author  haftmann 
Wed, 13 Feb 2008 09:35:31 +0100  
changeset 26062  16f334d7156a 
parent 25947  1f2f4d941e9e 
child 26072  f65a7fa2da6c 
permissions  rwrr 
3366  1 
(* Title: HOL/Divides.thy 
2 
ID: $Id$ 

3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

6865
5577ffe4c2f1
now div and mod are overloaded; dvd is polymorphic
paulson
parents:
3366
diff
changeset

4 
Copyright 1999 University of Cambridge 
18154  5 
*) 
3366  6 

18154  7 
header {* The division operators div, mod and the divides relation "dvd" *} 
3366  8 

15131  9 
theory Divides 
24268  10 
imports Power 
22993  11 
uses "~~/src/Provers/Arith/cancel_div_mod.ML" 
15131  12 
begin 
3366  13 

25942  14 
subsection {* Syntactic division operations *} 
15 

24993  16 
class div = times + 
25062  17 
fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70) 
18 
fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70) 

25942  19 
begin 
21408  20 

25942  21 
definition 
22 
dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) 

23 
where 

24 
[code func del]: "m dvd n \<longleftrightarrow> (\<exists>k. n = m * k)" 

25 

26 
end 

27 

28 
subsection {* Abstract divisibility in commutative semirings. *} 

29 

30 
class semiring_div = comm_semiring_1_cancel + div + 

31 
assumes mod_div_equality: "a div b * b + a mod b = a" 

32 
and div_by_0: "a div 0 = 0" 

33 
and mult_div: "b \<noteq> 0 \<Longrightarrow> a * b div b = a" 

34 
begin 

35 

36 
lemma div_by_1: "a div 1 = a" 

26062  37 
using mult_div [of 1 a] zero_neq_one by simp 
25942  38 

39 
lemma mod_by_1: "a mod 1 = 0" 

40 
proof  

41 
from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp 

42 
then have "a + a mod 1 = a + 0" by simp 

43 
then show ?thesis by (rule add_left_imp_eq) 

44 
qed 

45 

46 
lemma mod_by_0: "a mod 0 = a" 

47 
using mod_div_equality [of a zero] by simp 

48 

49 
lemma mult_mod: "a * b mod b = 0" 

50 
proof (cases "b = 0") 

51 
case True then show ?thesis by (simp add: mod_by_0) 

52 
next 

53 
case False with mult_div have abb: "a * b div b = a" . 

54 
from mod_div_equality have "a * b div b * b + a * b mod b = a * b" . 

55 
with abb have "a * b + a * b mod b = a * b + 0" by simp 

56 
then show ?thesis by (rule add_left_imp_eq) 

57 
qed 

58 

59 
lemma mod_self: "a mod a = 0" 

60 
using mult_mod [of one] by simp 

61 

62 
lemma div_self: "a \<noteq> 0 \<Longrightarrow> a div a = 1" 

63 
using mult_div [of _ one] by simp 

64 

65 
lemma div_0: "0 div a = 0" 

66 
proof (cases "a = 0") 

67 
case True then show ?thesis by (simp add: div_by_0) 

68 
next 

69 
case False with mult_div have "0 * a div a = 0" . 

70 
then show ?thesis by simp 

71 
qed 

72 

73 
lemma mod_0: "0 mod a = 0" 

74 
using mod_div_equality [of zero a] div_0 by simp 

75 

26062  76 
lemma mod_div_equality2: "b * (a div b) + a mod b = a" 
77 
unfolding mult_commute [of b] 

78 
by (rule mod_div_equality) 

79 

80 
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c" 

81 
by (simp add: mod_div_equality) 

82 

83 
lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c" 

84 
by (simp add: mod_div_equality2) 

85 

86 
lemma dvdI [intro?]: "a = b * c \<Longrightarrow> b dvd a" 

87 
unfolding dvd_def .. 

88 

89 
lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>c. a = b * c \<Longrightarrow> P) \<Longrightarrow> P" 

90 
unfolding dvd_def by blast 

91 

25942  92 
lemma dvd_def_mod [code func]: "a dvd b \<longleftrightarrow> b mod a = 0" 
93 
proof 

94 
assume "b mod a = 0" 

95 
with mod_div_equality [of b a] have "b div a * a = b" by simp 

96 
then have "b = a * (b div a)" unfolding mult_commute .. 

97 
then have "\<exists>c. b = a * c" .. 

98 
then show "a dvd b" unfolding dvd_def . 

99 
next 

100 
assume "a dvd b" 

101 
then have "\<exists>c. b = a * c" unfolding dvd_def . 

102 
then obtain c where "b = a * c" .. 

103 
then have "b mod a = a * c mod a" by simp 

104 
then have "b mod a = c * a mod a" by (simp add: mult_commute) 

105 
then show "b mod a = 0" by (simp add: mult_mod) 

106 
qed 

107 

108 
lemma dvd_refl: "a dvd a" 

109 
unfolding dvd_def_mod mod_self .. 

110 

111 
lemma dvd_trans: 

112 
assumes "a dvd b" and "b dvd c" 

113 
shows "a dvd c" 

114 
proof  

115 
from assms obtain v where "b = a * v" unfolding dvd_def by auto 

116 
moreover from assms obtain w where "c = b * w" unfolding dvd_def by auto 

117 
ultimately have "c = a * (v * w)" by (simp add: mult_assoc) 

118 
then show ?thesis unfolding dvd_def .. 

