paulson@3366
|
1 |
(* Title: HOL/Divides.thy
|
paulson@3366
|
2 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
|
paulson@6865
|
3 |
Copyright 1999 University of Cambridge
|
huffman@18154
|
4 |
*)
|
paulson@3366
|
5 |
|
haftmann@27651
|
6 |
header {* The division operators div and mod *}
|
paulson@3366
|
7 |
|
nipkow@15131
|
8 |
theory Divides
|
haftmann@33301
|
9 |
imports Nat_Numeral Nat_Transfer
|
haftmann@33335
|
10 |
uses "~~/src/Provers/Arith/cancel_div_mod.ML"
|
nipkow@15131
|
11 |
begin
|
paulson@3366
|
12 |
|
haftmann@25942
|
13 |
subsection {* Syntactic division operations *}
|
haftmann@25942
|
14 |
|
haftmann@27651
|
15 |
class div = dvd +
|
haftmann@27651
|
16 |
fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
|
haftmann@27651
|
17 |
and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
|
haftmann@21408
|
18 |
|
haftmann@25942
|
19 |
|
haftmann@27651
|
20 |
subsection {* Abstract division in commutative semirings. *}
|
haftmann@25942
|
21 |
|
haftmann@30930
|
22 |
class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
|
haftmann@25942
|
23 |
assumes mod_div_equality: "a div b * b + a mod b = a"
|
haftmann@27651
|
24 |
and div_by_0 [simp]: "a div 0 = 0"
|
haftmann@27651
|
25 |
and div_0 [simp]: "0 div a = 0"
|
haftmann@27651
|
26 |
and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
|
haftmann@30930
|
27 |
and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
|
haftmann@25942
|
28 |
begin
|
haftmann@25942
|
29 |
|
haftmann@26100
|
30 |
text {* @{const div} and @{const mod} *}
|
haftmann@26100
|
31 |
|
haftmann@26062
|
32 |
lemma mod_div_equality2: "b * (a div b) + a mod b = a"
|
haftmann@26062
|
33 |
unfolding mult_commute [of b]
|
haftmann@26062
|
34 |
by (rule mod_div_equality)
|
haftmann@26062
|
35 |
|
huffman@29400
|
36 |
lemma mod_div_equality': "a mod b + a div b * b = a"
|
huffman@29400
|
37 |
using mod_div_equality [of a b]
|
huffman@29400
|
38 |
by (simp only: add_ac)
|
huffman@29400
|
39 |
|
haftmann@26062
|
40 |
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
|
haftmann@30934
|
41 |
by (simp add: mod_div_equality)
|
haftmann@26062
|
42 |
|
haftmann@26062
|
43 |
lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
|
haftmann@30934
|
44 |
by (simp add: mod_div_equality2)
|
haftmann@26062
|
45 |
|
haftmann@27651
|
46 |
lemma mod_by_0 [simp]: "a mod 0 = a"
|
haftmann@30934
|
47 |
using mod_div_equality [of a zero] by simp
|
haftmann@26100
|
48 |
|
haftmann@27651
|
49 |
lemma mod_0 [simp]: "0 mod a = 0"
|
haftmann@30934
|
50 |
using mod_div_equality [of zero a] div_0 by simp
|
haftmann@26062
|
51 |
|
haftmann@27651
|
52 |
lemma div_mult_self2 [simp]:
|
haftmann@27651
|
53 |
assumes "b \<noteq> 0"
|
haftmann@27651
|
54 |
shows "(a + b * c) div b = c + a div b"
|
haftmann@27651
|
55 |
using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
|
haftmann@26062
|
56 |
|
haftmann@27651
|
57 |
lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
|
haftmann@27651
|
58 |
proof (cases "b = 0")
|
haftmann@27651
|
59 |
case True then show ?thesis by simp
|
haftmann@27651
|
60 |
next
|
haftmann@27651
|
61 |
case False
|
haftmann@27651
|
62 |
have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
|
haftmann@27651
|
63 |
by (simp add: mod_div_equality)
|
haftmann@27651
|
64 |
also from False div_mult_self1 [of b a c] have
|
haftmann@27651
|
65 |
"\<dots> = (c + a div b) * b + (a + c * b) mod b"
|
nipkow@29667
|
66 |
by (simp add: algebra_simps)
|
haftmann@27651
|
67 |
finally have "a = a div b * b + (a + c * b) mod b"
|
haftmann@27651
|
68 |
by (simp add: add_commute [of a] add_assoc left_distrib)
|
haftmann@27651
|
69 |
then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
|
haftmann@27651
|
70 |
by (simp add: mod_div_equality)
|
haftmann@27651
|
71 |
then show ?thesis by simp
|
haftmann@27651
|
72 |
qed
|
haftmann@27651
|
73 |
|
haftmann@27651
|
74 |
lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
|
haftmann@30934
|
75 |
by (simp add: mult_commute [of b])
|
haftmann@27651
|
76 |
|
haftmann@27651
|
77 |
lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
|
haftmann@27651
|
78 |
using div_mult_self2 [of b 0 a] by simp
|
haftmann@27651
|
79 |
|
haftmann@27651
|
80 |
lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
|
haftmann@27651
|
81 |
using div_mult_self1 [of b 0 a] by simp
|
haftmann@27651
|
82 |
|
haftmann@27651
|
83 |
lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
|
haftmann@27651
|
84 |
using mod_mult_self2 [of 0 b a] by simp
|
haftmann@27651
|
85 |
|
haftmann@27651
|
86 |
lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
|
haftmann@27651
|
87 |
using mod_mult_self1 [of 0 a b] by simp
|
haftmann@27651
|
88 |
|
haftmann@27651
|
89 |
lemma div_by_1 [simp]: "a div 1 = a"
|
haftmann@27651
|
90 |
using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
|
haftmann@27651
|
91 |
|
haftmann@27651
|
92 |
lemma mod_by_1 [simp]: "a mod 1 = 0"
|
haftmann@27651
|
93 |
proof -
|
haftmann@27651
|
94 |
from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
|
haftmann@27651
|
95 |
then have "a + a mod 1 = a + 0" by simp
|
haftmann@27651
|
96 |
then show ?thesis by (rule add_left_imp_eq)
|
haftmann@27651
|
97 |
qed
|
haftmann@27651
|
98 |
|
haftmann@27651
|
99 |
lemma mod_self [simp]: "a mod a = 0"
|
haftmann@27651
|
100 |
using mod_mult_self2_is_0 [of 1] by simp
|
haftmann@27651
|
101 |
|
haftmann@27651
|
102 |
lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
|
haftmann@27651
|
103 |
using div_mult_self2_is_id [of _ 1] by simp
|
haftmann@27651
|
104 |
|
haftmann@27676
|
105 |
lemma div_add_self1 [simp]:
|
haftmann@27651
|
106 |
assumes "b \<noteq> 0"
|
haftmann@27651
|
107 |
shows "(b + a) div b = a div b + 1"
|
haftmann@27651
|
108 |
using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
|
haftmann@27651
|
109 |
|
haftmann@27676
|
110 |
lemma div_add_self2 [simp]:
|
haftmann@27651
|
111 |
assumes "b \<noteq> 0"
|
haftmann@27651
|
112 |
shows "(a + b) div b = a div b + 1"
|
haftmann@27651
|
113 |
using assms div_add_self1 [of b a] by (simp add: add_commute)
|
haftmann@27651
|
114 |
|
haftmann@27676
|
115 |
lemma mod_add_self1 [simp]:
|
haftmann@27651
|
116 |
"(b + a) mod b = a mod b"
|
haftmann@27651
|
117 |
using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
|
haftmann@27651
|
118 |
|
haftmann@27676
|
119 |
lemma mod_add_self2 [simp]:
|
haftmann@27651
|
120 |
"(a + b) mod b = a mod b"
|
haftmann@27651
|
121 |
using mod_mult_self1 [of a 1 b] by simp
|
haftmann@27651
|
122 |
|
haftmann@27651
|
123 |
lemma mod_div_decomp:
|
haftmann@27651
|
124 |
fixes a b
|
haftmann@27651
|
125 |
obtains q r where "q = a div b" and "r = a mod b"
|
haftmann@27651
|
126 |
and "a = q * b + r"
|
haftmann@27651
|
127 |
proof -
|
haftmann@27651
|
128 |
from mod_div_equality have "a = a div b * b + a mod b" by simp
|
haftmann@27651
|
129 |
moreover have "a div b = a div b" ..
|
haftmann@27651
|
130 |
moreover have "a mod b = a mod b" ..
|
haftmann@27651
|
131 |
note that ultimately show thesis by blast
|
haftmann@27651
|
132 |
qed
|
haftmann@27651
|
133 |
|
bulwahn@46100
|
134 |
lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0"
|
haftmann@25942
|
135 |
proof
|
haftmann@25942
|
136 |
assume "b mod a = 0"
|
haftmann@25942
|
137 |
with mod_div_equality [of b a] have "b div a * a = b" by simp
|
haftmann@25942
|
138 |
then have "b = a * (b div a)" unfolding mult_commute ..
|
haftmann@25942
|
139 |
then have "\<exists>c. b = a * c" ..
|
haftmann@25942
|
140 |
then show "a dvd b" unfolding dvd_def .
|
haftmann@25942
|
141 |
next
|
haftmann@25942
|
142 |
assume "a dvd b"
|
haftmann@25942
|
143 |
then have "\<exists>c. b = a * c" unfolding dvd_def .
|
haftmann@25942
|
144 |
then obtain c where "b = a * c" ..
|
haftmann@25942
|
145 |
then have "b mod a = a * c mod a" by simp
|
haftmann@25942
|
146 |
then have "b mod a = c * a mod a" by (simp add: mult_commute)
|
haftmann@27651
|
147 |
then show "b mod a = 0" by simp
|
haftmann@25942
|
148 |
qed
|
haftmann@25942
|
149 |
|
huffman@29400
|
150 |
lemma mod_div_trivial [simp]: "a mod b div b = 0"
|
huffman@29400
|
151 |
proof (cases "b = 0")
|
huffman@29400
|
152 |
assume "b = 0"
|
huffman@29400
|
153 |
thus ?thesis by simp
|
huffman@29400
|
154 |
next
|
huffman@29400
|
155 |
assume "b \<noteq> 0"
|
huffman@29400
|
156 |
hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
|
huffman@29400
|
157 |
by (rule div_mult_self1 [symmetric])
|
huffman@29400
|
158 |
also have "\<dots> = a div b"
|
huffman@29400
|
159 |
by (simp only: mod_div_equality')
|
huffman@29400
|
160 |
also have "\<dots> = a div b + 0"
|
huffman@29400
|
161 |
by simp
|
huffman@29400
|
162 |
finally show ?thesis
|
huffman@29400
|
163 |
by (rule add_left_imp_eq)
|
huffman@29400
|
164 |
qed
|
huffman@29400
|
165 |
|
huffman@29400
|
166 |
lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
|
huffman@29400
|
167 |
proof -
|
huffman@29400
|
168 |
have "a mod b mod b = (a mod b + a div b * b) mod b"
|
huffman@29400
|
169 |
by (simp only: mod_mult_self1)
|
huffman@29400
|
170 |
also have "\<dots> = a mod b"
|
huffman@29400
|
171 |
by (simp only: mod_div_equality')
|
huffman@29400
|
172 |
finally show ?thesis .
|
huffman@29400
|
173 |
qed
|
huffman@29400
|
174 |
|
nipkow@29862
|
175 |
lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
|
nipkow@29885
|
176 |
by (rule dvd_eq_mod_eq_0[THEN iffD1])
|
nipkow@29862
|
177 |
|
nipkow@29862
|
178 |
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
|
nipkow@29862
|
179 |
by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
|
nipkow@29862
|
180 |
|
haftmann@33274
|
181 |
lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
|
haftmann@33274
|
182 |
by (drule dvd_div_mult_self) (simp add: mult_commute)
|
haftmann@33274
|
183 |
|
nipkow@29989
|
184 |
lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
|
nipkow@29989
|
185 |
apply (cases "a = 0")
|
nipkow@29989
|
186 |
apply simp
|
nipkow@29989
|
187 |
apply (auto simp: dvd_def mult_assoc)
|
nipkow@29989
|
188 |
done
|
nipkow@29989
|
189 |
|
nipkow@29862
|
190 |
lemma div_dvd_div[simp]:
|
nipkow@29862
|
191 |
"a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
|
nipkow@29862
|
192 |
apply (cases "a = 0")
|
nipkow@29862
|
193 |
apply simp
|
nipkow@29862
|
194 |
apply (unfold dvd_def)
|
nipkow@29862
|
195 |
apply auto
|
nipkow@29862
|
196 |
apply(blast intro:mult_assoc[symmetric])
|
nipkow@45761
|
197 |
apply(fastforce simp add: mult_assoc)
|
nipkow@29862
|
198 |
done
|
nipkow@29862
|
199 |
|
huffman@30015
|
200 |
lemma dvd_mod_imp_dvd: "[| k dvd m mod n; k dvd n |] ==> k dvd m"
|
huffman@30015
|
201 |
apply (subgoal_tac "k dvd (m div n) *n + m mod n")
|
huffman@30015
|
202 |
apply (simp add: mod_div_equality)
|
huffman@30015
|
203 |
apply (simp only: dvd_add dvd_mult)
|
huffman@30015
|
204 |
done
|
huffman@30015
|
205 |
|
huffman@29400
|
206 |
text {* Addition respects modular equivalence. *}
|
huffman@29400
|
207 |
|
huffman@29400
|
208 |
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
|
huffman@29400
|
209 |
proof -
|
huffman@29400
|
210 |
have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
|
huffman@29400
|
211 |
by (simp only: mod_div_equality)
|
huffman@29400
|
212 |
also have "\<dots> = (a mod c + b + a div c * c) mod c"
|
huffman@29400
|
213 |
by (simp only: add_ac)
|
huffman@29400
|
214 |
also have "\<dots> = (a mod c + b) mod c"
|
huffman@29400
|
215 |
by (rule mod_mult_self1)
|
huffman@29400
|
216 |
finally show ?thesis .
|
huffman@29400
|
217 |
qed
|
huffman@29400
|
218 |
|
huffman@29400
|
219 |
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
|
huffman@29400
|
220 |
proof -
|
huffman@29400
|
221 |
have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
|
huffman@29400
|
222 |
by (simp only: mod_div_equality)
|
huffman@29400
|
223 |
also have "\<dots> = (a + b mod c + b div c * c) mod c"
|
huffman@29400
|
224 |
by (simp only: add_ac)
|
huffman@29400
|
225 |
also have "\<dots> = (a + b mod c) mod c"
|
huffman@29400
|
226 |
by (rule mod_mult_self1)
|
huffman@29400
|
227 |
finally show ?thesis .
|
huffman@29400
|
228 |
qed
|
huffman@29400
|
229 |
|
huffman@29400
|
230 |
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
|
huffman@29400
|
231 |
by (rule trans [OF mod_add_left_eq mod_add_right_eq])
|
huffman@29400
|
232 |
|
huffman@29400
|
233 |
lemma mod_add_cong:
|
huffman@29400
|
234 |
assumes "a mod c = a' mod c"
|
huffman@29400
|
235 |
assumes "b mod c = b' mod c"
|
huffman@29400
|
236 |
shows "(a + b) mod c = (a' + b') mod c"
|
huffman@29400
|
237 |
proof -
|
huffman@29400
|
238 |
have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
|
huffman@29400
|
239 |
unfolding assms ..
|
huffman@29400
|
240 |
thus ?thesis
|
huffman@29400
|
241 |
by (simp only: mod_add_eq [symmetric])
|
huffman@29400
|
242 |
qed
|
huffman@29400
|
243 |
|
haftmann@30923
|
244 |
lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
|
nipkow@30837
|
245 |
\<Longrightarrow> (x + y) div z = x div z + y div z"
|
haftmann@30923
|
246 |
by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
|
nipkow@30837
|
247 |
|
huffman@29400
|
248 |
text {* Multiplication respects modular equivalence. *}
|
huffman@29400
|
249 |
|
huffman@29400
|
250 |
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
|
huffman@29400
|
251 |
proof -
|
huffman@29400
|
252 |
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
|
huffman@29400
|
253 |
by (simp only: mod_div_equality)
|
huffman@29400
|
254 |
also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
|
nipkow@29667
|
255 |
by (simp only: algebra_simps)
|
huffman@29400
|
256 |
also have "\<dots> = (a mod c * b) mod c"
|
huffman@29400
|
257 |
by (rule mod_mult_self1)
|
huffman@29400
|
258 |
finally show ?thesis .
|
huffman@29400
|
259 |
qed
|
huffman@29400
|
260 |
|
huffman@29400
|
261 |
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
|
huffman@29400
|
262 |
proof -
|
huffman@29400
|
263 |
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
|
huffman@29400
|
264 |
by (simp only: mod_div_equality)
|
huffman@29400
|
265 |
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
|
nipkow@29667
|
266 |
by (simp only: algebra_simps)
|
huffman@29400
|
267 |
also have "\<dots> = (a * (b mod c)) mod c"
|
huffman@29400
|
268 |
by (rule mod_mult_self1)
|
huffman@29400
|
269 |
finally show ?thesis .
|
huffman@29400
|
270 |
qed
|
huffman@29400
|
271 |
|
huffman@29400
|
272 |
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
|
huffman@29400
|
273 |
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
|
huffman@29400
|
274 |
|
huffman@29400
|
275 |
lemma mod_mult_cong:
|
huffman@29400
|
276 |
assumes "a mod c = a' mod c"
|
huffman@29400
|
277 |
assumes "b mod c = b' mod c"
|
huffman@29400
|
278 |
shows "(a * b) mod c = (a' * b') mod c"
|
huffman@29400
|
279 |
proof -
|
huffman@29400
|
280 |
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
|
huffman@29400
|
281 |
unfolding assms ..
|
huffman@29400
|
282 |
thus ?thesis
|
huffman@29400
|
283 |
by (simp only: mod_mult_eq [symmetric])
|
huffman@29400
|
284 |
qed
|
huffman@29400
|
285 |
|
huffman@29401
|
286 |
lemma mod_mod_cancel:
|
huffman@29401
|
287 |
assumes "c dvd b"
|
huffman@29401
|
288 |
shows "a mod b mod c = a mod c"
|
huffman@29401
|
289 |
proof -
|
huffman@29401
|
290 |
from `c dvd b` obtain k where "b = c * k"
|
huffman@29401
|
291 |
by (rule dvdE)
|
huffman@29401
|
292 |
have "a mod b mod c = a mod (c * k) mod c"
|
huffman@29401
|
293 |
by (simp only: `b = c * k`)
|
huffman@29401
|
294 |
also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
|
huffman@29401
|
295 |
by (simp only: mod_mult_self1)
|
huffman@29401
|
296 |
also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
|
huffman@29401
|
297 |
by (simp only: add_ac mult_ac)
|
huffman@29401
|
298 |
also have "\<dots> = a mod c"
|
huffman@29401
|
299 |
by (simp only: mod_div_equality)
|
huffman@29401
|
300 |
finally show ?thesis .
