author | hoelzl |
Mon, 03 Dec 2012 18:19:08 +0100 | |
changeset 50328 | 25b1e8686ce0 |
parent 49962 | a8cc904a6820 |
child 50422 | ee729dbd1b7f |
permissions | -rw-r--r-- |
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(* Title: HOL/Divides.thy |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1999 University of Cambridge |
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*) |
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header {* The division operators div and mod *} |
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theory Divides |
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imports Nat_Transfer |
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begin |
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ML_file "~~/src/Provers/Arith/cancel_div_mod.ML" |
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subsection {* Syntactic division operations *} |
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class div = dvd + |
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and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70) |
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subsection {* Abstract division in commutative semirings. *} |
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class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div + |
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assumes mod_div_equality: "a div b * b + a mod b = a" |
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and div_by_0 [simp]: "a div 0 = 0" |
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and div_0 [simp]: "0 div a = 0" |
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and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b" |
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and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b" |
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begin |
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text {* @{const div} and @{const mod} *} |
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lemma mod_div_equality2: "b * (a div b) + a mod b = a" |
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unfolding mult_commute [of b] |
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by (rule mod_div_equality) |
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lemma mod_div_equality': "a mod b + a div b * b = a" |
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using mod_div_equality [of a b] |
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by (simp only: add_ac) |
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c" |
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by (simp add: mod_div_equality) |
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c" |
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by (simp add: mod_div_equality2) |
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lemma mod_by_0 [simp]: "a mod 0 = a" |
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using mod_div_equality [of a zero] by simp |
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lemma mod_0 [simp]: "0 mod a = 0" |
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using mod_div_equality [of zero a] div_0 by simp |
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lemma div_mult_self2 [simp]: |
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assumes "b \<noteq> 0" |
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shows "(a + b * c) div b = c + a div b" |
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using assms div_mult_self1 [of b a c] by (simp add: mult_commute) |
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b" |
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proof (cases "b = 0") |
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case True then show ?thesis by simp |
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next |
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case False |
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have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b" |
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by (simp add: mod_div_equality) |
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also from False div_mult_self1 [of b a c] have |
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"\<dots> = (c + a div b) * b + (a + c * b) mod b" |
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by (simp add: algebra_simps) |
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finally have "a = a div b * b + (a + c * b) mod b" |
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by (simp add: add_commute [of a] add_assoc distrib_right) |
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then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b" |
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by (simp add: mod_div_equality) |
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then show ?thesis by simp |
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qed |
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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b" |
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by (simp add: mult_commute [of b]) |
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a" |
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using div_mult_self2 [of b 0 a] by simp |
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a" |
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using div_mult_self1 [of b 0 a] by simp |
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83 |
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0" |
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using mod_mult_self2 [of 0 b a] by simp |
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86 |
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0" |
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using mod_mult_self1 [of 0 a b] by simp |
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|
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lemma div_by_1 [simp]: "a div 1 = a" |
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using div_mult_self2_is_id [of 1 a] zero_neq_one by simp |
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92 |
|
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lemma mod_by_1 [simp]: "a mod 1 = 0" |
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proof - |
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from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp |
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96 |
then have "a + a mod 1 = a + 0" by simp |
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then show ?thesis by (rule add_left_imp_eq) |
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98 |
qed |
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99 |
|
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lemma mod_self [simp]: "a mod a = 0" |
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101 |
using mod_mult_self2_is_0 [of 1] by simp |
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102 |
|
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1" |
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104 |
using div_mult_self2_is_id [of _ 1] by simp |
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105 |
|
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lemma div_add_self1 [simp]: |
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assumes "b \<noteq> 0" |
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108 |
shows "(b + a) div b = a div b + 1" |
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109 |
using assms div_mult_self1 [of b a 1] by (simp add: add_commute) |
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|
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lemma div_add_self2 [simp]: |
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assumes "b \<noteq> 0" |
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113 |
shows "(a + b) div b = a div b + 1" |
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using assms div_add_self1 [of b a] by (simp add: add_commute) |
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115 |
|
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lemma mod_add_self1 [simp]: |
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"(b + a) mod b = a mod b" |
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118 |
using mod_mult_self1 [of a 1 b] by (simp add: add_commute) |
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119 |
|
27676 | 120 |
lemma mod_add_self2 [simp]: |
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121 |
"(a + b) mod b = a mod b" |
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122 |
using mod_mult_self1 [of a 1 b] by simp |
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123 |
|
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lemma mod_div_decomp: |
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125 |
fixes a b |
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126 |
obtains q r where "q = a div b" and "r = a mod b" |
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127 |
and "a = q * b + r" |
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128 |
proof - |
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129 |
from mod_div_equality have "a = a div b * b + a mod b" by simp |
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130 |
moreover have "a div b = a div b" .. |
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131 |
moreover have "a mod b = a mod b" .. |
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132 |
note that ultimately show thesis by blast |
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133 |
qed |
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134 |
|
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135 |
lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0" |
25942 | 136 |
proof |
137 |
assume "b mod a = 0" |
|
138 |
with mod_div_equality [of b a] have "b div a * a = b" by simp |
|
139 |
then have "b = a * (b div a)" unfolding mult_commute .. |
|
140 |
then have "\<exists>c. b = a * c" .. |
|
141 |
then show "a dvd b" unfolding dvd_def . |
|
142 |
next |
|
143 |
assume "a dvd b" |
|
144 |
then have "\<exists>c. b = a * c" unfolding dvd_def . |
|
145 |
then obtain c where "b = a * c" .. |
|
146 |
then have "b mod a = a * c mod a" by simp |
|
147 |
then have "b mod a = c * a mod a" by (simp add: mult_commute) |
|
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148 |
then show "b mod a = 0" by simp |
25942 | 149 |
qed |
150 |
||
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changeset
|
151 |
lemma mod_div_trivial [simp]: "a mod b div b = 0" |
fe17df4e4ab3
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changeset
|
152 |
proof (cases "b = 0") |
fe17df4e4ab3
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parents:
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changeset
|
153 |
assume "b = 0" |
fe17df4e4ab3
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huffman
parents:
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changeset
|
154 |
thus ?thesis by simp |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
155 |
next |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
156 |
assume "b \<noteq> 0" |
fe17df4e4ab3
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parents:
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diff
changeset
|
157 |
hence "a div b + a mod b div b = (a mod b + a div b * b) div b" |
fe17df4e4ab3
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parents:
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diff
changeset
|
158 |
by (rule div_mult_self1 [symmetric]) |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
159 |
also have "\<dots> = a div b" |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
160 |
by (simp only: mod_div_equality') |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
161 |
also have "\<dots> = a div b + 0" |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
162 |
by simp |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
163 |
finally show ?thesis |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
164 |
by (rule add_left_imp_eq) |
fe17df4e4ab3
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parents:
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changeset
|
165 |
qed |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
166 |
|
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
167 |
lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b" |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
168 |
proof - |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
169 |
have "a mod b mod b = (a mod b + a div b * b) mod b" |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
170 |
by (simp only: mod_mult_self1) |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
171 |
also have "\<dots> = a mod b" |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
172 |
by (simp only: mod_div_equality') |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
173 |
finally show ?thesis . |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
174 |
qed |
fe17df4e4ab3
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huffman
parents:
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changeset
|
175 |
|
29925 | 176 |
lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0" |
29948 | 177 |
by (rule dvd_eq_mod_eq_0[THEN iffD1]) |
29925 | 178 |
|
179 |
lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b" |
|
180 |
by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0) |
|
181 |
||
33274
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32010
diff
changeset
|
182 |
lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b" |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
32010
diff
changeset
|
183 |
by (drule dvd_div_mult_self) (simp add: mult_commute) |
b6ff7db522b5
moved lemmas for dvd on nat to theories Nat and Power
haftmann
parents:
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diff
changeset
|
184 |
|
30052 | 185 |
lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a" |
186 |
apply (cases "a = 0") |
|
187 |
apply simp |
|
188 |
apply (auto simp: dvd_def mult_assoc) |
|
189 |
done |
|
190 |
||
29925 | 191 |
lemma div_dvd_div[simp]: |
192 |
"a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)" |
|
193 |
apply (cases "a = 0") |
|
194 |
apply simp |
|
195 |
apply (unfold dvd_def) |
|
196 |
apply auto |
|
197 |
apply(blast intro:mult_assoc[symmetric]) |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44766
diff
changeset
|
198 |
apply(fastforce simp add: mult_assoc) |
29925 | 199 |
done |
200 |
||
30078
beee83623cc9
move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents:
30052
diff
changeset
|
201 |
lemma dvd_mod_imp_dvd: "[| k dvd m mod n; k dvd n |] ==> k dvd m" |
beee83623cc9
move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents:
30052
diff
changeset
|
202 |
apply (subgoal_tac "k dvd (m div n) *n + m mod n") |
beee83623cc9
move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents:
30052
diff
changeset
|
203 |
apply (simp add: mod_div_equality) |
beee83623cc9
move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents:
30052
diff
changeset
|
204 |
apply (simp only: dvd_add dvd_mult) |
beee83623cc9
move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents:
30052
diff
changeset
|
205 |
done |
beee83623cc9
move lemma dvd_mod_imp_dvd into class semiring_div
huffman
parents:
30052
diff
changeset
|
206 |
|
29403
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
207 |
text {* Addition respects modular equivalence. *} |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
208 |
|
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
209 |
lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c" |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
210 |
proof - |
fe17df4e4ab3
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huffman
parents:
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diff
changeset
|
211 |
have "(a + b) mod c = (a div c * c + a mod c + b) mod c" |
fe17df4e4ab3
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huffman
parents:
29252
diff
changeset
|
212 |
by (simp only: mod_div_equality) |
fe17df4e4ab3
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huffman
parents:
29252
diff
changeset
|
213 |
also have "\<dots> = (a mod c + b + a div c * c) mod c" |
fe17df4e4ab3
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huffman
parents:
29252
diff
changeset
|
214 |
by (simp only: add_ac) |
fe17df4e4ab3
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huffman
parents:
29252
diff
changeset
|
215 |
also have "\<dots> = (a mod c + b) mod c" |
fe17df4e4ab3
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huffman
parents:
29252
diff
changeset
|
216 |
by (rule mod_mult_self1) |
fe17df4e4ab3
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huffman
parents:
29252
diff
changeset
|
217 |
finally show ?thesis . |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
218 |
qed |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
219 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
220 |
lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c" |
fe17df4e4ab3
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huffman
parents:
29252
diff
changeset
|
221 |
proof - |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
222 |
have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
223 |
by (simp only: mod_div_equality) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
224 |
also have "\<dots> = (a + b mod c + b div c * c) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
225 |
by (simp only: add_ac) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
226 |
also have "\<dots> = (a + b mod c) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
227 |
by (rule mod_mult_self1) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
228 |
finally show ?thesis . |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
229 |
qed |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
230 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
231 |
lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
232 |
by (rule trans [OF mod_add_left_eq mod_add_right_eq]) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
233 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
234 |
