author | haftmann |
Tue, 16 Oct 2007 23:12:45 +0200 | |
changeset 25062 | af5ef0d4d655 |
parent 24748 | ee0a0eb6b738 |
child 25077 | c2ec5e589d78 |
permissions | -rw-r--r-- |
14770 | 1 |
(* Title: HOL/OrderedGroup.thy |
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ID: $Id$ |
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added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
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parents:
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changeset
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Author: Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel, |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
4 |
with contributions by Jeremy Avigad |
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*) |
6 |
||
7 |
header {* Ordered Groups *} |
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theory OrderedGroup |
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adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
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parents:
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imports Lattices |
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uses "~~/src/Provers/Arith/abel_cancel.ML" |
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begin |
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|
14 |
text {* |
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15 |
The theory of partially ordered groups is taken from the books: |
|
16 |
\begin{itemize} |
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17 |
\item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 |
|
18 |
\item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963 |
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19 |
\end{itemize} |
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Most of the used notions can also be looked up in |
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21 |
\begin{itemize} |
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\item \url{http://www.mathworld.com} by Eric Weisstein et. al. |
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\item \emph{Algebra I} by van der Waerden, Springer. |
24 |
\end{itemize} |
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25 |
*} |
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26 |
||
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subsection {* Semigroups and Monoids *} |
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|
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class semigroup_add = plus + |
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assumes add_assoc: "(a + b) + c = a + (b + c)" |
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|
32 |
class ab_semigroup_add = semigroup_add + |
|
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assumes add_commute: "a + b = b + a" |
34 |
begin |
|
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|
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lemma add_left_commute: "a + (b + c) = b + (a + c)" |
37 |
by (rule mk_left_commute [of "plus", OF add_assoc add_commute]) |
|
38 |
(*FIXME term_check*) |
|
39 |
||
40 |
theorems add_ac = add_assoc add_commute add_left_commute |
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41 |
||
42 |
end |
|
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|
44 |
theorems add_ac = add_assoc add_commute add_left_commute |
|
45 |
||
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class semigroup_mult = times + |
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assumes mult_assoc: "(a * b) * c = a * (b * c)" |
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|
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class ab_semigroup_mult = semigroup_mult + |
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assumes mult_commute: "a * b = b * a" |
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begin |
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|
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lemma mult_left_commute: "a * (b * c) = b * (a * c)" |
54 |
by (rule mk_left_commute [of "times", OF mult_assoc mult_commute]) |
|
55 |
(*FIXME term_check*) |
|
56 |
||
57 |
theorems mult_ac = mult_assoc mult_commute mult_left_commute |
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|
59 |
end |
|
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|
61 |
theorems mult_ac = mult_assoc mult_commute mult_left_commute |
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62 |
||
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class monoid_add = zero + semigroup_add + |
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assumes add_0_left [simp]: "0 + a = a" |
65 |
and add_0_right [simp]: "a + 0 = a" |
|
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|
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class comm_monoid_add = zero + ab_semigroup_add + |
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assumes add_0: "0 + a = a" |
69 |
begin |
|
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|
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subclass monoid_add |
72 |
by unfold_locales (insert add_0, simp_all add: add_commute) |
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73 |
||
74 |
end |
|
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|
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class monoid_mult = one + semigroup_mult + |
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assumes mult_1_left [simp]: "1 * a = a" |
78 |
assumes mult_1_right [simp]: "a * 1 = a" |
|
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|
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class comm_monoid_mult = one + ab_semigroup_mult + |
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assumes mult_1: "1 * a = a" |
82 |
begin |
|
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|
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subclass monoid_mult |
85 |
by unfold_locales (insert mult_1, simp_all add: mult_commute) |
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86 |
||
87 |
end |
|
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|
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class cancel_semigroup_add = semigroup_add + |
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assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
91 |
assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c" |
|
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|
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class cancel_ab_semigroup_add = ab_semigroup_add + |
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assumes add_imp_eq: "a + b = a + c \<Longrightarrow> b = c" |
95 |
begin |
|
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|
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subclass cancel_semigroup_add |
98 |
proof unfold_locales |
|
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fix a b c :: 'a |
100 |
assume "a + b = a + c" |
|
101 |
then show "b = c" by (rule add_imp_eq) |
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102 |
next |
|
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fix a b c :: 'a |
104 |
assume "b + a = c + a" |
|
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then have "a + b = a + c" by (simp only: add_commute) |
106 |
then show "b = c" by (rule add_imp_eq) |
|
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qed |
108 |
||
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end context cancel_ab_semigroup_add begin |
110 |
||
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lemma add_left_cancel [simp]: |
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"a + b = a + c \<longleftrightarrow> b = c" |
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by (blast dest: add_left_imp_eq) |
114 |
||
115 |
lemma add_right_cancel [simp]: |
|
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"b + a = c + a \<longleftrightarrow> b = c" |
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by (blast dest: add_right_imp_eq) |
118 |
||
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end |
120 |
||
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subsection {* Groups *} |
122 |
||
123 |
class group_add = minus + monoid_add + |
|
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assumes left_minus [simp]: "- a + a = 0" |