119 
qed 

120 

26062  121 
lemma zero_dvd_iff [noatp]: "0 dvd a \<longleftrightarrow> a = 0" 
25942  122 
unfolding dvd_def by simp 
123 

124 
lemma dvd_0: "a dvd 0" 

125 
unfolding dvd_def proof 

126 
show "0 = a * 0" by simp 

127 
qed 

128 

26062  129 
lemma one_dvd: "1 dvd a" 
130 
unfolding dvd_def by simp 

131 

132 
lemma dvd_mult: "a dvd c \<Longrightarrow> a dvd (b * c)" 

133 
unfolding dvd_def by (blast intro: mult_left_commute) 

134 

135 
lemma dvd_mult2: "a dvd b \<Longrightarrow> a dvd (b * c)" 

136 
apply (subst mult_commute) 

137 
apply (erule dvd_mult) 

138 
done 

139 

140 
lemma dvd_triv_right: "a dvd b * a" 

141 
by (rule dvd_mult) (rule dvd_refl) 

142 

143 
lemma dvd_triv_left: "a dvd a * b" 

144 
by (rule dvd_mult2) (rule dvd_refl) 

145 

146 
lemma mult_dvd_mono: "a dvd c \<Longrightarrow> b dvd d \<Longrightarrow> a * b dvd c * d" 

147 
apply (unfold dvd_def, clarify) 

148 
apply (rule_tac x = "k * ka" in exI) 

149 
apply (simp add: mult_ac) 

150 
done 

151 

152 
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c" 

153 
by (simp add: dvd_def mult_assoc, blast) 

154 

155 
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c" 

156 
unfolding mult_ac [of a] by (rule dvd_mult_left) 

157 

25942  158 
end 
159 

160 

161 
subsection {* Division on the natural numbers *} 

162 

163 
instantiation nat :: semiring_div 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

164 
begin 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

165 

c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

166 
definition 
22993  167 
div_def: "m div n == wfrec (pred_nat^+) 
168 
(%f j. if j<n  n=0 then 0 else Suc (f (jn))) m" 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

169 

25942  170 
lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+) 
171 
(%f j. if j<n  n=0 then 0 else Suc (f (jn)))" 

172 
by (simp add: div_def) 

173 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

174 
definition 
22261
9e185f78e7d4
Adapted to changes in Transitive_Closure theory.
berghofe
parents:
21911
diff
changeset

175 
mod_def: "m mod n == wfrec (pred_nat^+) 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

176 
(%f j. if j<n  n=0 then j else f (jn)) m" 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset

177 

22718  178 
lemma mod_eq: "(%m. m mod n) = 
22261
9e185f78e7d4
Adapted to changes in Transitive_Closure theory.
berghofe
parents:
21911
diff
changeset

179 
wfrec (pred_nat^+) (%f j. if j<n  n=0 then j else f (jn))" 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

180 
by (simp add: mod_def) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

181 

25942  182 
lemmas wf_less_trans = def_wfrec [THEN trans, 
183 
OF eq_reflection wf_pred_nat [THEN wf_trancl], standard] 

184 

185 
lemma div_less [simp]: "m < n \<Longrightarrow> m div n = (0\<Colon>nat)" 

186 
by (rule div_eq [THEN wf_less_trans]) simp 

187 

188 
lemma le_div_geq: "0 < n \<Longrightarrow> n \<le> m \<Longrightarrow> m div n = Suc ((m  n) div n)" 

189 
by (rule div_eq [THEN wf_less_trans]) (simp add: cut_apply less_eq) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

190 

25942  191 
lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a\<Colon>nat)" 
192 
by (rule mod_eq [THEN wf_less_trans]) simp 

193 

194 
lemma mod_less [simp]: "m < n \<Longrightarrow> m mod n = (m\<Colon>nat)" 

195 
by (rule mod_eq [THEN wf_less_trans]) simp 

196 

197 
lemma le_mod_geq: "(n\<Colon>nat) \<le> m \<Longrightarrow> m mod n = (m  n) mod n" 

198 
by (cases "n = 0", simp, rule mod_eq [THEN wf_less_trans]) 

199 
(simp add: cut_apply less_eq) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

200 

25942  201 
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m  n) mod n)" 
202 
by (simp add: le_mod_geq) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

203 

25942  204 
instance proof 
205 
fix n m :: nat 

206 
show "(m div n) * n + m mod n = m" 

207 
apply (cases "n = 0", simp) 

25947  208 
apply (induct m rule: less_induct) 
25942  209 
apply (subst mod_if) 
210 
apply (simp add: add_assoc add_diff_inverse le_div_geq) 

211 
done 

212 
next 

213 
fix n :: nat 

214 
show "n div 0 = 0" 

25947  215 
by (rule div_eq [THEN wf_less_trans]) simp 
25942  216 
next 
217 
fix n m :: nat 

218 
assume "n \<noteq> 0" 

219 
then show "m * n div n = m" 

220 
by (induct m) (simp_all add: le_div_geq) 

221 
qed 

222 

223 
end 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

224 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

225 

25942  226 
subsubsection{*Simproc for Cancelling Div and Mod*} 
227 

228 
lemmas mod_div_equality = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard] 