lemma mod_add_cong: |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
235 |
assumes "a mod c = a' mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
236 |
assumes "b mod c = b' mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
237 |
shows "(a + b) mod c = (a' + b') mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
238 |
proof - |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
239 |
have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
240 |
unfolding assms .. |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
241 |
thus ?thesis |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
242 |
by (simp only: mod_add_eq [symmetric]) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
243 |
qed |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
244 |
|
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
245 |
lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y |
30837
3d4832d9f7e4
added strong_setprod_cong[cong] (in analogy with setsum)
nipkow
parents:
30729
diff
changeset
|
246 |
\<Longrightarrow> (x + y) div z = x div z + y div z" |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
247 |
by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps) |
30837
3d4832d9f7e4
added strong_setprod_cong[cong] (in analogy with setsum)
nipkow
parents:
30729
diff
changeset
|
248 |
|
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
249 |
text {* Multiplication respects modular equivalence. *} |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
250 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
251 |
lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
252 |
proof - |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
253 |
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
254 |
by (simp only: mod_div_equality) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
255 |
also have "\<dots> = (a mod c * b + a div c * b * c) mod c" |
29667 | 256 |
by (simp only: algebra_simps) |
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
257 |
also have "\<dots> = (a mod c * b) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
258 |
by (rule mod_mult_self1) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
259 |
finally show ?thesis . |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
260 |
qed |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
261 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
262 |
lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
263 |
proof - |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
264 |
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
265 |
by (simp only: mod_div_equality) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
266 |
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c" |
29667 | 267 |
by (simp only: algebra_simps) |
29403
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
268 |
also have "\<dots> = (a * (b mod c)) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
269 |
by (rule mod_mult_self1) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
270 |
finally show ?thesis . |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
271 |
qed |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
272 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
273 |
lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
274 |
by (rule trans [OF mod_mult_left_eq mod_mult_right_eq]) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
275 |
|
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
276 |
lemma mod_mult_cong: |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
277 |
assumes "a mod c = a' mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
278 |
assumes "b mod c = b' mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
279 |
shows "(a * b) mod c = (a' * b') mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
280 |
proof - |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
281 |
have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c" |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
282 |
unfolding assms .. |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
283 |
thus ?thesis |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
284 |
by (simp only: mod_mult_eq [symmetric]) |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
285 |
qed |
fe17df4e4ab3
generalize some div/mod lemmas; remove type-specific proofs
huffman
parents:
29252
diff
changeset
|
286 |
|
47164 | 287 |
text {* Exponentiation respects modular equivalence. *} |
288 |
||
289 |
lemma power_mod: "(a mod b)^n mod b = a^n mod b" |
|
290 |
apply (induct n, simp_all) |
|
291 |
apply (rule mod_mult_right_eq [THEN trans]) |
|
292 |
apply (simp (no_asm_simp)) |
|
293 |
apply (rule mod_mult_eq [symmetric]) |
|
294 |
done |
|
295 |
||
29404 | 296 |
lemma mod_mod_cancel: |
297 |
assumes "c dvd b" |
|
298 |
shows "a mod b mod c = a mod c" |
|
299 |
proof - |
|
300 |
from `c dvd b` obtain k where "b = c * k" |
|
301 |
by (rule dvdE) |
|
302 |
have "a mod b mod c = a mod (c * k) mod c" |
|
303 |
by (simp only: `b = c * k`) |
|
304 |
also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c" |
|
305 |
by (simp only: mod_mult_self1) |
|
306 |
also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c" |
|
307 |
by (simp only: add_ac mult_ac) |
|
308 |
also have "\<dots> = a mod c" |
|
309 |
by (simp only: mod_div_equality) |
|
310 |
finally show ?thesis . |
|
311 |
qed |
|
312 |
||
30930 | 313 |
lemma div_mult_div_if_dvd: |
314 |
"y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)" |
|
315 |
apply (cases "y = 0", simp) |
|
316 |
apply (cases "z = 0", simp) |
|
317 |
apply (auto elim!: dvdE simp add: algebra_simps) |
|
30476 | 318 |
apply (subst mult_assoc [symmetric]) |
319 |
apply (simp add: no_zero_divisors) |
|
30930 | 320 |
done |
321 |
||
35367
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
322 |
lemma div_mult_swap: |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
323 |
assumes "c dvd b" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
324 |
shows "a * (b div c) = (a * b) div c" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
325 |
proof - |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
326 |
from assms have "b div c * (a div 1) = b * a div (c * 1)" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
327 |
by (simp only: div_mult_div_if_dvd one_dvd) |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
328 |
then show ?thesis by (simp add: mult_commute) |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
329 |
qed |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
330 |
|
30930 | 331 |
lemma div_mult_mult2 [simp]: |
332 |
"c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b" |
|
333 |
by (drule div_mult_mult1) (simp add: mult_commute) |
|
334 |
||
335 |
lemma div_mult_mult1_if [simp]: |
|
336 |
"(c * a) div (c * b) = (if c = 0 then 0 else a div b)" |
|
337 |
by simp_all |
|
30476 | 338 |
|
30930 | 339 |
lemma mod_mult_mult1: |
340 |
"(c * a) mod (c * b) = c * (a mod b)" |
|
341 |
proof (cases "c = 0") |
|
342 |
case True then show ?thesis by simp |
|
343 |
next |
|
344 |
case False |
|
345 |
from mod_div_equality |
|
346 |
have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" . |
|
347 |
with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b) |
|
348 |
= c * a + c * (a mod b)" by (simp add: algebra_simps) |
|
349 |
with mod_div_equality show ?thesis by simp |
|
350 |
qed |
|
351 |
||
352 |
lemma mod_mult_mult2: |
|
353 |
"(a * c) mod (b * c) = (a mod b) * c" |
|
354 |
using mod_mult_mult1 [of c a b] by (simp add: mult_commute) |
|
355 |
||
47159 | 356 |
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)" |
357 |
by (fact mod_mult_mult2 [symmetric]) |
|
358 |
||
359 |
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)" |
|
360 |
by (fact mod_mult_mult1 [symmetric]) |
|
361 |
||
31662
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset
|
362 |
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)" |
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset
|
363 |
unfolding dvd_def by (auto simp add: mod_mult_mult1) |
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset
|
364 |
|
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset
|
365 |
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m" |
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset
|
366 |
by (blast intro: dvd_mod_imp_dvd dvd_mod) |
57f7ef0dba8e
generalize lemmas dvd_mod and dvd_mod_iff to class semiring_div
huffman
parents:
31661
diff
changeset
|
367 |
|
31009
41fd307cab30
dropped reference to class recpower and lemma duplicate
haftmann
parents:
30934
diff
changeset
|
368 |
lemma div_power: |
31661
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset
|
369 |
"y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n" |
30476 | 370 |
apply (induct n) |
371 |
apply simp |
|
372 |
apply(simp add: div_mult_div_if_dvd dvd_power_same) |
|
373 |
done |
|
374 |
||
35367
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
375 |
lemma dvd_div_eq_mult: |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
376 |
assumes "a \<noteq> 0" and "a dvd b" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
377 |
shows "b div a = c \<longleftrightarrow> b = c * a" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
378 |
proof |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
379 |
assume "b = c * a" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
380 |
then show "b div a = c" by (simp add: assms) |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
381 |
next |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
382 |
assume "b div a = c" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
383 |
then have "b div a * a = c * a" by simp |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
384 |
moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self) |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
385 |
ultimately show "b = c * a" by simp |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
386 |
qed |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
387 |
|
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
388 |
lemma dvd_div_div_eq_mult: |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
389 |
assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
390 |
shows "b div a = d div c \<longleftrightarrow> b * c = a * d" |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
391 |
using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym) |
45a193f0ed0c
lemma div_mult_swap, dvd_div_eq_mult, dvd_div_div_eq_mult
haftmann
parents:
35216
diff
changeset
|
392 |
|
31661
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset
|
393 |
end |
1e252b8b2334
move lemma div_power into semiring_div context; class ring_div inherits from idom
huffman
parents:
31009
diff
changeset
|
394 |
|
35673 | 395 |
class ring_div = semiring_div + comm_ring_1 |
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
396 |
begin |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
397 |
|
36634 | 398 |
subclass ring_1_no_zero_divisors .. |
399 |
||
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
400 |
text {* Negation respects modular equivalence. *} |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
401 |
|
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
402 |
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
403 |
proof - |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
404 |
have "(- a) mod b = (- (a div b * b + a mod b)) mod b" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
405 |
by (simp only: mod_div_equality) |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
406 |
also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
407 |
by (simp only: minus_add_distrib minus_mult_left add_ac) |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
408 |
also have "\<dots> = (- (a mod b)) mod b" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
409 |
by (rule mod_mult_self1) |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
410 |
finally show ?thesis . |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
411 |
qed |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
412 |
|
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
413 |
lemma mod_minus_cong: |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
414 |
assumes "a mod b = a' mod b" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
415 |
shows "(- a) mod b = (- a') mod b" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
416 |
proof - |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
417 |
have "(- (a mod b)) mod b = (- (a' mod b)) mod b" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
418 |
unfolding assms .. |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
419 |
thus ?thesis |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
420 |
by (simp only: mod_minus_eq [symmetric]) |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
421 |
qed |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
422 |
|
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
423 |
text {* Subtraction respects modular equivalence. *} |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
424 |
|
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
425 |
lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
426 |
unfolding diff_minus |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
427 |
by (intro mod_add_cong mod_minus_cong) simp_all |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
428 |
|
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
429 |
lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
430 |
unfolding diff_minus |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
431 |
by (intro mod_add_cong mod_minus_cong) simp_all |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
432 |
|
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
433 |
lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
434 |
unfolding diff_minus |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
435 |
by (intro mod_add_cong mod_minus_cong) simp_all |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
436 |
|
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
437 |
lemma mod_diff_cong: |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
438 |
assumes "a mod c = a' mod c" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
439 |
assumes "b mod c = b' mod c" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
440 |
shows "(a - b) mod c = (a' - b') mod c" |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
441 |
unfolding diff_minus using assms |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
442 |
by (intro mod_add_cong mod_minus_cong) |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
443 |
|
30180 | 444 |
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)" |
445 |
apply (case_tac "y = 0") apply simp |
|
446 |
apply (auto simp add: dvd_def) |
|
447 |
apply (subgoal_tac "-(y * k) = y * - k") |
|
448 |
apply (erule ssubst) |
|
449 |
apply (erule div_mult_self1_is_id) |
|
450 |
apply simp |
|
451 |
done |
|
452 |
||
453 |
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)" |
|
454 |
apply (case_tac "y = 0") apply simp |
|
455 |
apply (auto simp add: dvd_def) |
|
456 |
apply (subgoal_tac "y * k = -y * -k") |
|
457 |
apply (erule ssubst) |
|
458 |
apply (rule div_mult_self1_is_id) |
|
459 |
apply simp |
|
460 |
apply simp |
|
461 |
done |
|
462 |
||
47159 | 463 |
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b" |
464 |
using div_mult_mult1 [of "- 1" a b] |
|
465 |
unfolding neg_equal_0_iff_equal by simp |
|
466 |
||
467 |
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)" |
|
468 |
using mod_mult_mult1 [of "- 1" a b] by simp |
|
469 |
||
470 |
lemma div_minus_right: "a div (-b) = (-a) div b" |
|
471 |
using div_minus_minus [of "-a" b] by simp |
|
472 |
||
473 |
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)" |
|
474 |
using mod_minus_minus [of "-a" b] by simp |
|
475 |
||
47160 | 476 |
lemma div_minus1_right [simp]: "a div (-1) = -a" |
477 |
using div_minus_right [of a 1] by simp |
|
478 |
||
479 |
lemma mod_minus1_right [simp]: "a mod (-1) = 0" |
|
480 |
using mod_minus_right [of a 1] by simp |
|
481 |
||
29405
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
482 |
end |
98ab21b14f09
add class ring_div; generalize mod/diff/minus proofs for class ring_div
huffman
parents:
29404
diff
changeset
|
483 |
|
25942 | 484 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
485 |
subsection {* Division on @{typ nat} *} |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
486 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
487 |
text {* |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
488 |
We define @{const div} and @{const mod} on @{typ nat} by means |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
489 |
of a characteristic relation with two input arguments |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
490 |
@{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
491 |
@{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder). |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
492 |
*} |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
493 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
494 |
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
495 |
"divmod_nat_rel m n qr \<longleftrightarrow> |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
496 |
m = fst qr * n + snd qr \<and> |
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
497 |
(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)" |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
498 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
499 |
text {* @{const divmod_nat_rel} is total: *} |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
500 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
501 |
lemma divmod_nat_rel_ex: |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
502 |
obtains q r where "divmod_nat_rel m n (q, r)" |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
503 |
proof (cases "n = 0") |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
504 |
case True with that show thesis |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
505 |
by (auto simp add: divmod_nat_rel_def) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
506 |
next |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
507 |
case False |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
508 |
have "\<exists>q r. m = q * n + r \<and> r < n" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