125 |
assumes diff_minus: "a - b = a + (- b)" |
|
126 |
begin |
|
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|
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lemma minus_add_cancel: "- a + (a + b) = b" |
129 |
by (simp add: add_assoc[symmetric]) |
|
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|
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lemma minus_zero [simp]: "- 0 = 0" |
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proof - |
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have "- 0 = - 0 + (0 + 0)" by (simp only: add_0_right) |
134 |
also have "\<dots> = 0" by (rule minus_add_cancel) |
|
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finally show ?thesis . |
136 |
qed |
|
137 |
||
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lemma minus_minus [simp]: "- (- a) = a" |
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proof - |
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have "- (- a) = - (- a) + (- a + a)" by simp |
141 |
also have "\<dots> = a" by (rule minus_add_cancel) |
|
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finally show ?thesis . |
143 |
qed |
|
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|
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lemma right_minus [simp]: "a + - a = 0" |
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proof - |
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have "a + - a = - (- a) + - a" by simp |
148 |
also have "\<dots> = 0" by (rule left_minus) |
|
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finally show ?thesis . |
150 |
qed |
|
151 |
||
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lemma right_minus_eq: "a - b = 0 \<longleftrightarrow> a = b" |
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proof |
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assume "a - b = 0" |
155 |
have "a = (a - b) + b" by (simp add:diff_minus add_assoc) |
|
156 |
also have "\<dots> = b" using `a - b = 0` by simp |
|
157 |
finally show "a = b" . |
|
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next |
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assume "a = b" thus "a - b = 0" by (simp add: diff_minus) |
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qed |
161 |
||
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lemma equals_zero_I: |
163 |
assumes "a + b = 0" |
|
164 |
shows "- a = b" |
|
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proof - |
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have "- a = - a + (a + b)" using assms by simp |
167 |
also have "\<dots> = b" by (simp add: add_assoc[symmetric]) |
|
23085 | 168 |
finally show ?thesis . |
169 |
qed |
|
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|
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lemma diff_self [simp]: "a - a = 0" |
172 |
by (simp add: diff_minus) |
|
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|
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lemma diff_0 [simp]: "0 - a = - a" |
175 |
by (simp add: diff_minus) |
|
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|
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lemma diff_0_right [simp]: "a - 0 = a" |
178 |
by (simp add: diff_minus) |
|
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|
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lemma diff_minus_eq_add [simp]: "a - - b = a + b" |
181 |
by (simp add: diff_minus) |
|
14738 | 182 |
|
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lemma neg_equal_iff_equal [simp]: |
184 |
"- a = - b \<longleftrightarrow> a = b" |
|
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proof |
186 |
assume "- a = - b" |
|
187 |
hence "- (- a) = - (- b)" |
|
188 |
by simp |
|
25062 | 189 |
thus "a = b" by simp |
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next |
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assume "a = b" |
192 |
thus "- a = - b" by simp |
|
14738 | 193 |
qed |
194 |
||
25062 | 195 |
lemma neg_equal_0_iff_equal [simp]: |
196 |
"- a = 0 \<longleftrightarrow> a = 0" |
|
197 |
by (subst neg_equal_iff_equal [symmetric], simp) |
|
14738 | 198 |
|
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lemma neg_0_equal_iff_equal [simp]: |
200 |
"0 = - a \<longleftrightarrow> 0 = a" |
|
201 |
by (subst neg_equal_iff_equal [symmetric], simp) |
|
14738 | 202 |
|
203 |
text{*The next two equations can make the simplifier loop!*} |
|
204 |
||
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lemma equation_minus_iff: |
206 |
"a = - b \<longleftrightarrow> b = - a" |
|
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proof - |
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have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal) |
209 |
thus ?thesis by (simp add: eq_commute) |
|
210 |
qed |
|
211 |
||
212 |
lemma minus_equation_iff: |
|
213 |
"- a = b \<longleftrightarrow> - b = a" |
|
214 |
proof - |
|
215 |
have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal) |
|
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thus ?thesis by (simp add: eq_commute) |
217 |
qed |
|
218 |
||
25062 | 219 |
end |
220 |
||
221 |
class ab_group_add = minus + comm_monoid_add + |
|
222 |
assumes ab_left_minus: "- a + a = 0" |
|
223 |
assumes ab_diff_minus: "a - b = a + (- b)" |
|
224 |
||
225 |
subclass (in ab_group_add) group_add |
|
226 |
by unfold_locales (simp_all add: ab_left_minus ab_diff_minus) |
|
227 |
||
228 |
subclass (in ab_group_add) cancel_semigroup_add |
|
229 |
proof unfold_locales |
|
230 |
fix a b c :: 'a |
|
231 |
assume "a + b = a + c" |
|
232 |
then have "- a + a + b = - a + a + c" |
|
233 |
unfolding add_assoc by simp |
|
234 |
then show "b = c" by simp |
|
235 |
next |
|
236 |
fix a b c :: 'a |
|
237 |
assume "b + a = c + a" |
|
238 |
then have "b + (a + - a) = c + (a + - a)" |
|
239 |
unfolding add_assoc [symmetric] by simp |
|
240 |
then show "b = c" by simp |
|
14738 | 241 |
qed |
242 |
||
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subclass (in ab_group_add) cancel_ab_semigroup_add |
244 |
proof unfold_locales |
|
245 |
fix a b c :: 'a |
|
246 |
assume "a + b = a + c" |
|
247 |
then have "- a + a + b = - a + a + c" |
|
248 |
unfolding add_assoc by simp |
|
249 |
then show "b = c" by simp |
|
250 |
qed |
|
251 |
||
252 |
context ab_group_add |
|
253 |
begin |
|
14738 | 254 |
|
25062 | 255 |
lemma uminus_add_conv_diff: |
256 |
"- a + b = b - a" |
|
257 |
by (simp add:diff_minus add_commute) |
|
258 |
||
259 |
lemma minus_add_distrib [simp]: |
|
260 |
"- (a + b) = - a + - b" |
|
261 |
by (rule equals_zero_I) (simp add: add_ac) |
|
262 |
||
263 |
lemma minus_diff_eq [simp]: |
|
264 |
"- (a - b) = b - a" |
|
265 |
by (simp add: diff_minus add_commute) |
|
266 |
||
267 |
end |
|
14738 | 268 |
|
269 |
subsection {* (Partially) Ordered Groups *} |
|
270 |
||
22390 | 271 |
class pordered_ab_semigroup_add = order + ab_semigroup_add + |
25062 | 272 |
assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b" |
273 |
begin |
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274 |
|
25062 | 275 |
lemma add_right_mono: |
276 |
"a \<le> b \<Longrightarrow> a + c \<le> b + c" |
|
22390 | 277 |
by (simp add: add_commute [of _ c] add_left_mono) |
14738 | 278 |
|
279 |
text {* non-strict, in both arguments *} |
|
280 |
lemma add_mono: |
|
25062 | 281 |
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d" |
14738 | 282 |
apply (erule add_right_mono [THEN order_trans]) |
283 |
apply (simp add: add_commute add_left_mono) |
|
284 |
done |
|
285 |
||
25062 | 286 |
end |
287 |
||
288 |
class pordered_cancel_ab_semigroup_add = |
|
289 |
pordered_ab_semigroup_add + cancel_ab_semigroup_add |
|
290 |
begin |
|
291 |
||
14738 | 292 |
lemma add_strict_left_mono: |
25062 | 293 |
"a < b \<Longrightarrow> c + a < c + b" |
294 |
by (auto simp add: less_le add_left_mono) |
|
14738 | 295 |
|
296 |
lemma add_strict_right_mono: |
|
25062 | 297 |
"a < b \<Longrightarrow> a + c < b + c" |
298 |
by (simp add: add_commute [of _ c] add_strict_left_mono) |
|
14738 | 299 |
|
300 |
text{*Strict monotonicity in both arguments*} |
|
25062 | 301 |
lemma add_strict_mono: |
302 |
"a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" |
|
303 |
apply (erule add_strict_right_mono [THEN less_trans]) |
|
14738 | 304 |
apply (erule add_strict_left_mono) |