|
huffman@29401
|
301 |
qed
|
huffman@29401
|
302 |
|
haftmann@30930
|
303 |
lemma div_mult_div_if_dvd:
|
haftmann@30930
|
304 |
"y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
|
haftmann@30930
|
305 |
apply (cases "y = 0", simp)
|
haftmann@30930
|
306 |
apply (cases "z = 0", simp)
|
haftmann@30930
|
307 |
apply (auto elim!: dvdE simp add: algebra_simps)
|
nipkow@30472
|
308 |
apply (subst mult_assoc [symmetric])
|
nipkow@30472
|
309 |
apply (simp add: no_zero_divisors)
|
haftmann@30930
|
310 |
done
|
nipkow@30472
|
311 |
|
haftmann@35367
|
312 |
lemma div_mult_swap:
|
haftmann@35367
|
313 |
assumes "c dvd b"
|
haftmann@35367
|
314 |
shows "a * (b div c) = (a * b) div c"
|
haftmann@35367
|
315 |
proof -
|
haftmann@35367
|
316 |
from assms have "b div c * (a div 1) = b * a div (c * 1)"
|
haftmann@35367
|
317 |
by (simp only: div_mult_div_if_dvd one_dvd)
|
haftmann@35367
|
318 |
then show ?thesis by (simp add: mult_commute)
|
haftmann@35367
|
319 |
qed
|
haftmann@35367
|
320 |
|
haftmann@30930
|
321 |
lemma div_mult_mult2 [simp]:
|
haftmann@30930
|
322 |
"c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
|
haftmann@30930
|
323 |
by (drule div_mult_mult1) (simp add: mult_commute)
|
haftmann@30930
|
324 |
|
haftmann@30930
|
325 |
lemma div_mult_mult1_if [simp]:
|
haftmann@30930
|
326 |
"(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
|
haftmann@30930
|
327 |
by simp_all
|
haftmann@30930
|
328 |
|
haftmann@30930
|
329 |
lemma mod_mult_mult1:
|
haftmann@30930
|
330 |
"(c * a) mod (c * b) = c * (a mod b)"
|
haftmann@30930
|
331 |
proof (cases "c = 0")
|
haftmann@30930
|
332 |
case True then show ?thesis by simp
|
haftmann@30930
|
333 |
next
|
haftmann@30930
|
334 |
case False
|
haftmann@30930
|
335 |
from mod_div_equality
|
haftmann@30930
|
336 |
have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
|
haftmann@30930
|
337 |
with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
|
haftmann@30930
|
338 |
= c * a + c * (a mod b)" by (simp add: algebra_simps)
|
haftmann@30930
|
339 |
with mod_div_equality show ?thesis by simp
|
haftmann@30930
|
340 |
qed
|
haftmann@30930
|
341 |
|
haftmann@30930
|
342 |
lemma mod_mult_mult2:
|
haftmann@30930
|
343 |
"(a * c) mod (b * c) = (a mod b) * c"
|
haftmann@30930
|
344 |
using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
|
haftmann@30930
|
345 |
|
huffman@31662
|
346 |
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
|
huffman@31662
|
347 |
unfolding dvd_def by (auto simp add: mod_mult_mult1)
|
huffman@31662
|
348 |
|
huffman@31662
|
349 |
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
|
huffman@31662
|
350 |
by (blast intro: dvd_mod_imp_dvd dvd_mod)
|
huffman@31662
|
351 |
|
haftmann@31009
|
352 |
lemma div_power:
|
huffman@31661
|
353 |
"y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
|
nipkow@30472
|
354 |
apply (induct n)
|
nipkow@30472
|
355 |
apply simp
|
nipkow@30472
|
356 |
apply(simp add: div_mult_div_if_dvd dvd_power_same)
|
nipkow@30472
|
357 |
done
|
nipkow@30472
|
358 |
|
haftmann@35367
|
359 |
lemma dvd_div_eq_mult:
|
haftmann@35367
|
360 |
assumes "a \<noteq> 0" and "a dvd b"
|
haftmann@35367
|
361 |
shows "b div a = c \<longleftrightarrow> b = c * a"
|
haftmann@35367
|
362 |
proof
|
haftmann@35367
|
363 |
assume "b = c * a"
|
haftmann@35367
|
364 |
then show "b div a = c" by (simp add: assms)
|
haftmann@35367
|
365 |
next
|
haftmann@35367
|
366 |
assume "b div a = c"
|
haftmann@35367
|
367 |
then have "b div a * a = c * a" by simp
|
haftmann@35367
|
368 |
moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
|
haftmann@35367
|
369 |
ultimately show "b = c * a" by simp
|
haftmann@35367
|
370 |
qed
|
haftmann@35367
|
371 |
|
haftmann@35367
|
372 |
lemma dvd_div_div_eq_mult:
|
haftmann@35367
|
373 |
assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
|
haftmann@35367
|
374 |
shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
|
haftmann@35367
|
375 |
using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
|
haftmann@35367
|
376 |
|
huffman@31661
|
377 |
end
|
huffman@31661
|
378 |
|
haftmann@35668
|
379 |
class ring_div = semiring_div + comm_ring_1
|
huffman@29402
|
380 |
begin
|
huffman@29402
|
381 |
|
haftmann@36622
|
382 |
subclass ring_1_no_zero_divisors ..
|
haftmann@36622
|
383 |
|
huffman@29402
|
384 |
text {* Negation respects modular equivalence. *}
|
huffman@29402
|
385 |
|
huffman@29402
|
386 |
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
|
huffman@29402
|
387 |
proof -
|
huffman@29402
|
388 |
have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
|
huffman@29402
|
389 |
by (simp only: mod_div_equality)
|
huffman@29402
|
390 |
also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
|
huffman@29402
|
391 |
by (simp only: minus_add_distrib minus_mult_left add_ac)
|
huffman@29402
|
392 |
also have "\<dots> = (- (a mod b)) mod b"
|
huffman@29402
|
393 |
by (rule mod_mult_self1)
|
huffman@29402
|
394 |
finally show ?thesis .
|
huffman@29402
|
395 |
qed
|
huffman@29402
|
396 |
|
huffman@29402
|
397 |
lemma mod_minus_cong:
|
huffman@29402
|
398 |
assumes "a mod b = a' mod b"
|
huffman@29402
|
399 |
shows "(- a) mod b = (- a') mod b"
|
huffman@29402
|
400 |
proof -
|
huffman@29402
|
401 |
have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
|
huffman@29402
|
402 |
unfolding assms ..
|
huffman@29402
|
403 |
thus ?thesis
|
huffman@29402
|
404 |
by (simp only: mod_minus_eq [symmetric])
|
huffman@29402
|
405 |
qed
|
huffman@29402
|
406 |
|
huffman@29402
|
407 |
text {* Subtraction respects modular equivalence. *}
|
huffman@29402
|
408 |
|
huffman@29402
|
409 |
lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
|
huffman@29402
|
410 |
unfolding diff_minus
|
huffman@29402
|
411 |
by (intro mod_add_cong mod_minus_cong) simp_all
|
huffman@29402
|
412 |
|
huffman@29402
|
413 |
lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
|
huffman@29402
|
414 |
unfolding diff_minus
|
huffman@29402
|
415 |
by (intro mod_add_cong mod_minus_cong) simp_all
|
huffman@29402
|
416 |
|
huffman@29402
|
417 |
lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
|
huffman@29402
|
418 |
unfolding diff_minus
|
huffman@29402
|
419 |
by (intro mod_add_cong mod_minus_cong) simp_all
|
huffman@29402
|
420 |
|
huffman@29402
|
421 |
lemma mod_diff_cong:
|
huffman@29402
|
422 |
assumes "a mod c = a' mod c"
|
huffman@29402
|
423 |
assumes "b mod c = b' mod c"
|
huffman@29402
|
424 |
shows "(a - b) mod c = (a' - b') mod c"
|
huffman@29402
|
425 |
unfolding diff_minus using assms
|
huffman@29402
|
426 |
by (intro mod_add_cong mod_minus_cong)
|
huffman@29402
|
427 |
|
nipkow@30180
|
428 |
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
|
nipkow@30180
|
429 |
apply (case_tac "y = 0") apply simp
|
nipkow@30180
|
430 |
apply (auto simp add: dvd_def)
|
nipkow@30180
|
431 |
apply (subgoal_tac "-(y * k) = y * - k")
|
nipkow@30180
|
432 |
apply (erule ssubst)
|
nipkow@30180
|
433 |
apply (erule div_mult_self1_is_id)
|
nipkow@30180
|
434 |
apply simp
|
nipkow@30180
|
435 |
done
|
nipkow@30180
|
436 |
|
nipkow@30180
|
437 |
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
|
nipkow@30180
|
438 |
apply (case_tac "y = 0") apply simp
|
nipkow@30180
|
439 |
apply (auto simp add: dvd_def)
|
nipkow@30180
|
440 |
apply (subgoal_tac "y * k = -y * -k")
|
nipkow@30180
|
441 |
apply (erule ssubst)
|
nipkow@30180
|
442 |
apply (rule div_mult_self1_is_id)
|
nipkow@30180
|
443 |
apply simp
|
nipkow@30180
|
444 |
apply simp
|
nipkow@30180
|
445 |
done
|
nipkow@30180
|
446 |
|
huffman@29402
|
447 |
end
|
huffman@29402
|
448 |
|
haftmann@25942
|
449 |
|
haftmann@26100
|
450 |
subsection {* Division on @{typ nat} *}
|
haftmann@26100
|
451 |
|
haftmann@26100
|
452 |
text {*
|
haftmann@26100
|
453 |
We define @{const div} and @{const mod} on @{typ nat} by means
|
haftmann@26100
|
454 |
of a characteristic relation with two input arguments
|
haftmann@26100
|
455 |
@{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
|
haftmann@26100
|
456 |
@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
|
haftmann@26100
|
457 |
*}
|
haftmann@26100
|
458 |
|
haftmann@33335
|
459 |
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
|
haftmann@33335
|
460 |
"divmod_nat_rel m n qr \<longleftrightarrow>
|
haftmann@30923
|
461 |
m = fst qr * n + snd qr \<and>
|
haftmann@30923
|
462 |
(if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
|
haftmann@26100
|
463 |
|
haftmann@33335
|
464 |
text {* @{const divmod_nat_rel} is total: *}
|
haftmann@26100
|
465 |
|
haftmann@33335
|
466 |
lemma divmod_nat_rel_ex:
|
haftmann@33335
|
467 |
obtains q r where "divmod_nat_rel m n (q, r)"
|
haftmann@26100
|
468 |
proof (cases "n = 0")
|
haftmann@30923
|
469 |
case True with that show thesis
|
haftmann@33335
|
470 |
by (auto simp add: divmod_nat_rel_def)
|
haftmann@26100
|
471 |
next
|
haftmann@26100
|
472 |
case False
|
haftmann@26100
|
473 |
have "\<exists>q r. m = q * n + r \<and> r < n"
|
haftmann@26100
|
474 |
proof (induct m)
|
haftmann@26100
|
475 |
case 0 with `n \<noteq> 0`
|
haftmann@26100
|
476 |
have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
|
haftmann@26100
|
477 |
then show ?case by blast
|
haftmann@26100
|
478 |
next
|
haftmann@26100
|
479 |
case (Suc m) then obtain q' r'
|
haftmann@26100
|
480 |
where m: "m = q' * n + r'" and n: "r' < n" by auto
|
haftmann@26100
|
481 |
then show ?case proof (cases "Suc r' < n")
|
haftmann@26100
|
482 |
case True
|
haftmann@26100
|
483 |
from m n have "Suc m = q' * n + Suc r'" by simp
|
haftmann@26100
|
484 |
with True show ?thesis by blast
|
haftmann@26100
|
485 |
next
|
haftmann@26100
|
486 |
case False then have "n \<le> Suc r'" by auto
|
haftmann@26100
|
487 |
moreover from n have "Suc r' \<le> n" by auto
|
haftmann@26100
|
488 |
ultimately have "n = Suc r'" by auto
|
haftmann@26100
|
489 |
with m have "Suc m = Suc q' * n + 0" by simp
|
haftmann@26100
|
490 |
with `n \<noteq> 0` show ?thesis by blast
|
haftmann@26100
|
491 |
qed
|
haftmann@26100
|
492 |
qed
|
haftmann@26100
|
493 |
with that show thesis
|
haftmann@33335
|
494 |
using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
|
haftmann@26100
|
495 |
qed
|
haftmann@26100
|
496 |
|
haftmann@33335
|
497 |
text {* @{const divmod_nat_rel} is injective: *}
|
haftmann@26100
|
498 |
|
haftmann@33335
|
499 |
lemma divmod_nat_rel_unique:
|
haftmann@33335
|
500 |
assumes "divmod_nat_rel m n qr"
|
haftmann@33335
|
501 |
and "divmod_nat_rel m n qr'"
|
haftmann@30923
|
502 |
shows "qr = qr'"
|
haftmann@26100
|
503 |
proof (cases "n = 0")
|
haftmann@26100
|
504 |
case True with assms show ?thesis
|
haftmann@30923
|
505 |
by (cases qr, cases qr')
|
haftmann@33335
|
506 |
(simp add: divmod_nat_rel_def)
|
haftmann@26100
|
507 |
next
|
haftmann@26100
|
508 |
case False
|
haftmann@26100
|
509 |
have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
|
haftmann@26100
|
510 |
apply (rule leI)
|
haftmann@26100
|
511 |
apply (subst less_iff_Suc_add)
|
haftmann@26100
|
512 |
apply (auto simp add: add_mult_distrib)
|
haftmann@26100
|
513 |
done
|
haftmann@30923
|
514 |
from `n \<noteq> 0` assms have "fst qr = fst qr'"
|
haftmann@33335
|
515 |
by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
|
haftmann@30923
|
516 |
moreover from this assms have "snd qr = snd qr'"
|
haftmann@33335
|
517 |
by (simp add: divmod_nat_rel_def)
|
haftmann@30923
|
518 |
ultimately show ?thesis by (cases qr, cases qr') simp
|
haftmann@26100
|
519 |
qed
|
haftmann@26100
|
520 |
|
haftmann@26100
|
521 |
text {*
|
haftmann@26100
|
522 |
We instantiate divisibility on the natural numbers by
|
haftmann@33335
|
523 |
means of @{const divmod_nat_rel}:
|
haftmann@26100
|
524 |
*}
|
haftmann@25942
|
525 |
|
haftmann@33335
|
526 |
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
|
haftmann@37767
|
527 |
"divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
|
haftmann@30923
|
528 |
|
haftmann@33335
|
529 |
lemma divmod_nat_rel_divmod_nat:
|
haftmann@33335
|
530 |
"divmod_nat_rel m n (divmod_nat m n)"
|
haftmann@30923
|
531 |
proof -
|
haftmann@33335
|
532 |
from divmod_nat_rel_ex
|
haftmann@33335
|
533 |
obtain qr where rel: "divmod_nat_rel m n qr" .
|
haftmann@30923
|
534 |
then show ?thesis
|
haftmann@33335
|
535 |
by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
|
haftmann@30923
|
536 |
qed
|
haftmann@30923
|
537 |
|
huffman@48006
|
538 |
lemma divmod_nat_unique:
|
haftmann@33335
|
539 |
assumes "divmod_nat_rel m n qr"
|
haftmann@33335
|
540 |
shows "divmod_nat m n = qr"
|
haftmann@33335
|
541 |
using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
|
haftmann@25571
|
542 |
|
huffman@47419
|
543 |
instantiation nat :: semiring_div
|
huffman@47419
|
544 |
begin
|
huffman@47419
|
545 |
|
haftmann@26100
|
546 |
definition div_nat where
|
haftmann@33335
|
547 |
"m div n = fst (divmod_nat m n)"
|
haftmann@25942
|
548 |
|
huffman@47419
|
549 |
lemma fst_divmod_nat [simp]:
|
huffman@47419
|
550 |
"fst (divmod_nat m n) = m div n"
|
huffman@47419
|
551 |
by (simp add: div_nat_def)
|
huffman@47419
|
552 |
|
haftmann@26100
|
553 |
definition mod_nat where
|
haftmann@33335
|
554 |
"m mod n = snd (divmod_nat m n)"
|
haftmann@25571
|
555 |
|
huffman@47419
|
556 |
lemma snd_divmod_nat [simp]:
|
huffman@47419
|
557 |
"snd (divmod_nat m n) = m mod n"
|
huffman@47419
|
558 |
by (simp add: mod_nat_def)
|
huffman@47419
|
559 |
|
haftmann@33335
|
560 |
lemma divmod_nat_div_mod:
|
haftmann@33335
|
561 |
"divmod_nat m n = (m div n, m mod n)"
|
huffman@47419
|
562 |
by (simp add: prod_eq_iff)
|
paulson@14267
|
563 |
|
huffman@48006
|
564 |
lemma div_nat_unique:
|
haftmann@33335
|
565 |
assumes "divmod_nat_rel m n (q, r)"
|
haftmann@26100
|
566 |
shows "m div n = q"
|
huffman@48006
|
567 |
using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
|
huffman@48006
|
568 |
|
huffman@48006
|
569 |
lemma mod_nat_unique:
|
haftmann@33335
|
570 |
assumes "divmod_nat_rel m n (q, r)"
|
haftmann@26100
|
571 |
shows "m mod n = r"
|
huffman@48006
|
572 |
using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
|
paulson@14267
|
573 |
|
haftmann@33335
|
574 |
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
|
huffman@47419
|
575 |
using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
|
paulson@14267
|
576 |
|
huffman@48007
|
577 |
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
|
huffman@48007
|
578 |
by (simp add: divmod_nat_unique divmod_nat_rel_def)
|
huffman@48007
|
579 |
|
huffman@48007
|
580 |
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
|
huffman@48007
|
581 |
by (simp add: divmod_nat_unique divmod_nat_rel_def)
|
haftmann@25942
|
582 |
|
huffman@48008
|
583 |
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
|
huffman@48008
|
584 |
by (simp add: divmod_nat_unique divmod_nat_rel_def)
|
haftmann@25942
|
585 |
|
haftmann@33335
|
586 |
lemma divmod_nat_step:
|
haftmann@26100
|
587 |
assumes "0 < n" and "n \<le> m"
|
haftmann@33335
|
588 |
shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
|
huffman@48006
|
589 |
proof (rule divmod_nat_unique)
|
huffman@48005
|
590 |
have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
|
huffman@48005
|
591 |
by (rule divmod_nat_rel)
|
huffman@48005
|
592 |
thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
|
huffman@48005
|
593 |
unfolding divmod_nat_rel_def using assms by auto
|
haftmann@26100
|
594 |
qed
|
haftmann@26100
|
595 |
|
wenzelm@26300
|
596 |
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
|
haftmann@26100
|
597 |
|
haftmann@26100
|
598 |
lemma div_less [simp]:
|
haftmann@26100
|
599 |
fixes m n :: nat
|
haftmann@26100
|
600 |
assumes "m < n"
|
haftmann@26100
|
601 |
shows "m div n = 0"
|
huffman@47419
|
602 |
using assms divmod_nat_base by (simp add: prod_eq_iff)
|
haftmann@26100
|
603 |
|
haftmann@26100
|
604 |
lemma le_div_geq:
|
haftmann@26100
|
605 |
fixes m n :: nat
|
haftmann@26100
|
606 |
assumes "0 < n" and "n \<le> m"
|
haftmann@26100
|
607 |
shows "m div n = Suc ((m - n) div n)"
|
huffman@47419
|
608 |
using assms divmod_nat_step by (simp add: prod_eq_iff)
|
haftmann@26100
|
609 |
|
haftmann@26100
|
610 |
lemma mod_less [simp]:
|
haftmann@26100
|
611 |
fixes m n :: nat
|
haftmann@26100
|
612 |
assumes "m < n"
|
haftmann@26100
|
613 |
shows "m mod n = m"
|
huffman@47419
|
614 |
using assms divmod_nat_base by (simp add: prod_eq_iff)
|
haftmann@26100
|
615 |
|
haftmann@26100
|
616 |
lemma le_mod_geq:
|
haftmann@26100
|
617 |
fixes m n :: nat
|
haftmann@26100
|
618 |
assumes "n \<le> m"
|
haftmann@26100
|
619 |
shows "m mod n = (m - n) mod n"
|
huffman@47419
|
620 |
using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
|
haftmann@25942
|
621 |
|
huffman@48007
|
622 |
instance proof
|
huffman@48007
|
623 |
fix m n :: nat
|
huffman@48007
|
624 |
show "m div n * n + m mod n = m"
|
huffman@48007
|
625 |
using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
|
huffman@48007
|
626 |
next
|
huffman@48007
|
627 |
fix m n q :: nat
|
huffman@48007
|
628 |
assume "n \<noteq> 0"
|
huffman@48007
|
629 |
then show "(q + m * n) div n = m + q div n"
|
huffman@48007
|
630 |
by (induct m) (simp_all add: le_div_geq)
|
huffman@48007
|
631 |
next
|
huffman@48007
|
632 |
fix m n q :: nat
|
huffman@48007
|
633 |
assume "m \<noteq> 0"
|
huffman@48007
|
634 |
hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
|
huffman@48007
|
635 |
unfolding divmod_nat_rel_def
|
huffman@48007
|
636 |
by (auto split: split_if_asm, simp_all add: algebra_simps)
|
huffman@48007
|
637 |
moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
|
huffman@48007
|
638 |
ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
|
huffman@48007
|
639 |
thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
|
huffman@48007
|
640 |
next
|
huffman@48007
|
641 |
fix n :: nat show "n div 0 = 0"
|
haftmann@33335
|
642 |
by (simp add: div_nat_def divmod_nat_zero)
|
huffman@48007
|
643 |
next
|
huffman@48007
|
644 |
fix n :: nat show "0 div n = 0"
|
huffman@48007
|
645 |
by (simp add: div_nat_def divmod_nat_zero_left)
|
haftmann@25942
|
646 |
qed
|
haftmann@26100
|
647 |
|
haftmann@25942
|
648 |
end
|
haftmann@25942
|
649 |
|
haftmann@33361
|
650 |
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
|
haftmann@33361
|
651 |
let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
|
huffman@47419
|
652 |
by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)
|
haftmann@33361
|
653 |
|
haftmann@26100
|
654 |
text {* Simproc for cancelling @{const div} and @{const mod} *}
|
haftmann@25942
|
655 |
|
haftmann@30934
|
656 |
ML {*
|
wenzelm@44467
|
657 |
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
|
wenzelm@41798
|
658 |
(
|
haftmann@30934
|
659 |
val div_name = @{const_name div};
|
haftmann@30934
|
660 |
val mod_name = @{const_name mod};
|
haftmann@30934
|
661 |
val mk_binop = HOLogic.mk_binop;
|
haftmann@30934
|
662 |
val mk_sum = Nat_Arith.mk_sum;
|
haftmann@30934
|
663 |
val dest_sum = Nat_Arith.dest_sum;
|
haftmann@25942
|
664 |
|
haftmann@30934
|
665 |
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
|
haftmann@25942
|
666 |
|
haftmann@30934
|
667 |
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
|
haftmann@35050
|
668 |
(@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
|
wenzelm@41798
|
669 |
)
|
haftmann@25942
|
670 |
*}
|
haftmann@25942
|
671 |
|
wenzelm@44467
|
672 |
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
|
wenzelm@44467
|
673 |
|
haftmann@26100
|
674 |
|
haftmann@26100
|
675 |
subsubsection {* Quotient *}
|
haftmann@26100
|
676 |
|
haftmann@26100
|
677 |
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
|
nipkow@29667
|
678 |
by (simp add: le_div_geq linorder_not_less)
|
haftmann@26100
|
679 |
|
haftmann@26100
|
680 |
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
|
nipkow@29667
|
681 |
by (simp add: div_geq)
|
haftmann@26100
|
682 |
|
haftmann@26100
|
683 |
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
|
nipkow@29667
|
684 |
by simp
|
haftmann@26100
|
685 |
|
haftmann@26100
|
686 |
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
|
nipkow@29667
|
687 |
by simp
|
haftmann@26100
|
688 |
|
haftmann@25942
|
689 |
|
haftmann@25942
|
690 |
subsubsection {* Remainder *}
|
haftmann@25942
|
691 |
|
haftmann@26100
|
692 |
lemma mod_less_divisor [simp]:
|
haftmann@26100
|
693 |
fixes m n :: nat
|
haftmann@26100
|
694 |
assumes "n > 0"
|
haftmann@26100
|
695 |
shows "m mod n < (n::nat)"
|
haftmann@33335
|
696 |
using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
|
haftmann@25942
|
697 |
|
haftmann@26100
|
698 |
lemma mod_less_eq_dividend [simp]:
|
haftmann@26100
|
699 |
fixes m n :: nat
|
haftmann@26100
|
700 |
shows "m mod n \<le> m"
|
haftmann@26100
|
701 |
proof (rule add_leD2)
|
haftmann@26100
|
702 |
from mod_div_equality have "m div n * n + m mod n = m" .