26062  229 
lemmas mod_div_equality2 = mod_div_equality2 [of "n\<Colon>nat" m, standard] 
230 
lemmas div_mod_equality = div_mod_equality [of "m\<Colon>nat" n k, standard] 

231 
lemmas div_mod_equality2 = div_mod_equality2 [of "m\<Colon>nat" n k, standard] 

25942  232 

233 
ML {* 

234 
structure CancelDivModData = 

235 
struct 

236 

237 
val div_name = @{const_name Divides.div}; 

238 
val mod_name = @{const_name Divides.mod}; 

239 
val mk_binop = HOLogic.mk_binop; 

240 
val mk_sum = NatArithUtils.mk_sum; 

241 
val dest_sum = NatArithUtils.dest_sum; 

242 

243 
(*logic*) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

244 

25942  245 
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}] 
246 

247 
val trans = trans 

248 

249 
val prove_eq_sums = 

250 
let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac} 

251 
in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end; 

252 

253 
end; 

254 

255 
structure CancelDivMod = CancelDivModFun(CancelDivModData); 

256 

257 
val cancel_div_mod_proc = NatArithUtils.prep_simproc 

258 
("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc); 

259 

260 
Addsimprocs[cancel_div_mod_proc]; 

261 
*} 

262 

263 

264 
subsubsection {* Remainder *} 

265 

266 
lemmas DIVISION_BY_ZERO_MOD [simp] = mod_by_0 [of "a\<Colon>nat", standard] 

267 

268 
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" 

269 
by (induct m) (simp_all add: le_div_geq) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

270 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

271 
lemma mod_geq: "~ m < (n::nat) ==> m mod n = (mn) mod n" 
25942  272 
by (simp add: le_mod_geq linorder_not_less) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

273 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

274 
lemma mod_1 [simp]: "m mod Suc 0 = 0" 
22718  275 
by (induct m) (simp_all add: mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

276 

25942  277 
lemmas mod_self [simp] = semiring_div_class.mod_self [of "n\<Colon>nat", standard] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

278 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

279 
lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)" 
22718  280 
apply (subgoal_tac "(n + m) mod n = (n+mn) mod n") 
281 
apply (simp add: add_commute) 

25942  282 
apply (subst le_mod_geq [symmetric], simp_all) 
22718  283 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

284 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

285 
lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)" 
22718  286 
by (simp add: add_commute mod_add_self2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

287 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

288 
lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)" 
22718  289 
by (induct k) (simp_all add: add_left_commute [of _ n]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

290 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

291 
lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)" 
22718  292 
by (simp add: mult_commute mod_mult_self1) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

293 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

294 
lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)" 
22718  295 
apply (cases "n = 0", simp) 
296 
apply (cases "k = 0", simp) 

297 
apply (induct m rule: nat_less_induct) 

298 
apply (subst mod_if, simp) 

299 
apply (simp add: mod_geq diff_mult_distrib) 

300 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

301 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

302 
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)" 
22718  303 
by (simp add: mult_commute [of k] mod_mult_distrib) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

304 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

305 
lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)" 
22718  306 
apply (cases "n = 0", simp) 
307 
apply (induct m, simp) 

308 
apply (rename_tac k) 

309 
apply (cut_tac m = "k * n" and n = n in mod_add_self2) 

310 
apply (simp add: add_commute) 

311 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

312 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

313 
lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)" 
22718  314 
by (simp add: mult_commute mod_mult_self_is_0) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

315 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

316 

25942  317 
subsubsection{*Quotient*} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

318 

25942  319 
lemmas DIVISION_BY_ZERO_DIV [simp] = div_by_0 [of "a\<Colon>nat", standard] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

320 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

321 
lemma div_geq: "[ 0<n; ~m<n ] ==> m div n = Suc((mn) div n)" 
25942  322 
by (simp add: le_div_geq linorder_not_less) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

323 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

324 
lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((mn) div n))" 
22718  325 
by (simp add: div_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

326 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

327 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

328 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

329 
(* a simple rearrangement of mod_div_equality: *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

330 
lemma mult_div_cancel: "(n::nat) * (m div n) = m  (m mod n)" 
22718  331 
by (cut_tac m = m and n = n in mod_div_equality2, arith) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

332 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

333 
lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)" 
22718  334 
apply (induct m rule: nat_less_induct) 
335 
apply (rename_tac m) 

336 
apply (case_tac "m<n", simp) 

337 
txt{*case @{term "n \<le> m"}*} 

338 
apply (simp add: mod_geq) 

339 
done 

15439  340 

341 
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" 

22718  342 
apply (drule mod_less_divisor [where m = m]) 
343 
apply simp 

344 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

345 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

346 
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" 
22718  347 
by (simp add: mult_commute div_mult_self_is_m) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

348 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

349 
(*mod_mult_distrib2 above is the counterpart for remainder*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

350 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

351 

25942  352 
subsubsection {* Proving advancedfacts about Quotient and Remainder *} 
353 

354 
definition 

355 
quorem :: "(nat*nat) * (nat*nat) => bool" where 

356 
(*This definition helps prove the harder properties of div and mod. 

357 
It is copied from IntDiv.thy; should it be overloaded?*) 

358 
"quorem = (%((a,b), (q,r)). 

