author  haftmann 
Fri, 23 Apr 2010 13:58:14 +0200  
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parent 35828  46cfc4b8112e 
child 36304  6984744e6b34 
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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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" 
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assumes field_divide_inverse: "a / b = a * inverse b" 

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

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

14738  30 
qed 
25230  31 

27516  32 
subclass idom .. 
25230  33 

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

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

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

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by (simp add: division_ring_inverse_add mult_ac) 

39 

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

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

44 
by (simp add: divide_inverse nonzero_inverse_mult_distrib) 

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

46 
by (simp only: mult_ac) 

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

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

49 
qed 

50 

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

54 

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

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by (simp add: divide_inverse mult_ac) 
30630  57 

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

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lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left 
30630  60 

61 
lemma add_frac_eq: 

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

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

64 
proof  

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

66 
using assms by simp 

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

68 
by (simp only: add_divide_distrib) 

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finally show ?thesis 

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by (simp only: mult_commute) 

71 
qed 

72 

73 
text{*Special Cancellation Simprules for Division*} 

74 

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lemma nonzero_mult_divide_cancel_right [simp, no_atp]: 
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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 
30630  78 

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

82 

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

86 

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

90 

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

94 

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

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lemma add_divide_eq_iff: 

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

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by (simp add: add_divide_distrib) 
30630  102 

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lemma divide_add_eq_iff: 

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

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by (simp add: add_divide_distrib) 
30630  106 

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lemma diff_divide_eq_iff: 

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

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by (simp add: diff_divide_distrib) 
30630  110 

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lemma divide_diff_eq_iff: 

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

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by (simp add: diff_divide_distrib) 
30630  114 

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lemmas field_eq_simps[no_atp] = algebra_simps 
30630  116 
(* pull / out*) 
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add_divide_eq_iff divide_add_eq_iff 

118 
diff_divide_eq_iff divide_diff_eq_iff 

119 
(* multiply eqn *) 

120 
nonzero_eq_divide_eq nonzero_divide_eq_eq 

121 
times_divide_eq_left times_divide_eq_right 

122 

123 
text{*An example:*} 

124 
lemma "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f\<rbrakk> \<Longrightarrow> ((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) = 1" 

125 
apply(subgoal_tac "(cd)*(ef)*(ab) \<noteq> 0") 

126 
apply(simp add:field_eq_simps) 

127 
apply(simp) 

128 
done 

129 

130 
lemma diff_frac_eq: 

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

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by (simp add: field_eq_simps times_divide_eq) 
30630  133 

134 
lemma frac_eq_eq: 

135 
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)" 

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by (simp add: field_eq_simps times_divide_eq) 
25230  137 

138 
end 

139 

14270  140 
text{*This version builds in division by zero while also reorienting 
141 
the righthand side.*} 

142 
lemma inverse_mult_distrib [simp]: 

143 
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})" 

144 
proof cases 

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

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thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac) 
14270  147 
next 
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assume "~ (a \<noteq> 0 & b \<noteq> 0)" 

29667  149 
thus ?thesis by force 
14270  150 
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  156 

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text {* Calculations with fractions *} 
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text{* There is a whole bunch of simprules just for class @{text 
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field} but none for class @{text field} and @{text nonzero_divides} 
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because the latter are covered by a simproc. *} 
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lemma mult_divide_mult_cancel_left: 
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"c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})" 
21328  165 
apply (cases "b = 0") 
35216  166 
apply simp_all 
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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  171 
apply (cases "b = 0") 
35216  172 
apply simp_all 
14321  173 
done 
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lemma divide_divide_eq_right [simp,no_atp]: 
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"a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})" 
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by (simp add: divide_inverse mult_ac) 
14288  178 

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lemma divide_divide_eq_left [simp,no_atp]: 
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"(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)" 
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by (simp add: divide_inverse mult_assoc) 
14288  182 

23389  183 

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text {*Special Cancellation Simprules for Division*} 
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lemma mult_divide_mult_cancel_left_if[simp,no_atp]: 
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fixes c :: "'a :: {field,division_by_zero}" 
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shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)" 
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by (simp add: mult_divide_mult_cancel_left) 
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text {* Division and Unary Minus *} 
14293  193 

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

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by (simp add: divide_inverse) 
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lemma divide_minus_right [simp, no_atp]: 
30630  198 
"a / (b::'a::{field,division_by_zero}) = (a / b)" 
199 
by (simp add: divide_inverse) 