509 |
proof (induct m) |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
510 |
case 0 with `n \<noteq> 0` |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
511 |
have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
512 |
then show ?case by blast |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
513 |
next |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
514 |
case (Suc m) then obtain q' r' |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
515 |
where m: "m = q' * n + r'" and n: "r' < n" by auto |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
516 |
then show ?case proof (cases "Suc r' < n") |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
517 |
case True |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
518 |
from m n have "Suc m = q' * n + Suc r'" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
519 |
with True show ?thesis by blast |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
520 |
next |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
521 |
case False then have "n \<le> Suc r'" by auto |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
522 |
moreover from n have "Suc r' \<le> n" by auto |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
523 |
ultimately have "n = Suc r'" by auto |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
524 |
with m have "Suc m = Suc q' * n + 0" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
525 |
with `n \<noteq> 0` show ?thesis by blast |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
526 |
qed |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
527 |
qed |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
528 |
with that show thesis |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
529 |
using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
530 |
qed |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
531 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
532 |
text {* @{const divmod_nat_rel} is injective: *} |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
533 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
534 |
lemma divmod_nat_rel_unique: |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
535 |
assumes "divmod_nat_rel m n qr" |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
536 |
and "divmod_nat_rel m n qr'" |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
537 |
shows "qr = qr'" |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
538 |
proof (cases "n = 0") |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
539 |
case True with assms show ?thesis |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
540 |
by (cases qr, cases qr') |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
541 |
(simp add: divmod_nat_rel_def) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
542 |
next |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
543 |
case False |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
544 |
have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
545 |
apply (rule leI) |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
546 |
apply (subst less_iff_Suc_add) |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
547 |
apply (auto simp add: add_mult_distrib) |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
548 |
done |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
549 |
from `n \<noteq> 0` assms have "fst qr = fst qr'" |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
550 |
by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym) |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
551 |
moreover from this assms have "snd qr = snd qr'" |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
552 |
by (simp add: divmod_nat_rel_def) |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
553 |
ultimately show ?thesis by (cases qr, cases qr') simp |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
554 |
qed |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
555 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
556 |
text {* |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
557 |
We instantiate divisibility on the natural numbers by |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
558 |
means of @{const divmod_nat_rel}: |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
559 |
*} |
25942 | 560 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
561 |
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where |
37767 | 562 |
"divmod_nat m n = (THE qr. divmod_nat_rel m n qr)" |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
563 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
564 |
lemma divmod_nat_rel_divmod_nat: |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
565 |
"divmod_nat_rel m n (divmod_nat m n)" |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
566 |
proof - |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
567 |
from divmod_nat_rel_ex |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
568 |
obtain qr where rel: "divmod_nat_rel m n qr" . |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
569 |
then show ?thesis |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
570 |
by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique) |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
571 |
qed |
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
572 |
|
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
573 |
lemma divmod_nat_unique: |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
574 |
assumes "divmod_nat_rel m n qr" |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
575 |
shows "divmod_nat m n = qr" |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
576 |
using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
577 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
578 |
instantiation nat :: semiring_div |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
579 |
begin |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
580 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
581 |
definition div_nat where |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
582 |
"m div n = fst (divmod_nat m n)" |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
583 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
584 |
lemma fst_divmod_nat [simp]: |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
585 |
"fst (divmod_nat m n) = m div n" |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
586 |
by (simp add: div_nat_def) |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
587 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
588 |
definition mod_nat where |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
589 |
"m mod n = snd (divmod_nat m n)" |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset
|
590 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
591 |
lemma snd_divmod_nat [simp]: |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
592 |
"snd (divmod_nat m n) = m mod n" |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
593 |
by (simp add: mod_nat_def) |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
594 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
595 |
lemma divmod_nat_div_mod: |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
596 |
"divmod_nat m n = (m div n, m mod n)" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
597 |
by (simp add: prod_eq_iff) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
598 |
|
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
599 |
lemma div_nat_unique: |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
600 |
assumes "divmod_nat_rel m n (q, r)" |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
601 |
shows "m div n = q" |
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
602 |
using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff) |
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
603 |
|
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
604 |
lemma mod_nat_unique: |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
605 |
assumes "divmod_nat_rel m n (q, r)" |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
606 |
shows "m mod n = r" |
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
607 |
using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff) |
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25162
diff
changeset
|
608 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
609 |
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
610 |
using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
611 |
|
47136 | 612 |
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)" |
613 |
by (simp add: divmod_nat_unique divmod_nat_rel_def) |
|
614 |
||
615 |
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)" |
|
616 |
by (simp add: divmod_nat_unique divmod_nat_rel_def) |
|
25942 | 617 |
|
47137 | 618 |
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)" |
619 |
by (simp add: divmod_nat_unique divmod_nat_rel_def) |
|
25942 | 620 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
621 |
lemma divmod_nat_step: |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
622 |
assumes "0 < n" and "n \<le> m" |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
623 |
shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)" |
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
624 |
proof (rule divmod_nat_unique) |
47134 | 625 |
have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)" |
626 |
by (rule divmod_nat_rel) |
|
627 |
thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)" |
|
628 |
unfolding divmod_nat_rel_def using assms by auto |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
629 |
qed |
25942 | 630 |
|
26300 | 631 |
text {* The ''recursion'' equations for @{const div} and @{const mod} *} |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
632 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
633 |
lemma div_less [simp]: |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
634 |
fixes m n :: nat |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
635 |
assumes "m < n" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
636 |
shows "m div n = 0" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
637 |
using assms divmod_nat_base by (simp add: prod_eq_iff) |
25942 | 638 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
639 |
lemma le_div_geq: |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
640 |
fixes m n :: nat |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
641 |
assumes "0 < n" and "n \<le> m" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
642 |
shows "m div n = Suc ((m - n) div n)" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
643 |
using assms divmod_nat_step by (simp add: prod_eq_iff) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
644 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
645 |
lemma mod_less [simp]: |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
646 |
fixes m n :: nat |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
647 |
assumes "m < n" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
648 |
shows "m mod n = m" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
649 |
using assms divmod_nat_base by (simp add: prod_eq_iff) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
650 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
651 |
lemma le_mod_geq: |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
652 |
fixes m n :: nat |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
653 |
assumes "n \<le> m" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
654 |
shows "m mod n = (m - n) mod n" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
655 |
using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
656 |
|
47136 | 657 |
instance proof |
658 |
fix m n :: nat |
|
659 |
show "m div n * n + m mod n = m" |
|
660 |
using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def) |
|
661 |
next |
|
662 |
fix m n q :: nat |
|
663 |
assume "n \<noteq> 0" |
|
664 |
then show "(q + m * n) div n = m + q div n" |
|
665 |
by (induct m) (simp_all add: le_div_geq) |
|
666 |
next |
|
667 |
fix m n q :: nat |
|
668 |
assume "m \<noteq> 0" |
|
669 |
hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)" |
|
670 |
unfolding divmod_nat_rel_def |
|
671 |
by (auto split: split_if_asm, simp_all add: algebra_simps) |
|
672 |
moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" . |
|
673 |
ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" . |
|
674 |
thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique) |
|
675 |
next |
|
676 |
fix n :: nat show "n div 0 = 0" |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
677 |
by (simp add: div_nat_def divmod_nat_zero) |
47136 | 678 |
next |
679 |
fix n :: nat show "0 div n = 0" |
|
680 |
by (simp add: div_nat_def divmod_nat_zero_left) |
|
25942 | 681 |
qed |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
682 |
|
25942 | 683 |
end |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
684 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
685 |
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
686 |
let (q, r) = divmod_nat (m - n) n in (Suc q, r))" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
687 |
by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
688 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
689 |
text {* Simproc for cancelling @{const div} and @{const mod} *} |
25942 | 690 |
|
30934 | 691 |
ML {* |
43594 | 692 |
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod |
41550 | 693 |
( |
30934 | 694 |
val div_name = @{const_name div}; |
695 |
val mod_name = @{const_name mod}; |
|
696 |
val mk_binop = HOLogic.mk_binop; |
|
48561
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
697 |
val mk_plus = HOLogic.mk_binop @{const_name Groups.plus}; |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
698 |
val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT; |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
699 |
fun mk_sum [] = HOLogic.zero |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
700 |
| mk_sum [t] = t |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
701 |
| mk_sum (t :: ts) = mk_plus (t, mk_sum ts); |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
702 |
fun dest_sum tm = |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
703 |
if HOLogic.is_zero tm then [] |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
704 |
else |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
705 |
(case try HOLogic.dest_Suc tm of |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
706 |
SOME t => HOLogic.Suc_zero :: dest_sum t |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
707 |
| NONE => |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
708 |
(case try dest_plus tm of |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
709 |
SOME (t, u) => dest_sum t @ dest_sum u |
12aa0cb2b447
move ML functions from nat_arith.ML to Divides.thy, which is the only place they are used
huffman
parents:
47268
diff
changeset
|
710 |
| NONE => [tm])); |
25942 | 711 |
|
30934 | 712 |
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]; |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
713 |
|
30934 | 714 |
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac |
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
34982
diff
changeset
|
715 |
(@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac})) |
41550 | 716 |
) |
25942 | 717 |
*} |
718 |
||
43594 | 719 |
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *} |
720 |
||
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
721 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
722 |
subsubsection {* Quotient *} |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
723 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
724 |
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)" |
29667 | 725 |
by (simp add: le_div_geq linorder_not_less) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
726 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
727 |
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))" |
29667 | 728 |
by (simp add: div_geq) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
729 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
730 |
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)" |
29667 | 731 |
by simp |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
732 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
733 |
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)" |
29667 | 734 |
by simp |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
735 |
|
25942 | 736 |
|
737 |
subsubsection {* Remainder *} |
|
738 |
||
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
739 |
lemma mod_less_divisor [simp]: |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
740 |
fixes m n :: nat |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
741 |
assumes "n > 0" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
742 |
shows "m mod n < (n::nat)" |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
743 |
using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
744 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
745 |
lemma mod_less_eq_dividend [simp]: |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
746 |
fixes m n :: nat |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
747 |
shows "m mod n \<le> m" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
748 |
proof (rule add_leD2) |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
749 |
from mod_div_equality have "m div n * n + m mod n = m" . |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
750 |
then show "m div n * n + m mod n \<le> m" by auto |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
751 |
qed |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
752 |
|
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
753 |
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n" |
29667 | 754 |
by (simp add: le_mod_geq linorder_not_less) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
755 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
756 |
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)" |
29667 | 757 |
by (simp add: le_mod_geq) |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
758 |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
759 |
lemma mod_1 [simp]: "m mod Suc 0 = 0" |
29667 | 760 |
by (induct m) (simp_all add: mod_geq) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
