author  haftmann 
Wed, 10 Feb 2010 14:12:02 +0100  
changeset 35090  88cc65ae046e 
parent 35084  e25eedfc15ce 
child 35216  7641e8d831d2 
permissions  rwrr 
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(* Title: HOL/Fields.thy 
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Author: Gertrud Bauer 
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3 
Author: Steven Obua 
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Author: Tobias Nipkow 
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5 
Author: Lawrence C Paulson 
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Author: Markus Wenzel 
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Author: Jeremy Avigad 
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*) 
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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 

68 
lemma nonzero_mult_divide_mult_cancel_left [simp, noatp]: 

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 

78 

79 
lemma nonzero_mult_divide_mult_cancel_right [simp, noatp]: 

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 

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

93 
lemmas times_divide_eq [noatp] = times_divide_eq_right times_divide_eq_left 

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)" 

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by (simp only: add_divide_distrib) 

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finally show ?thesis 

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by (simp only: mult_commute) 

105 
qed 

106 

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text{*Special Cancellation Simprules for Division*} 

108 

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lemma nonzero_mult_divide_cancel_right [simp, noatp]: 

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"b \<noteq> 0 \<Longrightarrow> a * b / b = a" 

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using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp 

112 

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lemma nonzero_mult_divide_cancel_left [simp, noatp]: 

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"a \<noteq> 0 \<Longrightarrow> a * b / a = b" 

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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, noatp]: 

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"\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a" 

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using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp 

120 

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lemma nonzero_divide_mult_cancel_left [simp, noatp]: 

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"\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b" 

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using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp 

124 

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lemma nonzero_mult_divide_mult_cancel_left2 [simp, noatp]: 

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"\<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) 

128 

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lemma nonzero_mult_divide_mult_cancel_right2 [simp, noatp]: 

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"\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b" 

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using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac) 

132 

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

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by (simp add: divide_inverse) 

135 

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

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by (simp add: divide_inverse nonzero_inverse_minus_eq) 

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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, noatp]: "(a) / b =  (a / b)" 

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by (simp add: divide_inverse) 

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

146 
by (simp add: diff_minus add_divide_distrib) 

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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 

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lemma divide_add_eq_iff: 

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

154 
by (simp add: add_divide_distrib) 

155 

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lemma diff_divide_eq_iff: 

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

158 
by (simp add: diff_divide_distrib) 

159 

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lemma divide_diff_eq_iff: 

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

162 
by (simp add: diff_divide_distrib) 

163 

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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 

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

173 
proof  

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

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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 

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

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by (erule subst, simp) 

185 

186 
lemmas field_eq_simps[noatp] = algebra_simps 

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:*} 

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

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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 

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class division_by_zero = zero + inverse + 
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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 
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assume "a=0" thus ?thesis by (simp add: inverse_zero) 

234 
next 

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assume "a\<noteq>0" 

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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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lemma mult_divide_mult_cancel_left: 
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"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" 
21328  285 
apply (cases "b = 0") 
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apply (simp_all add: nonzero_mult_divide_mult_cancel_left) 
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done 
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lemma mult_divide_mult_cancel_right: 
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"c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})" 
21328  291 
apply (cases "b = 0") 
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apply (simp_all add: nonzero_mult_divide_mult_cancel_right) 
14321  293 
done 
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lemma divide_divide_eq_right [simp,noatp]: 
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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 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

299 
lemma divide_divide_eq_left [simp,noatp]: 
23477
f4b83f03cac9
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 

24427  306 
lemma mult_divide_mult_cancel_left_if[simp,noatp]: 
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 

30630  317 
lemma divide_minus_right [simp, noatp]: 
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:
14267
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" 
5cf13e80be0e
Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents:
14267
diff
changeset

342 
by (simp add: a_gt_0 [THEN order_less_imp_not_eq2] zero_less_one) 
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:
14272
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.*} 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

443 
lemma inverse_less_iff_less [simp,noatp]: 
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 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

451 
lemma inverse_le_iff_le [simp,noatp]: 
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 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

485 
lemma inverse_less_iff_less_neg [simp,noatp]: 
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 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

496 
lemma inverse_le_iff_le_neg [simp,noatp]: 
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}))" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

527 
by (force simp add: order_le_less one_less_inverse_iff zero_less_one 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

528 
eq_commute [of 1]) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

529 

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

530 
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

531 
"(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

532 
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

533 

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

534 
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

535 
"(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

536 
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

537 

23389  538 

14288  539 
subsection{*Simplification of Inequalities Involving Literal Divisors*} 
540 

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

541 
lemma pos_le_divide_eq: "0 < (c::'a::linordered_field) ==> (a \<le> b/c) = (a*c \<le> b)" 
14288  542 
proof  
543 
assume less: "0<c" 

544 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 

545 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

546 
also have "... = (a*c \<le> b)" 

547 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

548 
finally show ?thesis . 