200 

201 
lemma minus_divide_divide: 

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"(a)/(b) = a / (b::'a::{field,division_by_zero})" 
21328  203 
apply (cases "b=0", simp) 
14293  204 
apply (simp add: nonzero_minus_divide_divide) 
205 
done 

206 

23482  207 
lemma eq_divide_eq: 
208 
"((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)" 

30630  209 
by (simp add: nonzero_eq_divide_eq) 
23482  210 

211 
lemma divide_eq_eq: 

212 
"(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)" 

30630  213 
by (force simp add: nonzero_divide_eq_eq) 
14293  214 

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lemma inverse_eq_1_iff [simp]: 
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"(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))" 
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by (insert inverse_eq_iff_eq [of x 1], simp) 
23389  218 

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lemma divide_eq_0_iff [simp,no_atp]: 
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"(a/b = 0) = (a=0  b=(0::'a::{field,division_by_zero}))" 
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by (simp add: divide_inverse) 
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lemma divide_cancel_right [simp,no_atp]: 
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224 
"(a/c = b/c) = (c = 0  a = (b::'a::{field,division_by_zero}))" 
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225 
apply (cases "c=0", simp) 
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226 
apply (simp add: divide_inverse) 
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227 
done 
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228 

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229 
lemma divide_cancel_left [simp,no_atp]: 
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230 
"(c/a = c/b) = (c = 0  a = (b::'a::{field,division_by_zero}))" 
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231 
apply (cases "c=0", simp) 
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232 
apply (simp add: divide_inverse) 
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233 
done 
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234 

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235 
lemma divide_eq_1_iff [simp,no_atp]: 
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236 
"(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" 
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237 
apply (cases "b=0", simp) 
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238 
apply (simp add: right_inverse_eq) 
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239 
done 
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240 

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241 
lemma one_eq_divide_iff [simp,no_atp]: 
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242 
"(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))" 
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243 
by (simp add: eq_commute [of 1]) 
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244 

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245 

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246 
text {* Ordered Fields *} 
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247 

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248 
class linordered_field = field + linordered_idom 
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249 
begin 
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250 

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251 
lemma positive_imp_inverse_positive: 
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252 
assumes a_gt_0: "0 < a" 
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253 
shows "0 < inverse a" 
23482  254 
proof  
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255 
have "0 < a * inverse a" 
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256 
by (simp add: a_gt_0 [THEN less_imp_not_eq2]) 
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257 
thus "0 < inverse a" 
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258 
by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff) 
23482  259 
qed 
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260 

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261 
lemma negative_imp_inverse_negative: 
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262 
"a < 0 \<Longrightarrow> inverse a < 0" 
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263 
by (insert positive_imp_inverse_positive [of "a"], 
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264 
simp add: nonzero_inverse_minus_eq less_imp_not_eq) 
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265 

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266 
lemma inverse_le_imp_le: 
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267 
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a" 
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268 
shows "b \<le> a" 
23482  269 
proof (rule classical) 
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270 
assume "~ b \<le> a" 
23482  271 
hence "a < b" by (simp add: linorder_not_le) 
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272 
hence bpos: "0 < b" by (blast intro: apos less_trans) 
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273 
hence "a * inverse a \<le> a * inverse b" 
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274 
by (simp add: apos invle less_imp_le mult_left_mono) 
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275 
hence "(a * inverse a) * b \<le> (a * inverse b) * b" 
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276 
by (simp add: bpos less_imp_le mult_right_mono) 
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277 
thus "b \<le> a" by (simp add: mult_assoc apos bpos less_imp_not_eq2) 
23482  278 
qed 
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279 

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280 
lemma inverse_positive_imp_positive: 
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281 
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0" 
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282 
shows "0 < a" 
23389  283 
proof  
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284 
have "0 < inverse (inverse a)" 
23389  285 
using inv_gt_0 by (rule positive_imp_inverse_positive) 
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286 
thus "0 < a" 
23389  287 
using nz by (simp add: nonzero_inverse_inverse_eq) 
288 
qed 

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289 

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290 
lemma inverse_negative_imp_negative: 
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291 
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0" 
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292 
shows "a < 0" 
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293 
proof  
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294 
have "inverse (inverse a) < 0" 
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295 
using inv_less_0 by (rule negative_imp_inverse_negative) 
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296 
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq) 
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297 
qed 
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298 