|
haftmann@26100
|
703 |
then show "m div n * n + m mod n \<le> m" by auto
|
haftmann@26100
|
704 |
qed
|
haftmann@26100
|
705 |
|
haftmann@26100
|
706 |
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
|
nipkow@29667
|
707 |
by (simp add: le_mod_geq linorder_not_less)
|
paulson@14267
|
708 |
|
haftmann@26100
|
709 |
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
|
nipkow@29667
|
710 |
by (simp add: le_mod_geq)
|
haftmann@26100
|
711 |
|
paulson@14267
|
712 |
lemma mod_1 [simp]: "m mod Suc 0 = 0"
|
nipkow@29667
|
713 |
by (induct m) (simp_all add: mod_geq)
|
paulson@14267
|
714 |
|
haftmann@26100
|
715 |
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
|
wenzelm@22718
|
716 |
apply (cases "n = 0", simp)
|
wenzelm@22718
|
717 |
apply (cases "k = 0", simp)
|
wenzelm@22718
|
718 |
apply (induct m rule: nat_less_induct)
|
wenzelm@22718
|
719 |
apply (subst mod_if, simp)
|
wenzelm@22718
|
720 |
apply (simp add: mod_geq diff_mult_distrib)
|
wenzelm@22718
|
721 |
done
|
paulson@14267
|
722 |
|
paulson@14267
|
723 |
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
|
nipkow@29667
|
724 |
by (simp add: mult_commute [of k] mod_mult_distrib)
|
paulson@14267
|
725 |
|
paulson@14267
|
726 |
(* a simple rearrangement of mod_div_equality: *)
|
paulson@14267
|
727 |
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
|
nipkow@29667
|
728 |
by (cut_tac a = m and b = n in mod_div_equality2, arith)
|
paulson@14267
|
729 |
|
nipkow@15439
|
730 |
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
|
wenzelm@22718
|
731 |
apply (drule mod_less_divisor [where m = m])
|
wenzelm@22718
|
732 |
apply simp
|
wenzelm@22718
|
733 |
done
|
paulson@14267
|
734 |
|
haftmann@26100
|
735 |
subsubsection {* Quotient and Remainder *}
|
paulson@14267
|
736 |
|
haftmann@33335
|
737 |
lemma divmod_nat_rel_mult1_eq:
|
bulwahn@47420
|
738 |
"divmod_nat_rel b c (q, r)
|
haftmann@33335
|
739 |
\<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
|
haftmann@33335
|
740 |
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
|
paulson@14267
|
741 |
|
haftmann@30923
|
742 |
lemma div_mult1_eq:
|
haftmann@30923
|
743 |
"(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
|
huffman@48006
|
744 |
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
|
paulson@14267
|
745 |
|
haftmann@33335
|
746 |
lemma divmod_nat_rel_add1_eq:
|
bulwahn@47420
|
747 |
"divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
|
haftmann@33335
|
748 |
\<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
|
haftmann@33335
|
749 |
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
|
paulson@14267
|
750 |
|
paulson@14267
|
751 |
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
|
paulson@14267
|
752 |
lemma div_add1_eq:
|
nipkow@25134
|
753 |
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
|
huffman@48006
|
754 |
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
|
paulson@14267
|
755 |
|
paulson@14267
|
756 |
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
|
wenzelm@22718
|
757 |
apply (cut_tac m = q and n = c in mod_less_divisor)
|
wenzelm@22718
|
758 |
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
|
wenzelm@22718
|
759 |
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
|
wenzelm@22718
|
760 |
apply (simp add: add_mult_distrib2)
|
wenzelm@22718
|
761 |
done
|
paulson@14267
|
762 |
|
haftmann@33335
|
763 |
lemma divmod_nat_rel_mult2_eq:
|
bulwahn@47420
|
764 |
"divmod_nat_rel a b (q, r)
|
haftmann@33335
|
765 |
\<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
|
haftmann@33335
|
766 |
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
|
paulson@14267
|
767 |
|
paulson@14267
|
768 |
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
|
huffman@48006
|
769 |
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
|
paulson@14267
|
770 |
|
paulson@14267
|
771 |
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
|
huffman@48006
|
772 |
by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
|
paulson@14267
|
773 |
|
paulson@14267
|
774 |
|
huffman@47419
|
775 |
subsubsection {* Further Facts about Quotient and Remainder *}
|
paulson@14267
|
776 |
|
paulson@14267
|
777 |
lemma div_1 [simp]: "m div Suc 0 = m"
|
nipkow@29667
|
778 |
by (induct m) (simp_all add: div_geq)
|
paulson@14267
|
779 |
|
paulson@14267
|
780 |
(* Monotonicity of div in first argument *)
|
haftmann@30923
|
781 |
lemma div_le_mono [rule_format (no_asm)]:
|
wenzelm@22718
|
782 |
"\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
|
paulson@14267
|
783 |
apply (case_tac "k=0", simp)
|
paulson@15251
|
784 |
apply (induct "n" rule: nat_less_induct, clarify)
|
paulson@14267
|
785 |
apply (case_tac "n<k")
|
paulson@14267
|
786 |
(* 1 case n<k *)
|
paulson@14267
|
787 |
apply simp
|
paulson@14267
|
788 |
(* 2 case n >= k *)
|
paulson@14267
|
789 |
apply (case_tac "m<k")
|
paulson@14267
|
790 |
(* 2.1 case m<k *)
|
paulson@14267
|
791 |
apply simp
|
paulson@14267
|
792 |
(* 2.2 case m>=k *)
|
nipkow@15439
|
793 |
apply (simp add: div_geq diff_le_mono)
|
paulson@14267
|
794 |
done
|
paulson@14267
|
795 |
|
paulson@14267
|
796 |
(* Antimonotonicity of div in second argument *)
|
paulson@14267
|
797 |
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
|
paulson@14267
|
798 |
apply (subgoal_tac "0<n")
|
wenzelm@22718
|
799 |
prefer 2 apply simp
|
paulson@15251
|
800 |
apply (induct_tac k rule: nat_less_induct)
|
paulson@14267
|
801 |
apply (rename_tac "k")
|
paulson@14267
|
802 |
apply (case_tac "k<n", simp)
|
paulson@14267
|
803 |
apply (subgoal_tac "~ (k<m) ")
|
wenzelm@22718
|
804 |
prefer 2 apply simp
|
paulson@14267
|
805 |
apply (simp add: div_geq)
|
paulson@15251
|
806 |
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
|
paulson@14267
|
807 |
prefer 2
|
paulson@14267
|
808 |
apply (blast intro: div_le_mono diff_le_mono2)
|
paulson@14267
|
809 |
apply (rule le_trans, simp)
|
nipkow@15439
|
810 |
apply (simp)
|
paulson@14267
|
811 |
done
|
paulson@14267
|
812 |
|
paulson@14267
|
813 |
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
|
paulson@14267
|
814 |
apply (case_tac "n=0", simp)
|
paulson@14267
|
815 |
apply (subgoal_tac "m div n \<le> m div 1", simp)
|
paulson@14267
|
816 |
apply (rule div_le_mono2)
|
paulson@14267
|
817 |
apply (simp_all (no_asm_simp))
|
paulson@14267
|
818 |
done
|
paulson@14267
|
819 |
|
wenzelm@22718
|
820 |
(* Similar for "less than" *)
|
paulson@17085
|
821 |
lemma div_less_dividend [rule_format]:
|
paulson@14267
|
822 |
"!!n::nat. 1<n ==> 0 < m --> m div n < m"
|
paulson@15251
|
823 |
apply (induct_tac m rule: nat_less_induct)
|
paulson@14267
|
824 |
apply (rename_tac "m")
|
paulson@14267
|
825 |
apply (case_tac "m<n", simp)
|
paulson@14267
|
826 |
apply (subgoal_tac "0<n")
|
wenzelm@22718
|
827 |
prefer 2 apply simp
|
paulson@14267
|
828 |
apply (simp add: div_geq)
|
paulson@14267
|
829 |
apply (case_tac "n<m")
|
paulson@15251
|
830 |
apply (subgoal_tac "(m-n) div n < (m-n) ")
|
paulson@14267
|
831 |
apply (rule impI less_trans_Suc)+
|
paulson@14267
|
832 |
apply assumption
|
nipkow@15439
|
833 |
apply (simp_all)
|
paulson@14267
|
834 |
done
|
paulson@14267
|
835 |
|
paulson@17085
|
836 |
declare div_less_dividend [simp]
|
paulson@17085
|
837 |
|
paulson@14267
|
838 |
text{*A fact for the mutilated chess board*}
|
paulson@14267
|
839 |
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
|
paulson@14267
|
840 |
apply (case_tac "n=0", simp)
|
paulson@15251
|
841 |
apply (induct "m" rule: nat_less_induct)
|
paulson@14267
|
842 |
apply (case_tac "Suc (na) <n")
|
paulson@14267
|
843 |
(* case Suc(na) < n *)
|
paulson@14267
|
844 |
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
|
paulson@14267
|
845 |
(* case n \<le> Suc(na) *)
|
paulson@16796
|
846 |
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
|
nipkow@15439
|
847 |
apply (auto simp add: Suc_diff_le le_mod_geq)
|
paulson@14267
|
848 |
done
|
paulson@14267
|
849 |
|
paulson@14267
|
850 |
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
|
nipkow@29667
|
851 |
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
|
paulson@17084
|
852 |
|
wenzelm@22718
|
853 |
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
|
paulson@14267
|
854 |
|
paulson@14267
|
855 |
(*Loses information, namely we also have r<d provided d is nonzero*)
|
paulson@14267
|
856 |
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
|
haftmann@27651
|
857 |
apply (cut_tac a = m in mod_div_equality)
|
wenzelm@22718
|
858 |
apply (simp only: add_ac)
|
wenzelm@22718
|
859 |
apply (blast intro: sym)
|
wenzelm@22718
|
860 |
done
|
paulson@14267
|
861 |
|
nipkow@13152
|
862 |
lemma split_div:
|
nipkow@13189
|
863 |
"P(n div k :: nat) =
|
nipkow@13189
|
864 |
((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
|
nipkow@13189
|
865 |
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
|
nipkow@13189
|
866 |
proof
|
nipkow@13189
|
867 |
assume P: ?P
|
nipkow@13189
|
868 |
show ?Q
|
nipkow@13189
|
869 |
proof (cases)
|
nipkow@13189
|
870 |
assume "k = 0"
|
haftmann@27651
|
871 |
with P show ?Q by simp
|
nipkow@13189
|
872 |
next
|
nipkow@13189
|
873 |
assume not0: "k \<noteq> 0"
|
nipkow@13189
|
874 |
thus ?Q
|
nipkow@13189
|
875 |
proof (simp, intro allI impI)
|
nipkow@13189
|
876 |
fix i j
|
nipkow@13189
|
877 |
assume n: "n = k*i + j" and j: "j < k"
|
nipkow@13189
|
878 |
show "P i"
|
nipkow@13189
|
879 |
proof (cases)
|
wenzelm@22718
|
880 |
assume "i = 0"
|
wenzelm@22718
|
881 |
with n j P show "P i" by simp
|
nipkow@13189
|
882 |
next
|
wenzelm@22718
|
883 |
assume "i \<noteq> 0"
|
wenzelm@22718
|
884 |
with not0 n j P show "P i" by(simp add:add_ac)
|
nipkow@13189
|
885 |
qed
|
nipkow@13189
|
886 |
qed
|
nipkow@13189
|
887 |
qed
|
nipkow@13189
|
888 |
next
|
nipkow@13189
|
889 |
assume Q: ?Q
|
nipkow@13189
|
890 |
show ?P
|
nipkow@13189
|
891 |
proof (cases)
|
nipkow@13189
|
892 |
assume "k = 0"
|
haftmann@27651
|
893 |
with Q show ?P by simp
|
nipkow@13189
|
894 |
next
|
nipkow@13189
|
895 |
assume not0: "k \<noteq> 0"
|
nipkow@13189
|
896 |
with Q have R: ?R by simp
|
nipkow@13189
|
897 |
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
|
nipkow@13517
|
898 |
show ?P by simp
|
nipkow@13189
|
899 |
qed
|
nipkow@13189
|
900 |
qed
|
nipkow@13189
|
901 |
|
berghofe@13882
|
902 |
lemma split_div_lemma:
|
haftmann@26100
|
903 |
assumes "0 < n"
|
haftmann@26100
|
904 |
shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
|
haftmann@26100
|
905 |
proof
|
haftmann@26100
|
906 |
assume ?rhs
|
haftmann@26100
|
907 |
with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
|
haftmann@26100
|
908 |
then have A: "n * q \<le> m" by simp
|
haftmann@26100
|
909 |
have "n - (m mod n) > 0" using mod_less_divisor assms by auto
|
haftmann@26100
|
910 |
then have "m < m + (n - (m mod n))" by simp
|
haftmann@26100
|
911 |
then have "m < n + (m - (m mod n))" by simp
|
haftmann@26100
|
912 |
with nq have "m < n + n * q" by simp
|
haftmann@26100
|
913 |
then have B: "m < n * Suc q" by simp
|
haftmann@26100
|
914 |
from A B show ?lhs ..
|
haftmann@26100
|
915 |
next
|
haftmann@26100
|
916 |
assume P: ?lhs
|
haftmann@33335
|
917 |
then have "divmod_nat_rel m n (q, m - n * q)"
|
haftmann@33335
|
918 |
unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
|
haftmann@33335
|
919 |
with divmod_nat_rel_unique divmod_nat_rel [of m n]
|
haftmann@30923
|
920 |
have "(q, m - n * q) = (m div n, m mod n)" by auto
|
haftmann@30923
|
921 |
then show ?rhs by simp
|
haftmann@26100
|
922 |
qed
|
berghofe@13882
|
923 |
|
berghofe@13882
|
924 |
theorem split_div':
|
berghofe@13882
|
925 |
"P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
|
paulson@14267
|
926 |
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
|
berghofe@13882
|
927 |
apply (case_tac "0 < n")
|
berghofe@13882
|
928 |
apply (simp only: add: split_div_lemma)
|
haftmann@27651
|
929 |
apply simp_all
|
berghofe@13882
|
930 |
done
|
berghofe@13882
|
931 |
|
nipkow@13189
|
932 |
lemma split_mod:
|
nipkow@13189
|
933 |
"P(n mod k :: nat) =
|
nipkow@13189
|
934 |
((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
|
nipkow@13189
|
935 |
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
|
nipkow@13189
|
936 |
proof
|
nipkow@13189
|
937 |
assume P: ?P
|
nipkow@13189
|
938 |
show ?Q
|
nipkow@13189
|
939 |
proof (cases)
|
nipkow@13189
|
940 |
assume "k = 0"
|
haftmann@27651
|
941 |
with P show ?Q by simp
|
nipkow@13189
|
942 |
next
|
nipkow@13189
|
943 |
assume not0: "k \<noteq> 0"
|
nipkow@13189
|
944 |
thus ?Q
|
nipkow@13189
|
945 |
proof (simp, intro allI impI)
|
nipkow@13189
|
946 |
fix i j
|
nipkow@13189
|
947 |
assume "n = k*i + j" "j < k"
|
nipkow@13189
|
948 |
thus "P j" using not0 P by(simp add:add_ac mult_ac)
|
nipkow@13189
|
949 |
qed
|
nipkow@13189
|
950 |
qed
|
nipkow@13189
|
951 |
next
|
nipkow@13189
|
952 |
assume Q: ?Q
|
nipkow@13189
|
953 |
show ?P
|
nipkow@13189
|
954 |
proof (cases)
|
nipkow@13189
|
955 |
assume "k = 0"
|
haftmann@27651
|
956 |
with Q show ?P by simp
|
nipkow@13189
|
957 |
next
|
nipkow@13189
|
958 |
assume not0: "k \<noteq> 0"
|
nipkow@13189
|
959 |
with Q have R: ?R by simp
|
nipkow@13189
|
960 |
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
|
nipkow@13517
|
961 |
show ?P by simp
|
nipkow@13189
|
962 |
qed
|
nipkow@13189
|
963 |
qed
|
nipkow@13189
|
964 |
|
berghofe@13882
|
965 |
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
|
berghofe@13882
|
966 |
apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
|
berghofe@13882
|
967 |
subst [OF mod_div_equality [of _ n]])
|
berghofe@13882
|
968 |
apply arith
|
berghofe@13882
|
969 |
done
|
berghofe@13882
|
970 |
|
haftmann@22800
|
971 |
lemma div_mod_equality':
|
haftmann@22800
|
972 |
fixes m n :: nat
|
haftmann@22800
|
973 |
shows "m div n * n = m - m mod n"
|
haftmann@22800
|
974 |
proof -
|
haftmann@22800
|
975 |
have "m mod n \<le> m mod n" ..
