author  blanchet 
Thu, 18 Mar 2010 12:58:52 +0100  
changeset 35828  46cfc4b8112e 
parent 35579  cc9a5a0ab5ea 
child 36301  72f4d079ebf8 
permissions  rwrr 
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(* Title: HOL/Fields.thy 
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Author: Gertrud Bauer 
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Author: Steven Obua 
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Author: Tobias Nipkow 
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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 
28 
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 

34 
lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b" 

35 
proof 

36 
assume neq: "b \<noteq> 0" 

37 
{ 

38 
hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac) 

39 
also assume "a / b = 1" 

40 
finally show "a = b" by simp 

41 
next 

42 
assume "a = b" 

43 
with neq show "a / b = 1" by (simp add: divide_inverse) 

44 
} 

45 
qed 

46 

47 
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a" 

29667  48 
by (simp add: divide_inverse) 
25230  49 

50 
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1" 

29667  51 
by (simp add: divide_inverse) 
25230  52 

53 
lemma divide_zero_left [simp]: "0 / a = 0" 

29667  54 
by (simp add: divide_inverse) 
25230  55 

56 
lemma inverse_eq_divide: "inverse a = 1 / a" 

29667  57 
by (simp add: divide_inverse) 
25230  58 

59 
lemma add_divide_distrib: "(a+b) / c = a/c + b/c" 

30630  60 
by (simp add: divide_inverse algebra_simps) 
61 

62 
text{*There is no slick version using division by zero.*} 

63 
lemma inverse_add: 

64 
"[ a \<noteq> 0; b \<noteq> 0 ] 

65 
==> inverse a + inverse b = (a + b) * inverse a * inverse b" 

66 
by (simp add: division_ring_inverse_add mult_ac) 

67 

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lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]: 
30630  69 
assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b" 
70 
proof  

71 
have "(c*a)/(c*b) = c * a * (inverse b * inverse c)" 

72 
by (simp add: divide_inverse nonzero_inverse_mult_distrib) 

73 
also have "... = a * inverse b * (inverse c * c)" 

74 
by (simp only: mult_ac) 

75 
also have "... = a * inverse b" by simp 

76 
finally show ?thesis by (simp add: divide_inverse) 

77 
qed 

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lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]: 
30630  80 
"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b" 
81 
by (simp add: mult_commute [of _ c]) 

82 

83 
lemma divide_1 [simp]: "a / 1 = a" 

84 
by (simp add: divide_inverse) 

85 

86 
lemma times_divide_eq_right: "a * (b / c) = (a * b) / c" 

87 
by (simp add: divide_inverse mult_assoc) 

88 

89 
lemma times_divide_eq_left: "(b / c) * a = (b * a) / c" 

90 
by (simp add: divide_inverse mult_ac) 

91 

92 
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  94 

95 
lemma add_frac_eq: 

96 
assumes "y \<noteq> 0" and "z \<noteq> 0" 

97 
shows "x / y + w / z = (x * z + w * y) / (y * z)" 

98 
proof  

99 
have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)" 

100 
using assms by simp 

101 
also have "\<dots> = (x * z + y * w) / (y * z)" 

102 
by (simp only: add_divide_distrib) 

103 
finally show ?thesis 

104 
by (simp only: mult_commute) 

105 
qed 

106 

107 
text{*Special Cancellation Simprules for Division*} 

108 

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lemma nonzero_mult_divide_cancel_right [simp, no_atp]: 
30630  110 
"b \<noteq> 0 \<Longrightarrow> a * b / b = a" 
111 
using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp 

112 

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lemma nonzero_mult_divide_cancel_left [simp, no_atp]: 
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"a \<noteq> 0 \<Longrightarrow> a * b / a = b" 
115 
using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp 

116 

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lemma nonzero_divide_mult_cancel_right [simp, no_atp]: 
30630  118 
"\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a" 
119 
using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp 

120 

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lemma nonzero_divide_mult_cancel_left [simp, no_atp]: 
30630  122 
"\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b" 
123 
using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp 

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lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]: 
30630  126 
"\<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) 

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lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]: 
30630  130 
"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b" 
131 
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac) 

132 

133 
lemma minus_divide_left: " (a / b) = (a) / b" 

134 
by (simp add: divide_inverse) 

135 

136 
lemma nonzero_minus_divide_right: "b \<noteq> 0 ==>  (a / b) = a / ( b)" 

137 
by (simp add: divide_inverse nonzero_inverse_minus_eq) 

138 

139 
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (a) / (b) = a / b" 

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by (simp add: divide_inverse nonzero_inverse_minus_eq) 

141 

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lemma divide_minus_left [simp, no_atp]: "(a) / b =  (a / b)" 
30630  143 
by (simp add: divide_inverse) 
144 

145 
lemma diff_divide_distrib: "(a  b) / c = a / c  b / c" 

146 
by (simp add: diff_minus add_divide_distrib) 

147 

148 
lemma add_divide_eq_iff: 

149 
"z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z" 

150 
by (simp add: add_divide_distrib) 

151 

152 
lemma divide_add_eq_iff: 

153 
"z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z" 

154 
by (simp add: add_divide_distrib) 

155 

156 
lemma diff_divide_eq_iff: 

157 
"z \<noteq> 0 \<Longrightarrow> x  y / z = (z * x  y) / z" 

158 
by (simp add: diff_divide_distrib) 

159 

160 
lemma divide_diff_eq_iff: 

161 
"z \<noteq> 0 \<Longrightarrow> x / z  y = (x  z * y) / z" 

162 
by (simp add: diff_divide_distrib) 

163 

164 
lemma nonzero_eq_divide_eq: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b" 

165 
proof  

166 
assume [simp]: "c \<noteq> 0" 

167 
have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp 

168 
also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc) 

169 
finally show ?thesis . 