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299 
lemma linordered_field_no_lb: 
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300 
"\<forall>x. \<exists>y. y < x" 
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301 
proof 
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302 
fix x::'a 
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303 
have m1: " (1::'a) < 0" by simp 
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304 
from add_strict_right_mono[OF m1, where c=x] 
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305 
have "( 1) + x < x" by simp 
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306 
thus "\<exists>y. y < x" by blast 
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307 
qed 
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308 

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309 
lemma linordered_field_no_ub: 
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310 
"\<forall> x. \<exists>y. y > x" 
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311 
proof 
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312 
fix x::'a 
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313 
have m1: " (1::'a) > 0" by simp 
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314 
from add_strict_right_mono[OF m1, where c=x] 
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315 
have "1 + x > x" by simp 
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316 
thus "\<exists>y. y > x" by blast 
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317 
qed 
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318 

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319 
lemma less_imp_inverse_less: 
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320 
assumes less: "a < b" and apos: "0 < a" 
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321 
shows "inverse b < inverse a" 
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322 
proof (rule ccontr) 
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323 
assume "~ inverse b < inverse a" 
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324 
hence "inverse a \<le> inverse b" by simp 
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325 
hence "~ (a < b)" 
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326 
by (simp add: not_less inverse_le_imp_le [OF _ apos]) 
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327 
thus False by (rule notE [OF _ less]) 
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328 
qed 
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329 

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330 
lemma inverse_less_imp_less: 
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331 
"inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a" 
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332 
apply (simp add: less_le [of "inverse a"] less_le [of "b"]) 
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333 
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
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334 
done 
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335 

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336 
text{*Both premises are essential. Consider 1 and 1.*} 
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337 
lemma inverse_less_iff_less [simp,no_atp]: 
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338 
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a" 
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339 
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
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340 

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341 
lemma le_imp_inverse_le: 
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342 
"a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a" 
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343 
by (force simp add: le_less less_imp_inverse_less) 
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344 

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345 
lemma inverse_le_iff_le [simp,no_atp]: 
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346 
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a" 
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347 
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
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348 

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349 

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350 
text{*These results refer to both operands being negative. The oppositesign 
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351 
case is trivial, since inverse preserves signs.*} 
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352 
lemma inverse_le_imp_le_neg: 
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353 
"inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a" 
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354 
apply (rule classical) 
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355 
apply (subgoal_tac "a < 0") 
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356 
prefer 2 apply force 
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357 
apply (insert inverse_le_imp_le [of "b" "a"]) 
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358 
apply (simp add: nonzero_inverse_minus_eq) 
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359 
done 
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360 

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361 
lemma less_imp_inverse_less_neg: 
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362 
"a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a" 
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more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

363 
apply (subgoal_tac "a < 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

364 
prefer 2 apply (blast intro: less_trans) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

365 
apply (insert less_imp_inverse_less [of "b" "a"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

366 
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

367 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

368 

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

369 
lemma inverse_less_imp_less_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

370 
"inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

371 
apply (rule classical) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

372 
apply (subgoal_tac "a < 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

373 
prefer 2 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

374 
apply force 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

375 
apply (insert inverse_less_imp_less [of "b" "a"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

376 
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

377 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

378 

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

379 
lemma inverse_less_iff_less_neg [simp,no_atp]: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

380 
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

381 
apply (insert inverse_less_iff_less [of "b" "a"]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

382 
apply (simp del: inverse_less_iff_less 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

383 
add: nonzero_inverse_minus_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

384 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

385 

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

386 
lemma le_imp_inverse_le_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

387 
"a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

388 
by (force simp add: le_less less_imp_inverse_less_neg) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

389 

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

390 
lemma inverse_le_iff_le_neg [simp,no_atp]: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

391 
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

392 
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

393 

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

394 
lemma pos_le_divide_eq: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

395 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

396 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

397 
hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

398 
by (simp add: mult_le_cancel_right less_not_sym [OF less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

399 
also have "... = (a*c \<le> b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

400 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

401 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

402 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

403 

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

404 
lemma neg_le_divide_eq: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

405 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

406 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

407 
hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

408 
by (simp add: mult_le_cancel_right less_not_sym [OF less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

409 
also have "... = (b \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

410 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

411 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

412 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

413 

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

414 
lemma pos_less_divide_eq: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

415 
"0 < c ==> (a < b/c) = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

416 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

417 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

418 
hence "(a < b/c) = (a*c < (b/c)*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

