author  haftmann 
Thu, 06 May 2010 23:11:56 +0200  
changeset 36719  d396f6f63d94 
parent 36425  a0297b98728c 
child 36774  9e444b09fbef 
permissions  rwrr 
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(* Title: HOL/Fields.thy 
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Author: Gertrud Bauer 
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3 
Author: Steven Obua 
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Author: Tobias Nipkow 
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5 
Author: Lawrence C Paulson 
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Author: Markus Wenzel 
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Author: Jeremy Avigad 
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HOL: installation of Ring_and_Field as the basis for Naturals and Reals
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*) 
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header {* Fields *} 
25152  11 

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theory Fields 
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imports Rings 
25186  14 
begin 
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class field = comm_ring_1 + inverse + 
35084  17 
assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" 
18 
assumes field_divide_inverse: "a / b = a * inverse b" 

25267  19 
begin 
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25267  21 
subclass division_ring 
28823  22 
proof 
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fix a :: 'a 
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assume "a \<noteq> 0" 
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thus "inverse a * a = 1" by (rule field_inverse) 
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thus "a * inverse a = 1" by (simp only: mult_commute) 
35084  27 
next 
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fix a b :: 'a 

29 
show "a / b = a * inverse b" by (rule field_divide_inverse) 

14738  30 
qed 
25230  31 

27516  32 
subclass idom .. 
25230  33 

30630  34 
text{*There is no slick version using division by zero.*} 
35 
lemma inverse_add: 

36 
"[ a \<noteq> 0; b \<noteq> 0 ] 

37 
==> inverse a + inverse b = (a + b) * inverse a * inverse b" 

38 
by (simp add: division_ring_inverse_add mult_ac) 

39 

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lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]: 
30630  41 
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b" 
42 
proof  

43 
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" 

44 
by (simp add: divide_inverse nonzero_inverse_mult_distrib) 

45 
also have "... = a * inverse b * (inverse c * c)" 

46 
by (simp only: mult_ac) 

47 
also have "... = a * inverse b" by simp 

48 
finally show ?thesis by (simp add: divide_inverse) 

49 
qed 

50 

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lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]: 
30630  52 
"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b" 
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by (simp add: mult_commute [of _ c]) 

54 

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lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c" 
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by (simp add: divide_inverse mult_ac) 
30630  57 

58 
text {* These are later declared as simp rules. *} 

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lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left 
30630  60 

61 
lemma add_frac_eq: 

62 
assumes "y \<noteq> 0" and "z \<noteq> 0" 

63 
shows "x / y + w / z = (x * z + w * y) / (y * z)" 

64 
proof  

65 
have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)" 

66 
using assms by simp 

67 
also have "\<dots> = (x * z + y * w) / (y * z)" 

68 
by (simp only: add_divide_distrib) 

69 
finally show ?thesis 

70 
by (simp only: mult_commute) 

71 
qed 

72 

73 
text{*Special Cancellation Simprules for Division*} 

74 

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lemma nonzero_mult_divide_cancel_right [simp, no_atp]: 
30630  76 
"b \<noteq> 0 \<Longrightarrow> a * b / b = a" 
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using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp 
30630  78 

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lemma nonzero_mult_divide_cancel_left [simp, no_atp]: 
30630  80 
"a \<noteq> 0 \<Longrightarrow> a * b / a = b" 
81 
using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp 

82 

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lemma nonzero_divide_mult_cancel_right [simp, no_atp]: 
30630  84 
"\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a" 
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using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp 

86 

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lemma nonzero_divide_mult_cancel_left [simp, no_atp]: 
30630  88 
"\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b" 
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using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp 

90 

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lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]: 
30630  92 
"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b" 
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using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac) 

94 

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lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]: 
30630  96 
"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b" 
97 
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac) 

98 

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lemma add_divide_eq_iff [field_simps]: 
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"z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z" 
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by (simp add: add_divide_distrib) 
30630  102 

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lemma divide_add_eq_iff [field_simps]: 
30630  104 
"z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z" 
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by (simp add: add_divide_distrib) 
30630  106 

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lemma diff_divide_eq_iff [field_simps]: 
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"z \<noteq> 0 \<Longrightarrow> x  y / z = (z * x  y) / z" 
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by (simp add: diff_divide_distrib) 
30630  110 