305 |
done |
|
306 |
||
307 |
lemma add_less_le_mono: |
|
25062 | 308 |
"a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d" |
309 |
apply (erule add_strict_right_mono [THEN less_le_trans]) |
|
310 |
apply (erule add_left_mono) |
|
14738 | 311 |
done |
312 |
||
313 |
lemma add_le_less_mono: |
|
25062 | 314 |
"a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" |
315 |
apply (erule add_right_mono [THEN le_less_trans]) |
|
14738 | 316 |
apply (erule add_strict_left_mono) |
317 |
done |
|
318 |
||
25062 | 319 |
end |
320 |
||
321 |
class pordered_ab_semigroup_add_imp_le = |
|
322 |
pordered_cancel_ab_semigroup_add + |
|
323 |
assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b" |
|
324 |
begin |
|
325 |
||
14738 | 326 |
lemma add_less_imp_less_left: |
25062 | 327 |
assumes less: "c + a < c + b" |
328 |
shows "a < b" |
|
14738 | 329 |
proof - |
330 |
from less have le: "c + a <= c + b" by (simp add: order_le_less) |
|
331 |
have "a <= b" |
|
332 |
apply (insert le) |
|
333 |
apply (drule add_le_imp_le_left) |
|
334 |
by (insert le, drule add_le_imp_le_left, assumption) |
|
335 |
moreover have "a \<noteq> b" |
|
336 |
proof (rule ccontr) |
|
337 |
assume "~(a \<noteq> b)" |
|
338 |
then have "a = b" by simp |
|
339 |
then have "c + a = c + b" by simp |
|
340 |
with less show "False"by simp |
|
341 |
qed |
|
342 |
ultimately show "a < b" by (simp add: order_le_less) |
|
343 |
qed |
|
344 |
||
345 |
lemma add_less_imp_less_right: |
|
25062 | 346 |
"a + c < b + c \<Longrightarrow> a < b" |
14738 | 347 |
apply (rule add_less_imp_less_left [of c]) |
348 |
apply (simp add: add_commute) |
|
349 |
done |
|
350 |
||
351 |
lemma add_less_cancel_left [simp]: |
|
25062 | 352 |
"c + a < c + b \<longleftrightarrow> a < b" |
353 |
by (blast intro: add_less_imp_less_left add_strict_left_mono) |
|
14738 | 354 |
|
355 |
lemma add_less_cancel_right [simp]: |
|
25062 | 356 |
"a + c < b + c \<longleftrightarrow> a < b" |
357 |
by (blast intro: add_less_imp_less_right add_strict_right_mono) |
|
14738 | 358 |
|
359 |
lemma add_le_cancel_left [simp]: |
|
25062 | 360 |
"c + a \<le> c + b \<longleftrightarrow> a \<le> b" |
361 |
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) |
|
14738 | 362 |
|
363 |
lemma add_le_cancel_right [simp]: |
|
25062 | 364 |
"a + c \<le> b + c \<longleftrightarrow> a \<le> b" |
365 |
by (simp add: add_commute [of a c] add_commute [of b c]) |
|
14738 | 366 |
|
367 |
lemma add_le_imp_le_right: |
|
25062 | 368 |
"a + c \<le> b + c \<Longrightarrow> a \<le> b" |
369 |
by simp |
|
370 |
||
371 |
end |
|
372 |
||
373 |
class pordered_ab_group_add = |
|
374 |
ab_group_add + pordered_ab_semigroup_add |
|
375 |
begin |
|
376 |
||
377 |
subclass pordered_cancel_ab_semigroup_add |
|
378 |
by unfold_locales |
|
379 |
||
380 |
subclass pordered_ab_semigroup_add_imp_le |
|
381 |
proof unfold_locales |
|
382 |
fix a b c :: 'a |
|
383 |
assume "c + a \<le> c + b" |
|
384 |
hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono) |
|
385 |
hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc) |
|
386 |
thus "a \<le> b" by simp |
|
387 |
qed |
|
388 |
||
389 |
end |
|
390 |
||
391 |
class ordered_ab_semigroup_add = |
|
392 |
linorder + pordered_ab_semigroup_add |
|
393 |
||
394 |
class ordered_cancel_ab_semigroup_add = |
|
395 |
linorder + pordered_cancel_ab_semigroup_add |
|
396 |
||
397 |
subclass (in ordered_cancel_ab_semigroup_add) ordered_ab_semigroup_add |
|
398 |
by unfold_locales |
|
399 |
||
400 |
subclass (in ordered_cancel_ab_semigroup_add) pordered_ab_semigroup_add_imp_le |
|
401 |
proof unfold_locales |
|
402 |
fix a b c :: 'a |
|
403 |
assume le: "c + a <= c + b" |
|
404 |
show "a <= b" |
|
405 |
proof (rule ccontr) |
|
406 |
assume w: "~ a \<le> b" |
|
407 |
hence "b <= a" by (simp add: linorder_not_le) |
|
408 |
hence le2: "c + b <= c + a" by (rule add_left_mono) |
|
409 |
have "a = b" |
|
410 |
apply (insert le) |
|
411 |
apply (insert le2) |
|
412 |
apply (drule antisym, simp_all) |
|
413 |
done |
|
414 |
with w show False |
|
415 |
by (simp add: linorder_not_le [symmetric]) |
|
416 |
qed |
|
417 |
qed |
|
418 |
||
419 |
-- {* FIXME continue localization here *} |
|
420 |
||
421 |
lemma max_add_distrib_left: |
|
422 |
fixes z :: "'a::pordered_ab_semigroup_add_imp_le" |
|
423 |
shows "(max x y) + z = max (x+z) (y+z)" |
|
424 |
by (rule max_of_mono [THEN sym], rule add_le_cancel_right) |
|
425 |
||
426 |
lemma min_add_distrib_left: |
|
427 |
fixes z :: "'a::pordered_ab_semigroup_add_imp_le" |
|
428 |
shows "(min x y) + z = min (x+z) (y+z)" |
|
429 |
by (rule min_of_mono [THEN sym], rule add_le_cancel_right) |
|
14738 | 430 |
|
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
431 |
lemma add_increasing: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
432 |
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
433 |
shows "[|0\<le>a; b\<le>c|] ==> b \<le> a + c" |
14738 | 434 |
by (insert add_mono [of 0 a b c], simp) |
435 |
||
15539 | 436 |
lemma add_increasing2: |
437 |
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
|
438 |
shows "[|0\<le>c; b\<le>a|] ==> b \<le> a + c" |
|
439 |
by (simp add:add_increasing add_commute[of a]) |
|
440 |
||
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
441 |
lemma add_strict_increasing: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
442 |
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
443 |
shows "[|0<a; b\<le>c|] ==> b < a + c" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
444 |
by (insert add_less_le_mono [of 0 a b c], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
445 |
|
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
446 |
lemma add_strict_increasing2: |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
447 |
fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
448 |
shows "[|0\<le>a; b<c|] ==> b < a + c" |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
449 |
by (insert add_le_less_mono [of 0 a b c], simp) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
450 |
|
19527 | 451 |
lemma max_diff_distrib_left: |
452 |
fixes z :: "'a::pordered_ab_group_add" |
|
453 |
shows "(max x y) - z = max (x-z) (y-z)" |
|
454 |
by (simp add: diff_minus, rule max_add_distrib_left) |
|
455 |
||
456 |
lemma min_diff_distrib_left: |
|
457 |
fixes z :: "'a::pordered_ab_group_add" |
|
458 |
shows "(min x y) - z = min (x-z) (y-z)" |
|
459 |
by (simp add: diff_minus, rule min_add_distrib_left) |
|
460 |
||
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
461 |
|
14738 | 462 |
subsection {* Ordering Rules for Unary Minus *} |
463 |
||
464 |
lemma le_imp_neg_le: |
|
23389 | 465 |
assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a" |
14738 | 466 |
proof - |
467 |
have "-a+a \<le> -a+b" |
|
23389 | 468 |
using `a \<le> b` by (rule add_left_mono) |
14738 | 469 |
hence "0 \<le> -a+b" |
470 |
by simp |
|
471 |
hence "0 + (-b) \<le> (-a + b) + (-b)" |
|
472 |
by (rule add_right_mono) |
|
473 |
thus ?thesis |
|
474 |
by (simp add: add_assoc) |
|
475 |
qed |
|
476 |
||
477 |
lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))" |
|
478 |
proof |
|
479 |
assume "- b \<le> - a" |
|
480 |
hence "- (- a) \<le> - (- b)" |
|
481 |
by (rule le_imp_neg_le) |
|
482 |
thus "a\<le>b" by simp |
|
483 |
next |
|
484 |
assume "a\<le>b" |
|
485 |
thus "-b \<le> -a" by (rule le_imp_neg_le) |
|
486 |
qed |
|
487 |
||
488 |
lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))" |
|
489 |
by (subst neg_le_iff_le [symmetric], simp) |
|
490 |
||
491 |
lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))" |
|
492 |
by (subst neg_le_iff_le [symmetric], simp) |
|
493 |
||
494 |
lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))" |
|
495 |
by (force simp add: order_less_le) |
|
496 |
||
497 |
lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))" |
|
498 |
by (subst neg_less_iff_less [symmetric], simp) |
|
499 |
||
500 |
lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))" |
|
501 |
by (subst neg_less_iff_less [symmetric], simp) |
|
502 |
||
503 |
text{*The next several equations can make the simplifier loop!*} |
|
504 |
||
505 |
lemma less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))" |
|
506 |
proof - |
|
507 |
have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less) |
|
508 |
thus ?thesis by simp |
|
509 |
qed |
|
510 |
||
511 |