170 
qed 

171 

172 
lemma nonzero_divide_eq_eq: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c" 

173 
proof  

174 
assume [simp]: "c \<noteq> 0" 

175 
have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp 

176 
also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 

177 
finally show ?thesis . 

178 
qed 

179 

180 
lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a" 

181 
by simp 

182 

183 
lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c" 

184 
by (erule subst, simp) 

185 

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lemmas field_eq_simps[no_atp] = algebra_simps 
30630  187 
(* pull / out*) 
188 
add_divide_eq_iff divide_add_eq_iff 

189 
diff_divide_eq_iff divide_diff_eq_iff 

190 
(* multiply eqn *) 

191 
nonzero_eq_divide_eq nonzero_divide_eq_eq 

192 
(* is added later: 

193 
times_divide_eq_left times_divide_eq_right 

194 
*) 

195 

196 
text{*An example:*} 

197 
lemma "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f\<rbrakk> \<Longrightarrow> ((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) = 1" 

198 
apply(subgoal_tac "(cd)*(ef)*(ab) \<noteq> 0") 

199 
apply(simp add:field_eq_simps) 

200 
apply(simp) 

201 
done 

202 

203 
lemma diff_frac_eq: 

204 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y  w / z = (x * z  w * y) / (y * z)" 

205 
by (simp add: field_eq_simps times_divide_eq) 

206 

207 
lemma frac_eq_eq: 

208 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)" 

209 
by (simp add: field_eq_simps times_divide_eq) 

25230  210 

211 
end 

212 

22390  213 
class division_by_zero = zero + inverse + 
25062  214 
assumes inverse_zero [simp]: "inverse 0 = 0" 
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25230  216 
lemma divide_zero [simp]: 
217 
"a / 0 = (0::'a::{field,division_by_zero})" 

29667  218 
by (simp add: divide_inverse) 
25230  219 

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lemma divide_self_if [simp]: 

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"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)" 

29667  222 
by simp 
25230  223 

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class linordered_field = field + linordered_idom 
25230  225 

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lemma inverse_nonzero_iff_nonzero [simp]: 
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"(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))" 
26274  228 
by (force dest: inverse_zero_imp_zero) 
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lemma inverse_minus_eq [simp]: 
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"inverse(a) = inverse(a::'a::{division_ring,division_by_zero})" 
14377  232 
proof cases 
35216  233 
assume "a=0" thus ?thesis by simp 
14377  234 
next 
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assume "a\<noteq>0" 

236 
thus ?thesis by (simp add: nonzero_inverse_minus_eq) 

237 
qed 

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lemma inverse_eq_imp_eq: 
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"inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})" 
21328  241 
apply (cases "a=0  b=0") 
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apply (force dest!: inverse_zero_imp_zero 
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simp add: eq_commute [of "0::'a"]) 
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apply (force dest!: nonzero_inverse_eq_imp_eq) 
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done 
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lemma inverse_eq_iff_eq [simp]: 
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"(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))" 
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by (force dest!: inverse_eq_imp_eq) 
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14270  251 
lemma inverse_inverse_eq [simp]: 
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"inverse(inverse (a::'a::{division_ring,division_by_zero})) = a" 
14270  253 
proof cases 
254 
assume "a=0" thus ?thesis by simp 

255 
next 

256 
assume "a\<noteq>0" 

257 
thus ?thesis by (simp add: nonzero_inverse_inverse_eq) 

258 
qed 

259 

260 
text{*This version builds in division by zero while also reorienting 

261 
the righthand side.*} 

262 
lemma inverse_mult_distrib [simp]: 

263 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

264 
proof cases 

265 
assume "a \<noteq> 0 & b \<noteq> 0" 

29667  266 
thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute) 
14270  267 
next 
268 
assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

29667  269 
thus ?thesis by force 
14270  270 
qed 
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lemma inverse_divide [simp]: 
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"inverse (a/b) = b / (a::'a::{field,division_by_zero})" 
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by (simp add: divide_inverse mult_commute) 
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23389  276 

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subsection {* 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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282 

5caa2710dd5b
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parents:
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283 
lemma mult_divide_mult_cancel_left: 
23477
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tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
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diff
changeset

284 
"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" 
21328  285 
apply (cases "b = 0") 
35216  286 
apply simp_all 
14277
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parents:
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diff
changeset

287 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
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diff
changeset

288 

23413
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tuned laws for cancellation in divisions for fields.
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parents:
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289 
lemma mult_divide_mult_cancel_right: 
23477
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tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

290 
"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 
21328  291 
apply (cases "b = 0") 
35216  292 
apply simp_all 
14321  293 
done 
23413
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tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

294 

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

295 
lemma divide_divide_eq_right [simp,no_atp]: 
23477
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tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

296 
"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

297 
by (simp add: divide_inverse mult_ac) 
14288  298 

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

299 
lemma divide_divide_eq_left [simp,no_atp]: 
23477
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tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