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lemma divide_diff_eq_iff [field_simps]: 
30630  112 
"z \<noteq> 0 \<Longrightarrow> x / z  y = (x  z * y) / z" 
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by (simp add: diff_divide_distrib) 
30630  114 

115 
lemma diff_frac_eq: 

116 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y  w / z = (x * z  w * y) / (y * z)" 

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by (simp add: field_simps) 
30630  118 

119 
lemma frac_eq_eq: 

120 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)" 

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by (simp add: field_simps) 
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end 
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class field_inverse_zero = field + 
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assumes field_inverse_zero: "inverse 0 = 0" 
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begin 
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subclass division_ring_inverse_zero proof 
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qed (fact field_inverse_zero) 
25230  131 

14270  132 
text{*This version builds in division by zero while also reorienting 
133 
the righthand side.*} 

134 
lemma inverse_mult_distrib [simp]: 

36409  135 
"inverse (a * b) = inverse a * inverse b" 
136 
proof cases 

137 
assume "a \<noteq> 0 & b \<noteq> 0" 

138 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac) 

139 
next 

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assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

141 
thus ?thesis by force 

142 
qed 

14270  143 

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lemma inverse_divide [simp]: 
36409  145 
"inverse (a / b) = b / a" 
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by (simp add: divide_inverse mult_commute) 
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23389  148 

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text {* Calculations with fractions *} 
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text{* There is a whole bunch of simprules just for class @{text 
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field} but none for class @{text field} and @{text nonzero_divides} 
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because the latter are covered by a simproc. *} 
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lemma mult_divide_mult_cancel_left: 
36409  156 
"c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b" 
21328  157 
apply (cases "b = 0") 
35216  158 
apply simp_all 
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done 
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lemma mult_divide_mult_cancel_right: 
36409  162 
"c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b" 
21328  163 
apply (cases "b = 0") 
35216  164 
apply simp_all 
14321  165 
done 
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36409  167 
lemma divide_divide_eq_right [simp, no_atp]: 
168 
"a / (b / c) = (a * c) / b" 

169 
by (simp add: divide_inverse mult_ac) 

14288  170 

36409  171 
lemma divide_divide_eq_left [simp, no_atp]: 
172 
"(a / b) / c = a / (b * c)" 

173 
by (simp add: divide_inverse mult_assoc) 

14288  174 

23389  175 

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text {*Special Cancellation Simprules for Division*} 
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36409  178 
lemma mult_divide_mult_cancel_left_if [simp,no_atp]: 
179 
shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)" 

180 
by (simp add: mult_divide_mult_cancel_left) 

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text {* Division and Unary Minus *} 
14293  184 

36409  185 
lemma minus_divide_right: 
186 
" (a / b) = a /  b" 

187 
by (simp add: divide_inverse) 

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lemma divide_minus_right [simp, no_atp]: 
36409  190 
"a /  b =  (a / b)" 
191 
by (simp add: divide_inverse) 

30630  192 

193 
lemma minus_divide_divide: 

36409  194 
"( a) / ( b) = a / b" 
21328  195 
apply (cases "b=0", simp) 
14293  196 
apply (simp add: nonzero_minus_divide_divide) 
197 
done 

198 

23482  199 
lemma eq_divide_eq: 
36409  200 
"a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)" 
201 
by (simp add: nonzero_eq_divide_eq) 

23482  202 

203 
lemma divide_eq_eq: 

36409  204 
"b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)" 
205 
by (force simp add: nonzero_divide_eq_eq) 

14293  206 

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lemma inverse_eq_1_iff [simp]: 
36409  208 
"inverse x = 1 \<longleftrightarrow> x = 1" 
209 
by (insert inverse_eq_iff_eq [of x 1], simp) 

23389  210 

36409  211 
lemma divide_eq_0_iff [simp, no_atp]: 
212 
"a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" 

213 
by (simp add: divide_inverse) 

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36409  215 
lemma divide_cancel_right [simp, no_atp]: 
216 
"a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b" 

217 
apply (cases "c=0", simp) 

218 
apply (simp add: divide_inverse) 

219 
done 

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36409  221 
lemma divide_cancel_left [simp, no_atp]: 
222 
"c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 

223 
apply (cases "c=0", simp) 

224 
apply (simp add: divide_inverse) 

225 
done 

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36409  227 
lemma divide_eq_1_iff [simp, no_atp]: 
228 
"a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b" 

229 
apply (cases "b=0", simp) 

230 
apply (simp add: right_inverse_eq) 

