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1195 |
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u|] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1196 |
==> z \<in> (\<Union>X \<in> P. Rep_preal(X))" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1197 |
by (auto elim: Rep_preal [THEN preal_downwards_closed]) |
14335 | 1198 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1199 |
text{*Part 4 of Dedekind sections definition*} |
14335 | 1200 |
lemma preal_sup_set_lemma4: |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1201 |
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; y \<in> (\<Union>X \<in> P. Rep_preal(X)) |] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1202 |
==> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1203 |
by (blast dest: Rep_preal [THEN preal_exists_greater]) |
14335 | 1204 |
|
1205 |
lemma preal_sup: |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1206 |
"[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y|] ==> (\<Union>X \<in> P. Rep_preal(X)) \<in> preal" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1207 |
apply (unfold preal_def cut_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1208 |
apply (blast intro!: preal_sup_set_not_empty preal_sup_set_not_rat_set |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1209 |
preal_sup_set_lemma3 preal_sup_set_lemma4) |
14335 | 1210 |
done |
1211 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1212 |
lemma preal_psup_le: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1213 |
"[| \<forall>X \<in> P. X \<le> Y; x \<in> P |] ==> x \<le> psup P" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1214 |
apply (simp (no_asm_simp) add: preal_le_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1215 |
apply (subgoal_tac "P \<noteq> {}") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1216 |
apply (auto simp add: psup_def preal_sup) |
14335 | 1217 |
done |
1218 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1219 |
lemma psup_le_ub: "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1220 |
apply (simp (no_asm_simp) add: preal_le_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1221 |
apply (simp add: psup_def preal_sup) |
14335 | 1222 |
apply (auto simp add: preal_le_def) |
1223 |
done |
|
1224 |
||
1225 |
text{*Supremum property*} |
|
1226 |
lemma preal_complete: |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1227 |
"[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> (\<exists>X \<in> P. Z < X) = (Z < psup P)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1228 |
apply (simp add: preal_less_def psup_def preal_sup) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1229 |
apply (auto simp add: preal_le_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1230 |
apply (rename_tac U) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1231 |
apply (cut_tac x = U and y = Z in linorder_less_linear) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1232 |
apply (auto simp add: preal_less_def) |
14335 | 1233 |
done |
1234 |
||
1235 |
||
20495 | 1236 |
subsection{*The Embedding from @{typ rat} into @{typ preal}*} |
14335 | 1237 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1238 |
lemma preal_of_rat_add_lemma1: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1239 |
"[|x < y + z; 0 < x; 0 < y|] ==> x * y * inverse (y + z) < (y::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1240 |
apply (frule_tac c = "y * inverse (y + z) " in mult_strict_right_mono) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1241 |
apply (simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1242 |
apply (simp add: mult_ac) |
14335 | 1243 |
done |
1244 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1245 |
lemma preal_of_rat_add_lemma2: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1246 |
assumes "u < x + y" |
19765 | 1247 |
and "0 < x" |
1248 |
and "0 < y" |
|
1249 |
and "0 < u" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1250 |
shows "\<exists>v w::rat. w < y & 0 < v & v < x & 0 < w & u = v + w" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1251 |
proof (intro exI conjI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1252 |
show "u * x * inverse(x+y) < x" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1253 |
by (simp add: preal_of_rat_add_lemma1) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1254 |
show "u * y * inverse(x+y) < y" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1255 |
by (simp add: preal_of_rat_add_lemma1 add_commute [of x]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1256 |
show "0 < u * x * inverse (x + y)" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1257 |
by (simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1258 |
show "0 < u * y * inverse (x + y)" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1259 |
by (simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1260 |
show "u = u * x * inverse (x + y) + u * y * inverse (x + y)" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1261 |
by (simp add: left_distrib [symmetric] right_distrib [symmetric] mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1262 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1263 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1264 |
lemma preal_of_rat_add: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1265 |
"[| 0 < x; 0 < y|] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1266 |
==> preal_of_rat ((x::rat) + y) = preal_of_rat x + preal_of_rat y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1267 |
apply (unfold preal_of_rat_def preal_add_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1268 |
apply (simp add: rat_mem_preal) |
14335 | 1269 |
apply (rule_tac f = Abs_preal in arg_cong) |
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1270 |
apply (auto simp add: add_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1271 |
apply (blast dest: preal_of_rat_add_lemma2) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1272 |
done |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1273 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1274 |