300 
"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)" 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

301 
by (simp add: divide_inverse mult_assoc) 
14288  302 

23389  303 

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

304 
subsubsection{*Special Cancellation Simprules for Division*} 
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

305 

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

306 
lemma mult_divide_mult_cancel_left_if[simp,no_atp]: 
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

307 
fixes c :: "'a :: {field,division_by_zero}" 
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

308 
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)" 
23413
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

309 
by (simp add: mult_divide_mult_cancel_left) 
5caa2710dd5b
tuned laws for cancellation in divisions for fields.
nipkow
parents:
23406
diff
changeset

310 

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

311 

14293  312 
subsection {* Division and Unary Minus *} 
313 

314 
lemma minus_divide_right: " (a/b) = a / (b::'a::{field,division_by_zero})" 

29407
5ef7e97fd9e4
move lemmas mult_minus{left,right} inside class ring
huffman
parents:
29406
diff
changeset

315 
by (simp add: divide_inverse) 
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset

316 

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

317 
lemma divide_minus_right [simp, no_atp]: 
30630  318 
"a / (b::'a::{field,division_by_zero}) = (a / b)" 
319 
by (simp add: divide_inverse) 

320 

321 
lemma minus_divide_divide: 

23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23413
diff
changeset

322 
"(a)/(b) = a / (b::'a::{field,division_by_zero})" 
21328  323 
apply (cases "b=0", simp) 
14293  324 
apply (simp add: nonzero_minus_divide_divide) 
325 
done 

326 

23482  327 
lemma eq_divide_eq: 
328 
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" 

30630  329 
by (simp add: nonzero_eq_divide_eq) 
23482  330 

331 
lemma divide_eq_eq: 

332 
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" 

30630  333 
by (force simp add: nonzero_divide_eq_eq) 
14293  334 

23389  335 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

336 
subsection {* Ordered Fields *} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
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diff
changeset

337 

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

338 
lemma positive_imp_inverse_positive: 
35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

339 
assumes a_gt_0: "0 < a" shows "0 < inverse (a::'a::linordered_field)" 
23482  340 
proof  
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

341 
have "0 < a * inverse a" 
35216  342 
by (simp add: a_gt_0 [THEN order_less_imp_not_eq2]) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

343 
thus "0 < inverse a" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

344 
by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff) 
23482  345 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

346 

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

347 
lemma negative_imp_inverse_negative: 
35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

348 
"a < 0 ==> inverse a < (0::'a::linordered_field)" 
23482  349 
by (insert positive_imp_inverse_positive [of "a"], 
350 
simp add: nonzero_inverse_minus_eq order_less_imp_not_eq) 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

351 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

352 
lemma inverse_le_imp_le: 
23482  353 
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

354 
shows "b \<le> (a::'a::linordered_field)" 
23482  355 
proof (rule classical) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

356 
assume "~ b \<le> a" 
23482  357 
hence "a < b" by (simp add: linorder_not_le) 
358 
hence bpos: "0 < b" by (blast intro: apos order_less_trans) 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

359 
hence "a * inverse a \<le> a * inverse b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

360 
by (simp add: apos invle order_less_imp_le mult_left_mono) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

361 
hence "(a * inverse a) * b \<le> (a * inverse b) * b" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

362 
by (simp add: bpos order_less_imp_le mult_right_mono) 
23482  363 
thus "b \<le> a" by (simp add: mult_assoc apos bpos order_less_imp_not_eq2) 
364 
qed 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

365 

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

366 
lemma inverse_positive_imp_positive: 
23482  367 
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

368 
shows "0 < (a::'a::linordered_field)" 
23389  369 
proof  
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

370 
have "0 < inverse (inverse a)" 
23389  371 
using inv_gt_0 by (rule positive_imp_inverse_positive) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

372 
thus "0 < a" 
23389  373 
using nz by (simp add: nonzero_inverse_inverse_eq) 
374 
qed 

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

375 

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

376 
lemma inverse_positive_iff_positive [simp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

377 
"(0 < inverse a) = (0 < (a::'a::{linordered_field,division_by_zero}))" 
21328  378 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

379 
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

380 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

381 

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

382 
lemma inverse_negative_imp_negative: 
23482  383 
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0" 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

384 
shows "a < (0::'a::linordered_field)" 
23389  385 
proof  
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

386 
have "inverse (inverse a) < 0" 
23389  387 
using inv_less_0 by (rule negative_imp_inverse_negative) 
23482  388 
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq) 
23389  389 
qed 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

390 

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

391 
lemma inverse_negative_iff_negative [simp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

392 
"(inverse a < 0) = (a < (0::'a::{linordered_field,division_by_zero}))" 
21328  393 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

394 
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

395 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

396 

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

397 
lemma inverse_nonnegative_iff_nonnegative [simp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

398 
"(0 \<le> inverse a) = (0 \<le> (a::'a::{linordered_field,division_by_zero}))" 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

399 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

400 

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

401 
lemma inverse_nonpositive_iff_nonpositive [simp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

402 
"(inverse a \<le> 0) = (a \<le> (0::'a::{linordered_field,division_by_zero}))" 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

403 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

404 

35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset

405 
lemma linordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::linordered_field)" 
23406
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

406 
proof 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

407 
fix x::'a 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

408 
have m1: " (1::'a) < 0" by simp 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