761 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
762 |
(* a simple rearrangement of mod_div_equality: *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
763 |
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)" |
47138 | 764 |
using mod_div_equality2 [of n m] by arith |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
765 |
|
15439 | 766 |
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)" |
22718 | 767 |
apply (drule mod_less_divisor [where m = m]) |
768 |
apply simp |
|
769 |
done |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
770 |
|
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
771 |
subsubsection {* Quotient and Remainder *} |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
772 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
773 |
lemma divmod_nat_rel_mult1_eq: |
46552 | 774 |
"divmod_nat_rel b c (q, r) |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
775 |
\<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)" |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
776 |
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
777 |
|
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
778 |
lemma div_mult1_eq: |
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
779 |
"(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)" |
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
780 |
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
781 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
782 |
lemma divmod_nat_rel_add1_eq: |
46552 | 783 |
"divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
784 |
\<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)" |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
785 |
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
786 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
787 |
(*NOT suitable for rewriting: the RHS has an instance of the LHS*) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
788 |
lemma div_add1_eq: |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
789 |
"(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)" |
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
790 |
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
791 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
792 |
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c" |
22718 | 793 |
apply (cut_tac m = q and n = c in mod_less_divisor) |
794 |
apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto) |
|
795 |
apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst) |
|
796 |
apply (simp add: add_mult_distrib2) |
|
797 |
done |
|
10559
d3fd54fc659b
many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents:
10214
diff
changeset
|
798 |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
799 |
lemma divmod_nat_rel_mult2_eq: |
46552 | 800 |
"divmod_nat_rel a b (q, r) |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
801 |
\<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)" |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
802 |
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
803 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
804 |
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)" |
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
805 |
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique]) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
806 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
807 |
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)" |
47135
fb67b596067f
rename lemmas {div,mod}_eq -> {div,mod}_nat_unique, for consistency with minus_unique, inverse_unique, etc.
huffman
parents:
47134
diff
changeset
|
808 |
by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique]) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
809 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
810 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
811 |
subsubsection {* Further Facts about Quotient and Remainder *} |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
812 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
813 |
lemma div_1 [simp]: "m div Suc 0 = m" |
29667 | 814 |
by (induct m) (simp_all add: div_geq) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
815 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
816 |
(* Monotonicity of div in first argument *) |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
817 |
lemma div_le_mono [rule_format (no_asm)]: |
22718 | 818 |
"\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)" |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
819 |
apply (case_tac "k=0", simp) |
15251 | 820 |
apply (induct "n" rule: nat_less_induct, clarify) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
821 |
apply (case_tac "n<k") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
822 |
(* 1 case n<k *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
823 |
apply simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
824 |
(* 2 case n >= k *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
825 |
apply (case_tac "m<k") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
826 |
(* 2.1 case m<k *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
827 |
apply simp |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
828 |
(* 2.2 case m>=k *) |
15439 | 829 |
apply (simp add: div_geq diff_le_mono) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
830 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
831 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
832 |
(* Antimonotonicity of div in second argument *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
833 |
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
834 |
apply (subgoal_tac "0<n") |
22718 | 835 |
prefer 2 apply simp |
15251 | 836 |
apply (induct_tac k rule: nat_less_induct) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
837 |
apply (rename_tac "k") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
838 |
apply (case_tac "k<n", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
839 |
apply (subgoal_tac "~ (k<m) ") |
22718 | 840 |
prefer 2 apply simp |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
841 |
apply (simp add: div_geq) |
15251 | 842 |
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n") |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
843 |
prefer 2 |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
844 |
apply (blast intro: div_le_mono diff_le_mono2) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
845 |
apply (rule le_trans, simp) |
15439 | 846 |
apply (simp) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
847 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
848 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
849 |
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
850 |
apply (case_tac "n=0", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
851 |
apply (subgoal_tac "m div n \<le> m div 1", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
852 |
apply (rule div_le_mono2) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
853 |
apply (simp_all (no_asm_simp)) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
854 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
855 |
|
22718 | 856 |
(* Similar for "less than" *) |
47138 | 857 |
lemma div_less_dividend [simp]: |
858 |
"\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m" |
|
859 |
apply (induct m rule: nat_less_induct) |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
860 |
apply (rename_tac "m") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
861 |
apply (case_tac "m<n", simp) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
862 |
apply (subgoal_tac "0<n") |
22718 | 863 |
prefer 2 apply simp |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
864 |
apply (simp add: div_geq) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
865 |
apply (case_tac "n<m") |
15251 | 866 |
apply (subgoal_tac "(m-n) div n < (m-n) ") |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
867 |
apply (rule impI less_trans_Suc)+ |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
868 |
apply assumption |
15439 | 869 |
apply (simp_all) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
870 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
871 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
872 |
text{*A fact for the mutilated chess board*} |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
873 |
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))" |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
874 |
apply (case_tac "n=0", simp) |
15251 | 875 |
apply (induct "m" rule: nat_less_induct) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
876 |
apply (case_tac "Suc (na) <n") |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
877 |
(* case Suc(na) < n *) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
878 |
apply (frule lessI [THEN less_trans], simp add: less_not_refl3) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
879 |
(* case n \<le> Suc(na) *) |
16796 | 880 |
apply (simp add: linorder_not_less le_Suc_eq mod_geq) |
15439 | 881 |
apply (auto simp add: Suc_diff_le le_mod_geq) |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
882 |
done |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
883 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
884 |
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)" |
29667 | 885 |
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def) |
17084
fb0a80aef0be
classical rules must have names for ATP integration
paulson
parents:
16796
diff
changeset
|
886 |
|
22718 | 887 |
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1] |
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
888 |
|
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
889 |
(*Loses information, namely we also have r<d provided d is nonzero*) |
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
890 |
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
891 |
apply (cut_tac a = m in mod_div_equality) |
22718 | 892 |
apply (simp only: add_ac) |
893 |
apply (blast intro: sym) |
|
894 |
done |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
895 |
|
13152 | 896 |
lemma split_div: |
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
897 |
"P(n div k :: nat) = |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
898 |
((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
899 |
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
900 |
proof |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
901 |
assume P: ?P |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
902 |
show ?Q |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
903 |
proof (cases) |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
904 |
assume "k = 0" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
905 |
with P show ?Q by simp |
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
906 |
next |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
907 |
assume not0: "k \<noteq> 0" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
908 |
thus ?Q |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
909 |
proof (simp, intro allI impI) |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
910 |
fix i j |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
911 |
assume n: "n = k*i + j" and j: "j < k" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
912 |
show "P i" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
913 |
proof (cases) |
22718 | 914 |
assume "i = 0" |
915 |
with n j P show "P i" by simp |
|
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
916 |
next |
22718 | 917 |
assume "i \<noteq> 0" |
918 |
with not0 n j P show "P i" by(simp add:add_ac) |
|
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
919 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
920 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
921 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
922 |
next |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
923 |
assume Q: ?Q |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
924 |
show ?P |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
925 |
proof (cases) |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
926 |
assume "k = 0" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
927 |
with Q show ?P by simp |
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
928 |
next |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
929 |
assume not0: "k \<noteq> 0" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
930 |
with Q have R: ?R by simp |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
931 |
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] |
13517 | 932 |
show ?P by simp |
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
933 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
934 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
935 |
|
13882 | 936 |
lemma split_div_lemma: |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
937 |
assumes "0 < n" |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
938 |
shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs") |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
939 |
proof |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
940 |
assume ?rhs |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
941 |
with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
942 |
then have A: "n * q \<le> m" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
943 |
have "n - (m mod n) > 0" using mod_less_divisor assms by auto |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
944 |
then have "m < m + (n - (m mod n))" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
945 |
then have "m < n + (m - (m mod n))" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
946 |
with nq have "m < n + n * q" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
947 |
then have B: "m < n * Suc q" by simp |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
948 |
from A B show ?lhs .. |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
949 |
next |
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
950 |
assume P: ?lhs |
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
951 |
then have "divmod_nat_rel m n (q, m - n * q)" |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
952 |
unfolding divmod_nat_rel_def by (auto simp add: mult_ac) |
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
953 |
with divmod_nat_rel_unique divmod_nat_rel [of m n] |
30923
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
954 |
have "(q, m - n * q) = (m div n, m mod n)" by auto |
2697a1d1d34a
more coherent developement in Divides.thy and IntDiv.thy
haftmann
parents:
30840
diff
changeset
|
955 |
then show ?rhs by simp |
26100
fbc60cd02ae2
using only an relation predicate to construct div and mod
haftmann
parents:
26072
diff
changeset
|
956 |
qed |
13882 | 957 |
|
958 |
theorem split_div': |
|
959 |
"P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or> |
|
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14208
diff
changeset
|
960 |
(\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))" |
13882 | 961 |
apply (case_tac "0 < n") |
962 |
apply (simp only: add: split_div_lemma) |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
963 |
apply simp_all |
13882 | 964 |
done |
965 |
||
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
966 |
lemma split_mod: |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
967 |
"P(n mod k :: nat) = |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
968 |
((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
969 |
(is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))") |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
970 |
proof |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
971 |
assume P: ?P |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
972 |
show ?Q |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
973 |
proof (cases) |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
974 |
assume "k = 0" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
975 |
with P show ?Q by simp |
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
976 |
next |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
977 |
assume not0: "k \<noteq> 0" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
978 |
thus ?Q |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
979 |
proof (simp, intro allI impI) |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
980 |
fix i j |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
981 |
assume "n = k*i + j" "j < k" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
982 |
thus "P j" using not0 P by(simp add:add_ac mult_ac) |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
983 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
984 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
985 |
next |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
986 |
assume Q: ?Q |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
987 |
show ?P |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
988 |
proof (cases) |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
989 |
assume "k = 0" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
990 |
with Q show ?P by simp |
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
991 |
next |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
992 |
assume not0: "k \<noteq> 0" |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
993 |
with Q have R: ?R by simp |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
994 |
from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"] |
13517 | 995 |
show ?P by simp |
13189
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
996 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
997 |
qed |
81ed5c6de890
Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents:
13152
diff
changeset
|
998 |
|
13882 | 999 |
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n" |
47138 | 1000 |
using mod_div_equality [of m n] by arith |
1001 |
||
1002 |
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n" |
|
1003 |
using mod_div_equality [of m n] by arith |
|
1004 |
(* FIXME: very similar to mult_div_cancel *) |
|
22800 | 1005 |
|
1006 |
||
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1007 |
subsubsection {* An ``induction'' law for modulus arithmetic. *} |
14640 | 1008 |
|
1009 |
lemma mod_induct_0: |
|
1010 |