359 
a = b*q + r & 

360 
(if 0<b then 0\<le>r & r<b else b<r & r \<le>0))" 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

361 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

362 
lemma unique_quotient_lemma: 
22718  363 
"[ b*q' + r' \<le> b*q + r; x < b; r < b ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

364 
==> q' \<le> (q::nat)" 
22718  365 
apply (rule leI) 
366 
apply (subst less_iff_Suc_add) 

367 
apply (auto simp add: add_mult_distrib2) 

368 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

369 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

370 
lemma unique_quotient: 
22718  371 
"[ quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); 0 < b ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

372 
==> q = q'" 
22718  373 
apply (simp add: split_ifs quorem_def) 
374 
apply (blast intro: order_antisym 

375 
dest: order_eq_refl [THEN unique_quotient_lemma] sym) 

376 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

377 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

378 
lemma unique_remainder: 
22718  379 
"[ quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); 0 < b ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

380 
==> r = r'" 
22718  381 
apply (subgoal_tac "q = q'") 
382 
prefer 2 apply (blast intro: unique_quotient) 

383 
apply (simp add: quorem_def) 

384 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

385 

25162  386 
lemma quorem_div_mod: "b > 0 ==> quorem ((a, b), (a div b, a mod b))" 
387 
unfolding quorem_def by simp 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

388 

25162  389 
lemma quorem_div: "[ quorem((a,b),(q,r)); b > 0 ] ==> a div b = q" 
390 
by (simp add: quorem_div_mod [THEN unique_quotient]) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

391 

25162  392 
lemma quorem_mod: "[ quorem((a,b),(q,r)); b > 0 ] ==> a mod b = r" 
393 
by (simp add: quorem_div_mod [THEN unique_remainder]) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

394 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

395 
(** A dividend of zero **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

396 

25942  397 
lemmas div_0 [simp] = semiring_div_class.div_0 [of "n\<Colon>nat", standard] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

398 

25942  399 
lemmas mod_0 [simp] = semiring_div_class.mod_0 [of "n\<Colon>nat", standard] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

400 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

401 
(** proving (a*b) div c = a * (b div c) + a * (b mod c) **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

402 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

403 
lemma quorem_mult1_eq: 
25162  404 
"[ quorem((b,c),(q,r)); c > 0 ] 
405 
==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))" 

406 
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

407 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

408 
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

409 
apply (cases "c = 0", simp) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

410 
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div]) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

411 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

412 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

413 
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

414 
apply (cases "c = 0", simp) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

415 
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod]) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

416 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

417 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

418 
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c" 
22718  419 
apply (rule trans) 
420 
apply (rule_tac s = "b*a mod c" in trans) 

421 
apply (rule_tac [2] mod_mult1_eq) 

422 
apply (simp_all add: mult_commute) 

423 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

424 

25162  425 
lemma mod_mult_distrib_mod: 
426 
"(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c" 

427 
apply (rule mod_mult1_eq' [THEN trans]) 

428 
apply (rule mod_mult1_eq) 

429 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

430 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

431 
(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

432 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

433 
lemma quorem_add1_eq: 
25162  434 
"[ quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); c > 0 ] 
435 
==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))" 

436 
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

437 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

438 
(*NOT suitable for rewriting: the RHS has an instance of the LHS*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

439 
lemma div_add1_eq: 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

440 
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

441 
apply (cases "c = 0", simp) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

442 
apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

443 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

444 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

445 
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

446 
apply (cases "c = 0", simp) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

447 
apply (blast intro: quorem_div_mod quorem_add1_eq [THEN quorem_mod]) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset

448 
done 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

449 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

450 

25942  451 
subsubsection {* Proving @{prop "a div (b*c) = (a div b) div c"} *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

452 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

453 
(** first, a lemma to bound the remainder **) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

454 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

455 
lemma mod_lemma: "[ (0::nat) < c; r < b ] ==> b * (q mod c) + r < b * c" 
22718  456 
apply (cut_tac m = q and n = c in mod_less_divisor) 
457 
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) 

458 
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) 

459 
apply (simp add: add_mult_distrib2) 

460 
done 

10559
d3fd54fc659b
many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents:
10214
diff
changeset

461 

22718  462 
lemma quorem_mult2_eq: "[ quorem ((a,b), (q,r)); 0 < b; 0 < c ] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

463 
==> quorem ((a, b*c), (q div c, b*(q mod c) + r))" 
22718  464 
by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

465 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

466 
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" 
22718  467 
apply (cases "b = 0", simp) 
468 
apply (cases "c = 0", simp) 

469 
apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div]) 

470 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

471 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

472 
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" 
22718  473 
apply (cases "b = 0", simp) 
474 
apply (cases "c = 0", simp) 

475 
apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod]) 

476 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

477 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

478 

25942  479 
subsubsection{*Cancellation of Common Factors in Division*} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

480 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

481 
lemma div_mult_mult_lemma: 
22718  482 
"[ (0::nat) < b; 0 < c ] ==> (c*a) div (c*b) = a div b" 
483 
by (auto simp add: div_mult2_eq) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

484 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

485 
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b" 
22718  486 
apply (cases "b = 0") 
487 
apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma) 

488 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

489 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

490 
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b" 
22718  491 
apply (drule div_mult_mult1) 
492 
apply (auto simp add: mult_commute) 

493 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

494 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

495 

25942  496 
subsubsection{*Further Facts about Quotient and Remainder*} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