|
haftmann@22800
|
976 |
from div_mod_equality have
|
haftmann@22800
|
977 |
"m div n * n + m mod n - m mod n = m - m mod n" by simp
|
haftmann@22800
|
978 |
with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
|
haftmann@22800
|
979 |
"m div n * n + (m mod n - m mod n) = m - m mod n"
|
haftmann@22800
|
980 |
by simp
|
haftmann@22800
|
981 |
then show ?thesis by simp
|
haftmann@22800
|
982 |
qed
|
haftmann@22800
|
983 |
|
haftmann@22800
|
984 |
|
huffman@47419
|
985 |
subsubsection {* An ``induction'' law for modulus arithmetic. *}
|
paulson@14640
|
986 |
|
paulson@14640
|
987 |
lemma mod_induct_0:
|
paulson@14640
|
988 |
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
|
paulson@14640
|
989 |
and base: "P i" and i: "i<p"
|
paulson@14640
|
990 |
shows "P 0"
|
paulson@14640
|
991 |
proof (rule ccontr)
|
paulson@14640
|
992 |
assume contra: "\<not>(P 0)"
|
paulson@14640
|
993 |
from i have p: "0<p" by simp
|
paulson@14640
|
994 |
have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
|
paulson@14640
|
995 |
proof
|
paulson@14640
|
996 |
fix k
|
paulson@14640
|
997 |
show "?A k"
|
paulson@14640
|
998 |
proof (induct k)
|
paulson@14640
|
999 |
show "?A 0" by simp -- "by contradiction"
|
paulson@14640
|
1000 |
next
|
paulson@14640
|
1001 |
fix n
|
paulson@14640
|
1002 |
assume ih: "?A n"
|
paulson@14640
|
1003 |
show "?A (Suc n)"
|
paulson@14640
|
1004 |
proof (clarsimp)
|
wenzelm@22718
|
1005 |
assume y: "P (p - Suc n)"
|
wenzelm@22718
|
1006 |
have n: "Suc n < p"
|
wenzelm@22718
|
1007 |
proof (rule ccontr)
|
wenzelm@22718
|
1008 |
assume "\<not>(Suc n < p)"
|
wenzelm@22718
|
1009 |
hence "p - Suc n = 0"
|
wenzelm@22718
|
1010 |
by simp
|
wenzelm@22718
|
1011 |
with y contra show "False"
|
wenzelm@22718
|
1012 |
by simp
|
wenzelm@22718
|
1013 |
qed
|
wenzelm@22718
|
1014 |
hence n2: "Suc (p - Suc n) = p-n" by arith
|
wenzelm@22718
|
1015 |
from p have "p - Suc n < p" by arith
|
wenzelm@22718
|
1016 |
with y step have z: "P ((Suc (p - Suc n)) mod p)"
|
wenzelm@22718
|
1017 |
by blast
|
wenzelm@22718
|
1018 |
show "False"
|
wenzelm@22718
|
1019 |
proof (cases "n=0")
|
wenzelm@22718
|
1020 |
case True
|
wenzelm@22718
|
1021 |
with z n2 contra show ?thesis by simp
|
wenzelm@22718
|
1022 |
next
|
wenzelm@22718
|
1023 |
case False
|
wenzelm@22718
|
1024 |
with p have "p-n < p" by arith
|
wenzelm@22718
|
1025 |
with z n2 False ih show ?thesis by simp
|
wenzelm@22718
|
1026 |
qed
|
paulson@14640
|
1027 |
qed
|
paulson@14640
|
1028 |
qed
|
paulson@14640
|
1029 |
qed
|
paulson@14640
|
1030 |
moreover
|
paulson@14640
|
1031 |
from i obtain k where "0<k \<and> i+k=p"
|
paulson@14640
|
1032 |
by (blast dest: less_imp_add_positive)
|
paulson@14640
|
1033 |
hence "0<k \<and> i=p-k" by auto
|
paulson@14640
|
1034 |
moreover
|
paulson@14640
|
1035 |
note base
|
paulson@14640
|
1036 |
ultimately
|
paulson@14640
|
1037 |
show "False" by blast
|
paulson@14640
|
1038 |
qed
|
paulson@14640
|
1039 |
|
paulson@14640
|
1040 |
lemma mod_induct:
|
paulson@14640
|
1041 |
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
|
paulson@14640
|
1042 |
and base: "P i" and i: "i<p" and j: "j<p"
|
paulson@14640
|
1043 |
shows "P j"
|
paulson@14640
|
1044 |
proof -
|
paulson@14640
|
1045 |
have "\<forall>j<p. P j"
|
paulson@14640
|
1046 |
proof
|
paulson@14640
|
1047 |
fix j
|
paulson@14640
|
1048 |
show "j<p \<longrightarrow> P j" (is "?A j")
|
paulson@14640
|
1049 |
proof (induct j)
|
paulson@14640
|
1050 |
from step base i show "?A 0"
|
wenzelm@22718
|
1051 |
by (auto elim: mod_induct_0)
|
paulson@14640
|
1052 |
next
|
paulson@14640
|
1053 |
fix k
|
paulson@14640
|
1054 |
assume ih: "?A k"
|
paulson@14640
|
1055 |
show "?A (Suc k)"
|
paulson@14640
|
1056 |
proof
|
wenzelm@22718
|
1057 |
assume suc: "Suc k < p"
|
wenzelm@22718
|
1058 |
hence k: "k<p" by simp
|
wenzelm@22718
|
1059 |
with ih have "P k" ..
|
wenzelm@22718
|
1060 |
with step k have "P (Suc k mod p)"
|
wenzelm@22718
|
1061 |
by blast
|
wenzelm@22718
|
1062 |
moreover
|
wenzelm@22718
|
1063 |
from suc have "Suc k mod p = Suc k"
|
wenzelm@22718
|
1064 |
by simp
|
wenzelm@22718
|
1065 |
ultimately
|
wenzelm@22718
|
1066 |
show "P (Suc k)" by simp
|
paulson@14640
|
1067 |
qed
|
paulson@14640
|
1068 |
qed
|
paulson@14640
|
1069 |
qed
|
paulson@14640
|
1070 |
with j show ?thesis by blast
|
paulson@14640
|
1071 |
qed
|
paulson@14640
|
1072 |
|
haftmann@33296
|
1073 |
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
|
haftmann@33296
|
1074 |
by (auto simp add: numeral_2_eq_2 le_div_geq)
|
haftmann@33296
|
1075 |
|
haftmann@33296
|
1076 |
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
|
haftmann@33296
|
1077 |
by (simp add: nat_mult_2 [symmetric])
|
haftmann@33296
|
1078 |
|
haftmann@33296
|
1079 |
lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
|
haftmann@33296
|
1080 |
apply (subgoal_tac "m mod 2 < 2")
|
haftmann@33296
|
1081 |
apply (erule less_2_cases [THEN disjE])
|
huffman@35208
|
1082 |
apply (simp_all (no_asm_simp) add: Let_def mod_Suc)
|
haftmann@33296
|
1083 |
done
|
haftmann@33296
|
1084 |
|
haftmann@33296
|
1085 |
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
|
haftmann@33296
|
1086 |
proof -
|
boehmes@35815
|
1087 |
{ fix n :: nat have "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
|
haftmann@33296
|
1088 |
moreover have "m mod 2 < 2" by simp
|
haftmann@33296
|
1089 |
ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
|
haftmann@33296
|
1090 |
then show ?thesis by auto
|
haftmann@33296
|
1091 |
qed
|
haftmann@33296
|
1092 |
|
haftmann@33296
|
1093 |
text{*These lemmas collapse some needless occurrences of Suc:
|
haftmann@33296
|
1094 |
at least three Sucs, since two and fewer are rewritten back to Suc again!
|
haftmann@33296
|
1095 |
We already have some rules to simplify operands smaller than 3.*}
|
haftmann@33296
|
1096 |
|
haftmann@33296
|
1097 |
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
|
haftmann@33296
|
1098 |
by (simp add: Suc3_eq_add_3)
|
haftmann@33296
|
1099 |
|
haftmann@33296
|
1100 |
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
|
haftmann@33296
|
1101 |
by (simp add: Suc3_eq_add_3)
|
haftmann@33296
|
1102 |
|
haftmann@33296
|
1103 |
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
|
haftmann@33296
|
1104 |
by (simp add: Suc3_eq_add_3)
|
haftmann@33296
|
1105 |
|
haftmann@33296
|
1106 |
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
|
haftmann@33296
|
1107 |
by (simp add: Suc3_eq_add_3)
|
haftmann@33296
|
1108 |
|
huffman@47978
|
1109 |
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
|
huffman@47978
|
1110 |
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
|
haftmann@33296
|
1111 |
|
haftmann@33361
|
1112 |
|
haftmann@33361
|
1113 |
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
|
haftmann@33361
|
1114 |
apply (induct "m")
|
haftmann@33361
|
1115 |
apply (simp_all add: mod_Suc)
|
haftmann@33361
|
1116 |
done
|
haftmann@33361
|
1117 |
|
huffman@47978
|
1118 |
declare Suc_times_mod_eq [of "numeral w", simp] for w
|
haftmann@33361
|
1119 |
|
haftmann@33361
|
1120 |
lemma [simp]: "n div k \<le> (Suc n) div k"
|
haftmann@33361
|
1121 |
by (simp add: div_le_mono)
|
haftmann@33361
|
1122 |
|
haftmann@33361
|
1123 |
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
|
haftmann@33361
|
1124 |
by (cases n) simp_all
|
haftmann@33361
|
1125 |
|
boehmes@35815
|
1126 |
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
|
boehmes@35815
|
1127 |
proof -
|
boehmes@35815
|
1128 |
from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
|
boehmes@35815
|
1129 |
from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp
|
boehmes@35815
|
1130 |
qed
|
haftmann@33361
|
1131 |
|
haftmann@33361
|
1132 |
(* Potential use of algebra : Equality modulo n*)
|
haftmann@33361
|
1133 |
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
|
haftmann@33361
|
1134 |
by (simp add: mult_ac add_ac)
|
haftmann@33361
|
1135 |
|
haftmann@33361
|
1136 |
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
|
haftmann@33361
|
1137 |
proof -
|
haftmann@33361
|
1138 |
have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
|
haftmann@33361
|
1139 |
also have "... = Suc m mod n" by (rule mod_mult_self3)
|
haftmann@33361
|
1140 |
finally show ?thesis .
|
haftmann@33361
|
1141 |
qed
|
haftmann@33361
|
1142 |
|
haftmann@33361
|
1143 |
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
|
haftmann@33361
|
1144 |
apply (subst mod_Suc [of m])
|
haftmann@33361
|
1145 |
apply (subst mod_Suc [of "m mod n"], simp)
|
haftmann@33361
|
1146 |
done
|
haftmann@33361
|
1147 |
|
huffman@47978
|
1148 |
lemma mod_2_not_eq_zero_eq_one_nat:
|
huffman@47978
|
1149 |
fixes n :: nat
|
huffman@47978
|
1150 |
shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
|
huffman@47978
|
1151 |
by simp
|
huffman@47978
|
1152 |
|
haftmann@33361
|
1153 |
|
haftmann@33361
|
1154 |
subsection {* Division on @{typ int} *}
|
haftmann@33361
|
1155 |
|
haftmann@33361
|
1156 |
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
|
haftmann@33361
|
1157 |
--{*definition of quotient and remainder*}
|
huffman@47978
|
1158 |
"divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
|
haftmann@33361
|
1159 |
(if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
|
haftmann@33361
|
1160 |
|
haftmann@33361
|
1161 |
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
|
haftmann@33361
|
1162 |
--{*for the division algorithm*}
|
huffman@47978
|
1163 |
"adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
|
haftmann@33361
|
1164 |
else (2 * q, r))"
|
haftmann@33361
|
1165 |
|
haftmann@33361
|
1166 |
text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
|
haftmann@33361
|
1167 |
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
|
haftmann@33361
|
1168 |
"posDivAlg a b = (if a < b \<or> b \<le> 0 then (0, a)
|
haftmann@33361
|
1169 |
else adjust b (posDivAlg a (2 * b)))"
|
haftmann@33361
|
1170 |
by auto
|
haftmann@33361
|
1171 |
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
|
haftmann@33361
|
1172 |
(auto simp add: mult_2)
|
haftmann@33361
|
1173 |
|
haftmann@33361
|
1174 |
text{*algorithm for the case @{text "a<0, b>0"}*}
|
haftmann@33361
|
1175 |
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
|
haftmann@33361
|
1176 |
"negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0 then (-1, a + b)
|
haftmann@33361
|
1177 |
else adjust b (negDivAlg a (2 * b)))"
|
haftmann@33361
|
1178 |
by auto
|
haftmann@33361
|
1179 |
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
|
haftmann@33361
|
1180 |
(auto simp add: mult_2)
|
haftmann@33361
|
1181 |
|
haftmann@33361
|
1182 |
text{*algorithm for the general case @{term "b\<noteq>0"}*}
|
haftmann@33361
|
1183 |
|
haftmann@33361
|
1184 |
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
|
haftmann@33361
|
1185 |
--{*The full division algorithm considers all possible signs for a, b
|
haftmann@33361
|
1186 |
including the special case @{text "a=0, b<0"} because
|
haftmann@33361
|
1187 |
@{term negDivAlg} requires @{term "a<0"}.*}
|
haftmann@33361
|
1188 |
"divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
|
haftmann@33361
|
1189 |
else if a = 0 then (0, 0)
|
huffman@47428
|
1190 |
else apsnd uminus (negDivAlg (-a) (-b))
|
haftmann@33361
|
1191 |
else
|
haftmann@33361
|
1192 |
if 0 < b then negDivAlg a b
|
huffman@47428
|
1193 |
else apsnd uminus (posDivAlg (-a) (-b)))"
|
haftmann@33361
|
1194 |
|
haftmann@33361
|
1195 |
instantiation int :: Divides.div
|
haftmann@33361
|
1196 |
begin
|
haftmann@33361
|
1197 |
|
huffman@47419
|
1198 |
definition div_int where
|
haftmann@33361
|
1199 |
"a div b = fst (divmod_int a b)"
|
haftmann@33361
|
1200 |
|
huffman@47419
|
1201 |
lemma fst_divmod_int [simp]:
|
huffman@47419
|
1202 |
"fst (divmod_int a b) = a div b"
|
huffman@47419
|
1203 |
by (simp add: div_int_def)
|
huffman@47419
|
1204 |
|
huffman@47419
|
1205 |
definition mod_int where
|
huffman@47428
|
1206 |
"a mod b = snd (divmod_int a b)"
|
haftmann@33361
|
1207 |
|
huffman@47419
|
1208 |
lemma snd_divmod_int [simp]:
|
huffman@47419
|
1209 |
"snd (divmod_int a b) = a mod b"
|
huffman@47419
|
1210 |
by (simp add: mod_int_def)
|
huffman@47419
|
1211 |
|
haftmann@33361
|
1212 |
instance ..