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))" |
|
512 |
proof - |
|
513 |
have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less) |
|
514 |
thus ?thesis by simp |
|
515 |
qed |
|
516 |
||
517 |
lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))" |
|
518 |
proof - |
|
519 |
have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff) |
|
520 |
have "(- (- a) <= -b) = (b <= - a)" |
|
521 |
apply (auto simp only: order_le_less) |
|
522 |
apply (drule mm) |
|
523 |
apply (simp_all) |
|
524 |
apply (drule mm[simplified], assumption) |
|
525 |
done |
|
526 |
then show ?thesis by simp |
|
527 |
qed |
|
528 |
||
529 |
lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))" |
|
530 |
by (auto simp add: order_le_less minus_less_iff) |
|
531 |
||
532 |
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)" |
|
533 |
by (simp add: diff_minus add_ac) |
|
534 |
||
535 |
lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)" |
|
536 |
by (simp add: diff_minus add_ac) |
|
537 |
||
538 |
lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))" |
|
539 |
by (auto simp add: diff_minus add_assoc) |
|
540 |
||
541 |
lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)" |
|
542 |
by (auto simp add: diff_minus add_assoc) |
|
543 |
||
544 |
lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))" |
|
545 |
by (simp add: diff_minus add_ac) |
|
546 |
||
547 |
lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)" |
|
548 |
by (simp add: diff_minus add_ac) |
|
549 |
||
550 |
lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)" |
|
551 |
by (simp add: diff_minus add_ac) |
|
552 |
||
553 |
lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)" |
|
554 |
by (simp add: diff_minus add_ac) |
|
555 |
||
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
556 |
text{*Further subtraction laws*} |
14738 | 557 |
|
558 |
lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))" |
|
559 |
proof - |
|
560 |
have "(a < b) = (a + (- b) < b + (-b))" |
|
561 |
by (simp only: add_less_cancel_right) |
|
562 |
also have "... = (a - b < 0)" by (simp add: diff_minus) |
|
563 |
finally show ?thesis . |
|
564 |
qed |
|
565 |
||
566 |
lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))" |
|
15481 | 567 |
apply (subst less_iff_diff_less_0 [of a]) |
14738 | 568 |
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst]) |
569 |
apply (simp add: diff_minus add_ac) |
|
570 |
done |
|
571 |
||
572 |
lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)" |
|
15481 | 573 |
apply (subst less_iff_diff_less_0 [of "a+b"]) |
574 |
apply (subst less_iff_diff_less_0 [of a]) |
|
14738 | 575 |
apply (simp add: diff_minus add_ac) |
576 |
done |
|
577 |
||
578 |
lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))" |
|
579 |
by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel) |
|
580 |
||
581 |
lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)" |
|
582 |
by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel) |
|
583 |
||
584 |
text{*This list of rewrites simplifies (in)equalities by bringing subtractions |
|
585 |
to the top and then moving negative terms to the other side. |
|
586 |
Use with @{text add_ac}*} |
|
587 |
lemmas compare_rls = |
|
588 |
diff_minus [symmetric] |
|
589 |
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 |
|
590 |
diff_less_eq less_diff_eq diff_le_eq le_diff_eq |
|
591 |
diff_eq_eq eq_diff_eq |
|
592 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
593 |
subsection {* Support for reasoning about signs *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
594 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
595 |
lemma add_pos_pos: "0 < |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
596 |
(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
597 |
==> 0 < y ==> 0 < x + y" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
598 |
apply (subgoal_tac "0 + 0 < x + y") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
599 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
600 |
apply (erule add_less_le_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
601 |
apply (erule order_less_imp_le) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
602 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
603 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
604 |
lemma add_pos_nonneg: "0 < |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
605 |
(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
606 |
==> 0 <= y ==> 0 < x + y" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
607 |
apply (subgoal_tac "0 + 0 < x + y") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
608 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
609 |
apply (erule add_less_le_mono, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
610 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
611 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
612 |
lemma add_nonneg_pos: "0 <= |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
613 |
(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
614 |
==> 0 < y ==> 0 < x + y" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
615 |
apply (subgoal_tac "0 + 0 < x + y") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
616 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
617 |
apply (erule add_le_less_mono, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
618 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
619 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
620 |
lemma add_nonneg_nonneg: "0 <= |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
621 |
(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
622 |
==> 0 <= y ==> 0 <= x + y" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
623 |
apply (subgoal_tac "0 + 0 <= x + y") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
624 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
625 |
apply (erule add_mono, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
626 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
627 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
628 |
lemma add_neg_neg: "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
629 |
< 0 ==> y < 0 ==> x + y < 0" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
630 |
apply (subgoal_tac "x + y < 0 + 0") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
631 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
632 |
apply (erule add_less_le_mono) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
633 |
apply (erule order_less_imp_le) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
634 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
635 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
636 |
lemma add_neg_nonpos: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
637 |
"(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) < 0 |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
638 |
==> y <= 0 ==> x + y < 0" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
639 |
apply (subgoal_tac "x + y < 0 + 0") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
640 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
641 |
apply (erule add_less_le_mono, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
642 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
643 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
644 |
lemma add_nonpos_neg: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
645 |
"(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
646 |
==> y < 0 ==> x + y < 0" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
647 |
apply (subgoal_tac "x + y < 0 + 0") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
648 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
649 |
apply (erule add_le_less_mono, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
650 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
651 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
652 |
lemma add_nonpos_nonpos: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
653 |
"(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
654 |
==> y <= 0 ==> x + y <= 0" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
655 |
apply (subgoal_tac "x + y <= 0 + 0") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
656 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
657 |
apply (erule add_mono, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