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" |
|
1011 |
and base: "P i" and i: "i<p" |
|
1012 |
shows "P 0" |
|
1013 |
proof (rule ccontr) |
|
1014 |
assume contra: "\<not>(P 0)" |
|
1015 |
from i have p: "0<p" by simp |
|
1016 |
have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k") |
|
1017 |
proof |
|
1018 |
fix k |
|
1019 |
show "?A k" |
|
1020 |
proof (induct k) |
|
1021 |
show "?A 0" by simp -- "by contradiction" |
|
1022 |
next |
|
1023 |
fix n |
|
1024 |
assume ih: "?A n" |
|
1025 |
show "?A (Suc n)" |
|
1026 |
proof (clarsimp) |
|
22718 | 1027 |
assume y: "P (p - Suc n)" |
1028 |
have n: "Suc n < p" |
|
1029 |
proof (rule ccontr) |
|
1030 |
assume "\<not>(Suc n < p)" |
|
1031 |
hence "p - Suc n = 0" |
|
1032 |
by simp |
|
1033 |
with y contra show "False" |
|
1034 |
by simp |
|
1035 |
qed |
|
1036 |
hence n2: "Suc (p - Suc n) = p-n" by arith |
|
1037 |
from p have "p - Suc n < p" by arith |
|
1038 |
with y step have z: "P ((Suc (p - Suc n)) mod p)" |
|
1039 |
by blast |
|
1040 |
show "False" |
|
1041 |
proof (cases "n=0") |
|
1042 |
case True |
|
1043 |
with z n2 contra show ?thesis by simp |
|
1044 |
next |
|
1045 |
case False |
|
1046 |
with p have "p-n < p" by arith |
|
1047 |
with z n2 False ih show ?thesis by simp |
|
1048 |
qed |
|
14640 | 1049 |
qed |
1050 |
qed |
|
1051 |
qed |
|
1052 |
moreover |
|
1053 |
from i obtain k where "0<k \<and> i+k=p" |
|
1054 |
by (blast dest: less_imp_add_positive) |
|
1055 |
hence "0<k \<and> i=p-k" by auto |
|
1056 |
moreover |
|
1057 |
note base |
|
1058 |
ultimately |
|
1059 |
show "False" by blast |
|
1060 |
qed |
|
1061 |
||
1062 |
lemma mod_induct: |
|
1063 |
assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)" |
|
1064 |
and base: "P i" and i: "i<p" and j: "j<p" |
|
1065 |
shows "P j" |
|
1066 |
proof - |
|
1067 |
have "\<forall>j<p. P j" |
|
1068 |
proof |
|
1069 |
fix j |
|
1070 |
show "j<p \<longrightarrow> P j" (is "?A j") |
|
1071 |
proof (induct j) |
|
1072 |
from step base i show "?A 0" |
|
22718 | 1073 |
by (auto elim: mod_induct_0) |
14640 | 1074 |
next |
1075 |
fix k |
|
1076 |
assume ih: "?A k" |
|
1077 |
show "?A (Suc k)" |
|
1078 |
proof |
|
22718 | 1079 |
assume suc: "Suc k < p" |
1080 |
hence k: "k<p" by simp |
|
1081 |
with ih have "P k" .. |
|
1082 |
with step k have "P (Suc k mod p)" |
|
1083 |
by blast |
|
1084 |
moreover |
|
1085 |
from suc have "Suc k mod p = Suc k" |
|
1086 |
by simp |
|
1087 |
ultimately |
|
1088 |
show "P (Suc k)" by simp |
|
14640 | 1089 |
qed |
1090 |
qed |
|
1091 |
qed |
|
1092 |
with j show ?thesis by blast |
|
1093 |
qed |
|
1094 |
||
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1095 |
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)" |
47138 | 1096 |
by (simp add: numeral_2_eq_2 le_div_geq) |
1097 |
||
1098 |
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2" |
|
1099 |
by (simp add: numeral_2_eq_2 le_mod_geq) |
|
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1100 |
|
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1101 |
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)" |
47217
501b9bbd0d6e
removed redundant nat-specific copies of theorems
huffman
parents:
47167
diff
changeset
|
1102 |
by (simp add: mult_2 [symmetric]) |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1103 |
|
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1104 |
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1105 |
proof - |
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset
|
1106 |
{ fix n :: nat have "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all } |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1107 |
moreover have "m mod 2 < 2" by simp |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1108 |
ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" . |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1109 |
then show ?thesis by auto |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1110 |
qed |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1111 |
|
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1112 |
text{*These lemmas collapse some needless occurrences of Suc: |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1113 |
at least three Sucs, since two and fewer are rewritten back to Suc again! |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1114 |
We already have some rules to simplify operands smaller than 3.*} |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1115 |
|
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1116 |
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1117 |
by (simp add: Suc3_eq_add_3) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1118 |
|
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1119 |
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1120 |
by (simp add: Suc3_eq_add_3) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1121 |
|
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1122 |
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1123 |
by (simp add: Suc3_eq_add_3) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1124 |
|
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1125 |
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n" |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1126 |
by (simp add: Suc3_eq_add_3) |
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1127 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1128 |
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1129 |
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v |
33296
a3924d1069e5
moved theory Divides after theory Nat_Numeral; tuned some proof texts
haftmann
parents:
33274
diff
changeset
|
1130 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1131 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1132 |
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1133 |
apply (induct "m") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1134 |
apply (simp_all add: mod_Suc) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1135 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1136 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1137 |
declare Suc_times_mod_eq [of "numeral w", simp] for w |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1138 |
|
47138 | 1139 |
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k" |
1140 |
by (simp add: div_le_mono) |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1141 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1142 |
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1143 |
by (cases n) simp_all |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1144 |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset
|
1145 |
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2" |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset
|
1146 |
proof - |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset
|
1147 |
from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset
|
1148 |
from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset
|
1149 |
qed |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1150 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1151 |
(* Potential use of algebra : Equality modulo n*) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1152 |
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1153 |
by (simp add: mult_ac add_ac) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1154 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1155 |
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1156 |
proof - |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1157 |
have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1158 |
also have "... = Suc m mod n" by (rule mod_mult_self3) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1159 |
finally show ?thesis . |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1160 |
qed |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1161 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1162 |
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1163 |
apply (subst mod_Suc [of m]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1164 |
apply (subst mod_Suc [of "m mod n"], simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1165 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1166 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1167 |
lemma mod_2_not_eq_zero_eq_one_nat: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1168 |
fixes n :: nat |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1169 |
shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1170 |
by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1171 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1172 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1173 |
subsection {* Division on @{typ int} *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1174 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1175 |
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1176 |
--{*definition of quotient and remainder*} |
47139
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1177 |
"divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and> |
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1178 |
(if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1179 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1180 |
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1181 |
--{*for the division algorithm*} |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1182 |
"adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1183 |
else (2 * q, r))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1184 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1185 |
text{*algorithm for the case @{text "a\<ge>0, b>0"}*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1186 |
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1187 |
"posDivAlg a b = (if a < b \<or> b \<le> 0 then (0, a) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1188 |
else adjust b (posDivAlg a (2 * b)))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1189 |
by auto |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1190 |
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1191 |
(auto simp add: mult_2) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1192 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1193 |
text{*algorithm for the case @{text "a<0, b>0"}*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1194 |
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1195 |
"negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0 then (-1, a + b) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1196 |
else adjust b (negDivAlg a (2 * b)))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1197 |
by auto |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1198 |
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1199 |
(auto simp add: mult_2) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1200 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1201 |
text{*algorithm for the general case @{term "b\<noteq>0"}*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1202 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1203 |
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1204 |
--{*The full division algorithm considers all possible signs for a, b |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1205 |
including the special case @{text "a=0, b<0"} because |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1206 |
@{term negDivAlg} requires @{term "a<0"}.*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1207 |
"divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1208 |
else if a = 0 then (0, 0) |
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset
|
1209 |
else apsnd uminus (negDivAlg (-a) (-b)) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1210 |
else |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1211 |
if 0 < b then negDivAlg a b |
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset
|
1212 |
else apsnd uminus (posDivAlg (-a) (-b)))" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1213 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1214 |
instantiation int :: Divides.div |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1215 |
begin |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1216 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1217 |
definition div_int where |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1218 |
"a div b = fst (divmod_int a b)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1219 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1220 |
lemma fst_divmod_int [simp]: |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1221 |
"fst (divmod_int a b) = a div b" |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1222 |
by (simp add: div_int_def) |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1223 |
|
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1224 |
definition mod_int where |
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset
|
1225 |
"a mod b = snd (divmod_int a b)" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1226 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1227 |
lemma snd_divmod_int [simp]: |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1228 |
"snd (divmod_int a b) = a mod b" |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1229 |
by (simp add: mod_int_def) |
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1230 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1231 |
instance .. |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1232 |
|
3366 | 1233 |
end |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1234 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1235 |
lemma divmod_int_mod_div: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1236 |
"divmod_int p q = (p div q, p mod q)" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1237 |
by (simp add: prod_eq_iff) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1238 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1239 |
text{* |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1240 |
Here is the division algorithm in ML: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1241 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1242 |
\begin{verbatim} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1243 |
fun posDivAlg (a,b) = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1244 |
if a<b then (0,a) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1245 |
else let val (q,r) = posDivAlg(a, 2*b) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1246 |
in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1247 |
end |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1248 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1249 |
fun negDivAlg (a,b) = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1250 |
if 0\<le>a+b then (~1,a+b) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1251 |
else let val (q,r) = negDivAlg(a, 2*b) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1252 |
in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1253 |
end; |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1254 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1255 |
fun negateSnd (q,r:int) = (q,~r); |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1256 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1257 |
fun divmod (a,b) = if 0\<le>a then |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1258 |
if b>0 then posDivAlg (a,b) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1259 |
else if a=0 then (0,0) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1260 |
else negateSnd (negDivAlg (~a,~b)) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1261 |
else |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1262 |
if 0<b then negDivAlg (a,b) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1263 |
else negateSnd (posDivAlg (~a,~b)); |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1264 |
\end{verbatim} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1265 |
*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1266 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1267 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1268 |
subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1269 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1270 |
lemma unique_quotient_lemma: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1271 |
"[| b*q' + r' \<le> b*q + r; 0 \<le> r'; r' < b; r < b |] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1272 |
==> q' \<le> (q::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1273 |
apply (subgoal_tac "r' + b * (q'-q) \<le> r") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1274 |
prefer 2 apply (simp add: right_diff_distrib) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1275 |
apply (subgoal_tac "0 < b * (1 + q - q') ") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1276 |
apply (erule_tac [2] order_le_less_trans) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1277 |
prefer 2 apply (simp add: right_diff_distrib distrib_left) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1278 |
apply (subgoal_tac "b * q' < b * (1 + q) ") |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1279 |
prefer 2 apply (simp add: right_diff_distrib distrib_left) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1280 |
apply (simp add: mult_less_cancel_left) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1281 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1282 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1283 |
lemma unique_quotient_lemma_neg: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1284 |
"[| b*q' + r' \<le> b*q + r; r \<le> 0; b < r; b < r' |] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1285 |
==> q \<le> (q'::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1286 |
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1287 |
auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1288 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1289 |
lemma unique_quotient: |
46552 | 1290 |
"[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |] |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1291 |
==> q = q'" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1292 |
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1293 |
apply (blast intro: order_antisym |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1294 |