549 
qed 

550 

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

551 
lemma neg_le_divide_eq: "c < (0::'a::linordered_field) ==> (a \<le> b/c) = (b \<le> a*c)" 
14288  552 
proof  
553 
assume less: "c<0" 

554 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 

555 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

556 
also have "... = (b \<le> a*c)" 

557 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

558 
finally show ?thesis . 

559 
qed 

560 

561 
lemma le_divide_eq: 

562 
"(a \<le> b/c) = 

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

564 
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

565 
else a \<le> (0::'a::{linordered_field,division_by_zero}))" 
21328  566 
apply (cases "c=0", simp) 
14288  567 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
568 
done 

569 

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

570 
lemma pos_divide_le_eq: "0 < (c::'a::linordered_field) ==> (b/c \<le> a) = (b \<le> a*c)" 
14288  571 
proof  
572 
assume less: "0<c" 

573 
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" 

574 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

575 
also have "... = (b \<le> a*c)" 

576 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

577 
finally show ?thesis . 

578 
qed 

579 

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

580 
lemma neg_divide_le_eq: "c < (0::'a::linordered_field) ==> (b/c \<le> a) = (a*c \<le> b)" 
14288  581 
proof  
582 
assume less: "c<0" 

583 
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" 

584 
by (simp add: mult_le_cancel_right order_less_not_sym [OF less]) 

585 
also have "... = (a*c \<le> b)" 

586 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

587 
finally show ?thesis . 

588 
qed 

589 

590 
lemma divide_le_eq: 

591 
"(b/c \<le> a) = 

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

593 
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

594 
else 0 \<le> (a::'a::{linordered_field,division_by_zero}))" 
21328  595 
apply (cases "c=0", simp) 
14288  596 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
597 
done 

598 

599 
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

600 
"0 < (c::'a::linordered_field) ==> (a < b/c) = (a*c < b)" 
14288  601 
proof  
602 
assume less: "0<c" 

603 
hence "(a < b/c) = (a*c < (b/c)*c)" 

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

604 
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) 
14288  605 
also have "... = (a*c < b)" 
606 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

607 
finally show ?thesis . 

608 
qed 

609 

610 
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

611 
"c < (0::'a::linordered_field) ==> (a < b/c) = (b < a*c)" 
14288  612 
proof  
613 
assume less: "c<0" 

614 
hence "(a < b/c) = ((b/c)*c < a*c)" 

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

615 
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) 
14288  616 
also have "... = (b < a*c)" 
617 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

618 
finally show ?thesis . 

619 
qed 

620 

621 
lemma less_divide_eq: 

622 
"(a < b/c) = 

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

624 
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

625 
else a < (0::'a::{linordered_field,division_by_zero}))" 
21328  626 
apply (cases "c=0", simp) 
14288  627 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
628 
done 

629 

630 
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

631 
"0 < (c::'a::linordered_field) ==> (b/c < a) = (b < a*c)" 
14288  632 
proof  
633 
assume less: "0<c" 

634 
hence "(b/c < a) = ((b/c)*c < a*c)" 

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

635 
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) 
14288  636 
also have "... = (b < a*c)" 
637 
by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 

638 
finally show ?thesis . 

639 
qed 

640 

641 
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

642 
"c < (0::'a::linordered_field) ==> (b/c < a) = (a*c < b)" 
14288  643 
proof  
644 
assume less: "c<0" 

645 
hence "(b/c < a) = (a*c < (b/c)*c)" 

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

646 
by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less]) 
14288  647 
also have "... = (a*c < b)" 
648 
by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 

649 
finally show ?thesis . 