231 
done 

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232 

36409  233 
lemma one_eq_divide_iff [simp, no_atp]: 
234 
"1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b" 

235 
by (simp add: eq_commute [of 1]) 

236 

36719  237 
lemma times_divide_times_eq: 
238 
"(x / y) * (z / w) = (x * z) / (y * w)" 

239 
by simp 

240 

241 
lemma add_frac_num: 

242 
"y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y" 

243 
by (simp add: add_divide_distrib) 

244 

245 
lemma add_num_frac: 

246 
"y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y" 

247 
by (simp add: add_divide_distrib add.commute) 

248 

36409  249 
end 
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250 

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251 

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252 
text {* Ordered Fields *} 
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253 

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class linordered_field = field + linordered_idom 
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begin 
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256 

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lemma positive_imp_inverse_positive: 
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258 
assumes a_gt_0: "0 < a" 
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259 
shows "0 < inverse a" 
23482  260 
proof  
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261 
have "0 < a * inverse a" 
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262 
by (simp add: a_gt_0 [THEN less_imp_not_eq2]) 
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263 
thus "0 < inverse a" 
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264 
by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff) 
23482  265 
qed 
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266 

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267 
lemma negative_imp_inverse_negative: 
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268 
"a < 0 \<Longrightarrow> inverse a < 0" 
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269 
by (insert positive_imp_inverse_positive [of "a"], 
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270 
simp add: nonzero_inverse_minus_eq less_imp_not_eq) 
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271 

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272 
lemma inverse_le_imp_le: 
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273 
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" 
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274 
shows "b \<le> a" 
23482  275 
proof (rule classical) 
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assume "~ b \<le> a" 
23482  277 
hence "a < b" by (simp add: linorder_not_le) 
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hence bpos: "0 < b" by (blast intro: apos less_trans) 
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hence "a * inverse a \<le> a * inverse b" 
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by (simp add: apos invle less_imp_le mult_left_mono) 
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281 
hence "(a * inverse a) * b \<le> (a * inverse b) * b" 
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282 
by (simp add: bpos less_imp_le mult_right_mono) 
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283 
thus "b \<le> a" by (simp add: mult_assoc apos bpos less_imp_not_eq2) 
23482  284 
qed 
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285 

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286 
lemma inverse_positive_imp_positive: 
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287 
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" 
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288 
shows "0 < a" 
23389  289 
proof  
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290 
have "0 < inverse (inverse a)" 
23389  291 
using inv_gt_0 by (rule positive_imp_inverse_positive) 
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292 
thus "0 < a" 
23389  293 
using nz by (simp add: nonzero_inverse_inverse_eq) 
294 
qed 

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295 

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296 
lemma inverse_negative_imp_negative: 
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297 
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0" 
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298 
shows "a < 0" 
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299 
proof  
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300 
have "inverse (inverse a) < 0" 
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301 
using inv_less_0 by (rule negative_imp_inverse_negative) 
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302 
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq) 
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303 
qed 
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304 

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305 
lemma linordered_field_no_lb: 
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306 
"\<forall>x. \<exists>y. y < x" 
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307 
proof 
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308 
fix x::'a 
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309 
have m1: " (1::'a) < 0" by simp 
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310 
from add_strict_right_mono[OF m1, where c=x] 
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311 
have "( 1) + x < x" by simp 
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312 
thus "\<exists>y. y < x" by blast 
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313 
qed 
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314 

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315 
lemma linordered_field_no_ub: 
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316 
"\<forall> x. \<exists>y. y > x" 
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317 
proof 
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318 
fix x::'a 
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319 
have m1: " (1::'a) > 0" by simp 
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320 
from add_strict_right_mono[OF m1, where c=x] 
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321 
have "1 + x > x" by simp 
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322 
thus "\<exists>y. y > x" by blast 
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323 
qed 
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324 

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325 
lemma less_imp_inverse_less: 
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326 
assumes less: "a < b" and apos: "0 < a" 
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327 
shows "inverse b < inverse a" 
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328 
proof (rule ccontr) 
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329 
assume "~ inverse b < inverse a" 
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330 
hence "inverse a \<le> inverse b" by simp 
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331 
hence "~ (a < b)" 
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332 
by (simp add: not_less inverse_le_imp_le [OF _ apos]) 
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333 
thus False by (rule notE [OF _ less]) 
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334 
qed 
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335 