409 
from add_strict_right_mono[OF m1, where c=x] 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

410 
have "( 1) + x < x" by simp 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

411 
thus "\<exists>y. y < x" by blast 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

412 
qed 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

413 

35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35032
diff
changeset

414 
lemma linordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::linordered_field)" 
23406
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

415 
proof 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

416 
fix x::'a 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

417 
have m1: " (1::'a) > 0" by simp 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

418 
from add_strict_right_mono[OF m1, where c=x] 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

419 
have "1 + x > x" by simp 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

420 
thus "\<exists>y. y > x" by blast 
167b53019d6f
added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents:
23400
diff
changeset

421 
qed 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

422 

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

423 
subsection{*AntiMonotonicity of @{term inverse}*} 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

424 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

425 
lemma less_imp_inverse_less: 
23482  426 
assumes less: "a < b" and apos: "0 < a" 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

427 
shows "inverse b < inverse (a::'a::linordered_field)" 
23482  428 
proof (rule ccontr) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

429 
assume "~ inverse b < inverse a" 
29667  430 
hence "inverse a \<le> inverse b" by (simp add: linorder_not_less) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

431 
hence "~ (a < b)" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

432 
by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos]) 
29667  433 
thus False by (rule notE [OF _ less]) 
23482  434 
qed 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

435 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

436 
lemma inverse_less_imp_less: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

437 
"[inverse a < inverse b; 0 < a] ==> b < (a::'a::linordered_field)" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

438 
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"]) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

439 
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

440 
done 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

441 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

442 
text{*Both premises are essential. Consider 1 and 1.*} 
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

443 
lemma inverse_less_iff_less [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

444 
"[0 < a; 0 < b] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

445 
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

446 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

447 
lemma le_imp_inverse_le: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

448 
"[a \<le> b; 0 < a] ==> inverse b \<le> inverse (a::'a::linordered_field)" 
23482  449 
by (force simp add: order_le_less less_imp_inverse_less) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

450 

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

451 
lemma inverse_le_iff_le [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

452 
"[0 < a; 0 < b] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

453 
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

454 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

455 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

456 
text{*These results refer to both operands being negative. The oppositesign 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

457 
case is trivial, since inverse preserves signs.*} 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

458 
lemma inverse_le_imp_le_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

459 
"[inverse a \<le> inverse b; b < 0] ==> b \<le> (a::'a::linordered_field)" 
23482  460 
apply (rule classical) 
461 
apply (subgoal_tac "a < 0") 

462 
prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 

463 
apply (insert inverse_le_imp_le [of "b" "a"]) 

464 
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

465 
done 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

466 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

467 
lemma less_imp_inverse_less_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

468 
"[a < b; b < 0] ==> inverse b < inverse (a::'a::linordered_field)" 
23482  469 
apply (subgoal_tac "a < 0") 
470 
prefer 2 apply (blast intro: order_less_trans) 

471 
apply (insert less_imp_inverse_less [of "b" "a"]) 

472 
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

473 
done 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

474 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

475 
lemma inverse_less_imp_less_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

476 
"[inverse a < inverse b; b < 0] ==> b < (a::'a::linordered_field)" 
23482  477 
apply (rule classical) 
478 
apply (subgoal_tac "a < 0") 

479 
prefer 2 

480 
apply (force simp add: linorder_not_less intro: order_le_less_trans) 

481 
apply (insert inverse_less_imp_less [of "b" "a"]) 

482 
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

483 
done 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

484 

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

485 
lemma inverse_less_iff_less_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

486 
"[a < 0; b < 0] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))" 
23482  487 
apply (insert inverse_less_iff_less [of "b" "a"]) 
488 
apply (simp del: inverse_less_iff_less 

489 
add: order_less_imp_not_eq nonzero_inverse_minus_eq) 

490 
done 

14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

491 

5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

492 
lemma le_imp_inverse_le_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

493 
"[a \<le> b; b < 0] ==> inverse b \<le> inverse (a::'a::linordered_field)" 
23482  494 
by (force simp add: order_le_less less_imp_inverse_less_neg) 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

495 

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

496 
lemma inverse_le_iff_le_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

497 
"[a < 0; b < 0] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))" 
14268
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

498 
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
14265
95b42e69436c
HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff
changeset

499 

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

500 

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

501 
subsection{*Inverses and the Number One*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

502 

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

503 
lemma one_less_inverse_iff: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

504 
"(1 < inverse x) = (0 < x & x < (1::'a::{linordered_field,division_by_zero}))" 
23482  505 
proof cases 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

506 
assume "0 < x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

507 
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

508 
show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

509 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

510 
assume notless: "~ (0 < x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

511 
have "~ (1 < inverse x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

512 
proof 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

513 
assume "1 < inverse x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

514 
also with notless have "... \<le> 0" by (simp add: linorder_not_less) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

515 
also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

516 
finally show False by auto 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

517 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

518 
with notless show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

519 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

520 

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

521 
lemma inverse_eq_1_iff [simp]: 
23482  522 
"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

523 
by (insert inverse_eq_iff_eq [of x 1], simp) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

524 

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

525 
lemma one_le_inverse_iff: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

526 
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{linordered_field,division_by_zero}))" 
35216  527 
by (force simp add: order_le_less one_less_inverse_iff) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

528 

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

529 
lemma inverse_less_1_iff: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

530 
"(inverse x < 1) = (x \<le> 0  1 < (x::'a::{linordered_field,division_by_zero}))" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

531 
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

532 

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

533 
lemma inverse_le_1_iff: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

534 
"(inverse x \<le> 1) = (x \<le> 0  1 \<le> (x::'a::{linordered_field,division_by_zero}))" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

535 
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

536 

23389  537 

14288  538 
subsection{*Simplification of Inequalities Involving Literal Divisors*} 
539 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

540 
lemma pos_le_divide_eq: "0 < (c::'a::linordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" 
14288  541 
proof  
542 
assume less: "0<c" 

543 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 

544 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

545 
also have "... = (a*c \<le> b)" 

546 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

547 
finally show ?thesis . 