497 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

498 
lemma div_1 [simp]: "m div Suc 0 = m" 
22718  499 
by (induct m) (simp_all add: div_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

500 

25942  501 
lemmas div_self [simp] = semiring_div_class.div_self [of "n\<Colon>nat", standard] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

502 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

503 
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)" 
22718  504 
apply (subgoal_tac "(n + m) div n = Suc ((n+mn) div n) ") 
505 
apply (simp add: add_commute) 

506 
apply (subst div_geq [symmetric], simp_all) 

507 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

508 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

509 
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)" 
22718  510 
by (simp add: add_commute div_add_self2) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

511 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

512 
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n" 
22718  513 
apply (subst div_add1_eq) 
514 
apply (subst div_mult1_eq, simp) 

515 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

516 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

517 
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)" 
22718  518 
by (simp add: mult_commute div_mult_self1) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

519 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

520 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

521 
(* Monotonicity of div in first argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

522 
lemma div_le_mono [rule_format (no_asm)]: 
22718  523 
"\<forall>m::nat. m \<le> n > (m div k) \<le> (n div k)" 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

524 
apply (case_tac "k=0", simp) 
15251  525 
apply (induct "n" rule: nat_less_induct, clarify) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

526 
apply (case_tac "n<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

527 
(* 1 case n<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

528 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

529 
(* 2 case n >= k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

530 
apply (case_tac "m<k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

531 
(* 2.1 case m<k *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

532 
apply simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

533 
(* 2.2 case m>=k *) 
15439  534 
apply (simp add: div_geq diff_le_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

535 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

536 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

537 
(* Antimonotonicity of div in second argument *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

538 
lemma div_le_mono2: "!!m::nat. [ 0<m; m\<le>n ] ==> (k div n) \<le> (k div m)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

539 
apply (subgoal_tac "0<n") 
22718  540 
prefer 2 apply simp 
15251  541 
apply (induct_tac k rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

542 
apply (rename_tac "k") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

543 
apply (case_tac "k<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

544 
apply (subgoal_tac "~ (k<m) ") 
22718  545 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

546 
apply (simp add: div_geq) 
15251  547 
apply (subgoal_tac "(kn) div n \<le> (km) div n") 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

548 
prefer 2 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

549 
apply (blast intro: div_le_mono diff_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

550 
apply (rule le_trans, simp) 
15439  551 
apply (simp) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

552 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

553 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

554 
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

555 
apply (case_tac "n=0", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

556 
apply (subgoal_tac "m div n \<le> m div 1", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

557 
apply (rule div_le_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

558 
apply (simp_all (no_asm_simp)) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

559 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

560 

22718  561 
(* Similar for "less than" *) 
17085  562 
lemma div_less_dividend [rule_format]: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

563 
"!!n::nat. 1<n ==> 0 < m > m div n < m" 
15251  564 
apply (induct_tac m rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

565 
apply (rename_tac "m") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

566 
apply (case_tac "m<n", simp) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

567 
apply (subgoal_tac "0<n") 
22718  568 
prefer 2 apply simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

569 
apply (simp add: div_geq) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

570 
apply (case_tac "n<m") 
15251  571 
apply (subgoal_tac "(mn) div n < (mn) ") 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

572 
apply (rule impI less_trans_Suc)+ 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

573 
apply assumption 
15439  574 
apply (simp_all) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

575 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

576 

17085  577 
declare div_less_dividend [simp] 
578 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

579 
text{*A fact for the mutilated chess board*} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

580 
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

581 
apply (case_tac "n=0", simp) 
15251  582 
apply (induct "m" rule: nat_less_induct) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

583 
apply (case_tac "Suc (na) <n") 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

584 
(* case Suc(na) < n *) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

585 
apply (frule lessI [THEN less_trans], simp add: less_not_refl3) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

586 
(* case n \<le> Suc(na) *) 
16796  587 
apply (simp add: linorder_not_less le_Suc_eq mod_geq) 
15439  588 
apply (auto simp add: Suc_diff_le le_mod_geq) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

589 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

590 

14437  591 
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)" 
22718  592 
by (cases "n = 0") auto 
14437  593 

594 
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)" 

22718  595 
by (cases "n = 0") auto 
14437  596 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

597 

25942  598 
subsubsection{*The Divides Relation*} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

599 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

600 
lemma dvdI [intro?]: "n = m * k ==> m dvd n" 
22718  601 
unfolding dvd_def by blast 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

602 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

603 
lemma dvdE [elim?]: "!!P. [m dvd n; !!k. n = m*k ==> P] ==> P" 
22718  604 
unfolding dvd_def by blast 
13152  605 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

606 
lemma dvd_0_right [iff]: "m dvd (0::nat)" 
22718  607 
unfolding dvd_def by (blast intro: mult_0_right [symmetric]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

608 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

609 
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)" 
22718  610 
by (force simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

611 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

612 
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)" 
22718  613 
by (blast intro: dvd_0_left) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

614 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24268
diff
changeset

615 
declare dvd_0_left_iff [noatp] 
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24268
diff
changeset

616 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

617 
lemma dvd_1_left [iff]: "Suc 0 dvd k" 
22718  618 
unfolding dvd_def by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

619 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

620 
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)" 
22718  621 
by (simp add: dvd_def) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