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336 
lemma inverse_less_imp_less: 
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337 
"inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a" 
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338 
apply (simp add: less_le [of "inverse a"] less_le [of "b"]) 
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339 
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
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340 
done 
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341 

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342 
text{*Both premises are essential. Consider 1 and 1.*} 
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343 
lemma inverse_less_iff_less [simp,no_atp]: 
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344 
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a" 
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345 
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
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346 

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347 
lemma le_imp_inverse_le: 
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348 
"a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a" 
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349 
by (force simp add: le_less less_imp_inverse_less) 
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350 

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351 
lemma inverse_le_iff_le [simp,no_atp]: 
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352 
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a" 
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353 
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
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354 

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355 

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356 
text{*These results refer to both operands being negative. The oppositesign 
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357 
case is trivial, since inverse preserves signs.*} 
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358 
lemma inverse_le_imp_le_neg: 
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359 
"inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a" 
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360 
apply (rule classical) 
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361 
apply (subgoal_tac "a < 0") 
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362 
prefer 2 apply force 
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363 
apply (insert inverse_le_imp_le [of "b" "a"]) 
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364 
apply (simp add: nonzero_inverse_minus_eq) 
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365 
done 
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366 

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367 
lemma less_imp_inverse_less_neg: 
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368 
"a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a" 
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369 
apply (subgoal_tac "a < 0") 
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370 
prefer 2 apply (blast intro: less_trans) 
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371 
apply (insert less_imp_inverse_less [of "b" "a"]) 
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372 
apply (simp add: nonzero_inverse_minus_eq) 
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373 
done 
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374 

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375 
lemma inverse_less_imp_less_neg: 
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376 
"inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a" 
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377 
apply (rule classical) 
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378 
apply (subgoal_tac "a < 0") 
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379 
prefer 2 
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380 
apply force 
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381 
apply (insert inverse_less_imp_less [of "b" "a"]) 
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382 
apply (simp add: nonzero_inverse_minus_eq) 
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383 
done 
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384 

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385 
lemma inverse_less_iff_less_neg [simp,no_atp]: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

386 
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

387 
apply (insert inverse_less_iff_less [of "b" "a"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

388 
apply (simp del: inverse_less_iff_less 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

389 
add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

390 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

391 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

392 
lemma le_imp_inverse_le_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

393 
"a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

394 
by (force simp add: le_less less_imp_inverse_less_neg) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

395 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

396 
lemma inverse_le_iff_le_neg [simp,no_atp]: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

397 
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

398 
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

399 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

400 
lemma pos_le_divide_eq [field_simps]: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

401 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

402 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

403 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

404 
by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

405 
also have "... = (a*c \<le> b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

406 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

407 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

408 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

409 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

410 
lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

411 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

412 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

413 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

414 
by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

415 
also have "... = (b \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

416 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

417 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

418 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

419 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

420 
lemma pos_less_divide_eq [field_simps]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

421 
"0 < c ==> (a < b/c) = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

422 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

423 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

424 
hence "(a < b/c) = (a*c < (b/c)*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

425 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

426 
also have "... = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

427 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

428 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

429 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

430 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

431 
lemma neg_less_divide_eq [field_simps]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

432 
"c < 0 ==> (a < b/c) = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

433 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

434 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

435 
hence "(a < b/c) = ((b/c)*c < a*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

436 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

437 
also have "... = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

438 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

439 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

440 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

441 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

442 
lemma pos_divide_less_eq [field_simps]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

443 
"0 < c ==> (b/c < a) = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

444 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

445 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

446 
hence "(b/c < a) = ((b/c)*c < a*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

447 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

448 
also have "... = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

449 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

450 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

451 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

452 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

453 
lemma neg_divide_less_eq [field_simps]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

454 
"c < 0 ==> (b/c < a) = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

455 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

456 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

457 
hence "(b/c < a) = (a*c < (b/c)*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

458 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

459 
also have "... = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

460 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

461 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

462 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

463 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

464 
lemma pos_divide_le_eq [field_simps]: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

465 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

466 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

467 
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

468 
by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

469 
also have "... = (b \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

470 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

471 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

472 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

473 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

474 
lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

475 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

476 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

477 
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" 
36304
6984744e6b34
less special treatment of times_divide_eq [simp]
haftmann
parents:
36301
diff
changeset

478 
by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq) 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

479 
also have "... = (a*c \<le> b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