|
haftmann@33361
|
1213 |
|
paulson@3366
|
1214 |
end
|
haftmann@33361
|
1215 |
|
haftmann@33361
|
1216 |
lemma divmod_int_mod_div:
|
haftmann@33361
|
1217 |
"divmod_int p q = (p div q, p mod q)"
|
huffman@47419
|
1218 |
by (simp add: prod_eq_iff)
|
haftmann@33361
|
1219 |
|
haftmann@33361
|
1220 |
text{*
|
haftmann@33361
|
1221 |
Here is the division algorithm in ML:
|
haftmann@33361
|
1222 |
|
haftmann@33361
|
1223 |
\begin{verbatim}
|
haftmann@33361
|
1224 |
fun posDivAlg (a,b) =
|
haftmann@33361
|
1225 |
if a<b then (0,a)
|
haftmann@33361
|
1226 |
else let val (q,r) = posDivAlg(a, 2*b)
|
haftmann@33361
|
1227 |
in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
|
haftmann@33361
|
1228 |
end
|
haftmann@33361
|
1229 |
|
haftmann@33361
|
1230 |
fun negDivAlg (a,b) =
|
haftmann@33361
|
1231 |
if 0\<le>a+b then (~1,a+b)
|
haftmann@33361
|
1232 |
else let val (q,r) = negDivAlg(a, 2*b)
|
haftmann@33361
|
1233 |
in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
|
haftmann@33361
|
1234 |
end;
|
haftmann@33361
|
1235 |
|
haftmann@33361
|
1236 |
fun negateSnd (q,r:int) = (q,~r);
|
haftmann@33361
|
1237 |
|
haftmann@33361
|
1238 |
fun divmod (a,b) = if 0\<le>a then
|
haftmann@33361
|
1239 |
if b>0 then posDivAlg (a,b)
|
haftmann@33361
|
1240 |
else if a=0 then (0,0)
|
haftmann@33361
|
1241 |
else negateSnd (negDivAlg (~a,~b))
|
haftmann@33361
|
1242 |
else
|
haftmann@33361
|
1243 |
if 0<b then negDivAlg (a,b)
|
haftmann@33361
|
1244 |
else negateSnd (posDivAlg (~a,~b));
|
haftmann@33361
|
1245 |
\end{verbatim}
|
haftmann@33361
|
1246 |
*}
|
haftmann@33361
|
1247 |
|
haftmann@33361
|
1248 |
|
huffman@47419
|
1249 |
subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
|
haftmann@33361
|
1250 |
|
haftmann@33361
|
1251 |
lemma unique_quotient_lemma:
|
haftmann@33361
|
1252 |
"[| b*q' + r' \<le> b*q + r; 0 \<le> r'; r' < b; r < b |]
|
haftmann@33361
|
1253 |
==> q' \<le> (q::int)"
|
haftmann@33361
|
1254 |
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
|
haftmann@33361
|
1255 |
prefer 2 apply (simp add: right_diff_distrib)
|
haftmann@33361
|
1256 |
apply (subgoal_tac "0 < b * (1 + q - q') ")
|
haftmann@33361
|
1257 |
apply (erule_tac [2] order_le_less_trans)
|
haftmann@33361
|
1258 |
prefer 2 apply (simp add: right_diff_distrib right_distrib)
|
haftmann@33361
|
1259 |
apply (subgoal_tac "b * q' < b * (1 + q) ")
|
haftmann@33361
|
1260 |
prefer 2 apply (simp add: right_diff_distrib right_distrib)
|
haftmann@33361
|
1261 |
apply (simp add: mult_less_cancel_left)
|
haftmann@33361
|
1262 |
done
|
haftmann@33361
|
1263 |
|
haftmann@33361
|
1264 |
lemma unique_quotient_lemma_neg:
|
haftmann@33361
|
1265 |
"[| b*q' + r' \<le> b*q + r; r \<le> 0; b < r; b < r' |]
|
haftmann@33361
|
1266 |
==> q \<le> (q'::int)"
|
haftmann@33361
|
1267 |
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,
|
haftmann@33361
|
1268 |
auto)
|
haftmann@33361
|
1269 |
|
haftmann@33361
|
1270 |
lemma unique_quotient:
|
bulwahn@47420
|
1271 |
"[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]
|
haftmann@33361
|
1272 |
==> q = q'"
|
haftmann@33361
|
1273 |
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
|
haftmann@33361
|
1274 |
apply (blast intro: order_antisym
|
haftmann@33361
|
1275 |
dest: order_eq_refl [THEN unique_quotient_lemma]
|
haftmann@33361
|
1276 |
order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
|
haftmann@33361
|
1277 |
done
|
haftmann@33361
|
1278 |
|
haftmann@33361
|
1279 |
|
haftmann@33361
|
1280 |
lemma unique_remainder:
|
bulwahn@47420
|
1281 |
"[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]
|
haftmann@33361
|
1282 |
==> r = r'"
|
haftmann@33361
|
1283 |
apply (subgoal_tac "q = q'")
|
haftmann@33361
|
1284 |
apply (simp add: divmod_int_rel_def)
|
haftmann@33361
|
1285 |
apply (blast intro: unique_quotient)
|
haftmann@33361
|
1286 |
done
|
haftmann@33361
|
1287 |
|
haftmann@33361
|
1288 |
|
huffman@47419
|
1289 |
subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
|
haftmann@33361
|
1290 |
|
haftmann@33361
|
1291 |
text{*And positive divisors*}
|
haftmann@33361
|
1292 |
|
haftmann@33361
|
1293 |
lemma adjust_eq [simp]:
|
huffman@47978
|
1294 |
"adjust b (q, r) =
|
huffman@47978
|
1295 |
(let diff = r - b in
|
huffman@47978
|
1296 |
if 0 \<le> diff then (2 * q + 1, diff)
|
haftmann@33361
|
1297 |
else (2*q, r))"
|
huffman@47978
|
1298 |
by (simp add: Let_def adjust_def)
|
haftmann@33361
|
1299 |
|
haftmann@33361
|
1300 |
declare posDivAlg.simps [simp del]
|
haftmann@33361
|
1301 |
|
haftmann@33361
|
1302 |
text{*use with a simproc to avoid repeatedly proving the premise*}
|
haftmann@33361
|
1303 |
lemma posDivAlg_eqn:
|
haftmann@33361
|
1304 |
"0 < b ==>
|
haftmann@33361
|
1305 |
posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
|
haftmann@33361
|
1306 |
by (rule posDivAlg.simps [THEN trans], simp)
|
haftmann@33361
|
1307 |
|
haftmann@33361
|
1308 |
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
|
haftmann@33361
|
1309 |
theorem posDivAlg_correct:
|
haftmann@33361
|
1310 |
assumes "0 \<le> a" and "0 < b"
|
haftmann@33361
|
1311 |
shows "divmod_int_rel a b (posDivAlg a b)"
|
wenzelm@41798
|
1312 |
using assms
|
wenzelm@41798
|
1313 |
apply (induct a b rule: posDivAlg.induct)
|
wenzelm@41798
|
1314 |
apply auto
|
wenzelm@41798
|
1315 |
apply (simp add: divmod_int_rel_def)
|
wenzelm@41798
|
1316 |
apply (subst posDivAlg_eqn, simp add: right_distrib)
|
wenzelm@41798
|
1317 |
apply (case_tac "a < b")
|
wenzelm@41798
|
1318 |
apply simp_all
|
wenzelm@41798
|
1319 |
apply (erule splitE)
|
wenzelm@41798
|
1320 |
apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
|
wenzelm@41798
|
1321 |
done
|
haftmann@33361
|
1322 |
|
haftmann@33361
|
1323 |
|
huffman@47419
|
1324 |
subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
|
haftmann@33361
|
1325 |
|
haftmann@33361
|
1326 |
text{*And positive divisors*}
|
haftmann@33361
|
1327 |
|
haftmann@33361
|
1328 |
declare negDivAlg.simps [simp del]
|
haftmann@33361
|
1329 |
|
haftmann@33361
|
1330 |
text{*use with a simproc to avoid repeatedly proving the premise*}
|
haftmann@33361
|
1331 |
lemma negDivAlg_eqn:
|
haftmann@33361
|
1332 |
"0 < b ==>
|
haftmann@33361
|
1333 |
negDivAlg a b =
|
haftmann@33361
|
1334 |
(if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
|
haftmann@33361
|
1335 |
by (rule negDivAlg.simps [THEN trans], simp)
|
haftmann@33361
|
1336 |
|
haftmann@33361
|
1337 |
(*Correctness of negDivAlg: it computes quotients correctly
|
haftmann@33361
|
1338 |
It doesn't work if a=0 because the 0/b equals 0, not -1*)
|
haftmann@33361
|
1339 |
lemma negDivAlg_correct:
|
haftmann@33361
|
1340 |
assumes "a < 0" and "b > 0"
|
haftmann@33361
|
1341 |
shows "divmod_int_rel a b (negDivAlg a b)"
|
wenzelm@41798
|
1342 |
using assms
|
wenzelm@41798
|
1343 |
apply (induct a b rule: negDivAlg.induct)
|
wenzelm@41798
|
1344 |
apply (auto simp add: linorder_not_le)
|
wenzelm@41798
|
1345 |
apply (simp add: divmod_int_rel_def)
|
wenzelm@41798
|
1346 |
apply (subst negDivAlg_eqn, assumption)
|
wenzelm@41798
|
1347 |
apply (case_tac "a + b < (0\<Colon>int)")
|
wenzelm@41798
|
1348 |
apply simp_all
|
wenzelm@41798
|
1349 |
apply (erule splitE)
|
wenzelm@41798
|
1350 |
apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
|
wenzelm@41798
|
1351 |
done
|
haftmann@33361
|
1352 |
|
haftmann@33361
|
1353 |
|
huffman@47419
|
1354 |
subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
|
haftmann@33361
|
1355 |
|
haftmann@33361
|
1356 |
(*the case a=0*)
|
haftmann@33361
|
1357 |
lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
|
haftmann@33361
|
1358 |
by (auto simp add: divmod_int_rel_def linorder_neq_iff)
|
haftmann@33361
|
1359 |
|
haftmann@33361
|
1360 |
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
|
haftmann@33361
|
1361 |
by (subst posDivAlg.simps, auto)
|
haftmann@33361
|
1362 |
|
haftmann@33361
|
1363 |
lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
|
haftmann@33361
|
1364 |
by (subst negDivAlg.simps, auto)
|
haftmann@33361
|
1365 |
|
huffman@47428
|
1366 |
lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
|
haftmann@33361
|
1367 |
by (auto simp add: split_ifs divmod_int_rel_def)
|
haftmann@33361
|
1368 |
|
haftmann@33361
|
1369 |
lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
|
haftmann@33361
|
1370 |
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
|
haftmann@33361
|
1371 |
posDivAlg_correct negDivAlg_correct)
|
haftmann@33361
|
1372 |
|
haftmann@33361
|
1373 |
text{*Arbitrary definitions for division by zero. Useful to simplify
|
haftmann@33361
|
1374 |
certain equations.*}
|
haftmann@33361
|
1375 |
|
haftmann@33361
|
1376 |
lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
|
haftmann@33361
|
1377 |
by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)
|
haftmann@33361
|
1378 |
|
haftmann@33361
|
1379 |
|
haftmann@33361
|
1380 |
text{*Basic laws about division and remainder*}
|
haftmann@33361
|
1381 |
|
haftmann@33361
|
1382 |
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
|
haftmann@33361
|
1383 |
apply (case_tac "b = 0", simp)
|
haftmann@33361
|
1384 |
apply (cut_tac a = a and b = b in divmod_int_correct)
|
huffman@47419
|
1385 |
apply (auto simp add: divmod_int_rel_def prod_eq_iff)
|
haftmann@33361
|
1386 |
done
|
haftmann@33361
|
1387 |
|
haftmann@33361
|
1388 |
lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
|
haftmann@33361
|
1389 |
by(simp add: zmod_zdiv_equality[symmetric])
|
haftmann@33361
|
1390 |
|
haftmann@33361
|
1391 |
lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
|
haftmann@33361
|
1392 |
by(simp add: mult_commute zmod_zdiv_equality[symmetric])
|
haftmann@33361
|
1393 |
|
haftmann@33361
|
1394 |
text {* Tool setup *}
|
haftmann@33361
|
1395 |
|
huffman@47978
|
1396 |
(* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
|
huffman@47978
|
1397 |
lemmas add_0s = add_0_left add_0_right
|
huffman@47978
|
1398 |
|
haftmann@33361
|
1399 |
ML {*
|
wenzelm@44467
|
1400 |
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
|
wenzelm@41798
|
1401 |
(
|
haftmann@33361
|
1402 |
val div_name = @{const_name div};
|
haftmann@33361
|
1403 |
val mod_name = @{const_name mod};
|
haftmann@33361
|
1404 |
val mk_binop = HOLogic.mk_binop;
|
haftmann@33361
|
1405 |
val mk_sum = Arith_Data.mk_sum HOLogic.intT;
|
haftmann@33361
|
1406 |
val dest_sum = Arith_Data.dest_sum;
|
haftmann@33361
|
1407 |
|
haftmann@33361
|
1408 |
val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
|
haftmann@33361
|
1409 |
|
haftmann@33361
|
1410 |
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
|
haftmann@33361
|
1411 |
(@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
|
wenzelm@41798
|
1412 |
)
|
haftmann@33361
|
1413 |
*}
|
haftmann@33361
|
1414 |
|
wenzelm@44467
|
1415 |
simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
|
wenzelm@44467
|
1416 |
|
haftmann@33361
|
1417 |
lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
|
haftmann@33361
|
1418 |
apply (cut_tac a = a and b = b in divmod_int_correct)
|
huffman@47419
|
1419 |
apply (auto simp add: divmod_int_rel_def prod_eq_iff)
|
haftmann@33361
|
1420 |
done
|
haftmann@33361
|
1421 |
|
wenzelm@46478
|
1422 |
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
|
wenzelm@46478
|
1423 |
and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
|
haftmann@33361
|
1424 |
|
haftmann@33361
|
1425 |
lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
|
haftmann@33361
|
1426 |
apply (cut_tac a = a and b = b in divmod_int_correct)
|
huffman@47419
|
1427 |
apply (auto simp add: divmod_int_rel_def prod_eq_iff)
|
haftmann@33361
|
1428 |
done
|
haftmann@33361
|
1429 |
|
wenzelm@46478
|
1430 |
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
|
wenzelm@46478
|
1431 |
and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
|
haftmann@33361
|
1432 |
|
haftmann@33361
|
1433 |
|
huffman@47419
|
1434 |
subsubsection {* General Properties of div and mod *}
|
haftmann@33361
|
1435 |
|
haftmann@33361
|
1436 |
lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
|
haftmann@33361
|
1437 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
|
haftmann@33361
|
1438 |
apply (force simp add: divmod_int_rel_def linorder_neq_iff)
|
haftmann@33361
|
1439 |
done
|
haftmann@33361
|
1440 |
|
bulwahn@47420
|
1441 |
lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r) |] ==> a div b = q"
|
bulwahn@47420
|
1442 |
apply (cases "b = 0")
|
bulwahn@47420
|
1443 |
apply (simp add: divmod_int_rel_def)
|
haftmann@33361
|
1444 |
by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
|
haftmann@33361
|
1445 |
|
bulwahn@47420
|
1446 |
lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r) |] ==> a mod b = r"
|
bulwahn@47420
|
1447 |
apply (cases "b = 0")
|
bulwahn@47420
|
1448 |
apply (simp add: divmod_int_rel_def)
|
haftmann@33361
|
1449 |
by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
|
haftmann@33361
|
1450 |
|
haftmann@33361
|
1451 |
lemma div_pos_pos_trivial: "[| (0::int) \<le> a; a < b |] ==> a div b = 0"
|
haftmann@33361
|
1452 |
apply (rule divmod_int_rel_div)
|
haftmann@33361
|
1453 |
apply (auto simp add: divmod_int_rel_def)
|
haftmann@33361
|
1454 |
done
|
haftmann@33361
|
1455 |
|
haftmann@33361
|
1456 |
lemma div_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a div b = 0"
|
haftmann@33361
|
1457 |
apply (rule divmod_int_rel_div)
|
haftmann@33361
|
1458 |
apply (auto simp add: divmod_int_rel_def)
|
haftmann@33361
|
1459 |
done
|
haftmann@33361
|
1460 |
|
haftmann@33361
|
1461 |
lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a div b = -1"
|
haftmann@33361
|
1462 |
apply (rule divmod_int_rel_div)
|
haftmann@33361
|
1463 |
apply (auto simp add: divmod_int_rel_def)
|
haftmann@33361
|
1464 |
done
|
haftmann@33361
|
1465 |
|
haftmann@33361
|
1466 |
(*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)
|
haftmann@33361
|
1467 |
|
haftmann@33361
|
1468 |
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a; a < b |] ==> a mod b = a"
|
haftmann@33361
|
1469 |
apply (rule_tac q = 0 in divmod_int_rel_mod)
|
haftmann@33361
|
1470 |
apply (auto simp add: divmod_int_rel_def)
|
haftmann@33361
|
1471 |
done
|
haftmann@33361
|
1472 |
|
haftmann@33361
|
1473 |
lemma mod_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a mod b = a"
|
haftmann@33361
|
1474 |
apply (rule_tac q = 0 in divmod_int_rel_mod)
|
haftmann@33361
|
1475 |
apply (auto simp add: divmod_int_rel_def)
|
haftmann@33361
|
1476 |
done
|
haftmann@33361
|
1477 |
|
haftmann@33361
|
1478 |
lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a mod b = a+b"
|
haftmann@33361
|
1479 |
apply (rule_tac q = "-1" in divmod_int_rel_mod)
|
haftmann@33361
|
1480 |
apply (auto simp add: divmod_int_rel_def)
|
haftmann@33361
|
1481 |
done
|
haftmann@33361
|
1482 |
|
haftmann@33361
|
1483 |
text{*There is no @{text mod_neg_pos_trivial}.*}
|
haftmann@33361
|
1484 |
|
haftmann@33361
|
1485 |
|
haftmann@33361
|
1486 |
(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
|
haftmann@33361
|
1487 |
lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
|
haftmann@33361
|
1488 |
apply (case_tac "b = 0", simp)
|
haftmann@33361
|
1489 |
apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified,
|
haftmann@33361
|
1490 |
THEN divmod_int_rel_div, THEN sym])
|
haftmann@33361
|
1491 |
|
haftmann@33361
|
1492 |
done
|
haftmann@33361
|
1493 |
|
haftmann@33361
|
1494 |
(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
|
haftmann@33361
|
1495 |
lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
|
haftmann@33361
|
1496 |
apply (case_tac "b = 0", simp)
|
haftmann@33361
|
1497 |
apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],
|
haftmann@33361
|
1498 |
auto)
|
haftmann@33361
|
1499 |
done
|
haftmann@33361
|
1500 |
|
haftmann@33361
|
1501 |
|
huffman@47419
|
1502 |
subsubsection {* Laws for div and mod with Unary Minus *}
|
haftmann@33361
|
1503 |
|
haftmann@33361
|
1504 |
lemma zminus1_lemma:
|
haftmann@33361
|
1505 |
"divmod_int_rel a b (q, r)
|
haftmann@33361
|
1506 |
==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,
|
haftmann@33361
|
1507 |
if r=0 then 0 else b-r)"
|
haftmann@33361
|
1508 |
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
|
haftmann@33361
|
1509 |
|
haftmann@33361
|
1510 |
|
haftmann@33361
|
1511 |
lemma zdiv_zminus1_eq_if:
|
haftmann@33361
|
1512 |
"b \<noteq> (0::int)
|
haftmann@33361
|
1513 |
==> (-a) div b =
|
haftmann@33361
|
1514 |
(if a mod b = 0 then - (a div b) else - (a div b) - 1)"
|
haftmann@33361
|
1515 |
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])
|
haftmann@33361
|
1516 |
|
haftmann@33361
|
1517 |
lemma zmod_zminus1_eq_if:
|
haftmann@33361
|
1518 |
"(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"
|
haftmann@33361
|
1519 |
apply (case_tac "b = 0", simp)
|
haftmann@33361
|
1520 |
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])
|
haftmann@33361
|
1521 |
done
|
haftmann@33361
|
1522 |
|
haftmann@33361
|
1523 |
lemma zmod_zminus1_not_zero:
|
haftmann@33361
|
1524 |
fixes k l :: int
|
haftmann@33361
|
1525 |
shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
|
haftmann@33361
|
1526 |
unfolding zmod_zminus1_eq_if by auto
|
haftmann@33361
|
1527 |
|
haftmann@33361
|
1528 |
lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
|
haftmann@33361
|
1529 |
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
|
haftmann@33361
|
1530 |
|
haftmann@33361
|
1531 |
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
|
haftmann@33361
|
1532 |
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
|
haftmann@33361
|
1533 |
|
haftmann@33361
|
1534 |
lemma zdiv_zminus2_eq_if:
|
haftmann@33361
|
1535 |
"b \<noteq> (0::int)
|
haftmann@33361
|
1536 |
==> a div (-b) =
|
haftmann@33361
|
1537 |
(if a mod b = 0 then - (a div b) else - (a div b) - 1)"
|
haftmann@33361
|
1538 |
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
|
haftmann@33361
|
1539 |
|
haftmann@33361
|
1540 |
lemma zmod_zminus2_eq_if:
|
haftmann@33361
|
1541 |
"a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"
|
haftmann@33361
|
1542 |
by (simp add: zmod_zminus1_eq_if zmod_zminus2)
|
haftmann@33361
|
1543 |
|
haftmann@33361
|
1544 |
lemma zmod_zminus2_not_zero:
|
haftmann@33361
|
1545 |
fixes k l :: int
|
haftmann@33361
|
1546 |
shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
|
haftmann@33361
|
1547 |
unfolding zmod_zminus2_eq_if by auto
|
haftmann@33361
|
1548 |
|
haftmann@33361
|
1549 |
|
huffman@47419
|
1550 |
subsubsection {* Division of a Number by Itself *}
|
haftmann@33361
|
1551 |
|
haftmann@33361
|
1552 |
lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
|
haftmann@33361
|
1553 |
apply (subgoal_tac "0 < a*q")
|
haftmann@33361
|
1554 |
apply (simp add: zero_less_mult_iff, arith)
|
haftmann@33361
|
1555 |
done
|
haftmann@33361
|
1556 |
|
haftmann@33361
|
1557 |
lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
|
haftmann@33361
|
1558 |
apply (subgoal_tac "0 \<le> a* (1-q) ")
|
haftmann@33361
|
1559 |
apply (simp add: zero_le_mult_iff)
|
haftmann@33361
|
1560 |
apply (simp add: right_diff_distrib)
|
haftmann@33361
|
1561 |
done
|
haftmann@33361
|
1562 |
|
bulwahn@47420
|
1563 |
lemma self_quotient: "[| divmod_int_rel a a (q, r) |] ==> q = 1"
|
haftmann@33361
|
1564 |
apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
|
haftmann@33361
|
1565 |
apply (rule order_antisym, safe, simp_all)
|
haftmann@33361
|
1566 |
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
|
haftmann@33361
|
1567 |
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
|
haftmann@33361
|
1568 |
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
|
haftmann@33361
|
1569 |
done
|
haftmann@33361
|
1570 |
|
bulwahn@47420
|
1571 |
lemma self_remainder: "[| divmod_int_rel a a (q, r) |] ==> r = 0"
|
bulwahn@47420
|
1572 |
apply (frule self_quotient)
|
haftmann@33361
|
1573 |
apply (simp add: divmod_int_rel_def)
|
haftmann@33361
|
1574 |
done
|
haftmann@33361
|
1575 |
|
haftmann@33361
|
1576 |
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
|
haftmann@33361
|
1577 |
by (simp add: divmod_int_rel_div_mod [THEN self_quotient])
|
haftmann@33361
|
1578 |
|
haftmann@33361
|
1579 |
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
|
haftmann@33361
|
1580 |
lemma zmod_self [simp]: "a mod a = (0::int)"
|
haftmann@33361
|
1581 |
apply (case_tac "a = 0", simp)
|
haftmann@33361
|
1582 |
apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])
|
haftmann@33361
|
1583 |
done
|
haftmann@33361
|
1584 |
|
haftmann@33361
|
1585 |
|
huffman@47419
|
1586 |
subsubsection {* Computation of Division and Remainder *}
|
haftmann@33361
|
1587 |
|
haftmann@33361
|
1588 |
lemma zdiv_zero [simp]: "(0::int) div b = 0"
|
haftmann@33361
|
1589 |
by (simp add: div_int_def divmod_int_def)
|
haftmann@33361
|
1590 |
|
haftmann@33361
|
1591 |
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
|
haftmann@33361
|
1592 |
by (simp add: div_int_def divmod_int_def)
|
haftmann@33361
|
1593 |
|
haftmann@33361
|
1594 |
lemma zmod_zero [simp]: "(0::int) mod b = 0"
|
haftmann@33361
|
1595 |
by (simp add: mod_int_def divmod_int_def)
|
haftmann@33361
|
1596 |
|
haftmann@33361
|
1597 |
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
|
haftmann@33361
|
1598 |
by (simp add: mod_int_def divmod_int_def)
|
haftmann@33361
|
1599 |
|
haftmann@33361
|
1600 |
text{*a positive, b positive *}
|
haftmann@33361
|
1601 |
|
haftmann@33361
|
1602 |
lemma div_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
|
haftmann@33361
|
1603 |
by (simp add: div_int_def divmod_int_def)
|
haftmann@33361
|
1604 |
|
haftmann@33361
|
1605 |
lemma mod_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
|
haftmann@33361
|
1606 |
by (simp add: mod_int_def divmod_int_def)
|
haftmann@33361
|
1607 |
|
haftmann@33361
|
1608 |
text{*a negative, b positive *}
|
haftmann@33361
|
1609 |
|
haftmann@33361
|
1610 |
lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"
|
haftmann@33361
|
1611 |
by (simp add: div_int_def divmod_int_def)
|
haftmann@33361
|
1612 |
|
haftmann@33361
|
1613 |
lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"
|
haftmann@33361
|
1614 |
by (simp add: mod_int_def divmod_int_def)
|
haftmann@33361
|
1615 |
|
haftmann@33361
|
1616 |
text{*a positive, b negative *}
|
haftmann@33361
|
1617 |
|
haftmann@33361
|
1618 |
lemma div_pos_neg:
|
huffman@47428
|
1619 |
"[| 0 < a; b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
|
haftmann@33361
|
1620 |
by (simp add: div_int_def divmod_int_def)
|
haftmann@33361
|
1621 |
|
haftmann@33361
|
1622 |
lemma mod_pos_neg:
|
huffman@47428
|
1623 |
"[| 0 < a; b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
|
haftmann@33361
|
1624 |
by (simp add: mod_int_def divmod_int_def)
|
haftmann@33361
|
1625 |
|
haftmann@33361
|
1626 |
text{*a negative, b negative *}
|
haftmann@33361
|
1627 |
|
haftmann@33361
|
1628 |
lemma div_neg_neg:
|
huffman@47428
|
1629 |
"[| a < 0; b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
|
haftmann@33361
|
1630 |
by (simp add: div_int_def divmod_int_def)
|
haftmann@33361
|
1631 |
|
haftmann@33361
|
1632 |
lemma mod_neg_neg:
|
huffman@47428
|
1633 |
"[| a < 0; b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
|
haftmann@33361
|
1634 |
by (simp add: mod_int_def divmod_int_def)
|
haftmann@33361
|
1635 |
|
haftmann@33361
|
1636 |
text {*Simplify expresions in which div and mod combine numerical constants*}
|
haftmann@33361
|
1637 |
|
huffman@46401
|
1638 |
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
|
bulwahn@47420
|
1639 |
by (rule divmod_int_rel_div [of a b q r]) (simp add: divmod_int_rel_def)
|
huffman@46401
|
1640 |
|
huffman@46401
|
1641 |
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
|
huffman@46401
|
1642 |
by (rule divmod_int_rel_div [of a b q r],
|
bulwahn@47420
|
1643 |
simp add: divmod_int_rel_def)
|
huffman@46401
|
1644 |
|
huffman@46401
|
1645 |
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
|
huffman@46401
|
1646 |
by (rule divmod_int_rel_mod [of a b q r],
|
bulwahn@47420
|
1647 |
simp add: divmod_int_rel_def)
|
huffman@46401
|
1648 |
|
huffman@46401
|
1649 |
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
|
huffman@46401
|
1650 |
by (rule divmod_int_rel_mod [of a b q r],
|
bulwahn@47420
|
1651 |
simp add: divmod_int_rel_def)
|
huffman@46401
|
1652 |
|
haftmann@33361
|
1653 |
(* simprocs adapted from HOL/ex/Binary.thy *)
|
haftmann@33361
|
1654 |
ML {*
|
haftmann@33361
|
1655 |
local
|
huffman@46401
|
1656 |
val mk_number = HOLogic.mk_number HOLogic.intT
|
huffman@46401
|
1657 |
val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
|
huffman@46401
|
1658 |
val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
|
huffman@46401
|
1659 |
val zero = @{term "0 :: int"}
|
huffman@46401
|
1660 |
val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
|
huffman@46401
|
1661 |
val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
|
huffman@46401
|
1662 |
val simps = @{thms arith_simps} @ @{thms rel_simps} @
|
huffman@47978
|
1663 |
map (fn th => th RS sym) [@{thm numeral_1_eq_1}]
|
huffman@46401
|
1664 |
fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
|
huffman@46401
|
1665 |
(K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));
|
haftmann@33361
|
1666 |
fun binary_proc proc ss ct =
|
haftmann@33361
|
1667 |
(case Thm.term_of ct of
|
haftmann@33361
|
1668 |
_ $ t $ u =>
|
haftmann@33361
|
1669 |
(case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
|
haftmann@33361
|
1670 |
SOME args => proc (Simplifier.the_context ss) args
|
haftmann@33361
|
1671 |
| NONE => NONE)
|
haftmann@33361
|
1672 |
| _ => NONE);
|
haftmann@33361
|
1673 |
in
|
huffman@46401
|
1674 |
fun divmod_proc posrule negrule =
|
huffman@46401
|
1675 |
binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
|
huffman@46401
|
1676 |
if b = 0 then NONE else let
|
huffman@46401
|
1677 |
val (q, r) = pairself mk_number (Integer.div_mod a b)
|
huffman@46401
|
1678 |
val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r)
|
huffman@46401
|
1679 |
val (goal2, goal3, rule) = if b > 0
|
huffman@46401
|
1680 |
then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection)
|
huffman@46401
|
1681 |
else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection)
|
huffman@46401
|
1682 |
in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
|
haftmann@33361
|
1683 |
end
|
haftmann@33361
|
1684 |
*}
|
haftmann@33361
|
1685 |
|
huffman@47978
|
1686 |
simproc_setup binary_int_div
|
huffman@47978
|
1687 |
("numeral m div numeral n :: int" |
|
huffman@47978
|
1688 |
"numeral m div neg_numeral n :: int" |
|
huffman@47978
|
1689 |
"neg_numeral m div numeral n :: int" |
|
huffman@47978
|
1690 |
"neg_numeral m div neg_numeral n :: int") =
|
huffman@46401
|
1691 |
{* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
|
haftmann@33361
|
1692 |
|
huffman@47978
|
1693 |
simproc_setup binary_int_mod
|
huffman@47978
|
1694 |
("numeral m mod numeral n :: int" |
|
huffman@47978
|
1695 |
"numeral m mod neg_numeral n :: int" |
|
huffman@47978
|
1696 |
"neg_numeral m mod numeral n :: int" |
|
huffman@47978
|
1697 |
"neg_numeral m mod neg_numeral n :: int") =
|
huffman@46401
|
1698 |
{* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
|
haftmann@33361
|
1699 |
|
huffman@47978
|
1700 |
lemmas posDivAlg_eqn_numeral [simp] =
|
huffman@47978
|
1701 |
posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
|
huffman@47978
|
1702 |
|
huffman@47978
|
1703 |
lemmas negDivAlg_eqn_numeral [simp] =
|
huffman@47978
|
1704 |
negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w
|
haftmann@33361
|
1705 |
|
haftmann@33361
|
1706 |
|
haftmann@33361
|
1707 |
text{*Special-case simplification *}
|
haftmann@33361
|
1708 |
|
haftmann@33361
|
1709 |
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
|
haftmann@33361
|
1710 |
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
|
haftmann@33361
|
1711 |
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
|
haftmann@33361
|
1712 |
apply (auto simp del: neg_mod_sign neg_mod_bound)
|
haftmann@33361
|
1713 |
done
|
haftmann@33361
|
1714 |
|
haftmann@33361
|
1715 |
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
|
haftmann@33361
|
1716 |
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
|
haftmann@33361
|
1717 |
|
haftmann@33361
|
1718 |
(** The last remaining special cases for constant arithmetic:
|
haftmann@33361
|
1719 |
1 div z and 1 mod z **)
|
haftmann@33361
|
1720 |
|
huffman@47978
|
1721 |
lemmas div_pos_pos_1_numeral [simp] =
|
huffman@47978
|
1722 |
div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
|
huffman@47978
|
1723 |
|
huffman@47978
|
1724 |
lemmas div_pos_neg_1_numeral [simp] =
|
huffman@47978
|
1725 |
div_pos_neg [OF zero_less_one, of "neg_numeral w",
|
huffman@47978
|
1726 |
OF neg_numeral_less_zero] for w
|
huffman@47978
|
1727 |
|
huffman@47978
|
1728 |
lemmas mod_pos_pos_1_numeral [simp] =
|
huffman@47978
|
1729 |
mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
|
huffman@47978
|
1730 |
|
huffman@47978
|
1731 |
lemmas mod_pos_neg_1_numeral [simp] =
|
huffman@47978
|
1732 |
mod_pos_neg [OF zero_less_one, of "neg_numeral w",
|
huffman@47978
|
1733 |
OF neg_numeral_less_zero] for w
|
huffman@47978
|
1734 |
|
huffman@47978
|
1735 |
lemmas posDivAlg_eqn_1_numeral [simp] =
|
huffman@47978
|
1736 |
posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
|
huffman@47978
|
1737 |
|
huffman@47978
|
1738 |
lemmas negDivAlg_eqn_1_numeral [simp] =
|
huffman@47978
|
1739 |
negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
|
haftmann@33361
|
1740 |
|
haftmann@33361
|
1741 |
|
huffman@47419
|
1742 |
subsubsection {* Monotonicity in the First Argument (Dividend) *}
|
haftmann@33361
|
1743 |
|
haftmann@33361
|
1744 |
lemma zdiv_mono1: "[| a \<le> a'; 0 < (b::int) |] ==> a div b \<le> a' div b"
|
haftmann@33361
|
1745 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
|
haftmann@33361
|
1746 |
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
|
haftmann@33361
|
1747 |
apply (rule unique_quotient_lemma)
|
haftmann@33361
|
1748 |
apply (erule subst)
|
haftmann@33361
|
1749 |
apply (erule subst, simp_all)
|
haftmann@33361
|
1750 |
done
|
haftmann@33361
|
1751 |
|
haftmann@33361
|
1752 |
lemma zdiv_mono1_neg: "[| a \<le> a'; (b::int) < 0 |] ==> a' div b \<le> a div b"
|
haftmann@33361
|
1753 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
|
haftmann@33361
|
1754 |
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
|
haftmann@33361
|
1755 |
apply (rule unique_quotient_lemma_neg)
|
haftmann@33361
|
1756 |
apply (erule subst)
|
haftmann@33361
|
1757 |
apply (erule subst, simp_all)
|
haftmann@33361
|
1758 |
done
|
haftmann@33361
|
1759 |
|
haftmann@33361
|
1760 |
|
huffman@47419
|
1761 |
subsubsection {* Monotonicity in the Second Argument (Divisor) *}
|
haftmann@33361
|
1762 |
|
haftmann@33361
|
1763 |
lemma q_pos_lemma:
|
haftmann@33361
|
1764 |
"[| 0 \<le> b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \<le> (q'::int)"
|
haftmann@33361
|
1765 |
apply (subgoal_tac "0 < b'* (q' + 1) ")
|
haftmann@33361
|
1766 |
apply (simp add: zero_less_mult_iff)
|
haftmann@33361
|
1767 |
apply (simp add: right_distrib)
|
haftmann@33361
|
1768 |
done
|
haftmann@33361
|
1769 |
|
haftmann@33361
|
1770 |
lemma zdiv_mono2_lemma:
|
haftmann@33361
|
1771 |
"[| b*q + r = b'*q' + r'; 0 \<le> b'*q' + r';
|
haftmann@33361
|
1772 |
r' < b'; 0 \<le> r; 0 < b'; b' \<le> b |]
|
haftmann@33361
|
1773 |
==> q \<le> (q'::int)"
|
haftmann@33361
|
1774 |
apply (frule q_pos_lemma, assumption+)
|
haftmann@33361
|
1775 |
apply (subgoal_tac "b*q < b* (q' + 1) ")
|
haftmann@33361
|
1776 |
apply (simp add: mult_less_cancel_left)
|
haftmann@33361
|
1777 |
apply (subgoal_tac "b*q = r' - r + b'*q'")
|
haftmann@33361
|
1778 |
prefer 2 apply simp
|
haftmann@33361
|
1779 |
apply (simp (no_asm_simp) add: right_distrib)
|
huffman@45637
|
1780 |
apply (subst add_commute, rule add_less_le_mono, arith)
|
haftmann@33361
|
1781 |
apply (rule mult_right_mono, auto)
|
haftmann@33361
|
1782 |
done
|
haftmann@33361
|
1783 |
|
haftmann@33361
|
1784 |
lemma zdiv_mono2:
|
haftmann@33361
|
1785 |
"[| (0::int) \<le> a; 0 < b'; b' \<le> b |] ==> a div b \<le> a div b'"
|
haftmann@33361
|
1786 |
apply (subgoal_tac "b \<noteq> 0")
|
haftmann@33361
|
1787 |
prefer 2 apply arith
|
haftmann@33361
|
1788 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
|
haftmann@33361
|
1789 |
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
|
haftmann@33361
|
1790 |
apply (rule zdiv_mono2_lemma)
|
haftmann@33361
|
1791 |
apply (erule subst)
|
haftmann@33361
|
1792 |
apply (erule subst, simp_all)
|
haftmann@33361
|
1793 |
done
|
haftmann@33361
|
1794 |
|
haftmann@33361
|
1795 |
lemma q_neg_lemma:
|
haftmann@33361
|
1796 |
"[| b'*q' + r' < 0; 0 \<le> r'; 0 < b' |] ==> q' \<le> (0::int)"
|
haftmann@33361
|
1797 |
apply (subgoal_tac "b'*q' < 0")
|
haftmann@33361
|
1798 |
apply (simp add: mult_less_0_iff, arith)
|
haftmann@33361
|
1799 |
done
|
haftmann@33361
|
1800 |
|
haftmann@33361
|
1801 |
lemma zdiv_mono2_neg_lemma:
|
haftmann@33361
|
1802 |
"[| b*q + r = b'*q' + r'; b'*q' + r' < 0;
|
haftmann@33361
|
1803 |
r < b; 0 \<le> r'; 0 < b'; b' \<le> b |]
|
haftmann@33361
|
1804 |
==> q' \<le> (q::int)"
|
haftmann@33361
|
1805 |
apply (frule q_neg_lemma, assumption+)
|
haftmann@33361
|
1806 |
apply (subgoal_tac "b*q' < b* (q + 1) ")
|
haftmann@33361
|
1807 |
apply (simp add: mult_less_cancel_left)
|
haftmann@33361
|
1808 |
apply (simp add: right_distrib)
|
haftmann@33361
|
1809 |
apply (subgoal_tac "b*q' \<le> b'*q'")
|
haftmann@33361
|
1810 |
prefer 2 apply (simp add: mult_right_mono_neg, arith)
|
haftmann@33361
|
1811 |
done
|
haftmann@33361
|
1812 |
|
haftmann@33361
|
1813 |
lemma zdiv_mono2_neg:
|
haftmann@33361
|
1814 |
"[| a < (0::int); 0 < b'; b' \<le> b |] ==> a div b' \<le> a div b"
|
haftmann@33361
|
1815 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
|
haftmann@33361
|
1816 |
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
|
haftmann@33361
|
1817 |
apply (rule zdiv_mono2_neg_lemma)
|
haftmann@33361
|
1818 |
apply (erule subst)
|
haftmann@33361
|
1819 |
apply (erule subst, simp_all)
|
haftmann@33361
|
1820 |
done
|
haftmann@33361
|
1821 |
|
haftmann@33361
|
1822 |
|
huffman@47419
|
1823 |
subsubsection {* More Algebraic Laws for div and mod *}
|
haftmann@33361
|
1824 |
|
haftmann@33361
|
1825 |
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
|
haftmann@33361
|
1826 |
|
haftmann@33361
|
1827 |
lemma zmult1_lemma:
|
bulwahn@47420
|
1828 |
"[| divmod_int_rel b c (q, r) |]
|
haftmann@33361
|
1829 |
==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
|
haftmann@33361
|
1830 |
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
|
haftmann@33361
|
1831 |
|
haftmann@33361
|
1832 |
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
|
haftmann@33361
|
1833 |
apply (case_tac "c = 0", simp)
|
haftmann@33361
|
1834 |
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])
|
haftmann@33361
|
1835 |
done
|
haftmann@33361
|
1836 |
|
haftmann@33361
|
1837 |
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
|
haftmann@33361
|
1838 |
apply (case_tac "c = 0", simp)
|
haftmann@33361
|
1839 |
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])
|
haftmann@33361
|
1840 |
done
|
haftmann@33361
|
1841 |
|
haftmann@33361
|
1842 |
lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
|
haftmann@33361
|
1843 |
apply (case_tac "b = 0", simp)
|
haftmann@33361
|
1844 |
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
|
haftmann@33361
|
1845 |
done
|
haftmann@33361
|
1846 |
|
haftmann@33361
|
1847 |
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
|
haftmann@33361
|
1848 |
|
haftmann@33361
|
1849 |
lemma zadd1_lemma:
|
bulwahn@47420
|
1850 |
"[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br) |]
|
haftmann@33361
|
1851 |
==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
|
haftmann@33361
|
1852 |
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
|
haftmann@33361
|
1853 |
|
haftmann@33361
|
1854 |
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
|
haftmann@33361
|
1855 |
lemma zdiv_zadd1_eq:
|
haftmann@33361
|
1856 |
"(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
|
haftmann@33361
|
1857 |
apply (case_tac "c = 0", simp)
|
haftmann@33361
|
1858 |
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)
|
haftmann@33361
|
1859 |
done
|
haftmann@33361
|
1860 |
|
haftmann@33361
|
1861 |
instance int :: ring_div
|
haftmann@33361
|
1862 |
proof
|
haftmann@33361
|
1863 |
fix a b c :: int
|
haftmann@33361
|
1864 |
assume not0: "b \<noteq> 0"
|
haftmann@33361
|
1865 |
show "(a + c * b) div b = c + a div b"
|
haftmann@33361
|
1866 |
unfolding zdiv_zadd1_eq [of a "c * b"] using not0
|
haftmann@33361
|
1867 |
by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
|
haftmann@33361
|
1868 |
next
|
haftmann@33361
|
1869 |
fix a b c :: int
|
haftmann@33361
|
1870 |
assume "a \<noteq> 0"
|
haftmann@33361
|
1871 |
then show "(a * b) div (a * c) = b div c"
|
haftmann@33361
|
1872 |
proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
|
haftmann@33361
|
1873 |