622 

25942  623 
lemmas dvd_refl [simp] = semiring_div_class.dvd_refl [of "m\<Colon>nat", standard] 
624 
lemmas dvd_trans [trans] = semiring_div_class.dvd_trans [of "m\<Colon>nat" n p, standard] 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

625 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

626 
lemma dvd_anti_sym: "[ m dvd n; n dvd m ] ==> m = (n::nat)" 
22718  627 
unfolding dvd_def 
628 
by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

629 

23684
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

630 
text {* @{term "op dvd"} is a partial order *} 
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

631 

25942  632 
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"] 
23684
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

633 
by unfold_locales (auto intro: dvd_trans dvd_anti_sym) 
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

634 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

635 
lemma dvd_add: "[ k dvd m; k dvd n ] ==> k dvd (m+n :: nat)" 
22718  636 
unfolding dvd_def 
637 
by (blast intro: add_mult_distrib2 [symmetric]) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

638 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

639 
lemma dvd_diff: "[ k dvd m; k dvd n ] ==> k dvd (mn :: nat)" 
22718  640 
unfolding dvd_def 
641 
by (blast intro: diff_mult_distrib2 [symmetric]) 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

642 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

643 
lemma dvd_diffD: "[ k dvd mn; k dvd n; n\<le>m ] ==> k dvd (m::nat)" 
22718  644 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 
645 
apply (blast intro: dvd_add) 

646 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

647 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

648 
lemma dvd_diffD1: "[ k dvd mn; k dvd m; n\<le>m ] ==> k dvd (n::nat)" 
22718  649 
by (drule_tac m = m in dvd_diff, auto) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

650 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

651 
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)" 
22718  652 
unfolding dvd_def by (blast intro: mult_left_commute) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

653 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

654 
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)" 
22718  655 
apply (subst mult_commute) 
656 
apply (erule dvd_mult) 

657 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

658 

17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

659 
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)" 
22718  660 
by (rule dvd_refl [THEN dvd_mult]) 
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

661 

fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

662 
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)" 
22718  663 
by (rule dvd_refl [THEN dvd_mult2]) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

664 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

665 
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))" 
22718  666 
apply (rule iffI) 
667 
apply (erule_tac [2] dvd_add) 

668 
apply (rule_tac [2] dvd_refl) 

669 
apply (subgoal_tac "n = (n+k) k") 

670 
prefer 2 apply simp 

671 
apply (erule ssubst) 

672 
apply (erule dvd_diff) 

673 
apply (rule dvd_refl) 

674 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

675 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

676 
lemma dvd_mod: "!!n::nat. [ f dvd m; f dvd n ] ==> f dvd m mod n" 
22718  677 
unfolding dvd_def 
678 
apply (case_tac "n = 0", auto) 

679 
apply (blast intro: mod_mult_distrib2 [symmetric]) 

680 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

681 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

682 
lemma dvd_mod_imp_dvd: "[ (k::nat) dvd m mod n; k dvd n ] ==> k dvd m" 
22718  683 
apply (subgoal_tac "k dvd (m div n) *n + m mod n") 
684 
apply (simp add: mod_div_equality) 

685 
apply (simp only: dvd_add dvd_mult) 

686 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

687 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

688 
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)" 
22718  689 
by (blast intro: dvd_mod_imp_dvd dvd_mod) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

690 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

691 
lemma dvd_mult_cancel: "!!k::nat. [ k*m dvd k*n; 0<k ] ==> m dvd n" 
22718  692 
unfolding dvd_def 
693 
apply (erule exE) 

694 
apply (simp add: mult_ac) 

695 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

696 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

697 
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))" 
22718  698 
apply auto 
699 
apply (subgoal_tac "m*n dvd m*1") 

700 
apply (drule dvd_mult_cancel, auto) 

701 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

702 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

703 
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))" 
22718  704 
apply (subst mult_commute) 
705 
apply (erule dvd_mult_cancel1) 

706 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

707 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

708 
lemma mult_dvd_mono: "[ i dvd m; j dvd n] ==> i*j dvd (m*n :: nat)" 
22718  709 
apply (unfold dvd_def, clarify) 
710 
apply (rule_tac x = "k*ka" in exI) 

711 
apply (simp add: mult_ac) 

712 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

713 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

714 
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k" 
22718  715 
by (simp add: dvd_def mult_assoc, blast) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

716 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

717 
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k" 
22718  718 
apply (unfold dvd_def, clarify) 
719 
apply (rule_tac x = "i*k" in exI) 

720 
apply (simp add: mult_ac) 

721 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

722 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

723 
lemma dvd_imp_le: "[ k dvd n; 0 < n ] ==> k \<le> (n::nat)" 
22718  724 
apply (unfold dvd_def, clarify) 
725 
apply (simp_all (no_asm_use) add: zero_less_mult_iff) 

726 
apply (erule conjE) 

727 
apply (rule le_trans) 

728 
apply (rule_tac [2] le_refl [THEN mult_le_mono]) 

729 
apply (erule_tac [2] Suc_leI, simp) 

730 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

731 

25942  732 
lemmas dvd_eq_mod_eq_0 = dvd_def_mod [of "k\<Colon>nat" n, standard] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

733 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

734 
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)" 
22718  735 
apply (subgoal_tac "m mod n = 0") 
736 
apply (simp add: mult_div_cancel) 

737 
apply (simp only: dvd_eq_mod_eq_0) 