650 
qed 

651 

652 
lemma divide_less_eq: 

653 
"(b/c < a) = 

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

655 
else if c < 0 then a*c < b 

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

656 
else 0 < (a::'a::{linordered_field,division_by_zero}))" 
21328  657 
apply (cases "c=0", simp) 
14288  658 
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
659 
done 

660 

23482  661 

662 
subsection{*Field simplification*} 

663 

29667  664 
text{* Lemmas @{text field_simps} multiply with denominators in in(equations) 
665 
if they can be proved to be nonzero (for equations) or positive/negative 

666 
(for inequations). Can be too aggressive and is therefore separate from the 

667 
more benign @{text algebra_simps}. *} 

14288  668 

29833  669 
lemmas field_simps[noatp] = field_eq_simps 
23482  670 
(* multiply ineqn *) 
671 
pos_divide_less_eq neg_divide_less_eq 

672 
pos_less_divide_eq neg_less_divide_eq 

673 
pos_divide_le_eq neg_divide_le_eq 

674 
pos_le_divide_eq neg_le_divide_eq 

14288  675 

23482  676 
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs 
23483  677 
of positivity/negativity needed for @{text field_simps}. Have not added @{text 
23482  678 
sign_simps} to @{text field_simps} because the former can lead to case 
679 
explosions. *} 

14288  680 

29833  681 
lemmas sign_simps[noatp] = group_simps 
23482  682 
zero_less_mult_iff mult_less_0_iff 
14288  683 

23482  684 
(* Only works once linear arithmetic is installed: 
685 
text{*An example:*} 

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

686 
lemma fixes a b c d e f :: "'a::linordered_field" 
23482  687 
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow> 
688 
((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) < 

689 
((ef)*(ab)*(cd))/((ef)*(ab)*(cd)) + u" 

690 
apply(subgoal_tac "(cd)*(ef)*(ab) > 0") 

691 
prefer 2 apply(simp add:sign_simps) 

692 
apply(subgoal_tac "(cd)*(ef)*(ab)*u > 0") 

693 
prefer 2 apply(simp add:sign_simps) 

694 
apply(simp add:field_simps) 

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

695 
done 
23482  696 
*) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

697 

23389  698 

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

699 
subsection{*Division and Signs*} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

700 

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

701 
lemma zero_less_divide_iff: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

702 
"((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

703 
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

704 

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

705 
lemma divide_less_0_iff: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

706 
"(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

707 
(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

708 
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

709 

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

710 
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

711 
"((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

712 
(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

713 
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

714 

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

715 
lemma divide_le_0_iff: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

716 
"(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

717 
(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

718 
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

719 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

720 
lemma divide_eq_0_iff [simp,noatp]: 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

721 
"(a/b = 0) = (a=0  b=(0::'a::{field,division_by_zero}))" 
23482  722 
by (simp add: divide_inverse) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

723 

23482  724 
lemma divide_pos_pos: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

725 
"0 < (x::'a::linordered_field) ==> 0 < y ==> 0 < x / y" 
23482  726 
by(simp add:field_simps) 
727 

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

728 

23482  729 
lemma divide_nonneg_pos: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

730 
"0 <= (x::'a::linordered_field) ==> 0 < y ==> 0 <= x / y" 
23482  731 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

732 

23482  733 
lemma divide_neg_pos: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

734 
"(x::'a::linordered_field) < 0 ==> 0 < y ==> x / y < 0" 
23482  735 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

736 

23482  737 
lemma divide_nonpos_pos: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

738 
"(x::'a::linordered_field) <= 0 ==> 0 < y ==> x / y <= 0" 
23482  739 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

740 

23482  741 
lemma divide_pos_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

742 
"0 < (x::'a::linordered_field) ==> y < 0 ==> x / y < 0" 
23482  743 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

744 

23482  745 
lemma divide_nonneg_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

746 
"0 <= (x::'a::linordered_field) ==> y < 0 ==> x / y <= 0" 
23482  747 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

748 

23482  749 
lemma divide_neg_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

750 
"(x::'a::linordered_field) < 0 ==> y < 0 ==> 0 < x / y" 
23482  751 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

752 

23482  753 
lemma divide_nonpos_neg: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

754 
"(x::'a::linordered_field) <= 0 ==> y < 0 ==> 0 <= x / y" 
23482  755 
by(simp add:field_simps) 
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

756 

23389  757 

14288  758 
subsection{*Cancellation Laws for Division*} 
759 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

760 
lemma divide_cancel_right [simp,noatp]: 
14288  761 
"(a/c = b/c) = (c = 0  a = (b::'a::{field,division_by_zero}))" 
23482  762 
apply (cases "c=0", simp) 
23496  763 
apply (simp add: divide_inverse) 
14288  764 
done 
765 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

766 
lemma divide_cancel_left [simp,noatp]: 
14288  767 
"(c/a = c/b) = (c = 0  a = (b::'a::{field,division_by_zero}))" 
23482  768 
apply (cases "c=0", simp) 
23496  769 
apply (simp add: divide_inverse) 
14288  770 
done 
771 

23389  772 

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

773 
subsection {* Division and the Number One *} 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

774 

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

775 
text{*Simplify expressions equated with 1*} 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