419 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

420 
also have "... = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

421 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

422 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

423 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

424 

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

425 
lemma neg_less_divide_eq: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

426 
"c < 0 ==> (a < b/c) = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

427 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

428 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

429 
hence "(a < b/c) = ((b/c)*c < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

430 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

431 
also have "... = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

432 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

433 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

434 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

435 

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

436 
lemma pos_divide_less_eq: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

438 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

439 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

440 
hence "(b/c < a) = ((b/c)*c < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

441 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

442 
also have "... = (b < a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

443 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

444 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

445 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

446 

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

447 
lemma neg_divide_less_eq: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

448 
"c < 0 ==> (b/c < a) = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

449 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

450 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

451 
hence "(b/c < a) = (a*c < (b/c)*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

452 
by (simp add: mult_less_cancel_right_disj less_not_sym [OF less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

453 
also have "... = (a*c < b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

454 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

455 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

456 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

457 

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

458 
lemma pos_divide_le_eq: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

459 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

460 
assume less: "0<c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

461 
hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

462 
by (simp add: mult_le_cancel_right less_not_sym [OF less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

463 
also have "... = (b \<le> a*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

464 
by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

465 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

466 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

467 

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

468 
lemma neg_divide_le_eq: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

469 
proof  
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

470 
assume less: "c<0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

471 
hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

472 
by (simp add: mult_le_cancel_right less_not_sym [OF less]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

473 
also have "... = (a*c \<le> b)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

474 
by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

475 
finally show ?thesis . 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

476 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

477 

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

478 
text{* Lemmas @{text field_simps} multiply with denominators in in(equations) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

479 
if they can be proved to be nonzero (for equations) or positive/negative 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

480 
(for inequations). Can be too aggressive and is therefore separate from the 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

481 
more benign @{text algebra_simps}. *} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

482 

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

483 
lemmas field_simps[no_atp] = field_eq_simps 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

484 
(* multiply ineqn *) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

485 
pos_divide_less_eq neg_divide_less_eq 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

486 
pos_less_divide_eq neg_less_divide_eq 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

487 
pos_divide_le_eq neg_divide_le_eq 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

488 
pos_le_divide_eq neg_le_divide_eq 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

489 

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

490 
thm field_eq_simps 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

491 

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

492 
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

493 
of positivity/negativity needed for @{text field_simps}. Have not added @{text 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

494 
sign_simps} to @{text field_simps} because the former can lead to case 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

495 
explosions. *} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

496 

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

497 
lemmas sign_simps[no_atp] = group_simps 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

498 
zero_less_mult_iff mult_less_0_iff 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

499 

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

500 
(* Only works once linear arithmetic is installed: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

501 
text{*An example:*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

502 
lemma fixes a b c d e f :: "'a::linordered_field" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

503 
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

504 
((ab)*(cd)*(ef))/((cd)*(ef)*(ab)) < 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

505 
((ef)*(ab)*(cd))/((ef)*(ab)*(cd)) + u" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

506 
apply(subgoal_tac "(cd)*(ef)*(ab) > 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

507 
prefer 2 apply(simp add:sign_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

508 
apply(subgoal_tac "(cd)*(ef)*(ab)*u > 0") 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

509 
prefer 2 apply(simp add:sign_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

510 
apply(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

511 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

512 
*) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

513 

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

514 
lemma divide_pos_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

515 
"0 < x ==> 0 < y ==> 0 < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

516 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

517 

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

518 
lemma divide_nonneg_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

519 
"0 <= x ==> 0 < y ==> 0 <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

520 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

521 

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

522 
lemma divide_neg_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

523 
"x < 0 ==> 0 < y ==> x / y < 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

524 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

525 

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

526 
lemma divide_nonpos_pos: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

527 
"x <= 0 ==> 0 < y ==> x / y <= 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

528 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

529 

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

530 
lemma divide_pos_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

531 
"0 < x ==> y < 0 ==> x / y < 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

532 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

533 

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

534 
lemma divide_nonneg_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

535 
"0 <= x ==> y < 0 ==> x / y <= 0" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

536 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

537 

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

538 
lemma divide_neg_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

539 
"x < 0 ==> y < 0 ==> 0 < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

540 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

541 

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

542 
lemma divide_nonpos_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

543 
"x <= 0 ==> y < 0 ==> 0 <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

544 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

545 

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

546 
lemma divide_strict_right_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