738 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

739 

21408  740 
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n" 
22718  741 
apply (unfold dvd_def) 
742 
apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst]) 

743 
apply (simp add: power_add) 

744 
done 

21408  745 

25162  746 
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat)  n=0)" 
22718  747 
by (induct n) auto 
21408  748 

749 
lemma power_le_dvd [rule_format]: "k^j dvd n > i\<le>j > k^i dvd (n::nat)" 

22718  750 
apply (induct j) 
751 
apply (simp_all add: le_Suc_eq) 

752 
apply (blast dest!: dvd_mult_right) 

753 
done 

21408  754 

755 
lemma power_dvd_imp_le: "[i^m dvd i^n; (1::nat) < i] ==> m \<le> n" 

22718  756 
apply (rule power_le_imp_le_exp, assumption) 
757 
apply (erule dvd_imp_le, simp) 

758 
done 

21408  759 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

760 
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" 
22718  761 
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def) 
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset

762 

22718  763 
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1] 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

764 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

765 
(*Loses information, namely we also have r<d provided d is nonzero*) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

766 
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d" 
22718  767 
apply (cut_tac m = m in mod_div_equality) 
768 
apply (simp only: add_ac) 

769 
apply (blast intro: sym) 

770 
done 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

771 

14131  772 

13152  773 
lemma split_div: 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

774 
"P(n div k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

775 
((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

776 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

777 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

778 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

779 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

780 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

781 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

782 
with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

783 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

784 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

785 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

786 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

787 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

788 
assume n: "n = k*i + j" and j: "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

789 
show "P i" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

790 
proof (cases) 
22718  791 
assume "i = 0" 
792 
with n j P show "P i" by simp 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

793 
next 
22718  794 
assume "i \<noteq> 0" 
795 
with not0 n j P show "P i" by(simp add:add_ac) 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

796 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

797 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

798 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

799 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

800 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

801 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

802 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

803 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

804 
with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

805 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

806 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

807 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

808 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  809 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

810 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

811 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

812 

13882  813 
lemma split_div_lemma: 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

814 
"0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))" 
25162  815 
apply (rule iffI) 
816 
apply (rule_tac a=m and r = "m  n * q" and r' = "m mod n" in unique_quotient) 

817 
prefer 3; apply assumption 

818 
apply (simp_all add: quorem_def) 

819 
apply arith 

820 
apply (rule conjI) 

821 
apply (rule_tac P="%x. n * (m div n) \<le> x" in 

13882  822 
subst [OF mod_div_equality [of _ n]]) 
25162  823 
apply (simp only: add: mult_ac) 
824 
apply (rule_tac P="%x. x < n + n * (m div n)" in 

13882  825 
subst [OF mod_div_equality [of _ n]]) 
25162  826 
apply (simp only: add: mult_ac add_ac) 
827 
apply (rule add_less_mono1, simp) 

828 
done 

13882  829 

830 
theorem split_div': 

831 
"P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or> 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

832 
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" 
13882  833 
apply (case_tac "0 < n") 
834 
apply (simp only: add: split_div_lemma) 

835 
apply (simp_all add: DIVISION_BY_ZERO_DIV) 

836 
done 

837 

13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

838 
lemma split_mod: 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

839 
"P(n mod k :: nat) = 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

840 
((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

841 
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

842 
proof 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

843 
assume P: ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

844 
show ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

845 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

846 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

847 
with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

848 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

849 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

850 
thus ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

851 
proof (simp, intro allI impI) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

852 
fix i j 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

853 
assume "n = k*i + j" "j < k" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

854 
thus "P j" using not0 P by(simp add:add_ac mult_ac) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

855 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

856 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

857 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

858 
assume Q: ?Q 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

859 
show ?P 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

860 
proof (cases) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

861 
assume "k = 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

862 
with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD) 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

863 
next 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

864 
assume not0: "k \<noteq> 0" 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

865 
with Q have R: ?R by simp 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

866 
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] 
13517  867 
show ?P by simp 
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

868 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

869 
qed 
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset

870 

13882  871 
theorem mod_div_equality': "(m::nat) mod n = m  (m div n) * n" 
872 
apply (rule_tac P="%x. m mod n = x  (m div n) * n" in 

873 
subst [OF mod_div_equality [of _ n]]) 

874 
apply arith 

875 
done 

876 

22800  877 
lemma div_mod_equality': 
878 
fixes m n :: nat 

879 
shows "m div n * n = m  m mod n" 

880 
proof  

881 
have "m mod n \<le> m mod n" .. 

882 
from div_mod_equality have 

883 
"m div n * n + m mod n  m mod n = m  m mod n" by simp 

884 
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have 

885 
"m div n * n + (m mod n  m mod n) = m  m mod n" 

886 
by simp 

887 
then show ?thesis by simp 

888 
qed 

889 

890 

25942  891 
subsubsection {*An ``induction'' law for modulus arithmetic.*} 
14640  892 

893 
lemma mod_induct_0: 

894 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

895 
and base: "P i" and i: "i<p" 

896 
shows "P 0" 

897 
proof (rule ccontr) 

898 
assume contra: "\<not>(P 0)" 

899 
from i have p: "0<p" by simp 

900 
have "\<forall>k. 0<k \<longrightarrow> \<not> P (pk)" (is "\<forall>k. ?A k") 