776 
lemma divide_eq_1_iff [simp,noatp]: 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

777 
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" 
23482  778 
apply (cases "b=0", simp) 
779 
apply (simp add: right_inverse_eq) 

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

780 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

781 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

782 
lemma one_eq_divide_iff [simp,noatp]: 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

783 
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" 
23482  784 
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

785 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

786 
lemma zero_eq_1_divide_iff [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

787 
"((0::'a::{linordered_field,division_by_zero}) = 1/a) = (a = 0)" 
23482  788 
apply (cases "a=0", simp) 
789 
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

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

791 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

792 
lemma one_divide_eq_0_iff [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

793 
"(1/a = (0::'a::{linordered_field,division_by_zero})) = (a = 0)" 
23482  794 
apply (cases "a=0", simp) 
795 
apply (insert zero_neq_one [THEN not_sym]) 

796 
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

797 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

798 

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

799 
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} 
18623  800 
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified] 
801 
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified] 

802 
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified] 

803 
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified] 

17085  804 

29833  805 
declare zero_less_divide_1_iff [simp,noatp] 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

806 
declare divide_less_0_1_iff [simp,noatp] 
29833  807 
declare zero_le_divide_1_iff [simp,noatp] 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

808 
declare divide_le_0_1_iff [simp,noatp] 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

809 

23389  810 

14293  811 
subsection {* Ordering Rules for Division *} 
812 

813 
lemma divide_strict_right_mono: 

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

814 
"[a < b; 0 < c] ==> a / c < b / (c::'a::linordered_field)" 
14293  815 
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
23482  816 
positive_imp_inverse_positive) 
14293  817 

818 
lemma divide_right_mono: 

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

819 
"[a \<le> b; 0 \<le> c] ==> a/c \<le> b/(c::'a::{linordered_field,division_by_zero})" 
23482  820 
by (force simp add: divide_strict_right_mono order_le_less) 
14293  821 

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

822 
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

823 
==> c <= 0 ==> b / c <= a / c" 
23482  824 
apply (drule divide_right_mono [of _ _ " c"]) 
825 
apply auto 

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

826 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

827 

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

828 
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

829 
"[b < a; c < 0] ==> a / c < b / (c::'a::linordered_field)" 
23482  830 
apply (drule divide_strict_right_mono [of _ _ "c"], simp) 
831 
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

832 
done 
14293  833 

834 
text{*The last premise ensures that @{term a} and @{term b} 

835 
have the same sign*} 

836 
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

837 
"[b < a; 0 < c; 0 < a*b] ==> c / a < c / (b::'a::linordered_field)" 
23482  838 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono) 
14293  839 

840 
lemma divide_left_mono: 

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

841 
"[b \<le> a; 0 \<le> c; 0 < a*b] ==> c / a \<le> c / (b::'a::linordered_field)" 
23482  842 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono) 
14293  843 

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

844 
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

845 
==> 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

846 
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

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

848 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

849 

14293  850 
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

851 
"[a < b; c < 0; 0 < a*b] ==> c / a < c / (b::'a::linordered_field)" 
23482  852 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg) 
853 

14293  854 

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

855 
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

856 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

857 
lemma le_divide_eq_1 [noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

858 
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

859 
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

860 
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

861 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

862 
lemma divide_le_eq_1 [noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

863 
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

864 
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

865 
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

866 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

867 
lemma less_divide_eq_1 [noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

868 
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

869 
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

870 
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

871 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

872 
lemma divide_less_eq_1 [noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

873 
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

874 
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

875 
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

876 

23389  877 

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

878 
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

879 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

880 
lemma le_divide_eq_1_pos [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

881 
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

882 
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

883 
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

884 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

885 
lemma le_divide_eq_1_neg [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

886 
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

887 
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

888 
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

889 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

890 
lemma divide_le_eq_1_pos [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

891 
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

892 
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

893 
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

894 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

895 
lemma divide_le_eq_1_neg [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

896 
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

897 
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

898 
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

899 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

900 
lemma less_divide_eq_1_pos [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

901 
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

902 
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

903 
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

904 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

905 
lemma less_divide_eq_1_neg [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

906 
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

907 
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

908 
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

909 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

910 
lemma divide_less_eq_1_pos [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

911 
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

912 
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

913 
by (auto simp add: divide_less_eq) 
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

914 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

915 
lemma divide_less_eq_1_neg [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

916 
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

917 
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

918 
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

919 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

920 
lemma eq_divide_eq_1 [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

921 
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

922 
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

923 
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

924 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23879
diff
changeset

925 
lemma divide_eq_eq_1 [simp,noatp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