case False then show ?thesis by auto
|
haftmann@33361
|
1874 |
next
|
haftmann@33361
|
1875 |
case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
|
haftmann@33361
|
1876 |
with `a \<noteq> 0`
|
haftmann@33361
|
1877 |
have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"
|
haftmann@33361
|
1878 |
apply (auto simp add: divmod_int_rel_def)
|
haftmann@33361
|
1879 |
apply (auto simp add: algebra_simps)
|
haftmann@33361
|
1880 |
apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)
|
haftmann@33361
|
1881 |
done
|
haftmann@33361
|
1882 |
moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
|
haftmann@33361
|
1883 |
ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
|
bulwahn@47420
|
1884 |
from this show ?thesis by (rule divmod_int_rel_div)
|
haftmann@33361
|
1885 |
qed
|
haftmann@33361
|
1886 |
qed auto
|
haftmann@33361
|
1887 |
|
haftmann@33361
|
1888 |
lemma posDivAlg_div_mod:
|
haftmann@33361
|
1889 |
assumes "k \<ge> 0"
|
haftmann@33361
|
1890 |
and "l \<ge> 0"
|
haftmann@33361
|
1891 |
shows "posDivAlg k l = (k div l, k mod l)"
|
haftmann@33361
|
1892 |
proof (cases "l = 0")
|
haftmann@33361
|
1893 |
case True then show ?thesis by (simp add: posDivAlg.simps)
|
haftmann@33361
|
1894 |
next
|
haftmann@33361
|
1895 |
case False with assms posDivAlg_correct
|
haftmann@33361
|
1896 |
have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
|
haftmann@33361
|
1897 |
by simp
|
bulwahn@47420
|
1898 |
from divmod_int_rel_div [OF this] divmod_int_rel_mod [OF this]
|
haftmann@33361
|
1899 |
show ?thesis by simp
|
haftmann@33361
|
1900 |
qed
|
haftmann@33361
|
1901 |
|
haftmann@33361
|
1902 |
lemma negDivAlg_div_mod:
|
haftmann@33361
|
1903 |
assumes "k < 0"
|
haftmann@33361
|
1904 |
and "l > 0"
|
haftmann@33361
|
1905 |
shows "negDivAlg k l = (k div l, k mod l)"
|
haftmann@33361
|
1906 |
proof -
|
haftmann@33361
|
1907 |
from assms have "l \<noteq> 0" by simp
|
haftmann@33361
|
1908 |
from assms negDivAlg_correct
|
haftmann@33361
|
1909 |
have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
|
haftmann@33361
|
1910 |
by simp
|
bulwahn@47420
|
1911 |
from divmod_int_rel_div [OF this] divmod_int_rel_mod [OF this]
|
haftmann@33361
|
1912 |
show ?thesis by simp
|
haftmann@33361
|
1913 |
qed
|
haftmann@33361
|
1914 |
|
haftmann@33361
|
1915 |
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
|
haftmann@33361
|
1916 |
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
|
haftmann@33361
|
1917 |
|
haftmann@33361
|
1918 |
(* REVISIT: should this be generalized to all semiring_div types? *)
|
haftmann@33361
|
1919 |
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
|
haftmann@33361
|
1920 |
|
huffman@47978
|
1921 |
lemma zmod_zdiv_equality':
|
huffman@47978
|
1922 |
"(m\<Colon>int) mod n = m - (m div n) * n"
|
huffman@47978
|
1923 |
by (rule_tac P="%x. m mod n = x - (m div n) * n" in subst [OF mod_div_equality [of _ n]])
|
huffman@47978
|
1924 |
arith
|
huffman@47978
|
1925 |
|
haftmann@33361
|
1926 |
|
huffman@47419
|
1927 |
subsubsection {* Proving @{term "a div (b*c) = (a div b) div c"} *}
|
haftmann@33361
|
1928 |
|
haftmann@33361
|
1929 |
(*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but
|
haftmann@33361
|
1930 |
7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems
|
haftmann@33361
|
1931 |
to cause particular problems.*)
|
haftmann@33361
|
1932 |
|
haftmann@33361
|
1933 |
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
|
haftmann@33361
|
1934 |
|
haftmann@33361
|
1935 |
lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \<le> 0 |] ==> b*c < b*(q mod c) + r"
|
haftmann@33361
|
1936 |
apply (subgoal_tac "b * (c - q mod c) < r * 1")
|
haftmann@33361
|
1937 |
apply (simp add: algebra_simps)
|
haftmann@33361
|
1938 |
apply (rule order_le_less_trans)
|
haftmann@33361
|
1939 |
apply (erule_tac [2] mult_strict_right_mono)
|
haftmann@33361
|
1940 |
apply (rule mult_left_mono_neg)
|
huffman@35208
|
1941 |
using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
|
haftmann@33361
|
1942 |
apply (simp)
|
haftmann@33361
|
1943 |
apply (simp)
|
haftmann@33361
|
1944 |
done
|
haftmann@33361
|
1945 |
|
haftmann@33361
|
1946 |
lemma zmult2_lemma_aux2:
|
haftmann@33361
|
1947 |
"[| (0::int) < c; b < r; r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
|
haftmann@33361
|
1948 |
apply (subgoal_tac "b * (q mod c) \<le> 0")
|
haftmann@33361
|
1949 |
apply arith
|
haftmann@33361
|
1950 |
apply (simp add: mult_le_0_iff)
|
haftmann@33361
|
1951 |
done
|
haftmann@33361
|
1952 |
|
haftmann@33361
|
1953 |
lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \<le> r; r < b |] ==> 0 \<le> b * (q mod c) + r"
|
haftmann@33361
|
1954 |
apply (subgoal_tac "0 \<le> b * (q mod c) ")
|
haftmann@33361
|
1955 |
apply arith
|
haftmann@33361
|
1956 |
apply (simp add: zero_le_mult_iff)
|
haftmann@33361
|
1957 |
done
|
haftmann@33361
|
1958 |
|
haftmann@33361
|
1959 |
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
|
haftmann@33361
|
1960 |
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
|
haftmann@33361
|
1961 |
apply (simp add: right_diff_distrib)
|
haftmann@33361
|
1962 |
apply (rule order_less_le_trans)
|
haftmann@33361
|
1963 |
apply (erule mult_strict_right_mono)
|
haftmann@33361
|
1964 |
apply (rule_tac [2] mult_left_mono)
|
haftmann@33361
|
1965 |
apply simp
|
huffman@35208
|
1966 |
using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
|
haftmann@33361
|
1967 |
apply simp
|
haftmann@33361
|
1968 |
done
|
haftmann@33361
|
1969 |
|
bulwahn@47420
|
1970 |
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]
|
haftmann@33361
|
1971 |
==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
|
haftmann@33361
|
1972 |
by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
|
haftmann@33361
|
1973 |
zero_less_mult_iff right_distrib [symmetric]
|
haftmann@33361
|
1974 |
zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
|
haftmann@33361
|
1975 |
|
haftmann@33361
|
1976 |
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
|
haftmann@33361
|
1977 |
apply (case_tac "b = 0", simp)
|
haftmann@33361
|
1978 |
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])
|
haftmann@33361
|
1979 |
done
|
haftmann@33361
|
1980 |
|
haftmann@33361
|
1981 |
lemma zmod_zmult2_eq:
|
haftmann@33361
|
1982 |
"(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
|
haftmann@33361
|
1983 |
apply (case_tac "b = 0", simp)
|
haftmann@33361
|
1984 |
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])
|
haftmann@33361
|
1985 |
done
|
haftmann@33361
|
1986 |
|
huffman@47978
|
1987 |
lemma div_pos_geq:
|
huffman@47978
|
1988 |
fixes k l :: int
|
huffman@47978
|
1989 |
assumes "0 < l" and "l \<le> k"
|
huffman@47978
|
1990 |
shows "k div l = (k - l) div l + 1"
|
huffman@47978
|
1991 |
proof -
|
huffman@47978
|
1992 |
have "k = (k - l) + l" by simp
|
huffman@47978
|
1993 |
then obtain j where k: "k = j + l" ..
|
huffman@47978
|
1994 |
with assms show ?thesis by simp
|
huffman@47978
|
1995 |
qed
|
huffman@47978
|
1996 |
|
huffman@47978
|
1997 |
lemma mod_pos_geq:
|
huffman@47978
|
1998 |
fixes k l :: int
|
huffman@47978
|
1999 |
assumes "0 < l" and "l \<le> k"
|
huffman@47978
|
2000 |
shows "k mod l = (k - l) mod l"
|
huffman@47978
|
2001 |
proof -
|
huffman@47978
|
2002 |
have "k = (k - l) + l" by simp
|
huffman@47978
|
2003 |
then obtain j where k: "k = j + l" ..
|
huffman@47978
|
2004 |
with assms show ?thesis by simp
|
huffman@47978
|
2005 |
qed
|
huffman@47978
|
2006 |
|
haftmann@33361
|
2007 |
|
huffman@47419
|
2008 |
subsubsection {* Splitting Rules for div and mod *}
|
haftmann@33361
|
2009 |
|
haftmann@33361
|
2010 |
text{*The proofs of the two lemmas below are essentially identical*}
|
haftmann@33361
|
2011 |
|
haftmann@33361
|
2012 |
lemma split_pos_lemma:
|
haftmann@33361
|
2013 |
"0<k ==>
|
haftmann@33361
|
2014 |
P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
|
haftmann@33361
|
2015 |
apply (rule iffI, clarify)
|
haftmann@33361
|
2016 |
apply (erule_tac P="P ?x ?y" in rev_mp)
|
haftmann@33361
|
2017 |
apply (subst mod_add_eq)
|
haftmann@33361
|
2018 |
apply (subst zdiv_zadd1_eq)
|
haftmann@33361
|
2019 |
apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
|
haftmann@33361
|
2020 |
txt{*converse direction*}
|
haftmann@33361
|
2021 |
apply (drule_tac x = "n div k" in spec)
|
haftmann@33361
|
2022 |
apply (drule_tac x = "n mod k" in spec, simp)
|
haftmann@33361
|
2023 |
done
|
haftmann@33361
|
2024 |
|
haftmann@33361
|
2025 |
lemma split_neg_lemma:
|
haftmann@33361
|
2026 |
"k<0 ==>
|
haftmann@33361
|
2027 |
P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
|
haftmann@33361
|
2028 |
apply (rule iffI, clarify)
|
haftmann@33361
|
2029 |
apply (erule_tac P="P ?x ?y" in rev_mp)
|
haftmann@33361
|
2030 |
apply (subst mod_add_eq)
|
haftmann@33361
|
2031 |
apply (subst zdiv_zadd1_eq)
|
haftmann@33361
|
2032 |
apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
|
haftmann@33361
|
2033 |
txt{*converse direction*}
|
haftmann@33361
|
2034 |
apply (drule_tac x = "n div k" in spec)
|
haftmann@33361
|
2035 |
apply (drule_tac x = "n mod k" in spec, simp)
|
haftmann@33361
|
2036 |
done
|
haftmann@33361
|
2037 |
|
haftmann@33361
|
2038 |
lemma split_zdiv:
|
haftmann@33361
|
2039 |
"P(n div k :: int) =
|
haftmann@33361
|
2040 |
((k = 0 --> P 0) &
|
haftmann@33361
|
2041 |
(0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
|
haftmann@33361
|
2042 |
(k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
|
haftmann@33361
|
2043 |
apply (case_tac "k=0", simp)
|
haftmann@33361
|
2044 |
apply (simp only: linorder_neq_iff)
|
haftmann@33361
|
2045 |
apply (erule disjE)
|
haftmann@33361
|
2046 |
apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
|
haftmann@33361
|
2047 |
split_neg_lemma [of concl: "%x y. P x"])
|
haftmann@33361
|
2048 |
done
|
haftmann@33361
|
2049 |
|
haftmann@33361
|
2050 |
lemma split_zmod:
|
haftmann@33361
|
2051 |
"P(n mod k :: int) =
|
haftmann@33361
|
2052 |
((k = 0 --> P n) &
|
haftmann@33361
|
2053 |
(0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
|
haftmann@33361
|
2054 |
(k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
|
haftmann@33361
|
2055 |
apply (case_tac "k=0", simp)
|
haftmann@33361
|
2056 |
apply (simp only: linorder_neq_iff)
|
haftmann@33361
|
2057 |
apply (erule disjE)
|
haftmann@33361
|
2058 |
apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
|
haftmann@33361
|
2059 |
split_neg_lemma [of concl: "%x y. P y"])
|
haftmann@33361
|
2060 |
done
|
haftmann@33361
|
2061 |
|
webertj@33725
|
2062 |
text {* Enable (lin)arith to deal with @{const div} and @{const mod}
|
webertj@33725
|
2063 |
when these are applied to some constant that is of the form
|
huffman@47978
|
2064 |
@{term "numeral k"}: *}
|
huffman@47978
|
2065 |
declare split_zdiv [of _ _ "numeral k", arith_split] for k
|
huffman@47978
|
2066 |
declare split_zmod [of _ _ "numeral k", arith_split] for k
|
haftmann@33361
|
2067 |
|
haftmann@33361
|
2068 |
|
huffman@47419
|
2069 |
subsubsection {* Speeding up the Division Algorithm with Shifting *}
|
haftmann@33361
|
2070 |
|
haftmann@33361
|
2071 |
text{*computing div by shifting *}
|
haftmann@33361
|
2072 |
|
haftmann@33361
|
2073 |
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
|
haftmann@33361
|
2074 |
proof cases
|
haftmann@33361
|
2075 |
assume "a=0"
|
haftmann@33361
|
2076 |
thus ?thesis by simp
|
haftmann@33361
|
2077 |
next
|
haftmann@33361
|
2078 |
assume "a\<noteq>0" and le_a: "0\<le>a"
|
haftmann@33361
|
2079 |
hence a_pos: "1 \<le> a" by arith
|
haftmann@33361
|
2080 |
hence one_less_a2: "1 < 2 * a" by arith
|
haftmann@33361
|
2081 |
hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
|
haftmann@33361
|
2082 |
unfolding mult_le_cancel_left
|
haftmann@33361
|
2083 |
by (simp add: add1_zle_eq add_commute [of 1])
|
haftmann@33361
|
2084 |
with a_pos have "0 \<le> b mod a" by simp
|
haftmann@33361
|
2085 |
hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
|
haftmann@33361
|
2086 |
by (simp add: mod_pos_pos_trivial one_less_a2)
|
haftmann@33361
|
2087 |
with le_2a
|
haftmann@33361
|
2088 |
have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
|
haftmann@33361
|
2089 |
by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
|
haftmann@33361
|
2090 |
right_distrib)
|
haftmann@33361
|
2091 |
thus ?thesis
|
haftmann@33361
|
2092 |
by (subst zdiv_zadd1_eq,
|
haftmann@33361
|
2093 |
simp add: mod_mult_mult1 one_less_a2
|
haftmann@33361
|
2094 |
div_pos_pos_trivial)
|
haftmann@33361
|
2095 |
qed
|
haftmann@33361
|
2096 |
|
boehmes@35815
|
2097 |
lemma neg_zdiv_mult_2:
|
boehmes@35815
|
2098 |
assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
|
boehmes@35815
|
2099 |
proof -
|
boehmes@35815
|
2100 |
have R: "1 + - (2 * (b + 1)) = - (1 + 2 * b)" by simp
|
boehmes@35815
|
2101 |
have "(1 + 2 * (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a)"
|
boehmes@35815
|
2102 |
by (rule pos_zdiv_mult_2, simp add: A)
|
boehmes@35815
|
2103 |
thus ?thesis
|
boehmes@35815
|
2104 |
by (simp only: R zdiv_zminus_zminus diff_minus
|
boehmes@35815
|
2105 |
minus_add_distrib [symmetric] mult_minus_right)
|
boehmes@35815
|
2106 |
qed
|
haftmann@33361
|
2107 |
|
huffman@47978
|
2108 |
(* FIXME: add rules for negative numerals *)
|
huffman@47978
|
2109 |
lemma zdiv_numeral_Bit0 [simp]:
|
huffman@47978
|
2110 |
"numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
|
huffman@47978
|
2111 |
numeral v div (numeral w :: int)"
|
huffman@47978
|
2112 |
unfolding numeral.simps unfolding mult_2 [symmetric]
|
huffman@47978
|
2113 |
by (rule div_mult_mult1, simp)
|
huffman@47978
|
2114 |
|
huffman@47978
|
2115 |
lemma zdiv_numeral_Bit1 [simp]:
|
huffman@47978
|
2116 |
"numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
|
huffman@47978
|
2117 |
(numeral v div (numeral w :: int))"
|
huffman@47978
|
2118 |
unfolding numeral.simps
|
huffman@47978
|
2119 |
unfolding mult_2 [symmetric] add_commute [of _ 1]
|
huffman@47978
|
2120 |
by (rule pos_zdiv_mult_2, simp)
|
haftmann@33361
|
2121 |
|
haftmann@33361
|
2122 |
|
huffman@47419
|
2123 |
subsubsection {* Computing mod by Shifting (proofs resemble those for div) *}
|
haftmann@33361
|
2124 |
|
haftmann@33361
|
2125 |
lemma pos_zmod_mult_2:
|
haftmann@33361
|
2126 |
fixes a b :: int
|
haftmann@33361
|
2127 |
assumes "0 \<le> a"
|
haftmann@33361
|
2128 |
shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
|
haftmann@33361
|
2129 |
proof (cases "0 < a")
|
haftmann@33361
|
2130 |
case False with assms show ?thesis by simp
|
haftmann@33361
|
2131 |
next
|
haftmann@33361
|
2132 |
case True
|
haftmann@33361
|
2133 |
then have "b mod a < a" by (rule pos_mod_bound)
|
haftmann@33361
|
2134 |
then have "1 + b mod a \<le> a" by simp
|
haftmann@33361
|
2135 |
then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp
|
haftmann@33361
|
2136 |
from `0 < a` have "0 \<le> b mod a" by (rule pos_mod_sign)
|
haftmann@33361
|
2137 |
then have B: "0 \<le> 1 + 2 * (b mod a)" by simp
|
haftmann@33361
|
2138 |
have "((1\<Colon>int) mod ((2\<Colon>int) * a) + (2\<Colon>int) * b mod ((2\<Colon>int) * a)) mod ((2\<Colon>int) * a) = (1\<Colon>int) + (2\<Colon>int) * (b mod a)"
|
haftmann@33361
|
2139 |
using `0 < a` and A
|
haftmann@33361
|
2140 |
by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)
|
haftmann@33361
|
2141 |
then show ?thesis by (subst mod_add_eq)
|
haftmann@33361
|
2142 |
qed
|
haftmann@33361
|
2143 |
|
haftmann@33361
|
2144 |
lemma neg_zmod_mult_2:
|
haftmann@33361
|
2145 |
fixes a b :: int
|
haftmann@33361
|
2146 |
assumes "a \<le> 0"
|
haftmann@33361
|
2147 |
shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
|
haftmann@33361
|
2148 |
proof -
|
haftmann@33361
|
2149 |
from assms have "0 \<le> - a" by auto
|
haftmann@33361
|
2150 |
then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"
|
haftmann@33361
|
2151 |
by (rule pos_zmod_mult_2)
|
haftmann@33361
|
2152 |
then show ?thesis by (simp add: zmod_zminus2 algebra_simps)
|
haftmann@33361
|
2153 |
(simp add: diff_minus add_ac)
|
haftmann@33361
|
2154 |
qed
|
haftmann@33361
|
2155 |
|
huffman@47978
|
2156 |
(* FIXME: add rules for negative numerals *)
|
huffman@47978
|
2157 |
lemma zmod_numeral_Bit0 [simp]:
|
huffman@47978
|
2158 |
"numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
|
huffman@47978
|
2159 |
(2::int) * (numeral v mod numeral w)"
|
huffman@47978
|
2160 |
unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
|
huffman@47978
|
2161 |
unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
|
huffman@47978
|
2162 |
|
huffman@47978
|
2163 |
lemma zmod_numeral_Bit1 [simp]:
|
huffman@47978
|
2164 |
"numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
|
huffman@47978
|
2165 |
2 * (numeral v mod numeral w) + (1::int)"
|
huffman@47978
|
2166 |
unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
|
huffman@47978
|
2167 |
unfolding mult_2 [symmetric] add_commute [of _ 1]
|
huffman@47978
|
2168 |
by (rule pos_zmod_mult_2, simp)
|
haftmann@33361
|
2169 |
|
nipkow@39729
|
2170 |
lemma zdiv_eq_0_iff:
|
nipkow@39729
|
2171 |
"(i::int) div k = 0 \<longleftrightarrow> k=0 \<or> 0\<le>i \<and> i<k \<or> i\<le>0 \<and> k<i" (is "?L = ?R")
|
nipkow@39729
|
2172 |
proof
|
nipkow@39729
|
2173 |
assume ?L
|
nipkow@39729
|
2174 |
have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
|
nipkow@39729
|
2175 |
with `?L` show ?R by blast
|
nipkow@39729
|
2176 |
next
|
nipkow@39729
|
2177 |
assume ?R thus ?L
|
nipkow@39729
|
2178 |
by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
|
nipkow@39729
|
2179 |
qed
|
nipkow@39729
|
2180 |
|
nipkow@39729
|
2181 |
|
huffman@47419
|
2182 |
subsubsection {* Quotients of Signs *}
|
haftmann@33361
|
2183 |
|
haftmann@33361
|
2184 |
lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"
|
haftmann@33361
|
2185 |
apply (subgoal_tac "a div b \<le> -1", force)
|
haftmann@33361
|
2186 |
apply (rule order_trans)
|
haftmann@33361
|
2187 |
apply (rule_tac a' = "-1" in zdiv_mono1)
|
haftmann@33361
|
2188 |
apply (auto simp add: div_eq_minus1)
|
haftmann@33361
|
2189 |
done
|
haftmann@33361
|
2190 |
|
haftmann@33361
|
2191 |
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
|
haftmann@33361
|
2192 |
by (drule zdiv_mono1_neg, auto)
|
haftmann@33361
|
2193 |
|
haftmann@33361
|
2194 |
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
|
haftmann@33361
|
2195 |
by (drule zdiv_mono1, auto)
|
haftmann@33361
|
2196 |
|
nipkow@33798
|
2197 |
text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
|
nipkow@33798
|
2198 |
conditional upon the sign of @{text a} or @{text b}. There are many more.