548 
qed 

549 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

550 
lemma neg_le_divide_eq: "c < (0::'a::linordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" 
14288  551 
proof  
552 
assume less: "c<0" 

553 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 

554 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

555 
also have "... = (b \<le> a*c)" 

556 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

557 
finally show ?thesis . 

558 
qed 

559 

560 
lemma le_divide_eq: 

561 
"(a \<le> b/c) = 

562 
(if 0 < c then a*c \<le> b 

563 
else if c < 0 then b \<le> a*c 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

564 
else a \<le> (0::'a::{linordered_field,division_by_zero}))" 
21328  565 
apply (cases "c=0", simp) 
14288  566 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
567 
done 

568 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

569 
lemma pos_divide_le_eq: "0 < (c::'a::linordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" 
14288  570 
proof  
571 
assume less: "0<c" 

572 
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" 

573 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

574 
also have "... = (b \<le> a*c)" 

575 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

576 
finally show ?thesis . 

577 
qed 

578 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

579 
lemma neg_divide_le_eq: "c < (0::'a::linordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" 
14288  580 
proof  
581 
assume less: "c<0" 

582 
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" 

583 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

584 
also have "... = (a*c \<le> b)" 

585 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

586 
finally show ?thesis . 

587 
qed 

588 

589 
lemma divide_le_eq: 

590 
"(b/c \<le> a) = 

591 
(if 0 < c then b \<le> a*c 

592 
else if c < 0 then a*c \<le> b 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

593 
else 0 \<le> (a::'a::{linordered_field,division_by_zero}))" 
21328  594 
apply (cases "c=0", simp) 
14288  595 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
596 
done 

597 

598 
lemma pos_less_divide_eq: 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

599 
"0 < (c::'a::linordered_field) ==> (a < b/c) = (a*c < b)" 
14288  600 
proof  
601 
assume less: "0<c" 

602 
hence "(a < b/c) = (a*c < (b/c)*c)" 

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

603 
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) 
14288  604 
also have "... = (a*c < b)" 
605 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

606 
finally show ?thesis . 

607 
qed 

608 

609 
lemma neg_less_divide_eq: 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

610 
"c < (0::'a::linordered_field) ==> (a < b/c) = (b < a*c)" 
14288  611 
proof  
612 
assume less: "c<0" 

613 
hence "(a < b/c) = ((b/c)*c < a*c)" 

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

614 
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) 
14288  615 
also have "... = (b < a*c)" 
616 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

617 
finally show ?thesis . 

618 
qed 

619 

620 
lemma less_divide_eq: 

621 
"(a < b/c) = 

622 
(if 0 < c then a*c < b 

623 
else if c < 0 then b < a*c 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

624 
else a < (0::'a::{linordered_field,division_by_zero}))" 
21328  625 
apply (cases "c=0", simp) 
14288  626 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
627 
done 

628 

629 
lemma pos_divide_less_eq: 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

630 
"0 < (c::'a::linordered_field) ==> (b/c < a) = (b < a*c)" 
14288  631 
proof  
632 
assume less: "0<c" 

633 
hence "(b/c < a) = ((b/c)*c < a*c)" 

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

634 
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) 
14288  635 
also have "... = (b < a*c)" 
636 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

637 
finally show ?thesis . 

638 
qed 

639 

640 
lemma neg_divide_less_eq: 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

641 
"c < (0::'a::linordered_field) ==> (b/c < a) = (a*c < b)" 
14288  642 
proof  
643 
assume less: "c<0" 

644 
hence "(b/c < a) = (a*c < (b/c)*c)" 

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

645 
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) 
14288  646 
also have "... = (a*c < b)" 
647 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

648 
finally show ?thesis . 

649 
qed 

650 

651 
lemma divide_less_eq: 

652 
"(b/c < a) = 

653 
(if 0 < c then b < a*c 

654 
else if c < 0 then a*c < b 

35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
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34146
diff
changeset

655 
else 0 < (a::'a::{linordered_field,division_by_zero}))" 
21328  656 
apply (cases "c=0", simp) 
14288  657 
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
658 
done 

659 

23482  660 

661 
subsection{*Field simplification*} 

662 

29667  663 
text{* Lemmas @{text field_simps} multiply with denominators in in(equations) 
664 
if they can be proved to be nonzero (for equations) or positive/negative 

665 
(for inequations). Can be too aggressive and is therefore separate from the 

666 
more benign @{text algebra_simps}. *} 

14288  667 

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

668 
lemmas field_simps[no_atp] = field_eq_simps 
23482  669 
(* multiply ineqn *) 
670 
pos_divide_less_eq neg_divide_less_eq 