547 
"[a < b; 0 < c] ==> a / c < b / c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

548 
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

549 
positive_imp_inverse_positive) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

550 

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

551 

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

552 
lemma divide_strict_right_mono_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

553 
"[b < a; c < 0] ==> a / c < b / c" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

554 
apply (drule divide_strict_right_mono [of _ _ "c"], simp) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

555 
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

556 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

557 

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

558 
text{*The last premise ensures that @{term a} and @{term b} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

559 
have the same sign*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

560 
lemma divide_strict_left_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

561 
"[b < a; 0 < c; 0 < a*b] ==> c / a < c / b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

562 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

563 

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

564 
lemma divide_left_mono: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

565 
"[b \<le> a; 0 \<le> c; 0 < a*b] ==> c / a \<le> c / b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

566 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

567 

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

568 
lemma divide_strict_left_mono_neg: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

569 
"[a < b; c < 0; 0 < a*b] ==> c / a < c / b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

570 
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

571 

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

572 
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

573 
x / y <= z" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

574 
by (subst pos_divide_le_eq, assumption+) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

575 

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

576 
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

577 
z <= x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

578 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

579 

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

580 
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

581 
x / y < z" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

582 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

583 

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

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

585 
z < x / y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

586 
by(simp add:field_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

587 

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

588 
lemma frac_le: "0 <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

589 
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

590 
apply (rule mult_imp_div_pos_le) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

592 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

593 
apply (rule mult_imp_le_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

594 
apply (rule mult_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

595 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

597 

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

598 
lemma frac_less: "0 <= x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

599 
x < y ==> 0 < w ==> w <= z ==> x / z < y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

600 
apply (rule mult_imp_div_pos_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

602 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

603 
apply (rule mult_imp_less_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

604 
apply (erule mult_less_le_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

605 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

607 

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

608 
lemma frac_less2: "0 < x ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

609 
x <= y ==> 0 < w ==> w < z ==> x / z < y / w" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

610 
apply (rule mult_imp_div_pos_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

611 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

612 
apply (subst times_divide_eq_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

613 
apply (rule mult_imp_less_div_pos, assumption) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

614 
apply (erule mult_le_less_imp_less) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

615 
apply simp_all 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

616 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

617 

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

618 
text{*It's not obvious whether these should be simprules or not. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

619 
Their effect is to gather terms into one big fraction, like 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

620 
a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

621 
seem to need them.*} 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

622 

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

623 
declare times_divide_eq [simp] 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

624 

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

625 
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

626 
by (simp add: field_simps zero_less_two) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

627 

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

628 
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

629 
by (simp add: field_simps zero_less_two) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

630 

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

631 
subclass dense_linorder 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

633 
fix x y :: 'a 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

634 
from less_add_one show "\<exists>y. x < y" .. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

635 
from less_add_one have "x + ( 1) < (x + 1) + ( 1)" by (rule add_strict_right_mono) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

636 
then have "x  1 < x + 1  1" by (simp only: diff_minus [symmetric]) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

637 
then have "x  1 < x" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

638 
then show "\<exists>y. y < x" .. 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

639 
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

640 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

641 

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

642 
lemma nonzero_abs_inverse: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

643 
"a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

644 
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

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

646 
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

647 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

648 

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

649 
lemma nonzero_abs_divide: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

650 
"b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

651 
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

652 

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

653 
lemma field_le_epsilon: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

654 
assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

655 
shows "x \<le> y" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

656 
proof (rule dense_le) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

657 
fix t assume "t < x" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

658 
hence "0 < x  t" by (simp add: less_diff_eq) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

659 
from e [OF this] have "x + 0 \<le> x + (y  t)" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

660 
then have "0 \<le> y  t" by (simp only: add_le_cancel_left) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

661 
then show "t \<le> y" by (simp add: algebra_simps) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

662 
qed 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

663 

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

664 
end 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

665 

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

666 
lemma le_divide_eq: 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

667 
"(a \<le> b/c) = 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

668 
(if 0 < c then a*c \<le> b 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

669 
else if c < 0 then b \<le> a*c 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

670 
else a \<le> (0::'a::{linordered_field,division_by_zero}))" 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

671 
apply (cases "c=0", simp) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

672 
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

673 
done 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

674 

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

675 
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

676 
"(0 < inverse a) = (0 < (a::'a::{linordered_field,division_by_zero}))" 
21328  677 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