480 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

481 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

482 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

483 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

484 
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

485 
of positivity/negativity needed for @{text field_simps}. Have not added @{text 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

486 
sign_simps} to @{text field_simps} because the former can lead to case 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

487 
explosions. *} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

488 

36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

489 
lemmas sign_simps [no_atp] = algebra_simps 
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

490 
zero_less_mult_iff mult_less_0_iff 
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

491 

89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

492 
lemmas (in ) sign_simps [no_atp] = algebra_simps 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

493 
zero_less_mult_iff mult_less_0_iff 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

494 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

495 
(* Only works once linear arithmetic is installed: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

496 
text{*An example:*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

497 
lemma fixes a b c d e f :: "'a::linordered_field" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

498 
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

499 
((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) < 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

500 
((ef)*(ab)*(cd))/((ef)*(ab)*(cd)) + u" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

501 
apply(subgoal_tac "(cd)*(ef)*(ab) > 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

502 
prefer 2 apply(simp add:sign_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

503 
apply(subgoal_tac "(cd)*(ef)*(ab)*u > 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

504 
prefer 2 apply(simp add:sign_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

505 
apply(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

506 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

507 
*) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

508 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

509 
lemma divide_pos_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

510 
"0 < x ==> 0 < y ==> 0 < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

511 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

512 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

513 
lemma divide_nonneg_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

514 
"0 <= x ==> 0 < y ==> 0 <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

515 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

516 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

517 
lemma divide_neg_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

518 
"x < 0 ==> 0 < y ==> x / y < 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

519 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

520 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

521 
lemma divide_nonpos_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

522 
"x <= 0 ==> 0 < y ==> x / y <= 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

523 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

524 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

525 
lemma divide_pos_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

526 
"0 < x ==> y < 0 ==> x / y < 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

527 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

528 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

529 
lemma divide_nonneg_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

530 
"0 <= x ==> y < 0 ==> x / y <= 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

531 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

532 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

533 
lemma divide_neg_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

534 
"x < 0 ==> y < 0 ==> 0 < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

535 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

536 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

537 
lemma divide_nonpos_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

538 
"x <= 0 ==> y < 0 ==> 0 <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

539 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

540 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

541 
lemma divide_strict_right_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

542 
"[a < b; 0 < c] ==> a / c < b / c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

543 
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

544 
positive_imp_inverse_positive) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

545 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

546 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

547 
lemma divide_strict_right_mono_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

548 
"[b < a; c < 0] ==> a / c < b / c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

549 
apply (drule divide_strict_right_mono [of _ _ "c"], simp) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

550 
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

551 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

552 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

553 
text{*The last premise ensures that @{term a} and @{term b} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

554 
have the same sign*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

555 
lemma divide_strict_left_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

556 
"[b < a; 0 < c; 0 < a*b] ==> c / a < c / b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

557 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

558 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

559 
lemma divide_left_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

560 
"[b \<le> a; 0 \<le> c; 0 < a*b] ==> c / a \<le> c / b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

561 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

562 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

563 
lemma divide_strict_left_mono_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

564 
"[a < b; c < 0; 0 < a*b] ==> c / a < c / b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

565 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

566 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

567 
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

568 
x / y <= z" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

569 
by (subst pos_divide_le_eq, assumption+) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

570 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

571 
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

572 
z <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

573 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

574 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

575 
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

576 
x / y < z" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

577 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

578 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

579 
lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

580 
z < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

581 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

582 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

583 
lemma frac_le: "0 <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

584 
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

585 
apply (rule mult_imp_div_pos_le) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

586 
apply simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

587 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

588 
apply (rule mult_imp_le_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

589 
apply (rule mult_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

590 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

591 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

592 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

593 
lemma frac_less: "0 <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

594 
x < y ==> 0 < w ==> w <= z ==> x / z < y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

595 
apply (rule mult_imp_div_pos_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

596 
apply simp 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

597 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

598 
apply (rule mult_imp_less_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

599 
apply (erule mult_less_le_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

600 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

601 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

602 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

603 
lemma frac_less2: "0 < x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

604 
x <= y ==> 0 < w ==> w < z ==> x / z < y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

605 
apply (rule mult_imp_div_pos_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

606 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

607 
apply (rule mult_imp_less_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

608 
apply (erule mult_le_less_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

609 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

610 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

611 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

612 
text{*It's not obvious whether these should be simprules or not. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