dest: order_eq_refl [THEN unique_quotient_lemma] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1295 |
order_eq_refl [THEN unique_quotient_lemma_neg] sym)+ |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1296 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1297 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1298 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1299 |
lemma unique_remainder: |
46552 | 1300 |
"[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |] |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1301 |
==> r = r'" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1302 |
apply (subgoal_tac "q = q'") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1303 |
apply (simp add: divmod_int_rel_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1304 |
apply (blast intro: unique_quotient) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1305 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1306 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1307 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1308 |
subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1309 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1310 |
text{*And positive divisors*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1311 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1312 |
lemma adjust_eq [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1313 |
"adjust b (q, r) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1314 |
(let diff = r - b in |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1315 |
if 0 \<le> diff then (2 * q + 1, diff) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1316 |
else (2*q, r))" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1317 |
by (simp add: Let_def adjust_def) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1318 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1319 |
declare posDivAlg.simps [simp del] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1320 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1321 |
text{*use with a simproc to avoid repeatedly proving the premise*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1322 |
lemma posDivAlg_eqn: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1323 |
"0 < b ==> |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1324 |
posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1325 |
by (rule posDivAlg.simps [THEN trans], simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1326 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1327 |
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1328 |
theorem posDivAlg_correct: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1329 |
assumes "0 \<le> a" and "0 < b" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1330 |
shows "divmod_int_rel a b (posDivAlg a b)" |
41550 | 1331 |
using assms |
1332 |
apply (induct a b rule: posDivAlg.induct) |
|
1333 |
apply auto |
|
1334 |
apply (simp add: divmod_int_rel_def) |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1335 |
apply (subst posDivAlg_eqn, simp add: distrib_left) |
41550 | 1336 |
apply (case_tac "a < b") |
1337 |
apply simp_all |
|
1338 |
apply (erule splitE) |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1339 |
apply (auto simp add: distrib_left Let_def mult_ac mult_2_right) |
41550 | 1340 |
done |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1341 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1342 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1343 |
subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1344 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1345 |
text{*And positive divisors*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1346 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1347 |
declare negDivAlg.simps [simp del] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1348 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1349 |
text{*use with a simproc to avoid repeatedly proving the premise*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1350 |
lemma negDivAlg_eqn: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1351 |
"0 < b ==> |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1352 |
negDivAlg a b = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1353 |
(if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1354 |
by (rule negDivAlg.simps [THEN trans], simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1355 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1356 |
(*Correctness of negDivAlg: it computes quotients correctly |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1357 |
It doesn't work if a=0 because the 0/b equals 0, not -1*) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1358 |
lemma negDivAlg_correct: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1359 |
assumes "a < 0" and "b > 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1360 |
shows "divmod_int_rel a b (negDivAlg a b)" |
41550 | 1361 |
using assms |
1362 |
apply (induct a b rule: negDivAlg.induct) |
|
1363 |
apply (auto simp add: linorder_not_le) |
|
1364 |
apply (simp add: divmod_int_rel_def) |
|
1365 |
apply (subst negDivAlg_eqn, assumption) |
|
1366 |
apply (case_tac "a + b < (0\<Colon>int)") |
|
1367 |
apply simp_all |
|
1368 |
apply (erule splitE) |
|
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1369 |
apply (auto simp add: distrib_left Let_def mult_ac mult_2_right) |
41550 | 1370 |
done |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1371 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1372 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1373 |
subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1374 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1375 |
(*the case a=0*) |
47139
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1376 |
lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1377 |
by (auto simp add: divmod_int_rel_def linorder_neq_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1378 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1379 |
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1380 |
by (subst posDivAlg.simps, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1381 |
|
47139
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1382 |
lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)" |
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1383 |
by (subst posDivAlg.simps, auto) |
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1384 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1385 |
lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1386 |
by (subst negDivAlg.simps, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1387 |
|
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset
|
1388 |
lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)" |
47139
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1389 |
by (auto simp add: divmod_int_rel_def) |
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1390 |
|
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1391 |
lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)" |
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1392 |
apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1393 |
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1394 |
posDivAlg_correct negDivAlg_correct) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1395 |
|
47141
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1396 |
lemma divmod_int_unique: |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1397 |
assumes "divmod_int_rel a b qr" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1398 |
shows "divmod_int a b = qr" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1399 |
using assms divmod_int_correct [of a b] |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1400 |
using unique_quotient [of a b] unique_remainder [of a b] |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1401 |
by (metis pair_collapse) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1402 |
|
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1403 |
lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1404 |
using divmod_int_correct by (simp add: divmod_int_mod_div) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1405 |
|
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1406 |
lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1407 |
by (simp add: divmod_int_rel_div_mod [THEN unique_quotient]) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1408 |
|
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1409 |
lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1410 |
by (simp add: divmod_int_rel_div_mod [THEN unique_remainder]) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1411 |
|
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1412 |
instance int :: ring_div |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1413 |
proof |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1414 |
fix a b :: int |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1415 |
show "a div b * b + a mod b = a" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1416 |
using divmod_int_rel_div_mod [of a b] |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1417 |
unfolding divmod_int_rel_def by (simp add: mult_commute) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1418 |
next |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1419 |
fix a b c :: int |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1420 |
assume "b \<noteq> 0" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1421 |
hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1422 |
using divmod_int_rel_div_mod [of a b] |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1423 |
unfolding divmod_int_rel_def by (auto simp: algebra_simps) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1424 |
thus "(a + c * b) div b = c + a div b" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1425 |
by (rule div_int_unique) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1426 |
next |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1427 |
fix a b c :: int |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1428 |
assume "c \<noteq> 0" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1429 |
hence "\<And>q r. divmod_int_rel a b (q, r) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1430 |
\<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1431 |
unfolding divmod_int_rel_def |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1432 |
by - (rule linorder_cases [of 0 b], auto simp: algebra_simps |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1433 |
mult_less_0_iff zero_less_mult_iff mult_strict_right_mono |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1434 |
mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1435 |
hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1436 |
using divmod_int_rel_div_mod [of a b] . |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1437 |
thus "(c * a) div (c * b) = a div b" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1438 |
by (rule div_int_unique) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1439 |
next |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1440 |
fix a :: int show "a div 0 = 0" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1441 |
by (rule div_int_unique, simp add: divmod_int_rel_def) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1442 |
next |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1443 |
fix a :: int show "0 div a = 0" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1444 |
by (rule div_int_unique, auto simp add: divmod_int_rel_def) |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1445 |
qed |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1446 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1447 |
text{*Basic laws about division and remainder*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1448 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1449 |
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)" |
47141
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1450 |
by (fact mod_div_equality2 [symmetric]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1451 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1452 |
text {* Tool setup *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1453 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1454 |
(* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1455 |
lemmas add_0s = add_0_left add_0_right |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1456 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1457 |
ML {* |
43594 | 1458 |
structure Cancel_Div_Mod_Int = Cancel_Div_Mod |
41550 | 1459 |
( |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1460 |
val div_name = @{const_name div}; |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1461 |
val mod_name = @{const_name mod}; |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1462 |
val mk_binop = HOLogic.mk_binop; |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1463 |
val mk_sum = Arith_Data.mk_sum HOLogic.intT; |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1464 |
val dest_sum = Arith_Data.dest_sum; |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1465 |
|
47165 | 1466 |
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]; |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1467 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1468 |
val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1469 |
(@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac})) |
41550 | 1470 |
) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1471 |
*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1472 |
|
43594 | 1473 |
simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *} |
1474 |
||
47141
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1475 |
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1476 |
using divmod_int_correct [of a b] |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1477 |
by (auto simp add: divmod_int_rel_def prod_eq_iff) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1478 |
|
45607 | 1479 |
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1] |
1480 |
and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2] |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1481 |
|
47141
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1482 |
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b" |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1483 |
using divmod_int_correct [of a b] |
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1484 |
by (auto simp add: divmod_int_rel_def prod_eq_iff) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1485 |
|
45607 | 1486 |
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1] |
1487 |
and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2] |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1488 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1489 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1490 |
subsubsection {* General Properties of div and mod *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1491 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1492 |
lemma div_pos_pos_trivial: "[| (0::int) \<le> a; a < b |] ==> a div b = 0" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1493 |
apply (rule div_int_unique) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1494 |
apply (auto simp add: divmod_int_rel_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1495 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1496 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1497 |
lemma div_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a div b = 0" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1498 |
apply (rule div_int_unique) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1499 |
apply (auto simp add: divmod_int_rel_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1500 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1501 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1502 |
lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a div b = -1" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1503 |
apply (rule div_int_unique) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1504 |
apply (auto simp add: divmod_int_rel_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1505 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1506 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1507 |
(*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1508 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1509 |
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a; a < b |] ==> a mod b = a" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1510 |
apply (rule_tac q = 0 in mod_int_unique) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1511 |
apply (auto simp add: divmod_int_rel_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1512 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1513 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1514 |
lemma mod_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a mod b = a" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1515 |
apply (rule_tac q = 0 in mod_int_unique) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1516 |
apply (auto simp add: divmod_int_rel_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1517 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1518 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1519 |
lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a mod b = a+b" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1520 |
apply (rule_tac q = "-1" in mod_int_unique) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1521 |
apply (auto simp add: divmod_int_rel_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1522 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1523 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1524 |
text{*There is no @{text mod_neg_pos_trivial}.*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1525 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1526 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1527 |
subsubsection {* Laws for div and mod with Unary Minus *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1528 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1529 |
lemma zminus1_lemma: |