658 |
done |
14738 | 659 |
|
660 |
subsection{*Lemmas for the @{text cancel_numerals} simproc*} |
|
661 |
||
662 |
lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))" |
|
663 |
by (simp add: compare_rls) |
|
664 |
||
665 |
lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))" |
|
666 |
by (simp add: compare_rls) |
|
667 |
||
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
668 |
|
14738 | 669 |
subsection {* Lattice Ordered (Abelian) Groups *} |
670 |
||
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
671 |
class lordered_ab_group_meet = pordered_ab_group_add + lower_semilattice |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
672 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
673 |
class lordered_ab_group_join = pordered_ab_group_add + upper_semilattice |
14738 | 674 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
675 |
class lordered_ab_group = pordered_ab_group_add + lattice |
14738 | 676 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
677 |
instance lordered_ab_group \<subseteq> lordered_ab_group_meet by default |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
678 |
instance lordered_ab_group \<subseteq> lordered_ab_group_join by default |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
679 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
680 |
lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + (c::'a::{pordered_ab_group_add, lower_semilattice}))" |
14738 | 681 |
apply (rule order_antisym) |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
682 |
apply (simp_all add: le_infI) |
14738 | 683 |
apply (rule add_le_imp_le_left [of "-a"]) |
684 |
apply (simp only: add_assoc[symmetric], simp) |
|
21312 | 685 |
apply rule |
686 |
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+ |
|
14738 | 687 |
done |
688 |
||
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
689 |
lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a+ (c::'a::{pordered_ab_group_add, upper_semilattice}))" |
14738 | 690 |
apply (rule order_antisym) |
691 |
apply (rule add_le_imp_le_left [of "-a"]) |
|
692 |
apply (simp only: add_assoc[symmetric], simp) |
|
21312 | 693 |
apply rule |
694 |
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+ |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
695 |
apply (rule le_supI) |
21312 | 696 |
apply (simp_all) |
14738 | 697 |
done |
698 |
||
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
699 |
lemma add_inf_distrib_right: "inf a b + (c::'a::lordered_ab_group) = inf (a+c) (b+c)" |
14738 | 700 |
proof - |
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
701 |
have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left) |
14738 | 702 |
thus ?thesis by (simp add: add_commute) |
703 |
qed |
|
704 |
||
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
705 |
lemma add_sup_distrib_right: "sup a b + (c::'a::lordered_ab_group) = sup (a+c) (b+c)" |
14738 | 706 |
proof - |
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
707 |
have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left) |
14738 | 708 |
thus ?thesis by (simp add: add_commute) |
709 |
qed |
|
710 |
||
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
711 |
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left |
14738 | 712 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
713 |
lemma inf_eq_neg_sup: "inf a (b\<Colon>'a\<Colon>lordered_ab_group) = - sup (-a) (-b)" |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
714 |
proof (rule inf_unique) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
715 |
fix a b :: 'a |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
716 |
show "- sup (-a) (-b) \<le> a" by (rule add_le_imp_le_right [of _ "sup (-a) (-b)"]) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
717 |
(simp, simp add: add_sup_distrib_left) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
718 |
next |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
719 |
fix a b :: 'a |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
720 |
show "- sup (-a) (-b) \<le> b" by (rule add_le_imp_le_right [of _ "sup (-a) (-b)"]) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
721 |
(simp, simp add: add_sup_distrib_left) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
722 |
next |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
723 |
fix a b c :: 'a |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
724 |
assume "a \<le> b" "a \<le> c" |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
725 |
then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric]) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
726 |
(simp add: le_supI) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
727 |
qed |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
728 |
|
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
729 |
lemma sup_eq_neg_inf: "sup a (b\<Colon>'a\<Colon>lordered_ab_group) = - inf (-a) (-b)" |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
730 |
proof (rule sup_unique) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
731 |
fix a b :: 'a |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
732 |
show "a \<le> - inf (-a) (-b)" by (rule add_le_imp_le_right [of _ "inf (-a) (-b)"]) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
733 |
(simp, simp add: add_inf_distrib_left) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
734 |
next |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
735 |
fix a b :: 'a |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
736 |
show "b \<le> - inf (-a) (-b)" by (rule add_le_imp_le_right [of _ "inf (-a) (-b)"]) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
737 |
(simp, simp add: add_inf_distrib_left) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
738 |
next |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
739 |
fix a b c :: 'a |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
740 |
assume "a \<le> c" "b \<le> c" |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
741 |
then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric]) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
742 |
(simp add: le_infI) |
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
743 |
qed |
14738 | 744 |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
745 |
lemma add_eq_inf_sup: "a + b = sup a b + inf a (b\<Colon>'a\<Colon>lordered_ab_group)" |
14738 | 746 |
proof - |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
747 |
have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
748 |
hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
749 |
hence "0 = (-a + sup a b) + (inf a b + (-b))" |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
750 |
apply (simp add: add_sup_distrib_left add_inf_distrib_right) |
14738 | 751 |
by (simp add: diff_minus add_commute) |
752 |
thus ?thesis |
|
753 |
apply (simp add: compare_rls) |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
754 |
apply (subst add_left_cancel[symmetric, of "a+b" "sup a b + inf a b" "-a"]) |
14738 | 755 |
apply (simp only: add_assoc, simp add: add_assoc[symmetric]) |
756 |
done |
|
757 |
qed |
|
758 |
||
759 |
subsection {* Positive Part, Negative Part, Absolute Value *} |
|
760 |
||
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
761 |
definition |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
762 |
nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" where |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
763 |
"nprt x = inf x 0" |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
764 |
|
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
765 |
definition |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
766 |
pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" where |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
767 |
"pprt x = sup x 0" |
14738 | 768 |
|
769 |
lemma prts: "a = pprt a + nprt a" |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
770 |
by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric]) |
14738 | 771 |
|
772 |
lemma zero_le_pprt[simp]: "0 \<le> pprt a" |
|
21312 | 773 |
by (simp add: pprt_def) |
14738 | 774 |
|
775 |
lemma nprt_le_zero[simp]: "nprt a \<le> 0" |
|
21312 | 776 |
by (simp add: nprt_def) |
14738 | 777 |
|
778 |
lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r") |
|
779 |
proof - |
|
780 |
have a: "?l \<longrightarrow> ?r" |
|
781 |