613 
Their effect is to gather terms into one big fraction, like 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

614 
a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

615 
seem to need them.*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

616 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

617 
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

618 
by (simp add: field_simps zero_less_two) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

619 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

620 
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

621 
by (simp add: field_simps zero_less_two) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

622 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

623 
subclass dense_linorder 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

624 
proof 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

625 
fix x y :: 'a 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

626 
from less_add_one show "\<exists>y. x < y" .. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

627 
from less_add_one have "x + ( 1) < (x + 1) + ( 1)" by (rule add_strict_right_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

628 
then have "x  1 < x + 1  1" by (simp only: diff_minus [symmetric]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

629 
then have "x  1 < x" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

630 
then show "\<exists>y. y < x" .. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

631 
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

632 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

633 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

634 
lemma nonzero_abs_inverse: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

635 
"a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

636 
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

637 
negative_imp_inverse_negative) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

638 
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

639 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

640 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

641 
lemma nonzero_abs_divide: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

642 
"b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

643 
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

644 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

645 
lemma field_le_epsilon: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

646 
assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

647 
shows "x \<le> y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

648 
proof (rule dense_le) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

649 
fix t assume "t < x" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

650 
hence "0 < x  t" by (simp add: less_diff_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

651 
from e [OF this] have "x + 0 \<le> x + (y  t)" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

652 
then have "0 \<le> y  t" by (simp only: add_le_cancel_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

653 
then show "t \<le> y" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

654 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

655 

72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

656 
end 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

657 

36414  658 
class linordered_field_inverse_zero = linordered_field + field_inverse_zero 
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

659 
begin 
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36343
diff
changeset

660 

36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

661 
lemma le_divide_eq: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

662 
"(a \<le> b/c) = 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

663 
(if 0 < c then a*c \<le> b 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

664 
else if c < 0 then b \<le> a*c 
36409  665 
else a \<le> 0)" 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

666 
apply (cases "c=0", simp) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

667 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

668 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

669 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

670 
lemma inverse_positive_iff_positive [simp]: 
36409  671 
"(0 < inverse a) = (0 < a)" 
21328  672 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

673 
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

674 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

675 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

676 
lemma inverse_negative_iff_negative [simp]: 
36409  677 
"(inverse a < 0) = (a < 0)" 
21328  678 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

679 
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

680 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

681 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

682 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
36409  683 
"0 \<le> inverse a \<longleftrightarrow> 0 \<le> a" 
684 
by (simp add: not_less [symmetric]) 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

685 

ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

686 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
36409  687 
"inverse a \<le> 0 \<longleftrightarrow> a \<le> 0" 
688 
by (simp add: not_less [symmetric]) 

14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

689 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

690 
lemma one_less_inverse_iff: 
36409  691 
"1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1" 
23482  692 
proof cases 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

693 
assume "0 < x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

694 
with inverse_less_iff_less [OF zero_less_one, of x] 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

695 
show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

696 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

697 
assume notless: "~ (0 < x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

698 
have "~ (1 < inverse x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

699 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

700 
assume "1 < inverse x" 
36409  701 
also with notless have "... \<le> 0" by simp 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

702 
also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

703 
finally show False by auto 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

704 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

705 
with notless show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

706 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

707 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

708 
lemma one_le_inverse_iff: 
36409  709 
"1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1" 
710 
proof (cases "x = 1") 

711 
case True then show ?thesis by simp 

712 
next 

713 
case False then have "inverse x \<noteq> 1" by simp 

714 
then have "1 \<noteq> inverse x" by blast 

715 
then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less) 

716 
with False show ?thesis by (auto simp add: one_less_inverse_iff) 

717 
qed 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

718 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

719 
lemma inverse_less_1_iff: 
36409  720 
"inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x" 
721 
by (simp add: not_le [symmetric] one_le_inverse_iff) 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

722 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

723 
lemma inverse_le_1_iff: 
36409  724 
"inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x" 
725 
by (simp add: not_less [symmetric] one_less_inverse_iff) 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

726 

14288  727 
lemma divide_le_eq: 
728 
"(b/c \<le> a) = 

729 
(if 0 < c then b \<le> a*c 

730 
else if c < 0 then a*c \<le> b 

36409  731 
else 0 \<le> a)" 
21328  732 
apply (cases "c=0", simp) 
36409  733 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq) 
14288  734 
done 
735 

736 
lemma less_divide_eq: 