671 
pos_less_divide_eq neg_less_divide_eq 

672 
pos_divide_le_eq neg_divide_le_eq 

673 
pos_le_divide_eq neg_le_divide_eq 

14288  674 

23482  675 
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs 
23483  676 
of positivity/negativity needed for @{text field_simps}. Have not added @{text 
23482  677 
sign_simps} to @{text field_simps} because the former can lead to case 
678 
explosions. *} 

14288  679 

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

680 
lemmas sign_simps[no_atp] = group_simps 
23482  681 
zero_less_mult_iff mult_less_0_iff 
14288  682 

23482  683 
(* Only works once linear arithmetic is installed: 
684 
text{*An example:*} 

35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
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parents:
34146
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changeset

685 
lemma fixes a b c d e f :: "'a::linordered_field" 
23482  686 
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow> 
687 
((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) < 

688 
((ef)*(ab)*(cd))/((ef)*(ab)*(cd)) + u" 

689 
apply(subgoal_tac "(cd)*(ef)*(ab) > 0") 

690 
prefer 2 apply(simp add:sign_simps) 

691 
apply(subgoal_tac "(cd)*(ef)*(ab)*u > 0") 

692 
prefer 2 apply(simp add:sign_simps) 

693 
apply(simp add:field_simps) 

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

694 
done 
23482  695 
*) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

696 

23389  697 

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

698 
subsection{*Division and Signs*} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

699 

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

700 
lemma zero_less_divide_iff: 
35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
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changeset

701 
"((0::'a::{linordered_field,division_by_zero}) < 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

702 
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

703 

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

704 
lemma divide_less_0_iff: 
35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

705 
"(a/b < (0::'a::{linordered_field,division_by_zero})) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

706 
(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

707 
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

708 

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

709 
lemma zero_le_divide_iff: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

710 
"((0::'a::{linordered_field,division_by_zero}) \<le> a/b) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

711 
(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

712 
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

713 

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

714 
lemma divide_le_0_iff: 
35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

715 
"(a/b \<le> (0::'a::{linordered_field,division_by_zero})) = 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

716 
(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

717 
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

718 

35828
46cfc4b8112e
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blanchet
parents:
35579
diff
changeset

719 
lemma divide_eq_0_iff [simp,no_atp]: 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

720 
"(a/b = 0) = (a=0  b=(0::'a::{field,division_by_zero}))" 
23482  721 
by (simp add: divide_inverse) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

722 

23482  723 
lemma divide_pos_pos: 
35028
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more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

724 
"0 < (x::'a::linordered_field) ==> 0 < y ==> 0 < x / y" 
23482  725 
by(simp add:field_simps) 
726 

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

727 

23482  728 
lemma divide_nonneg_pos: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

729 
"0 <= (x::'a::linordered_field) ==> 0 < y ==> 0 <= x / y" 
23482  730 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

731 

23482  732 
lemma divide_neg_pos: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

733 
"(x::'a::linordered_field) < 0 ==> 0 < y ==> x / y < 0" 
23482  734 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

735 

23482  736 
lemma divide_nonpos_pos: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

737 
"(x::'a::linordered_field) <= 0 ==> 0 < y ==> x / y <= 0" 
23482  738 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

739 

23482  740 
lemma divide_pos_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

741 
"0 < (x::'a::linordered_field) ==> y < 0 ==> x / y < 0" 
23482  742 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

743 

23482  744 
lemma divide_nonneg_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

745 
"0 <= (x::'a::linordered_field) ==> y < 0 ==> x / y <= 0" 
23482  746 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

747 

23482  748 
lemma divide_neg_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

749 
"(x::'a::linordered_field) < 0 ==> y < 0 ==> 0 < x / y" 
23482  750 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

751 

23482  752 
lemma divide_nonpos_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

753 
"(x::'a::linordered_field) <= 0 ==> y < 0 ==> 0 <= x / y" 
23482  754 
by(simp add:field_simps) 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

755 

23389  756 

14288  757 
subsection{*Cancellation Laws for Division*} 
758 

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

759 
lemma divide_cancel_right [simp,no_atp]: 
14288  760 
"(a/c = b/c) = (c = 0  a = (b::'a::{field,division_by_zero}))" 
23482  761 
apply (cases "c=0", simp) 
23496  762 
apply (simp add: divide_inverse) 
14288  763 
done 
764 

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

765 
lemma divide_cancel_left [simp,no_atp]: 
14288  766 
"(c/a = c/b) = (c = 0  a = (b::'a::{field,division_by_zero}))" 
23482  767 
apply (cases "c=0", simp) 
23496  768 
apply (simp add: divide_inverse) 
14288  769 
done 
770 

23389  771 

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

772 
subsection {* Division and the Number One *} 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

773 

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

774 
text{*Simplify expressions equated with 1*} 
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

775 
lemma divide_eq_1_iff [simp,no_atp]: 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

776 
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" 
23482  777 
apply (cases "b=0", simp) 
778 
apply (simp add: right_inverse_eq) 

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

779 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

780 

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

781 
lemma one_eq_divide_iff [simp,no_atp]: 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

782 
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" 
23482  783 
by (simp add: eq_commute [of 1]) 
14353
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 zero_eq_1_divide_iff [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

786 
"((0::'a::{linordered_field,division_by_zero}) = 1/a) = (a = 0)" 
23482  787 
apply (cases "a=0", simp) 
788 
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