apply (auto) |
|
782 |
apply (rule add_le_imp_le_right[of _ "-b" _]) |
|
783 |
apply (simp add: add_assoc) |
|
784 |
done |
|
785 |
have b: "?r \<longrightarrow> ?l" |
|
786 |
apply (auto) |
|
787 |
apply (rule add_le_imp_le_right[of _ "b" _]) |
|
788 |
apply (simp) |
|
789 |
done |
|
790 |
from a b show ?thesis by blast |
|
791 |
qed |
|
792 |
||
15580 | 793 |
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def) |
794 |
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def) |
|
795 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
796 |
lemma pprt_eq_id[simp,noatp]: "0 <= x \<Longrightarrow> pprt x = x" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
797 |
by (simp add: pprt_def le_iff_sup sup_aci) |
15580 | 798 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
799 |
lemma nprt_eq_id[simp,noatp]: "x <= 0 \<Longrightarrow> nprt x = x" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
800 |
by (simp add: nprt_def le_iff_inf inf_aci) |
15580 | 801 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
802 |
lemma pprt_eq_0[simp,noatp]: "x <= 0 \<Longrightarrow> pprt x = 0" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
803 |
by (simp add: pprt_def le_iff_sup sup_aci) |
15580 | 804 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
805 |
lemma nprt_eq_0[simp,noatp]: "0 <= x \<Longrightarrow> nprt x = 0" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
806 |
by (simp add: nprt_def le_iff_inf inf_aci) |
15580 | 807 |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
808 |
lemma sup_0_imp_0: "sup a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" |
14738 | 809 |
proof - |
810 |
{ |
|
811 |
fix a::'a |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
812 |
assume hyp: "sup a (-a) = 0" |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
813 |
hence "sup a (-a) + a = a" by (simp) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
814 |
hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
815 |
hence "sup (a+a) 0 <= a" by (simp) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
816 |
hence "0 <= a" by (blast intro: order_trans inf_sup_ord) |
14738 | 817 |
} |
818 |
note p = this |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
819 |
assume hyp:"sup a (-a) = 0" |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
820 |
hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute) |
14738 | 821 |
from p[OF hyp] p[OF hyp2] show "a = 0" by simp |
822 |
qed |
|
823 |
||
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
824 |
lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)" |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
825 |
apply (simp add: inf_eq_neg_sup) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
826 |
apply (simp add: sup_commute) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
827 |
apply (erule sup_0_imp_0) |
15481 | 828 |
done |
14738 | 829 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
830 |
lemma inf_0_eq_0[simp,noatp]: "(inf a (-a) = 0) = (a = (0::'a::lordered_ab_group))" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
831 |
by (auto, erule inf_0_imp_0) |
14738 | 832 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
833 |
lemma sup_0_eq_0[simp,noatp]: "(sup a (-a) = 0) = (a = (0::'a::lordered_ab_group))" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
834 |
by (auto, erule sup_0_imp_0) |
14738 | 835 |
|
836 |
lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))" |
|
837 |
proof |
|
838 |
assume "0 <= a + a" |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
839 |
hence a:"inf (a+a) 0 = 0" by (simp add: le_iff_inf inf_commute) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
840 |
have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_") by (simp add: add_sup_inf_distribs inf_aci) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
841 |
hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
842 |
hence "inf a 0 = 0" by (simp only: add_right_cancel) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
843 |
then show "0 <= a" by (simp add: le_iff_inf inf_commute) |
14738 | 844 |
next |
845 |
assume a: "0 <= a" |
|
846 |
show "0 <= a + a" by (simp add: add_mono[OF a a, simplified]) |
|
847 |
qed |
|
848 |
||
849 |
lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" |
|
850 |
proof - |
|
851 |
have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp) |
|
852 |
moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add) |
|
853 |
ultimately show ?thesis by blast |
|
854 |
qed |
|
855 |
||
856 |
lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s) |
|
857 |
proof cases |
|
858 |
assume a: "a < 0" |
|
859 |
thus ?s by (simp add: add_strict_mono[OF a a, simplified]) |
|
860 |
next |
|
861 |
assume "~(a < 0)" |
|
862 |
hence a:"0 <= a" by (simp) |
|
863 |
hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified]) |
|
864 |
hence "~(a+a < 0)" by simp |
|
865 |
with a show ?thesis by simp |
|
866 |
qed |
|
867 |
||
23879 | 868 |
class lordered_ab_group_abs = lordered_ab_group + abs + |
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
869 |
assumes abs_lattice: "abs x = sup x (uminus x)" |
14738 | 870 |
|
871 |
lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)" |
|
872 |
by (simp add: abs_lattice) |
|
873 |
||
874 |
lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))" |
|
875 |
by (simp add: abs_lattice) |
|
876 |
||
877 |
lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))" |
|
878 |
proof - |
|
879 |
have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac) |
|
880 |
thus ?thesis by simp |
|
881 |
qed |
|
882 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
883 |
declare abs_0_eq [noatp] (*essentially the same as the other one*) |
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
884 |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
885 |
lemma neg_inf_eq_sup[simp]: "- inf a (b::_::lordered_ab_group) = sup (-a) (-b)" |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
886 |
by (simp add: inf_eq_neg_sup) |
14738 | 887 |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
888 |
lemma neg_sup_eq_inf[simp]: "- sup a (b::_::lordered_ab_group) = inf (-a) (-b)" |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
889 |
by (simp del: neg_inf_eq_sup add: sup_eq_neg_inf) |
14738 | 890 |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
891 |
lemma sup_eq_if: "sup a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))" |
14738 | 892 |
proof - |
893 |
note b = add_le_cancel_right[of a a "-a",symmetric,simplified] |
|
894 |
have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp) |
|
22452
8a86fd2a1bf0
adjusted to new lattice theory developement in Lattices.thy / FixedPoint.thy
haftmann
parents:
22422
diff
changeset
|
895 |
show ?thesis by (auto simp add: max_def b linorder_not_less sup_max) |
14738 | 896 |
qed |
897 |
||
898 |
lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))" |
|
899 |
proof - |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
900 |
show ?thesis by (simp add: abs_lattice sup_eq_if) |
14738 | 901 |
qed |
902 |
||
903 |
lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)" |
|
904 |
proof - |
|
21312 | 905 |
have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice) |
14738 | 906 |
show ?thesis by (rule add_mono[OF a b, simplified]) |
907 |
qed |
|
908 |
||
909 |
lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" |
|
910 |
proof |
|
911 |
assume "abs a <= 0" |
|
912 |
hence "abs a = 0" by (auto dest: order_antisym) |
|
913 |
thus "a = 0" by simp |
|
914 |
next |
|
915 |
assume "a = 0" |
|
916 |
thus "abs a <= 0" by simp |
|
917 |
qed |
|
918 |
||
919 |
lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))" |
|
920 |
by (simp add: order_less_le) |
|
921 |
||
922 |
lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)" |
|
923 |
proof - |
|
924 |
have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto |
|
925 |
show ?thesis by (simp add: a) |
|
926 |
qed |
|
927 |
||
928 |
lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)" |
|
21312 | 929 |
by (simp add: abs_lattice) |
14738 | 930 |
|
931 |
lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)" |
|
21312 | 932 |
by (simp add: abs_lattice) |
14738 | 933 |
|
934 |
lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a" |
|
935 |
apply (simp add: pprt_def nprt_def diff_minus) |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