47139
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1530 |
"divmod_int_rel a b (q, r) ==> b \<noteq> 0 |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1531 |
==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1, |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1532 |
if r=0 then 0 else b-r)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1533 |
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1534 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1535 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1536 |
lemma zdiv_zminus1_eq_if: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1537 |
"b \<noteq> (0::int) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1538 |
==> (-a) div b = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1539 |
(if a mod b = 0 then - (a div b) else - (a div b) - 1)" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1540 |
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1541 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1542 |
lemma zmod_zminus1_eq_if: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1543 |
"(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1544 |
apply (case_tac "b = 0", simp) |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1545 |
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1546 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1547 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1548 |
lemma zmod_zminus1_not_zero: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1549 |
fixes k l :: int |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1550 |
shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1551 |
unfolding zmod_zminus1_eq_if by auto |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1552 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1553 |
lemma zdiv_zminus2_eq_if: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1554 |
"b \<noteq> (0::int) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1555 |
==> a div (-b) = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1556 |
(if a mod b = 0 then - (a div b) else - (a div b) - 1)" |
47159 | 1557 |
by (simp add: zdiv_zminus1_eq_if div_minus_right) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1558 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1559 |
lemma zmod_zminus2_eq_if: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1560 |
"a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)" |
47159 | 1561 |
by (simp add: zmod_zminus1_eq_if mod_minus_right) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1562 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1563 |
lemma zmod_zminus2_not_zero: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1564 |
fixes k l :: int |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1565 |
shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1566 |
unfolding zmod_zminus2_eq_if by auto |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1567 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1568 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1569 |
subsubsection {* Computation of Division and Remainder *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1570 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1571 |
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1572 |
by (simp add: div_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1573 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1574 |
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1575 |
by (simp add: mod_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1576 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1577 |
text{*a positive, b positive *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1578 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1579 |
lemma div_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a div b = fst (posDivAlg a b)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1580 |
by (simp add: div_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1581 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1582 |
lemma mod_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a mod b = snd (posDivAlg a b)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1583 |
by (simp add: mod_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1584 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1585 |
text{*a negative, b positive *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1586 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1587 |
lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1588 |
by (simp add: div_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1589 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1590 |
lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1591 |
by (simp add: mod_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1592 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1593 |
text{*a positive, b negative *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1594 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1595 |
lemma div_pos_neg: |
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset
|
1596 |
"[| 0 < a; b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1597 |
by (simp add: div_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1598 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1599 |
lemma mod_pos_neg: |
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset
|
1600 |
"[| 0 < a; b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1601 |
by (simp add: mod_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1602 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1603 |
text{*a negative, b negative *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1604 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1605 |
lemma div_neg_neg: |
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset
|
1606 |
"[| a < 0; b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1607 |
by (simp add: div_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1608 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1609 |
lemma mod_neg_neg: |
46560
8e252a608765
remove constant negateSnd in favor of 'apsnd uminus' (from Florian Haftmann)
huffman
parents:
46552
diff
changeset
|
1610 |
"[| a < 0; b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1611 |
by (simp add: mod_int_def divmod_int_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1612 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1613 |
text {*Simplify expresions in which div and mod combine numerical constants*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1614 |
|
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1615 |
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1616 |
by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def) |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1617 |
|
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1618 |
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1619 |
by (rule div_int_unique [of a b q r], |
46552 | 1620 |
simp add: divmod_int_rel_def) |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1621 |
|
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1622 |
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1623 |
by (rule mod_int_unique [of a b q r], |
46552 | 1624 |
simp add: divmod_int_rel_def) |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1625 |
|
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1626 |
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r" |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1627 |
by (rule mod_int_unique [of a b q r], |
46552 | 1628 |
simp add: divmod_int_rel_def) |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1629 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1630 |
(* simprocs adapted from HOL/ex/Binary.thy *) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1631 |
ML {* |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1632 |
local |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1633 |
val mk_number = HOLogic.mk_number HOLogic.intT |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1634 |
val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1635 |
val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"} |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1636 |
val zero = @{term "0 :: int"} |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1637 |
val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1638 |
val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1639 |
val simps = @{thms arith_simps} @ @{thms rel_simps} @ |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1640 |
map (fn th => th RS sym) [@{thm numeral_1_eq_1}] |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1641 |
fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal) |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1642 |
(K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps)))); |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1643 |
fun binary_proc proc ss ct = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1644 |
(case Thm.term_of ct of |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1645 |
_ $ t $ u => |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1646 |
(case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1647 |
SOME args => proc (Simplifier.the_context ss) args |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1648 |
| NONE => NONE) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1649 |
| _ => NONE); |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1650 |
in |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1651 |
fun divmod_proc posrule negrule = |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1652 |
binary_proc (fn ctxt => fn ((a, t), (b, u)) => |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1653 |
if b = 0 then NONE else let |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1654 |
val (q, r) = pairself mk_number (Integer.div_mod a b) |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1655 |
val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r) |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1656 |
val (goal2, goal3, rule) = if b > 0 |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1657 |
then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection) |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1658 |
else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection) |
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1659 |
in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1660 |
end |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1661 |
*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1662 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1663 |
simproc_setup binary_int_div |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1664 |
("numeral m div numeral n :: int" | |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1665 |
"numeral m div neg_numeral n :: int" | |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1666 |
"neg_numeral m div numeral n :: int" | |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1667 |
"neg_numeral m div neg_numeral n :: int") = |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1668 |
{* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1669 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1670 |
simproc_setup binary_int_mod |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1671 |
("numeral m mod numeral n :: int" | |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1672 |
"numeral m mod neg_numeral n :: int" | |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1673 |
"neg_numeral m mod numeral n :: int" | |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1674 |
"neg_numeral m mod neg_numeral n :: int") = |
45530
0c4853bb77bf
rewrite integer numeral div/mod simprocs to not return conditional rewrites; add regression tests
huffman
parents:
45231
diff
changeset
|
1675 |
{* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1676 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1677 |
lemmas posDivAlg_eqn_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1678 |
posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1679 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1680 |
lemmas negDivAlg_eqn_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1681 |
negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1682 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1683 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1684 |
text{*Special-case simplification *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1685 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1686 |
(** The last remaining special cases for constant arithmetic: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1687 |
1 div z and 1 mod z **) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1688 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1689 |
lemmas div_pos_pos_1_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1690 |
div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1691 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1692 |
lemmas div_pos_neg_1_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1693 |
div_pos_neg [OF zero_less_one, of "neg_numeral w", |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1694 |
OF neg_numeral_less_zero] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1695 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1696 |
lemmas mod_pos_pos_1_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1697 |
mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1698 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1699 |
lemmas mod_pos_neg_1_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1700 |
mod_pos_neg [OF zero_less_one, of "neg_numeral w", |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1701 |
OF neg_numeral_less_zero] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1702 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1703 |
lemmas posDivAlg_eqn_1_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1704 |
posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1705 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1706 |
lemmas negDivAlg_eqn_1_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1707 |
negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1708 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1709 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1710 |
subsubsection {* Monotonicity in the First Argument (Dividend) *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1711 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1712 |
lemma zdiv_mono1: "[| a \<le> a'; 0 < (b::int) |] ==> a div b \<le> a' div b" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1713 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1714 |
apply (cut_tac a = a' and b = b in zmod_zdiv_equality) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1715 |
apply (rule unique_quotient_lemma) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1716 |
apply (erule subst) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1717 |
apply (erule subst, simp_all) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1718 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1719 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1720 |
lemma zdiv_mono1_neg: "[| a \<le> a'; (b::int) < 0 |] ==> a' div b \<le> a div b" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1721 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1722 |
apply (cut_tac a = a' and b = b in zmod_zdiv_equality) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1723 |
apply (rule unique_quotient_lemma_neg) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1724 |
apply (erule subst) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1725 |
apply (erule subst, simp_all) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1726 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1727 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1728 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1729 |
subsubsection {* Monotonicity in the Second Argument (Divisor) *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1730 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1731 |
lemma q_pos_lemma: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1732 |
"[| 0 \<le> b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \<le> (q'::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1733 |
apply (subgoal_tac "0 < b'* (q' + 1) ") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1734 |
apply (simp add: zero_less_mult_iff) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1735 |
apply (simp add: distrib_left) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1736 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1737 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1738 |
lemma zdiv_mono2_lemma: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1739 |
"[| b*q + r = b'*q' + r'; 0 \<le> b'*q' + r'; |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1740 |
r' < b'; 0 \<le> r; 0 < b'; b' \<le> b |] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1741 |
==> q \<le> (q'::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1742 |
apply (frule q_pos_lemma, assumption+) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1743 |
apply (subgoal_tac "b*q < b* (q' + 1) ") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1744 |
apply (simp add: mult_less_cancel_left) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1745 |
apply (subgoal_tac "b*q = r' - r + b'*q'") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1746 |
prefer 2 apply simp |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1747 |
apply (simp (no_asm_simp) add: distrib_left) |
44766 | 1748 |
apply (subst add_commute, rule add_less_le_mono, arith) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1749 |
apply (rule mult_right_mono, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1750 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1751 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1752 |
lemma zdiv_mono2: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1753 |
"[| (0::int) \<le> a; 0 < b'; b' \<le> b |] ==> a div b \<le> a div b'" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1754 |
apply (subgoal_tac "b \<noteq> 0") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1755 |