789 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

790 

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

791 
lemma one_divide_eq_0_iff [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

792 
"(1/a = (0::'a::{linordered_field,division_by_zero})) = (a = 0)" 
23482  793 
apply (cases "a=0", simp) 
794 
apply (insert zero_neq_one [THEN not_sym]) 

795 
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

796 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

797 

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

798 
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} 
18623  799 
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified] 
800 
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified] 

801 
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified] 

802 
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified] 

17085  803 

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

804 
declare zero_less_divide_1_iff [simp,no_atp] 
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

805 
declare divide_less_0_1_iff [simp,no_atp] 
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

806 
declare zero_le_divide_1_iff [simp,no_atp] 
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

807 
declare divide_le_0_1_iff [simp,no_atp] 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

808 

23389  809 

14293  810 
subsection {* Ordering Rules for Division *} 
811 

812 
lemma divide_strict_right_mono: 

35028
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haftmann
parents:
34146
diff
changeset

813 
"[a < b; 0 < c] ==> a / c < b / (c::'a::linordered_field)" 
14293  814 
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
23482  815 
positive_imp_inverse_positive) 
14293  816 

817 
lemma divide_right_mono: 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

818 
"[a \<le> b; 0 \<le> c] ==> a/c \<le> b/(c::'a::{linordered_field,division_by_zero})" 
23482  819 
by (force simp add: divide_strict_right_mono order_le_less) 
14293  820 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

821 
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

822 
==> c <= 0 ==> b / c <= a / c" 
23482  823 
apply (drule divide_right_mono [of _ _ " c"]) 
824 
apply auto 

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

825 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

826 

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

827 
lemma divide_strict_right_mono_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

828 
"[b < a; c < 0] ==> a / c < b / (c::'a::linordered_field)" 
23482  829 
apply (drule divide_strict_right_mono [of _ _ "c"], simp) 
830 
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric]) 

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

831 
done 
14293  832 

833 
text{*The last premise ensures that @{term a} and @{term b} 

834 
have the same sign*} 

835 
lemma divide_strict_left_mono: 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

836 
"[b < a; 0 < c; 0 < a*b] ==> c / a < c / (b::'a::linordered_field)" 
23482  837 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono) 
14293  838 

839 
lemma divide_left_mono: 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

840 
"[b \<le> a; 0 \<le> c; 0 < a*b] ==> c / a \<le> c / (b::'a::linordered_field)" 
23482  841 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono) 
14293  842 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

843 
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

844 
==> 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

845 
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

846 
apply (auto simp add: mult_commute) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

847 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

848 

14293  849 
lemma divide_strict_left_mono_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

850 
"[a < b; c < 0; 0 < a*b] ==> c / a < c / (b::'a::linordered_field)" 
23482  851 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg) 
852 

14293  853 

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

854 
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

855 

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

856 
lemma le_divide_eq_1 [no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

857 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

858 
shows "(1 \<le> b / a) = ((0 < a & a \<le> b)  (a < 0 & b \<le> a))" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

859 
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

860 

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

861 
lemma divide_le_eq_1 [no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

862 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

863 
shows "(b / a \<le> 1) = ((0 < a & b \<le> a)  (a < 0 & a \<le> b)  a=0)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

864 
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

865 

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

866 
lemma less_divide_eq_1 [no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

867 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

868 
shows "(1 < b / a) = ((0 < a & a < b)  (a < 0 & b < a))" 
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 [no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

872 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

873 
shows "(b / a < 1) = ((0 < a & b < a)  (a < 0 & a < b)  a=0)" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

874 
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

875 

23389  876 

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

877 
subsection{*Conditional Simplification Rules: No Case Splits*} 
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 le_divide_eq_1_pos [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

880 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

881 
shows "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

882 
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

883 

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

884 
lemma le_divide_eq_1_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

885 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

886 
shows "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

887 
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

888 

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

889 
lemma divide_le_eq_1_pos [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

890 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

891 
shows "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

892 
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

893 

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

894 
lemma divide_le_eq_1_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

895 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

896 
shows "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

897 
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

898 

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

899 
lemma less_divide_eq_1_pos [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

900 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

901 
shows "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

902 
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

903 

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

904 
lemma less_divide_eq_1_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

905 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

906 
shows "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

907 
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

908 

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

909 
lemma divide_less_eq_1_pos [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

910 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

911 
shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)" 
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

912 
by (auto simp add: divide_less_eq) 
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

913 

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

914 
lemma divide_less_eq_1_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

915 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

916 
shows "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

917 
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

918 

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

919 
lemma eq_divide_eq_1 [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

920 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

921 
shows "(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

922 
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

923 

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

924 
lemma divide_eq_eq_1 [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

925 
fixes a :: "'a :: {linordered_field,division_by_zero}" 
18649
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

926 
shows "(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

927 
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

928 

23389  929 

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

930 
subsection {* Reasoning about inequalities with division *} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

931 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

932 
lemma mult_imp_div_pos_le: "0 < (y::'a::linordered_field) ==> x <= z * y ==> 
33319  933 
x / y <= z" 
934 
by (subst pos_divide_le_eq, assumption+) 

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

935 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

936 
lemma mult_imp_le_div_pos: "0 < (y::'a::linordered_field) ==> z * y <= x ==> 
23482  937 
z <= x / y" 
938 
by(simp add:field_simps) 