901 
proof 

902 
fix k 

903 
show "?A k" 

904 
proof (induct k) 

905 
show "?A 0" by simp  "by contradiction" 

906 
next 

907 
fix n 

908 
assume ih: "?A n" 

909 
show "?A (Suc n)" 

910 
proof (clarsimp) 

22718  911 
assume y: "P (p  Suc n)" 
912 
have n: "Suc n < p" 

913 
proof (rule ccontr) 

914 
assume "\<not>(Suc n < p)" 

915 
hence "p  Suc n = 0" 

916 
by simp 

917 
with y contra show "False" 

918 
by simp 

919 
qed 

920 
hence n2: "Suc (p  Suc n) = pn" by arith 

921 
from p have "p  Suc n < p" by arith 

922 
with y step have z: "P ((Suc (p  Suc n)) mod p)" 

923 
by blast 

924 
show "False" 

925 
proof (cases "n=0") 

926 
case True 

927 
with z n2 contra show ?thesis by simp 

928 
next 

929 
case False 

930 
with p have "pn < p" by arith 

931 
with z n2 False ih show ?thesis by simp 

932 
qed 

14640  933 
qed 
934 
qed 

935 
qed 

936 
moreover 

937 
from i obtain k where "0<k \<and> i+k=p" 

938 
by (blast dest: less_imp_add_positive) 

939 
hence "0<k \<and> i=pk" by auto 

940 
moreover 

941 
note base 

942 
ultimately 

943 
show "False" by blast 

944 
qed 

945 

946 
lemma mod_induct: 

947 
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" 

948 
and base: "P i" and i: "i<p" and j: "j<p" 

949 
shows "P j" 

950 
proof  

951 
have "\<forall>j<p. P j" 

952 
proof 

953 
fix j 

954 
show "j<p \<longrightarrow> P j" (is "?A j") 

955 
proof (induct j) 

956 
from step base i show "?A 0" 

22718  957 
by (auto elim: mod_induct_0) 
14640  958 
next 
959 
fix k 

960 
assume ih: "?A k" 

961 
show "?A (Suc k)" 

962 
proof 

22718  963 
assume suc: "Suc k < p" 
964 
hence k: "k<p" by simp 

965 
with ih have "P k" .. 

966 
with step k have "P (Suc k mod p)" 

967 
by blast 

968 
moreover 

969 
from suc have "Suc k mod p = Suc k" 

970 
by simp 

971 
ultimately 

972 
show "P (Suc k)" by simp 

14640  973 
qed 
974 
qed 

975 
qed 

976 
with j show ?thesis by blast 

977 
qed 

978 

979 

18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

980 
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c" 
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

981 
apply (rule trans [symmetric]) 
22718  982 
apply (rule mod_add1_eq, simp) 
18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

983 
apply (rule mod_add1_eq [symmetric]) 
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

984 
done 
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

985 

46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

986 
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c" 
22718  987 
apply (rule trans [symmetric]) 
988 
apply (rule mod_add1_eq, simp) 

989 
apply (rule mod_add1_eq [symmetric]) 

990 
done 

18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
18154
diff
changeset

991 

22800  992 
lemma mod_div_decomp: 
993 
fixes n k :: nat 

994 
obtains m q where "m = n div k" and "q = n mod k" 

995 
and "n = m * k + q" 

996 
proof  

997 
from mod_div_equality have "n = n div k * k + n mod k" by auto 

998 
moreover have "n div k = n div k" .. 

999 
moreover have "n mod k = n mod k" .. 

1000 
note that ultimately show thesis by blast 

1001 
qed 

1002 

20589  1003 

25942  1004 
subsubsection {* Code generation for div, mod and dvd on nat *} 
20589  1005 

22845  1006 
definition [code func del]: 
20589  1007 
"divmod (m\<Colon>nat) n = (m div n, m mod n)" 
1008 

22718  1009 
lemma divmod_zero [code]: "divmod m 0 = (0, m)" 
20589  1010 
unfolding divmod_def by simp 
1011 

1012 
lemma divmod_succ [code]: 

1013 
"divmod m (Suc k) = (if m < Suc k then (0, m) else 

1014 
let 

1015 
(p, q) = divmod (m  Suc k) (Suc k) 

22718  1016 
in (Suc p, q))" 
20589  1017 
unfolding divmod_def Let_def split_def 
1018 
by (auto intro: div_geq mod_geq) 

1019 

22718  1020 
lemma div_divmod [code]: "m div n = fst (divmod m n)" 
20589  1021 
unfolding divmod_def by simp 
1022 

22718  1023 
lemma mod_divmod [code]: "m mod n = snd (divmod m n)" 
20589  1024 
unfolding divmod_def by simp 
1025 

21191  1026 
code_modulename SML 
23017  1027 
Divides Nat 
20640  1028 

21911
e29bcab0c81c
added OCaml code generation (without dictionaries)
haftmann
parents:
21408
diff
changeset

1029 
code_modulename OCaml 
23017  1030 
Divides Nat 
1031 

1032 
code_modulename Haskell 

1033 
Divides Nat 

21911
e29bcab0c81c
added OCaml code generation (without dictionaries)
haftmann
parents:
21408
diff
changeset

1034 

23684
8c508c4dc53b
introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents:
23162
diff
changeset

1035 
hide (open) const divmod 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset

1036 

3366  1037 
end 