936 |
apply (simp add: add_sup_inf_distribs sup_aci abs_lattice[symmetric]) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
937 |
apply (subst sup_absorb2, auto) |
14738 | 938 |
done |
939 |
||
940 |
lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)" |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
941 |
by (simp add: abs_lattice sup_commute) |
14738 | 942 |
|
943 |
lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)" |
|
944 |
apply (simp add: abs_lattice[of "abs a"]) |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
945 |
apply (subst sup_absorb1) |
14738 | 946 |
apply (rule order_trans[of _ 0]) |
947 |
by auto |
|
948 |
||
15093
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
949 |
lemma abs_minus_commute: |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
950 |
fixes a :: "'a::lordered_ab_group_abs" |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
951 |
shows "abs (a-b) = abs(b-a)" |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
952 |
proof - |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
953 |
have "abs (a-b) = abs (- (a-b))" by (simp only: abs_minus_cancel) |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
954 |
also have "... = abs(b-a)" by simp |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
955 |
finally show ?thesis . |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
956 |
qed |
49ede01e9ee6
conversion of Integration and NSPrimes to Isar scripts
paulson
parents:
15010
diff
changeset
|
957 |
|
14738 | 958 |
lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
959 |
by (simp add: le_iff_inf nprt_def inf_commute) |
14738 | 960 |
|
961 |
lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)" |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
962 |
by (simp add: le_iff_sup pprt_def sup_commute) |
14738 | 963 |
|
964 |
lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)" |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
965 |
by (simp add: le_iff_sup pprt_def sup_commute) |
14738 | 966 |
|
967 |
lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)" |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
968 |
by (simp add: le_iff_inf nprt_def inf_commute) |
14738 | 969 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
970 |
lemma pprt_mono[simp,noatp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> pprt a <= pprt b" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
971 |
by (simp add: le_iff_sup pprt_def sup_aci) |
15580 | 972 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24137
diff
changeset
|
973 |
lemma nprt_mono[simp,noatp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> nprt a <= nprt b" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
974 |
by (simp add: le_iff_inf nprt_def inf_aci) |
15580 | 975 |
|
19404 | 976 |
lemma pprt_neg: "pprt (-x) = - nprt x" |
977 |
by (simp add: pprt_def nprt_def) |
|
978 |
||
979 |
lemma nprt_neg: "nprt (-x) = - pprt x" |
|
980 |
by (simp add: pprt_def nprt_def) |
|
981 |
||
14738 | 982 |
lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)" |
983 |
by (simp) |
|
984 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
985 |
lemma abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)" |
14738 | 986 |
by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts) |
987 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
988 |
lemma abs_of_pos: "0 < (x::'a::lordered_ab_group_abs) ==> abs x = x"; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
989 |
by (rule abs_of_nonneg, rule order_less_imp_le); |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
990 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
991 |
lemma abs_of_nonpos [simp]: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)" |
14738 | 992 |
by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts) |
993 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
994 |
lemma abs_of_neg: "(x::'a::lordered_ab_group_abs) < 0 ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
995 |
abs x = - x" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
996 |
by (rule abs_of_nonpos, rule order_less_imp_le) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
997 |
|
14738 | 998 |
lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)" |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
999 |
by (simp add: abs_lattice le_supI) |
14738 | 1000 |
|
1001 |
lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))" |
|
1002 |
proof - |
|
1003 |
from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)" |
|
1004 |
by (simp add: add_assoc[symmetric]) |
|
1005 |
thus ?thesis by simp |
|
1006 |
qed |
|
1007 |
||
1008 |
lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))" |
|
1009 |
proof - |
|
1010 |
from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)" |
|
1011 |
by (simp add: add_assoc[symmetric]) |
|
1012 |
thus ?thesis by simp |
|
1013 |
qed |
|
1014 |
||
1015 |
lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)" |
|
1016 |
by (insert abs_ge_self, blast intro: order_trans) |
|
1017 |
||
1018 |
lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)" |
|
1019 |
by (insert abs_le_D1 [of "-a"], simp) |
|
1020 |
||
1021 |
lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))" |
|
1022 |
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2) |
|
1023 |
||
15539 | 1024 |
lemma abs_triangle_ineq: "abs(a+b) \<le> abs a + abs(b::'a::lordered_ab_group_abs)" |
14738 | 1025 |
proof - |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
1026 |
have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n") |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
1027 |
by (simp add: abs_lattice add_sup_inf_distribs sup_aci diff_minus) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
1028 |
have a:"a+b <= sup ?m ?n" by (simp) |
21312 | 1029 |
have b:"-a-b <= ?n" by (simp) |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
1030 |
have c:"?n <= sup ?m ?n" by (simp) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
1031 |
from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans) |
14738 | 1032 |
have e:"-a-b = -(a+b)" by (simp add: diff_minus) |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
1033 |
from a d e have "abs(a+b) <= sup ?m ?n" |
14738 | 1034 |
by (drule_tac abs_leI, auto) |
1035 |
with g[symmetric] show ?thesis by simp |
|
1036 |
qed |
|
1037 |
||
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1038 |
lemma abs_triangle_ineq2: "abs (a::'a::lordered_ab_group_abs) - |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1039 |
abs b <= abs (a - b)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1040 |
apply (simp add: compare_rls) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1041 |
apply (subgoal_tac "abs a = abs (a - b + b)") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1042 |
apply (erule ssubst) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1043 |
apply (rule abs_triangle_ineq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1044 |
apply (rule arg_cong);back; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1045 |
apply (simp add: compare_rls) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1046 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1047 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1048 |
lemma abs_triangle_ineq3: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1049 |
"abs(abs (a::'a::lordered_ab_group_abs) - abs b) <= abs (a - b)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1050 |
apply (subst abs_le_iff) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1051 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1052 |
apply (rule abs_triangle_ineq2) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1053 |
apply (subst abs_minus_commute) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1054 |
apply (rule abs_triangle_ineq2) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1055 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1056 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1057 |
lemma abs_triangle_ineq4: "abs ((a::'a::lordered_ab_group_abs) - b) <= |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1058 |
abs a + abs b" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1059 |
proof -; |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1060 |
have "abs(a - b) = abs(a + - b)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1061 |