|
nipkow@33798
|
2199 |
They should all be simp rules unless that causes too much search. *}
|
nipkow@33798
|
2200 |
|
haftmann@33361
|
2201 |
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
|
haftmann@33361
|
2202 |
apply auto
|
haftmann@33361
|
2203 |
apply (drule_tac [2] zdiv_mono1)
|
haftmann@33361
|
2204 |
apply (auto simp add: linorder_neq_iff)
|
haftmann@33361
|
2205 |
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
|
haftmann@33361
|
2206 |
apply (blast intro: div_neg_pos_less0)
|
haftmann@33361
|
2207 |
done
|
haftmann@33361
|
2208 |
|
haftmann@33361
|
2209 |
lemma neg_imp_zdiv_nonneg_iff:
|
nipkow@33798
|
2210 |
"b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
|
haftmann@33361
|
2211 |
apply (subst zdiv_zminus_zminus [symmetric])
|
haftmann@33361
|
2212 |
apply (subst pos_imp_zdiv_nonneg_iff, auto)
|
haftmann@33361
|
2213 |
done
|
haftmann@33361
|
2214 |
|
haftmann@33361
|
2215 |
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
|
haftmann@33361
|
2216 |
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
|
haftmann@33361
|
2217 |
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
|
haftmann@33361
|
2218 |
|
nipkow@39729
|
2219 |
lemma pos_imp_zdiv_pos_iff:
|
nipkow@39729
|
2220 |
"0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
|
nipkow@39729
|
2221 |
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
|
nipkow@39729
|
2222 |
by arith
|
nipkow@39729
|
2223 |
|
haftmann@33361
|
2224 |
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
|
haftmann@33361
|
2225 |
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
|
haftmann@33361
|
2226 |
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
|
haftmann@33361
|
2227 |
|
nipkow@33798
|
2228 |
lemma nonneg1_imp_zdiv_pos_iff:
|
nipkow@33798
|
2229 |
"(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
|
nipkow@33798
|
2230 |
apply rule
|
nipkow@33798
|
2231 |
apply rule
|
nipkow@33798
|
2232 |
using div_pos_pos_trivial[of a b]apply arith
|
nipkow@33798
|
2233 |
apply(cases "b=0")apply simp
|
nipkow@33798
|
2234 |
using div_nonneg_neg_le0[of a b]apply arith
|
nipkow@33798
|
2235 |
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
|
nipkow@33798
|
2236 |
done
|
nipkow@33798
|
2237 |
|
nipkow@39729
|
2238 |
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
|
nipkow@39729
|
2239 |
apply (rule split_zmod[THEN iffD2])
|
nipkow@45761
|
2240 |
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
|
nipkow@39729
|
2241 |
done
|
nipkow@39729
|
2242 |
|
nipkow@39729
|
2243 |
|
haftmann@33361
|
2244 |
subsubsection {* The Divides Relation *}
|
haftmann@33361
|
2245 |
|
huffman@47978
|
2246 |
lemmas zdvd_iff_zmod_eq_0_numeral [simp] =
|
huffman@47978
|
2247 |
dvd_eq_mod_eq_0 [of "numeral x::int" "numeral y::int"]
|
huffman@47978
|
2248 |
dvd_eq_mod_eq_0 [of "numeral x::int" "neg_numeral y::int"]
|
huffman@47978
|
2249 |
dvd_eq_mod_eq_0 [of "neg_numeral x::int" "numeral y::int"]
|
huffman@47978
|
2250 |
dvd_eq_mod_eq_0 [of "neg_numeral x::int" "neg_numeral y::int"] for x y
|
haftmann@33361
|
2251 |
|
haftmann@33361
|
2252 |
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
|
haftmann@33361
|
2253 |
by (rule dvd_mod) (* TODO: remove *)
|
haftmann@33361
|
2254 |
|
haftmann@33361
|
2255 |
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
|
haftmann@33361
|
2256 |
by (rule dvd_mod_imp_dvd) (* TODO: remove *)
|
haftmann@33361
|
2257 |
|
huffman@47978
|
2258 |
lemmas dvd_eq_mod_eq_0_numeral [simp] =
|
huffman@47978
|
2259 |
dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
|
huffman@47978
|
2260 |
|
huffman@47978
|
2261 |
|
huffman@47978
|
2262 |
subsubsection {* Further properties *}
|
huffman@47978
|
2263 |
|
haftmann@33361
|
2264 |
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
|
haftmann@33361
|
2265 |
using zmod_zdiv_equality[where a="m" and b="n"]
|
haftmann@33361
|
2266 |
by (simp add: algebra_simps)
|
haftmann@33361
|
2267 |
|
haftmann@33361
|
2268 |
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
|
haftmann@33361
|
2269 |
apply (induct "y", auto)
|
haftmann@33361
|
2270 |
apply (rule zmod_zmult1_eq [THEN trans])
|
haftmann@33361
|
2271 |
apply (simp (no_asm_simp))
|
haftmann@33361
|
2272 |
apply (rule mod_mult_eq [symmetric])
|
haftmann@33361
|
2273 |
done
|
haftmann@33361
|
2274 |
|
haftmann@33361
|
2275 |
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
|
haftmann@33361
|
2276 |
apply (subst split_div, auto)
|
haftmann@33361
|
2277 |
apply (subst split_zdiv, auto)
|
haftmann@33361
|
2278 |
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
|
haftmann@33361
|
2279 |
apply (auto simp add: divmod_int_rel_def of_nat_mult)
|
haftmann@33361
|
2280 |
done
|
haftmann@33361
|
2281 |
|
haftmann@33361
|
2282 |
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
|
haftmann@33361
|
2283 |
apply (subst split_mod, auto)
|
haftmann@33361
|
2284 |
apply (subst split_zmod, auto)
|
haftmann@33361
|
2285 |
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia
|
haftmann@33361
|
2286 |
in unique_remainder)
|
haftmann@33361
|
2287 |
apply (auto simp add: divmod_int_rel_def of_nat_mult)
|
haftmann@33361
|
2288 |
done
|
haftmann@33361
|
2289 |
|
haftmann@33361
|
2290 |
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
|
haftmann@33361
|
2291 |
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
|
haftmann@33361
|
2292 |
|
haftmann@33361
|
2293 |
lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
|
haftmann@33361
|
2294 |
apply (subgoal_tac "m mod n = 0")
|
haftmann@33361
|
2295 |
apply (simp add: zmult_div_cancel)
|
haftmann@33361
|
2296 |
apply (simp only: dvd_eq_mod_eq_0)
|
haftmann@33361
|
2297 |
done
|
haftmann@33361
|
2298 |
|
haftmann@33361
|
2299 |
text{*Suggested by Matthias Daum*}
|
haftmann@33361
|
2300 |
lemma int_power_div_base:
|
haftmann@33361
|
2301 |
"\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
|
haftmann@33361
|
2302 |
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
|
haftmann@33361
|
2303 |
apply (erule ssubst)
|
haftmann@33361
|
2304 |
apply (simp only: power_add)
|
haftmann@33361
|
2305 |
apply simp_all
|
haftmann@33361
|
2306 |
done
|
haftmann@33361
|
2307 |
|
haftmann@33361
|
2308 |
text {* by Brian Huffman *}
|
haftmann@33361
|
2309 |
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
|
haftmann@33361
|
2310 |
by (rule mod_minus_eq [symmetric])
|
haftmann@33361
|
2311 |
|
haftmann@33361
|
2312 |
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
|
haftmann@33361
|
2313 |
by (rule mod_diff_left_eq [symmetric])
|
haftmann@33361
|
2314 |
|
haftmann@33361
|
2315 |
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
|
haftmann@33361
|
2316 |
by (rule mod_diff_right_eq [symmetric])
|
haftmann@33361
|
2317 |
|
haftmann@33361
|
2318 |
lemmas zmod_simps =
|
haftmann@33361
|
2319 |
mod_add_left_eq [symmetric]
|
haftmann@33361
|
2320 |
mod_add_right_eq [symmetric]
|
haftmann@33361
|
2321 |
zmod_zmult1_eq [symmetric]
|
haftmann@33361
|
2322 |
mod_mult_left_eq [symmetric]
|
haftmann@33361
|
2323 |
zpower_zmod
|
haftmann@33361
|
2324 |
zminus_zmod zdiff_zmod_left zdiff_zmod_right
|
haftmann@33361
|
2325 |
|
haftmann@33361
|
2326 |
text {* Distributive laws for function @{text nat}. *}
|
haftmann@33361
|
2327 |
|
haftmann@33361
|
2328 |
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
|
haftmann@33361
|
2329 |
apply (rule linorder_cases [of y 0])
|
haftmann@33361
|
2330 |
apply (simp add: div_nonneg_neg_le0)
|
haftmann@33361
|
2331 |
apply simp
|
haftmann@33361
|
2332 |
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
|
haftmann@33361
|
2333 |
done
|
haftmann@33361
|
2334 |
|
haftmann@33361
|
2335 |
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
|
haftmann@33361
|
2336 |
lemma nat_mod_distrib:
|
haftmann@33361
|
2337 |
"\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
|
haftmann@33361
|
2338 |
apply (case_tac "y = 0", simp)
|
haftmann@33361
|
2339 |
apply (simp add: nat_eq_iff zmod_int)
|
haftmann@33361
|
2340 |
done
|
haftmann@33361
|
2341 |
|
haftmann@33361
|
2342 |
text {* transfer setup *}
|
haftmann@33361
|
2343 |
|
haftmann@33361
|
2344 |
lemma transfer_nat_int_functions:
|
haftmann@33361
|
2345 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
|
haftmann@33361
|
2346 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
|
haftmann@33361
|
2347 |
by (auto simp add: nat_div_distrib nat_mod_distrib)
|
haftmann@33361
|
2348 |
|
haftmann@33361
|
2349 |
lemma transfer_nat_int_function_closures:
|
haftmann@33361
|
2350 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
|
haftmann@33361
|
2351 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
|
haftmann@33361
|
2352 |
apply (cases "y = 0")
|
haftmann@33361
|
2353 |
apply (auto simp add: pos_imp_zdiv_nonneg_iff)
|
haftmann@33361
|
2354 |
apply (cases "y = 0")
|
haftmann@33361
|
2355 |
apply auto
|
haftmann@33361
|
2356 |
done
|
haftmann@33361
|
2357 |
|
haftmann@35644
|
2358 |
declare transfer_morphism_nat_int [transfer add return:
|
haftmann@33361
|
2359 |
transfer_nat_int_functions
|
haftmann@33361
|
2360 |
transfer_nat_int_function_closures
|
haftmann@33361
|
2361 |
]
|
haftmann@33361
|
2362 |
|
haftmann@33361
|
2363 |
lemma transfer_int_nat_functions:
|
haftmann@33361
|
2364 |
"(int x) div (int y) = int (x div y)"
|
haftmann@33361
|
2365 |
"(int x) mod (int y) = int (x mod y)"
|
haftmann@33361
|
2366 |
by (auto simp add: zdiv_int zmod_int)
|
haftmann@33361
|
2367 |
|
haftmann@33361
|
2368 |
lemma transfer_int_nat_function_closures:
|
haftmann@33361
|
2369 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
|
haftmann@33361
|
2370 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
|
haftmann@33361
|
2371 |
by (simp_all only: is_nat_def transfer_nat_int_function_closures)
|
haftmann@33361
|
2372 |
|
haftmann@35644
|
2373 |
declare transfer_morphism_int_nat [transfer add return:
|
haftmann@33361
|
2374 |
transfer_int_nat_functions
|
haftmann@33361
|
2375 |
transfer_int_nat_function_closures
|
haftmann@33361
|
2376 |
]
|
haftmann@33361
|
2377 |
|
haftmann@33361
|
2378 |
text{*Suggested by Matthias Daum*}
|
haftmann@33361
|
2379 |
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
|
haftmann@33361
|
2380 |
apply (subgoal_tac "nat x div nat k < nat x")
|
nipkow@34225
|
2381 |
apply (simp add: nat_div_distrib [symmetric])
|
haftmann@33361
|
2382 |
apply (rule Divides.div_less_dividend, simp_all)
|
haftmann@33361
|
2383 |
done
|
haftmann@33361
|
2384 |
|
haftmann@35668
|
2385 |
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
|
haftmann@35668
|
2386 |
proof
|
haftmann@35668
|
2387 |
assume H: "x mod n = y mod n"
|
haftmann@35668
|
2388 |
hence "x mod n - y mod n = 0" by simp
|
haftmann@35668
|
2389 |
hence "(x mod n - y mod n) mod n = 0" by simp
|
haftmann@35668
|
2390 |
hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
|
haftmann@35668
|
2391 |
thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
|
haftmann@35668
|
2392 |
next
|
haftmann@35668
|
2393 |
assume H: "n dvd x - y"
|
haftmann@35668
|
2394 |
then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
|
haftmann@35668
|
2395 |
hence "x = n*k + y" by simp
|
haftmann@35668
|
2396 |
hence "x mod n = (n*k + y) mod n" by simp
|
haftmann@35668
|
2397 |
thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
|
haftmann@35668
|
2398 |
qed
|
haftmann@35668
|
2399 |
|
haftmann@35668
|
2400 |
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"
|
haftmann@35668
|
2401 |
shows "\<exists>q. x = y + n * q"
|
haftmann@35668
|
2402 |
proof-
|
haftmann@35668
|
2403 |
from xy have th: "int x - int y = int (x - y)" by simp
|
haftmann@35668
|
2404 |
from xyn have "int x mod int n = int y mod int n"
|
huffman@47419
|
2405 |
by (simp add: zmod_int [symmetric])
|
haftmann@35668
|
2406 |
hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])
|
haftmann@35668
|
2407 |
hence "n dvd x - y" by (simp add: th zdvd_int)
|
haftmann@35668
|
2408 |
then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
|
haftmann@35668
|
2409 |
qed
|
haftmann@35668
|
2410 |
|
haftmann@35668
|
2411 |
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
|
haftmann@35668
|
2412 |
(is "?lhs = ?rhs")
|
haftmann@35668
|
2413 |
proof
|
haftmann@35668
|
2414 |
assume H: "x mod n = y mod n"
|
haftmann@35668
|
2415 |
{assume xy: "x \<le> y"
|
haftmann@35668
|
2416 |
from H have th: "y mod n = x mod n" by simp
|
haftmann@35668
|
2417 |
from nat_mod_eq_lemma[OF th xy] have ?rhs
|
haftmann@35668
|
2418 |
apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
|
haftmann@35668
|
2419 |
moreover
|
haftmann@35668
|
2420 |
{assume xy: "y \<le> x"
|
haftmann@35668
|
2421 |
from nat_mod_eq_lemma[OF H xy] have ?rhs
|
haftmann@35668
|
2422 |
apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
|
haftmann@35668
|
2423 |
ultimately show ?rhs using linear[of x y] by blast
|
haftmann@35668
|
2424 |
next
|
haftmann@35668
|
2425 |
assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
|
haftmann@35668
|
2426 |
hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
|
haftmann@35668
|
2427 |
thus ?lhs by simp
|
haftmann@35668
|
2428 |
qed
|
haftmann@35668
|
2429 |
|
huffman@47978
|
2430 |
lemma div_nat_numeral [simp]:
|
huffman@47978
|
2431 |
"(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"
|
haftmann@35668
|
2432 |
by (simp add: nat_div_distrib)
|
haftmann@35668
|
2433 |
|
huffman@47978
|
2434 |
lemma one_div_nat_numeral [simp]:
|
huffman@47978
|
2435 |
"Suc 0 div numeral v' = nat (1 div numeral v')"
|
huffman@47978
|
2436 |
by (subst nat_div_distrib, simp_all)
|
huffman@47978
|
2437 |
|
huffman@47978
|
2438 |
lemma mod_nat_numeral [simp]:
|
huffman@47978
|
2439 |
"(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"
|
haftmann@35668
|
2440 |
by (simp add: nat_mod_distrib)
|
haftmann@35668
|
2441 |
|
huffman@47978
|
2442 |
lemma one_mod_nat_numeral [simp]:
|
huffman@47978
|
2443 |
"Suc 0 mod numeral v' = nat (1 mod numeral v')"
|
huffman@47978
|
2444 |
by (subst nat_mod_distrib) simp_all
|
huffman@47978
|
2445 |
|
huffman@47978
|
2446 |
lemma mod_2_not_eq_zero_eq_one_int:
|
huffman@47978
|
2447 |
fixes k :: int
|
huffman@47978
|
2448 |
shows "k mod 2 \<noteq> 0 \<longleftrightarrow> k mod 2 = 1"
|
huffman@47978
|
2449 |
by auto
|
huffman@47978
|
2450 |
|
huffman@47978
|
2451 |
|
huffman@47978
|
2452 |
subsubsection {* Tools setup *}
|
huffman@47978
|
2453 |
|
huffman@47978
|
2454 |
text {* Nitpick *}
|
haftmann@35668
|
2455 |
|
blanchet@42663
|
2456 |
lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'
|
haftmann@35668
|
2457 |
|
haftmann@35668
|
2458 |
|
haftmann@35668
|
2459 |
subsubsection {* Code generation *}
|
haftmann@35668
|
2460 |
|
haftmann@35668
|
2461 |
definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
|
haftmann@35668
|
2462 |
"pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
|
haftmann@35668
|
2463 |
|
haftmann@35668
|
2464 |
lemma pdivmod_posDivAlg [code]:
|
haftmann@35668
|
2465 |
"pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
|
haftmann@35668
|
2466 |
by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
|
haftmann@35668
|
2467 |
|
haftmann@35668
|
2468 |
lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
|
haftmann@35668
|
2469 |
apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
|
haftmann@35668
|
2470 |
then pdivmod k l
|
haftmann@35668
|
2471 |
else (let (r, s) = pdivmod k l in
|
huffman@47978
|
2472 |
if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
|
haftmann@35668
|
2473 |
proof -
|
haftmann@35668
|
2474 |
have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
|
haftmann@35668
|
2475 |
show ?thesis
|
haftmann@35668
|
2476 |
by (simp add: divmod_int_mod_div pdivmod_def)
|
haftmann@35668
|
2477 |
(auto simp add: aux not_less not_le zdiv_zminus1_eq_if
|
haftmann@35668
|
2478 |
zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
|
haftmann@35668
|
2479 |
qed
|
haftmann@35668
|
2480 |
|
haftmann@35668
|
2481 |
lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
|
haftmann@35668
|
2482 |
apsnd ((op *) (sgn l)) (if sgn k = sgn l
|
haftmann@35668
|
2483 |
then pdivmod k l
|
haftmann@35668
|
2484 |
else (let (r, s) = pdivmod k l in
|
haftmann@35668
|
2485 |
if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
|
haftmann@35668
|
2486 |
proof -
|
haftmann@35668
|
2487 |
have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
|
haftmann@35668
|
2488 |
by (auto simp add: not_less sgn_if)
|
haftmann@35668
|
2489 |
then show ?thesis by (simp add: divmod_int_pdivmod)
|
haftmann@35668
|
2490 |
qed
|
haftmann@33361
|
2491 |
|
haftmann@33364
|
2492 |
code_modulename SML
|
haftmann@33364
|
2493 |
Divides Arith
|
haftmann@33364
|
2494 |
|
haftmann@33364
|
2495 |
code_modulename OCaml
|
haftmann@33364
|
2496 |
Divides Arith
|
haftmann@33364
|
2497 |
|
haftmann@33364
|
2498 |
code_modulename Haskell
|
haftmann@33364
|
2499 |
Divides Arith
|
haftmann@33364
|
2500 |
|
haftmann@33361
|
2501 |
end
|