926 
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

927 
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

928 
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

929 

23389  930 

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

931 
subsection {* Reasoning about inequalities with division *} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

932 

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

933 
lemma mult_imp_div_pos_le: "0 < (y::'a::linordered_field) ==> x <= z * y ==> 
33319  934 
x / y <= z" 
935 
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

936 

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

937 
lemma mult_imp_le_div_pos: "0 < (y::'a::linordered_field) ==> z * y <= x ==> 
23482  938 
z <= x / y" 
939 
by(simp add:field_simps) 

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

940 

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

941 
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

942 
x / y < z" 
23482  943 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

944 

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

945 
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

946 
z < x / y" 
23482  947 
by(simp add:field_simps) 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

948 

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

949 
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

950 
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

951 
apply (rule mult_imp_div_pos_le) 
25230  952 
apply simp 
953 
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

954 
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

955 
apply (rule mult_mono) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

956 
apply simp_all 
14293  957 
done 
958 

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

959 
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

960 
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

961 
apply (rule mult_imp_div_pos_less) 
33319  962 
apply simp 
963 
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

964 
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

965 
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

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

967 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

968 

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

969 
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

970 
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

971 
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

972 
apply simp_all 
33319  973 
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

974 
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

975 
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

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

977 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

978 

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

979 
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

980 
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

981 
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

982 
seem to need them.*} 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

983 

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

984 
declare times_divide_eq [simp] 
14293  985 

23389  986 

14293  987 
subsection {* Ordered Fields are Dense *} 
988 

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

989 
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::linordered_field)" 
23482  990 
by (simp add: field_simps zero_less_two) 
14293  991 

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

992 
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::linordered_field) < b" 
23482  993 
by (simp add: field_simps zero_less_two) 
14293  994 

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

995 
instance linordered_field < dense_linorder 
24422  996 
proof 
997 
fix x y :: 'a 

998 
have "x < x + 1" by simp 

999 
then show "\<exists>y. x < y" .. 

1000 
have "x  1 < x" by simp 

1001 
then show "\<exists>y. y < x" .. 

1002 
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum) 

1003 
qed 

14293  1004 

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

1005 

14293  1006 
subsection {* Absolute Value *} 
1007 

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

1008 
lemma nonzero_abs_inverse: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1009 
"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

1010 
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

1011 
negative_imp_inverse_negative) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1012 
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

1013 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1014 

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

1015 
lemma abs_inverse [simp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1016 
"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

1017 
inverse (abs a)" 
21328  1018 
apply (cases "a=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1019 
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1020 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1021 

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

1022 
lemma nonzero_abs_divide: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1023 
"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

1024 
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

1025 

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

1026 
lemma abs_divide [simp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

1027 
"abs (a / (b::'a::{linordered_field,division_by_zero})) = abs a / abs b" 
21328  1028 
apply (cases "b=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1029 
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1030 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

1031 

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

1032 
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

1033 
abs x / y = abs (x / y)" 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1034 
apply (subst abs_divide) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1035 
apply (simp add: order_less_imp_le) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

1036 
done 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

1037 

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

1038 

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

1039 
lemma field_le_epsilon: 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1040 
fixes x y :: "'a :: {division_by_zero,linordered_field}" 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1041 
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

1042 
shows "x \<le> y" 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1043 
proof (rule ccontr) 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1044 
obtain two :: 'a where two: "two = 1 + 1" by simp 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1045 
assume "\<not> x \<le> y" 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1046 
then have yx: "y < x" by simp 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1047 
then have "y +  y < x +  y" by (rule add_strict_right_mono) 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1048 
then have "x  y > 0" by (simp add: diff_minus) 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1049 
then have "(x  y) / two > 0" 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1050 
by (rule divide_pos_pos) (simp add: two) 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1051 
then have "x \<le> y + (x  y) / two" by (rule e) 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1052 
also have "... = (x  y + two * y) / two" 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1053 
by (simp add: add_divide_distrib two) 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1054 
also have "... = (x + y) / two" 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1055 
by (simp add: two algebra_simps) 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1056 
also have "... < x" using yx 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1057 
by (simp add: two pos_divide_less_eq algebra_simps) 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1058 
finally have "x < x" . 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1059 
then show False .. 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1060 
qed 
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

1061 

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

1062 

33364  1063 
code_modulename SML 
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

1064 
Fields Arith 
33364  1065 

1066 
code_modulename OCaml 

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

1067 
Fields Arith 
33364  1068 

1069 
code_modulename Haskell 

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

1070 
Fields Arith 
33364  1071 

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

1072 
end 