lemma preal_of_rat_mult_lemma1: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1275 |
"[|x < y; 0 < x; 0 < z|] ==> x * z * inverse y < (z::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1276 |
apply (frule_tac c = "z * inverse y" in mult_strict_right_mono) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1277 |
apply (simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1278 |
apply (subgoal_tac "y * (z * inverse y) = z * (y * inverse y)") |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1279 |
apply (simp_all add: mult_ac) |
14335 | 1280 |
done |
1281 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1282 |
lemma preal_of_rat_mult_lemma2: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1283 |
assumes xless: "x < y * z" |
19765 | 1284 |
and xpos: "0 < x" |
1285 |
and ypos: "0 < y" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1286 |
shows "x * z * inverse y * inverse z < (z::rat)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1287 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1288 |
have "0 < y * z" using prems by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1289 |
hence zpos: "0 < z" using prems by (simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1290 |
have "x * z * inverse y * inverse z = x * inverse y * (z * inverse z)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1291 |
by (simp add: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1292 |
also have "... = x/y" using zpos |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14387
diff
changeset
|
1293 |
by (simp add: divide_inverse) |
23389 | 1294 |
also from xless have "... < z" |
1295 |
by (simp add: pos_divide_less_eq [OF ypos] mult_commute) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1296 |
finally show ?thesis . |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1297 |
qed |
14335 | 1298 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1299 |
lemma preal_of_rat_mult_lemma3: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1300 |
assumes uless: "u < x * y" |
19765 | 1301 |
and "0 < x" |
1302 |
and "0 < y" |
|
1303 |
and "0 < u" |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1304 |
shows "\<exists>v w::rat. v < x & w < y & 0 < v & 0 < w & u = v * w" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1305 |
proof - |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1306 |
from dense [OF uless] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1307 |
obtain r where "u < r" "r < x * y" by blast |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1308 |
thus ?thesis |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1309 |
proof (intro exI conjI) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1310 |
show "u * x * inverse r < x" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1311 |
by (simp add: preal_of_rat_mult_lemma1) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1312 |
show "r * y * inverse x * inverse y < y" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1313 |
by (simp add: preal_of_rat_mult_lemma2) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1314 |
show "0 < u * x * inverse r" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1315 |
by (simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1316 |
show "0 < r * y * inverse x * inverse y" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1317 |
by (simp add: zero_less_mult_iff) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1318 |
have "u * x * inverse r * (r * y * inverse x * inverse y) = |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1319 |
u * (r * inverse r) * (x * inverse x) * (y * inverse y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1320 |
by (simp only: mult_ac) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1321 |
thus "u = u * x * inverse r * (r * y * inverse x * inverse y)" using prems |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1322 |
by simp |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1323 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1324 |
qed |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1325 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1326 |
lemma preal_of_rat_mult: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1327 |
"[| 0 < x; 0 < y|] |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1328 |
==> preal_of_rat ((x::rat) * y) = preal_of_rat x * preal_of_rat y" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1329 |
apply (unfold preal_of_rat_def preal_mult_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1330 |
apply (simp add: rat_mem_preal) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1331 |
apply (rule_tac f = Abs_preal in arg_cong) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1332 |
apply (auto simp add: zero_less_mult_iff mult_strict_mono mult_set_def) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1333 |
apply (blast dest: preal_of_rat_mult_lemma3) |
14335 | 1334 |
done |
1335 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1336 |
lemma preal_of_rat_less_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1337 |
"[| 0 < x; 0 < y|] ==> (preal_of_rat x < preal_of_rat y) = (x < y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1338 |
by (force simp add: preal_of_rat_def preal_less_def rat_mem_preal) |
14335 | 1339 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1340 |
lemma preal_of_rat_le_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1341 |
"[| 0 < x; 0 < y|] ==> (preal_of_rat x \<le> preal_of_rat y) = (x \<le> y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1342 |
by (simp add: preal_of_rat_less_iff linorder_not_less [symmetric]) |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1343 |
|
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1344 |
lemma preal_of_rat_eq_iff: |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1345 |
"[| 0 < x; 0 < y|] ==> (preal_of_rat x = preal_of_rat y) = (x = y)" |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14335
diff
changeset
|
1346 |
by (simp add: preal_of_rat_le_iff order_eq_iff) |
14335 | 1347 |
|
5078 | 1348 |
end |