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

939 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

940 
lemma mult_imp_div_pos_less: "0 < (y::'a::linordered_field) ==> x < z * y ==> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

941 
x / y < z" 
23482  942 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

943 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

944 
lemma mult_imp_less_div_pos: "0 < (y::'a::linordered_field) ==> z * y < x ==> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

945 
z < x / y" 
23482  946 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

947 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

948 
lemma frac_le: "(0::'a::linordered_field) <= x ==> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

949 
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

950 
apply (rule mult_imp_div_pos_le) 
25230  951 
apply simp 
952 
apply (subst times_divide_eq_left) 

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

953 
apply (rule mult_imp_le_div_pos, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

954 
apply (rule mult_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

955 
apply simp_all 
14293  956 
done 
957 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

958 
lemma frac_less: "(0::'a::linordered_field) <= x ==> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

959 
x < y ==> 0 < w ==> w <= z ==> x / z < y / w" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

960 
apply (rule mult_imp_div_pos_less) 
33319  961 
apply simp 
962 
apply (subst times_divide_eq_left) 

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

963 
apply (rule mult_imp_less_div_pos, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

964 
apply (erule mult_less_le_imp_less) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

965 
apply simp_all 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

966 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

967 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

968 
lemma frac_less2: "(0::'a::linordered_field) < x ==> 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

969 
x <= y ==> 0 < w ==> w < z ==> x / z < y / w" 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

970 
apply (rule mult_imp_div_pos_less) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

971 
apply simp_all 
33319  972 
apply (subst times_divide_eq_left) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

973 
apply (rule mult_imp_less_div_pos, assumption) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

974 
apply (erule mult_le_less_imp_less) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

975 
apply simp_all 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

976 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

977 

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

978 
text{*It's not obvious whether these should be simprules or not. 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

979 
Their effect is to gather terms into one big fraction, like 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

980 
a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

981 
seem to need them.*} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

982 

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

983 
declare times_divide_eq [simp] 
14293  984 

23389  985 

14293  986 
subsection {* Ordered Fields are Dense *} 
987 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

988 
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::linordered_field)" 
23482  989 
by (simp add: field_simps zero_less_two) 
14293  990 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

991 
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::linordered_field) < b" 
23482  992 
by (simp add: field_simps zero_less_two) 
14293  993 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

994 
instance linordered_field < dense_linorder 
24422  995 
proof 
996 
fix x y :: 'a 

997 
have "x < x + 1" by simp 

998 
then show "\<exists>y. x < y" .. 

999 
have "x  1 < x" by simp 

1000 
then show "\<exists>y. y < x" .. 

1001 
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum) 

1002 
qed 

14293  1003 

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

1004 

14293  1005 
subsection {* Absolute Value *} 
1006 

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

1007 
lemma nonzero_abs_inverse: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1008 
"a \<noteq> 0 ==> abs (inverse (a::'a::linordered_field)) = inverse (abs a)" 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1009 
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1010 
negative_imp_inverse_negative) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1011 
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1012 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1013 

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

1014 
lemma abs_inverse [simp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1015 
"abs (inverse (a::'a::{linordered_field,division_by_zero})) = 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1016 
inverse (abs a)" 
21328  1017 
apply (cases "a=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1018 
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1019 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1020 

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

1021 
lemma nonzero_abs_divide: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1022 
"b \<noteq> 0 ==> abs (a / (b::'a::linordered_field)) = abs a / abs b" 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1023 
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1024 

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

1025 
lemma abs_divide [simp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1026 
"abs (a / (b::'a::{linordered_field,division_by_zero})) = abs a / abs b" 
21328  1027 
apply (cases "b=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1028 
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1029 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1030 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1031 
lemma abs_div_pos: "(0::'a::{division_by_zero,linordered_field}) < y ==> 
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1032 
abs x / y = abs (x / y)" 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1033 
apply (subst abs_divide) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1034 
apply (simp add: order_less_imp_le) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1035 
done 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1036 

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

1037 
lemma field_le_epsilon: 
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1038 
fixes x y :: "'a\<Colon>linordered_field" 
35090
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1039 
assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e" 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1040 
shows "x \<le> y" 
35579
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1041 
proof (rule dense_le) 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1042 
fix t assume "t < x" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1043 
hence "0 < x  t" by (simp add: less_diff_eq) 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1044 
from e[OF this] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1045 
show "t \<le> y" by (simp add: field_simps) 
35090
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1046 
qed 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1047 

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

1048 
lemma field_le_mult_one_interval: 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1049 
fixes x :: "'a\<Colon>{linordered_field,division_by_zero}" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1050 
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

1051 
shows "x \<le> y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1052 
proof (cases "0 < x") 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1053 
assume "0 < x" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1054 
thus ?thesis 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1055 
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

1056 
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

1057 
next 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1058 
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

1059 
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

1060 
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

1061 
also note *[OF s] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1062 
finally show ?thesis . 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

1063 
qed 
35090
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1064 

33364  1065 
code_modulename SML 
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

1066 
Fields Arith 
33364  1067 

1068 
code_modulename OCaml 

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

1069 
Fields Arith 
33364  1070 

1071 
code_modulename Haskell 

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

1072 
Fields Arith 
33364  1073 

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

1074 
end 