by (subst diff_minus, rule refl) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1062 |
also have "... <= abs a + abs (- b)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1063 |
by (rule abs_triangle_ineq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1064 |
finally show ?thesis |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1065 |
by simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1066 |
qed |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16417
diff
changeset
|
1067 |
|
14738 | 1068 |
lemma abs_diff_triangle_ineq: |
1069 |
"\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" |
|
1070 |
proof - |
|
1071 |
have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac) |
|
1072 |
also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq) |
|
1073 |
finally show ?thesis . |
|
1074 |
qed |
|
1075 |
||
15539 | 1076 |
lemma abs_add_abs[simp]: |
1077 |
fixes a:: "'a::{lordered_ab_group_abs}" |
|
1078 |
shows "abs(abs a + abs b) = abs a + abs b" (is "?L = ?R") |
|
1079 |
proof (rule order_antisym) |
|
1080 |
show "?L \<ge> ?R" by(rule abs_ge_self) |
|
1081 |
next |
|
1082 |
have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq) |
|
1083 |
also have "\<dots> = ?R" by simp |
|
1084 |
finally show "?L \<le> ?R" . |
|
1085 |
qed |
|
1086 |
||
14754
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1087 |
text {* Needed for abelian cancellation simprocs: *} |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1088 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1089 |
lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1090 |
apply (subst add_left_commute) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1091 |
apply (subst add_left_cancel) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1092 |
apply simp |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1093 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1094 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1095 |
lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1096 |
apply (subst add_cancel_21[of _ _ _ 0, simplified]) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1097 |
apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified]) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1098 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1099 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1100 |
lemma less_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1101 |
by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y']) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1102 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1103 |
lemma le_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1104 |
apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of y' x']) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1105 |
apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0]) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1106 |
done |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1107 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1108 |
lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1109 |
by (simp add: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y']) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1110 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1111 |
lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1112 |
by (simp add: diff_minus) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1113 |
|
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1114 |
lemma add_minus_cancel: "(a::'a::ab_group_add) + (-a + b) = b" |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1115 |
by (simp add: add_assoc[symmetric]) |
a080eeeaec14
Modification / Installation of Provers/Arith/abel_cancel.ML for OrderedGroup.thy
obua
parents:
14738
diff
changeset
|
1116 |
|
15178 | 1117 |
lemma le_add_right_mono: |
1118 |
assumes |
|
1119 |
"a <= b + (c::'a::pordered_ab_group_add)" |
|
1120 |
"c <= d" |
|
1121 |
shows "a <= b + d" |
|
1122 |
apply (rule_tac order_trans[where y = "b+c"]) |
|
1123 |
apply (simp_all add: prems) |
|
1124 |
done |
|
1125 |
||
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23389
diff
changeset
|
1126 |
lemmas group_simps = |
15178 | 1127 |
mult_ac |
1128 |
add_ac |
|
1129 |
add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2 |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23389
diff
changeset
|
1130 |
diff_eq_eq eq_diff_eq diff_minus[symmetric] uminus_add_conv_diff |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23389
diff
changeset
|
1131 |
diff_less_eq less_diff_eq diff_le_eq le_diff_eq |
15178 | 1132 |
|
1133 |
lemma estimate_by_abs: |
|
24380
c215e256beca
moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents:
24286
diff
changeset
|
1134 |
"a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b" |
15178 | 1135 |
proof - |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23389
diff
changeset
|
1136 |
assume "a+b <= c" |
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23389
diff
changeset
|
1137 |
hence 2: "a <= c+(-b)" by (simp add: group_simps) |
15178 | 1138 |
have 3: "(-b) <= abs b" by (rule abs_ge_minus_self) |
1139 |
show ?thesis by (rule le_add_right_mono[OF 2 3]) |
|
1140 |
qed |
|
1141 |
||
22482 | 1142 |
|
1143 |
subsection {* Tools setup *} |
|
1144 |
||
17085 | 1145 |
text{*Simplification of @{term "x-y < 0"}, etc.*} |
24380
c215e256beca
moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents:
24286
diff
changeset
|
1146 |
lemmas diff_less_0_iff_less [simp] = less_iff_diff_less_0 [symmetric] |
c215e256beca
moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents:
24286
diff
changeset
|
1147 |
lemmas diff_eq_0_iff_eq [simp, noatp] = eq_iff_diff_eq_0 [symmetric] |
c215e256beca
moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents:
24286
diff
changeset
|
1148 |
lemmas diff_le_0_iff_le [simp] = le_iff_diff_le_0 [symmetric] |
17085 | 1149 |
|
22482 | 1150 |
ML {* |
1151 |
structure ab_group_add_cancel = Abel_Cancel( |
|
1152 |
struct |
|
1153 |
||
1154 |
(* term order for abelian groups *) |
|
1155 |
||
1156 |
fun agrp_ord (Const (a, _)) = find_index (fn a' => a = a') |
|
22997 | 1157 |
[@{const_name HOL.zero}, @{const_name HOL.plus}, |
1158 |
@{const_name HOL.uminus}, @{const_name HOL.minus}] |
|
22482 | 1159 |
| agrp_ord _ = ~1; |
1160 |
||
1161 |
fun termless_agrp (a, b) = (Term.term_lpo agrp_ord (a, b) = LESS); |
|
1162 |
||
1163 |
local |
|
1164 |
val ac1 = mk_meta_eq @{thm add_assoc}; |
|
1165 |
val ac2 = mk_meta_eq @{thm add_commute}; |
|
1166 |
val ac3 = mk_meta_eq @{thm add_left_commute}; |
|
22997 | 1167 |
fun solve_add_ac thy _ (_ $ (Const (@{const_name HOL.plus},_) $ _ $ _) $ _) = |
22482 | 1168 |
SOME ac1 |
22997 | 1169 |
| solve_add_ac thy _ (_ $ x $ (Const (@{const_name HOL.plus},_) $ y $ z)) = |
22482 | 1170 |
if termless_agrp (y, x) then SOME ac3 else NONE |
1171 |
| solve_add_ac thy _ (_ $ x $ y) = |
|
1172 |
if termless_agrp (y, x) then SOME ac2 else NONE |
|
1173 |
| solve_add_ac thy _ _ = NONE |
|
1174 |
in |
|
1175 |
val add_ac_proc = Simplifier.simproc @{theory} |
|
1176 |
"add_ac_proc" ["x + y::'a::ab_semigroup_add"] solve_add_ac; |
|
1177 |
end; |
|
1178 |
||
1179 |
val cancel_ss = HOL_basic_ss settermless termless_agrp |
|
1180 |
addsimprocs [add_ac_proc] addsimps |
|
23085 | 1181 |
[@{thm add_0_left}, @{thm add_0_right}, @{thm diff_def}, |
22482 | 1182 |
@{thm minus_add_distrib}, @{thm minus_minus}, @{thm minus_zero}, |
1183 |
@{thm right_minus}, @{thm left_minus}, @{thm add_minus_cancel}, |
|
1184 |
@{thm minus_add_cancel}]; |
|
1185 |
||
22548 | 1186 |
val eq_reflection = @{thm eq_reflection}; |
22482 | 1187 |
|
24137
8d7896398147
replaced Theory.self_ref by Theory.check_thy, which now produces a checked ref;
wenzelm
parents:
23879
diff
changeset
|
1188 |
val thy_ref = Theory.check_thy @{theory}; |
22482 | 1189 |
|
22548 | 1190 |
val T = TFree("'a", ["OrderedGroup.ab_group_add"]); |
22482 | 1191 |
|
22548 | 1192 |
val eqI_rules = [@{thm less_eqI}, @{thm le_eqI}, @{thm eq_eqI}]; |
22482 | 1193 |
|
1194 |
val dest_eqI = |
|
1195 |
fst o HOLogic.dest_bin "op =" HOLogic.boolT o HOLogic.dest_Trueprop o concl_of; |
|
1196 |
||
1197 |
end); |
|
1198 |
*} |
|
1199 |
||
1200 |
ML_setup {* |
|
1201 |
Addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]; |
|
1202 |
*} |
|
17085 | 1203 |
|
14738 | 1204 |
end |