737 
"(a < b/c) = 

738 
(if 0 < c then a*c < b 

739 
else if c < 0 then b < a*c 

36409  740 
else a < 0)" 
21328  741 
apply (cases "c=0", simp) 
36409  742 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq) 
14288  743 
done 
744 

745 
lemma divide_less_eq: 

746 
"(b/c < a) = 

747 
(if 0 < c then b < a*c 

748 
else if c < 0 then a*c < b 

36409  749 
else 0 < a)" 
21328  750 
apply (cases "c=0", simp) 
36409  751 
apply (force simp add: pos_divide_less_eq neg_divide_less_eq) 
14288  752 
done 
753 

36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

754 
text {*Division and Signs*} 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

755 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

756 
lemma zero_less_divide_iff: 
36409  757 
"(0 < a/b) = (0 < a & 0 < b  a < 0 & b < 0)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

758 
by (simp add: divide_inverse zero_less_mult_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

759 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

760 
lemma divide_less_0_iff: 
36409  761 
"(a/b < 0) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

762 
(0 < a & b < 0  a < 0 & 0 < b)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

763 
by (simp add: divide_inverse mult_less_0_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

764 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

765 
lemma zero_le_divide_iff: 
36409  766 
"(0 \<le> a/b) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

767 
(0 \<le> a & 0 \<le> b  a \<le> 0 & b \<le> 0)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

768 
by (simp add: divide_inverse zero_le_mult_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

769 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

770 
lemma divide_le_0_iff: 
36409  771 
"(a/b \<le> 0) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

772 
(0 \<le> a & b \<le> 0  a \<le> 0 & 0 \<le> b)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

773 
by (simp add: divide_inverse mult_le_0_iff) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

774 

36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

775 
text {* Division and the Number One *} 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

776 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

777 
text{*Simplify expressions equated with 1*} 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

778 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

779 
lemma zero_eq_1_divide_iff [simp,no_atp]: 
36409  780 
"(0 = 1/a) = (a = 0)" 
23482  781 
apply (cases "a=0", simp) 
782 
apply (auto simp add: nonzero_eq_divide_eq) 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

783 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

784 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

785 
lemma one_divide_eq_0_iff [simp,no_atp]: 
36409  786 
"(1/a = 0) = (a = 0)" 
23482  787 
apply (cases "a=0", simp) 
788 
apply (insert zero_neq_one [THEN not_sym]) 

789 
apply (auto simp add: nonzero_divide_eq_eq) 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

790 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

791 

79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

792 
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} 
36423  793 

794 
lemma zero_le_divide_1_iff [simp, no_atp]: 

795 
"0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a" 

796 
by (simp add: zero_le_divide_iff) 

17085  797 

36423  798 
lemma zero_less_divide_1_iff [simp, no_atp]: 
799 
"0 < 1 / a \<longleftrightarrow> 0 < a" 

800 
by (simp add: zero_less_divide_iff) 

801 

802 
lemma divide_le_0_1_iff [simp, no_atp]: 

803 
"1 / a \<le> 0 \<longleftrightarrow> a \<le> 0" 

804 
by (simp add: divide_le_0_iff) 

805 

806 
lemma divide_less_0_1_iff [simp, no_atp]: 

807 
"1 / a < 0 \<longleftrightarrow> a < 0" 

808 
by (simp add: divide_less_0_iff) 

14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

809 

14293  810 
lemma divide_right_mono: 
36409  811 
"[a \<le> b; 0 \<le> c] ==> a/c \<le> b/c" 
812 
by (force simp add: divide_strict_right_mono le_less) 

14293  813 

36409  814 
lemma divide_right_mono_neg: "a <= b 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

815 
==> c <= 0 ==> b / c <= a / c" 
23482  816 
apply (drule divide_right_mono [of _ _ " c"]) 
817 
apply auto 

16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

818 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

819 

36409  820 
lemma divide_left_mono_neg: "a <= b 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

821 
==> c <= 0 ==> 0 < a * b ==> c / a <= c / b" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

822 
apply (drule divide_left_mono [of _ _ " c"]) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

823 
apply (auto simp add: mult_commute) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

824 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

825 

c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

826 
text{*Simplify quotients that are compared with the value 1.*} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

827 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

828 
lemma le_divide_eq_1 [no_atp]: 
36409  829 
"(1 \<le> b / a) = ((0 < a & a \<le> b)  (a < 0 & b \<le> a))" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