678 
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

679 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

680 

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

681 
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

682 
"(inverse a < 0) = (a < (0::'a::{linordered_field,division_by_zero}))" 
21328  683 
apply (cases "a = 0", simp) 
14277
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

684 
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

685 
done 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

686 

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

687 
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

688 
"(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

689 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

690 

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

691 
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

692 
"(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

693 
by (simp add: linorder_not_less [symmetric]) 
ad66687ece6e
more field division lemmas transferred from Real to Ring_and_Field
paulson
parents:
14272
diff
changeset

694 

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

695 
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

696 
"(1 < inverse x) = (0 < x & x < (1::'a::{linordered_field,division_by_zero}))" 
23482  697 
proof cases 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

698 
assume "0 < x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

699 
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

700 
show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

701 
next 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

702 
assume notless: "~ (0 < x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

703 
have "~ (1 < inverse x)" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

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

705 
assume "1 < inverse x" 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

706 
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

707 
also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

708 
finally show False by auto 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

709 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

710 
with notless show ?thesis by simp 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

711 
qed 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

712 

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

713 
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

714 
"(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{linordered_field,division_by_zero}))" 
35216  715 
by (force simp add: order_le_less one_less_inverse_iff) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14353
diff
changeset

716 

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

717 
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

718 
"(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

719 
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

720 

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

721 
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

722 
"(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

723 
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

724 

14288  725 
lemma divide_le_eq: 
726 
"(b/c \<le> a) = 

727 
(if 0 < c then b \<le> a*c 

728 
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

729 
else 0 \<le> (a::'a::{linordered_field,division_by_zero}))" 
21328  730 
apply (cases "c=0", simp) 
14288  731 
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
732 
done 

733 

734 
lemma less_divide_eq: 

735 
"(a < b/c) = 

736 
(if 0 < c then a*c < b 

737 
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

738 
else a < (0::'a::{linordered_field,division_by_zero}))" 
21328  739 
apply (cases "c=0", simp) 
14288  740 
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
741 
done 

742 

743 
lemma divide_less_eq: 

744 
"(b/c < a) = 

745 
(if 0 < c then b < a*c 

746 
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

747 
else 0 < (a::'a::{linordered_field,division_by_zero}))" 
21328  748 
apply (cases "c=0", simp) 
14288  749 
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
750 
done 

751 

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

752 
text {*Division and Signs*} 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

753 

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

754 
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

755 
"((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

756 
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

757 

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

758 
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

759 
"(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

760 
(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

761 
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

762 

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

763 
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

764 
"((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

765 
(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

766 
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

767 

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

768 
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

769 
"(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

770 
(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

771 
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

772 

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

773 
text {* Division and the Number One *} 
14353
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*} 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

776 

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

777 
lemma zero_eq_1_divide_iff [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

778 
"((0::'a::{linordered_field,division_by_zero}) = 1/a) = (a = 0)" 
23482  779 
apply (cases "a=0", simp) 
780 
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

781 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

782 

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

783 
lemma one_divide_eq_0_iff [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

784 
"(1/a = (0::'a::{linordered_field,division_by_zero})) = (a = 0)" 
23482  785 
apply (cases "a=0", simp) 
786 
apply (insert zero_neq_one [THEN not_sym]) 

787 
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

788 
done 
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

789 

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

790 
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*} 
18623  791 
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified] 
792 
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified] 

793 
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified] 

794 
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified] 

17085  795 

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

796 
declare zero_less_divide_1_iff [simp,no_atp] 
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

797 
declare divide_less_0_1_iff [simp,no_atp] 
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

798 
declare zero_le_divide_1_iff [simp,no_atp] 
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35579
diff
changeset

799 
declare divide_le_0_1_iff [simp,no_atp] 
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14348
diff
changeset

800 

14293  801 
lemma divide_right_mono: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

802 
"[a \<le> b; 0 \<le> c] ==> a/c \<le> b/(c::'a::{linordered_field,division_by_zero})" 
23482  803 
by (force simp add: divide_strict_right_mono order_le_less) 
14293  804 

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

805 
lemma divide_right_mono_neg: "(a::'a::{linordered_field,division_by_zero}) <= b 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

806 
==> c <= 0 ==> b / c <= a / c" 
23482  807 
apply (drule divide_right_mono [of _ _ " c"]) 
808 
apply auto 