830 
by (auto simp add: le_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

831 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

832 
lemma divide_le_eq_1 [no_atp]: 
36409  833 
"(b / a \<le> 1) = ((0 < a & b \<le> a)  (a < 0 & a \<le> b)  a=0)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

834 
by (auto simp add: divide_le_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

835 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

836 
lemma less_divide_eq_1 [no_atp]: 
36409  837 
"(1 < b / a) = ((0 < a & a < b)  (a < 0 & b < a))" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

838 
by (auto simp add: less_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

839 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

840 
lemma divide_less_eq_1 [no_atp]: 
36409  841 
"(b / a < 1) = ((0 < a & b < a)  (a < 0 & a < b)  a=0)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

842 
by (auto simp add: divide_less_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

843 

23389  844 

36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

845 
text {*Conditional Simplification Rules: No Case Splits*} 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

846 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

847 
lemma le_divide_eq_1_pos [simp,no_atp]: 
36409  848 
"0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

849 
by (auto simp add: le_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

850 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

851 
lemma le_divide_eq_1_neg [simp,no_atp]: 
36409  852 
"a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

853 
by (auto simp add: le_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

854 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

855 
lemma divide_le_eq_1_pos [simp,no_atp]: 
36409  856 
"0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

857 
by (auto simp add: divide_le_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

858 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

859 
lemma divide_le_eq_1_neg [simp,no_atp]: 
36409  860 
"a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

861 
by (auto simp add: divide_le_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

862 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

863 
lemma less_divide_eq_1_pos [simp,no_atp]: 
36409  864 
"0 < a \<Longrightarrow> (1 < b/a) = (a < b)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

865 
by (auto simp add: less_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

866 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

867 
lemma less_divide_eq_1_neg [simp,no_atp]: 
36409  868 
"a < 0 \<Longrightarrow> (1 < b/a) = (b < a)" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

869 
by (auto simp add: less_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

870 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

871 
lemma divide_less_eq_1_pos [simp,no_atp]: 
36409  872 
"0 < a \<Longrightarrow> (b/a < 1) = (b < a)" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

873 
by (auto simp add: divide_less_eq) 
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

874 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

875 
lemma divide_less_eq_1_neg [simp,no_atp]: 
36409  876 
"a < 0 \<Longrightarrow> b/a < 1 <> a < b" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

877 
by (auto simp add: divide_less_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

878 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

879 
lemma eq_divide_eq_1 [simp,no_atp]: 
36409  880 
"(1 = b/a) = ((a \<noteq> 0 & a = b))" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

881 
by (auto simp add: eq_divide_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

882 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

883 
lemma divide_eq_eq_1 [simp,no_atp]: 
36409  884 
"(b/a = 1) = ((a \<noteq> 0 & a = b))" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

885 
by (auto simp add: divide_eq_eq) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

886 

14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

887 
lemma abs_inverse [simp]: 
36409  888 
"\<bar>inverse a\<bar> = 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

889 
inverse \<bar>a\<bar>" 
21328  890 
apply (cases "a=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

891 
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

892 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

893 

15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

894 
lemma abs_divide [simp]: 
36409  895 
"\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>" 
21328  896 
apply (cases "b=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

897 
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

898 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

899 

36409  900 
lemma abs_div_pos: "0 < y ==> 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

901 
\<bar>x\<bar> / y = \<bar>x / y\<bar>" 
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

902 
apply (subst abs_divide) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

903 
apply (simp add: order_less_imp_le) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

904 
done 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

905 

35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

906 
lemma field_le_mult_one_interval: 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

907 
assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

908 
shows "x \<le> y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

909 
proof (cases "0 < x") 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

910 
assume "0 < x" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

911 
thus ?thesis 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

912 
using dense_le_bounded[of 0 1 "y/x"] * 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

913 
unfolding le_divide_eq if_P[OF `0 < x`] by simp 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

914 
next 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

915 
assume "\<not>0 < x" hence "x \<le> 0" by simp 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

916 
obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

917 
hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

918 
also note *[OF s] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

919 
finally show ?thesis . 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

920 
qed 
35090
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

921 

36409  922 
end 
923 

33364  924 
code_modulename SML 
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

925 
Fields Arith 
33364  926 

927 
code_modulename OCaml 

35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

928 
Fields Arith 
33364  929 

930 
code_modulename Haskell 

35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

931 
Fields Arith 
33364  932 

14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

933 
end 