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

809 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

810 

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

811 
lemma divide_left_mono_neg: "(a::'a::{linordered_field,division_by_zero}) <= b 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

812 
==> 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

813 
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

814 
apply (auto simp add: mult_commute) 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

815 
done 
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

816 

23482  817 

14293  818 

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

819 
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

820 

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

821 
lemma le_divide_eq_1 [no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

822 
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

823 
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

824 
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

825 

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

826 
lemma divide_le_eq_1 [no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

827 
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

828 
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

829 
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

830 

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

831 
lemma less_divide_eq_1 [no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

832 
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

833 
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

834 
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

835 

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

836 
lemma divide_less_eq_1 [no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

837 
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

838 
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

839 
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

840 

23389  841 

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

842 
text {*Conditional Simplification Rules: No Case Splits*} 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

843 

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

844 
lemma le_divide_eq_1_pos [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

845 
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

846 
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

847 
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

848 

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

849 
lemma le_divide_eq_1_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

850 
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

851 
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

852 
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

853 

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

854 
lemma divide_le_eq_1_pos [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

855 
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

856 
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

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

858 

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

859 
lemma divide_le_eq_1_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

860 
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

861 
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

862 
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

863 

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

864 
lemma less_divide_eq_1_pos [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

865 
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

866 
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

867 
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

868 

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

869 
lemma less_divide_eq_1_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

870 
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

871 
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

872 
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

873 

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

874 
lemma divide_less_eq_1_pos [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

875 
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

876 
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

877 
by (auto simp add: divide_less_eq) 
bb99c2e705ca
tidied, and added missing thm divide_less_eq_1_neg
paulson
parents:
18623
diff
changeset

878 

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

879 
lemma divide_less_eq_1_neg [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

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

881 
shows "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

882 
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

883 

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

884 
lemma eq_divide_eq_1 [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

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

886 
shows "(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

887 
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

888 

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

889 
lemma divide_eq_eq_1 [simp,no_atp]: 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34146
diff
changeset

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

891 
shows "(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

892 
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

893 

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

894 
lemma abs_inverse [simp]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

895 
"\<bar>inverse (a::'a::{linordered_field,division_by_zero})\<bar> = 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

896 
inverse \<bar>a\<bar>" 
21328  897 
apply (cases "a=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

898 
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

899 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

900 

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

901 
lemma abs_divide [simp]: 
36301
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

902 
"\<bar>a / (b::'a::{linordered_field,division_by_zero})\<bar> = \<bar>a\<bar> / \<bar>b\<bar>" 
21328  903 
apply (cases "b=0", simp) 
14294
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

904 
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

905 
done 
f4d806fd72ce
absolute value theorems moved to HOL/Ring_and_Field
paulson
parents:
14293
diff
changeset

906 

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

907 
lemma abs_div_pos: "(0::'a::{linordered_field,division_by_zero}) < y ==> 
72f4d079ebf8
more localization; factored out lemmas for division_ring
haftmann
parents:
35828
diff
changeset

908 
\<bar>x\<bar> / y = \<bar>x / y\<bar>" 
25304
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

909 
apply (subst abs_divide) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

910 
apply (simp add: order_less_imp_le) 
7491c00f0915
removed subclass edge ordered_ring < lordered_ring
haftmann
parents:
25267
diff
changeset

911 
done 
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16568
diff
changeset

912 

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

913 
lemma field_le_mult_one_interval: 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

914 
fixes x :: "'a\<Colon>{linordered_field,division_by_zero}" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

915 
assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

916 
shows "x \<le> y" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

917 
proof (cases "0 < x") 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

918 
assume "0 < x" 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

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

920 
using dense_le_bounded[of 0 1 "y/x"] * 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

921 
unfolding le_divide_eq if_P[OF `0 < x`] by simp 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

922 
next 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

923 
assume "\<not>0 < x" hence "x \<le> 0" by simp 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

924 
obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

925 
hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

926 
also note *[OF s] 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

927 
finally show ?thesis . 
cc9a5a0ab5ea
Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents:
35216
diff
changeset

928 
qed 
35090
88cc65ae046e
moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents:
35084
diff
changeset

929 

33364  930 
code_modulename SML 
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
35043
diff
changeset

931 
Fields Arith 
33364  932 

933 
code_modulename OCaml 

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

934 
Fields Arith 
33364  935 

936 
code_modulename Haskell 

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

937 
Fields Arith 
33364  938 

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

939 
end 