prefer 2 apply arith |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1756 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1757 |
apply (cut_tac a = a and b = b' in zmod_zdiv_equality) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1758 |
apply (rule zdiv_mono2_lemma) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1759 |
apply (erule subst) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1760 |
apply (erule subst, simp_all) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1761 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1762 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1763 |
lemma q_neg_lemma: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1764 |
"[| b'*q' + r' < 0; 0 \<le> r'; 0 < b' |] ==> q' \<le> (0::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1765 |
apply (subgoal_tac "b'*q' < 0") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1766 |
apply (simp add: mult_less_0_iff, arith) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1767 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1768 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1769 |
lemma zdiv_mono2_neg_lemma: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1770 |
"[| b*q + r = b'*q' + r'; b'*q' + r' < 0; |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1771 |
r < b; 0 \<le> r'; 0 < b'; b' \<le> b |] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1772 |
==> q' \<le> (q::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1773 |
apply (frule q_neg_lemma, assumption+) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1774 |
apply (subgoal_tac "b*q' < b* (q + 1) ") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1775 |
apply (simp add: mult_less_cancel_left) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1776 |
apply (simp add: distrib_left) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1777 |
apply (subgoal_tac "b*q' \<le> b'*q'") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1778 |
prefer 2 apply (simp add: mult_right_mono_neg, arith) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1779 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1780 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1781 |
lemma zdiv_mono2_neg: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1782 |
"[| a < (0::int); 0 < b'; b' \<le> b |] ==> a div b' \<le> a div b" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1783 |
apply (cut_tac a = a and b = b in zmod_zdiv_equality) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1784 |
apply (cut_tac a = a and b = b' in zmod_zdiv_equality) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1785 |
apply (rule zdiv_mono2_neg_lemma) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1786 |
apply (erule subst) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1787 |
apply (erule subst, simp_all) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1788 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1789 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1790 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1791 |
subsubsection {* More Algebraic Laws for div and mod *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1792 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1793 |
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1794 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1795 |
lemma zmult1_lemma: |
46552 | 1796 |
"[| divmod_int_rel b c (q, r) |] |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1797 |
==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1798 |
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left mult_ac) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1799 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1800 |
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1801 |
apply (case_tac "c = 0", simp) |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1802 |
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1803 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1804 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1805 |
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1806 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1807 |
lemma zadd1_lemma: |
46552 | 1808 |
"[| divmod_int_rel a c (aq, ar); divmod_int_rel b c (bq, br) |] |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1809 |
==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1810 |
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1811 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1812 |
(*NOT suitable for rewriting: the RHS has an instance of the LHS*) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1813 |
lemma zdiv_zadd1_eq: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1814 |
"(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1815 |
apply (case_tac "c = 0", simp) |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1816 |
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1817 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1818 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1819 |
lemma posDivAlg_div_mod: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1820 |
assumes "k \<ge> 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1821 |
and "l \<ge> 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1822 |
shows "posDivAlg k l = (k div l, k mod l)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1823 |
proof (cases "l = 0") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1824 |
case True then show ?thesis by (simp add: posDivAlg.simps) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1825 |
next |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1826 |
case False with assms posDivAlg_correct |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1827 |
have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1828 |
by simp |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1829 |
from div_int_unique [OF this] mod_int_unique [OF this] |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1830 |
show ?thesis by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1831 |
qed |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1832 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1833 |
lemma negDivAlg_div_mod: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1834 |
assumes "k < 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1835 |
and "l > 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1836 |
shows "negDivAlg k l = (k div l, k mod l)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1837 |
proof - |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1838 |
from assms have "l \<noteq> 0" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1839 |
from assms negDivAlg_correct |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1840 |
have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1841 |
by simp |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1842 |
from div_int_unique [OF this] mod_int_unique [OF this] |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1843 |
show ?thesis by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1844 |
qed |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1845 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1846 |
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1847 |
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1848 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1849 |
(* REVISIT: should this be generalized to all semiring_div types? *) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1850 |
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1851 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1852 |
lemma zmod_zdiv_equality': |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1853 |
"(m\<Colon>int) mod n = m - (m div n) * n" |
47141
02d6b816e4b3
move int::ring_div instance upward, simplify several proofs
huffman
parents:
47140
diff
changeset
|
1854 |
using mod_div_equality [of m n] by arith |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1855 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1856 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1857 |
subsubsection {* Proving @{term "a div (b*c) = (a div b) div c"} *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1858 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1859 |
(*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1860 |
7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1861 |
to cause particular problems.*) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1862 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1863 |
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1864 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1865 |
lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \<le> 0 |] ==> b*c < b*(q mod c) + r" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1866 |
apply (subgoal_tac "b * (c - q mod c) < r * 1") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1867 |
apply (simp add: algebra_simps) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1868 |
apply (rule order_le_less_trans) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1869 |
apply (erule_tac [2] mult_strict_right_mono) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1870 |
apply (rule mult_left_mono_neg) |
35216 | 1871 |
using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1872 |
apply (simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1873 |
apply (simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1874 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1875 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1876 |
lemma zmult2_lemma_aux2: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1877 |
"[| (0::int) < c; b < r; r \<le> 0 |] ==> b * (q mod c) + r \<le> 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1878 |
apply (subgoal_tac "b * (q mod c) \<le> 0") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1879 |
apply arith |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1880 |
apply (simp add: mult_le_0_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1881 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1882 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1883 |
lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \<le> r; r < b |] ==> 0 \<le> b * (q mod c) + r" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1884 |
apply (subgoal_tac "0 \<le> b * (q mod c) ") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1885 |
apply arith |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1886 |
apply (simp add: zero_le_mult_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1887 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1888 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1889 |
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1890 |
apply (subgoal_tac "r * 1 < b * (c - q mod c) ") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1891 |
apply (simp add: right_diff_distrib) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1892 |
apply (rule order_less_le_trans) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1893 |
apply (erule mult_strict_right_mono) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1894 |
apply (rule_tac [2] mult_left_mono) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1895 |
apply simp |
35216 | 1896 |
using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1897 |
apply simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1898 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1899 |
|
46552 | 1900 |
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |] |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1901 |
==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1902 |
by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
48891
diff
changeset
|
1903 |
zero_less_mult_iff distrib_left [symmetric] |
47139
98bddfa0f717
extend definition of divmod_int_rel to handle denominator=0 case
huffman
parents:
47138
diff
changeset
|
1904 |
zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1905 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1906 |
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1907 |
apply (case_tac "b = 0", simp) |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1908 |
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1909 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1910 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1911 |
lemma zmod_zmult2_eq: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1912 |
"(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1913 |
apply (case_tac "b = 0", simp) |
47140
97c3676c5c94
rename lemmas {divmod_int_rel_{div,mod} -> {div,mod}_int_unique, for consistency with nat lemma names
huffman
parents:
47139
diff
changeset
|
1914 |
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1915 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1916 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1917 |
lemma div_pos_geq: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1918 |
fixes k l :: int |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1919 |
assumes "0 < l" and "l \<le> k" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1920 |
shows "k div l = (k - l) div l + 1" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1921 |
proof - |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1922 |
have "k = (k - l) + l" by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1923 |
then obtain j where k: "k = j + l" .. |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1924 |
with assms show ?thesis by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1925 |
qed |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1926 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1927 |
lemma mod_pos_geq: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1928 |
fixes k l :: int |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1929 |
assumes "0 < l" and "l \<le> k" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1930 |
shows "k mod l = (k - l) mod l" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1931 |
proof - |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1932 |
have "k = (k - l) + l" by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1933 |
then obtain j where k: "k = j + l" .. |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1934 |
with assms show ?thesis by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1935 |
qed |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1936 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1937 |
|
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
1938 |
subsubsection {* Splitting Rules for div and mod *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1939 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1940 |
text{*The proofs of the two lemmas below are essentially identical*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1941 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1942 |
lemma split_pos_lemma: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1943 |
"0<k ==> |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1944 |
P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1945 |
apply (rule iffI, clarify) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1946 |
apply (erule_tac P="P ?x ?y" in rev_mp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1947 |
apply (subst mod_add_eq) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1948 |
apply (subst zdiv_zadd1_eq) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1949 |
apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1950 |
txt{*converse direction*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1951 |
apply (drule_tac x = "n div k" in spec) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1952 |
apply (drule_tac x = "n mod k" in spec, simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1953 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1954 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1955 |
lemma split_neg_lemma: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1956 |
"k<0 ==> |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1957 |
P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1958 |
apply (rule iffI, clarify) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1959 |
apply (erule_tac P="P ?x ?y" in rev_mp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1960 |
apply (subst mod_add_eq) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1961 |
apply (subst zdiv_zadd1_eq) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1962 |
apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1963 |
txt{*converse direction*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1964 |
apply (drule_tac x = "n div k" in spec) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1965 |
apply (drule_tac x = "n mod k" in spec, simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1966 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1967 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1968 |
lemma split_zdiv: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1969 |
"P(n div k :: int) = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1970 |
((k = 0 --> P 0) & |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1971 |
(0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1972 |
(k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1973 |
apply (case_tac "k=0", simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1974 |
apply (simp only: linorder_neq_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1975 |
apply (erule disjE) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1976 |
apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1977 |
split_neg_lemma [of concl: "%x y. P x"]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1978 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1979 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1980 |
lemma split_zmod: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1981 |
"P(n mod k :: int) = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1982 |
((k = 0 --> P n) & |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1983 |
(0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1984 |
(k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1985 |
apply (case_tac "k=0", simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1986 |
apply (simp only: linorder_neq_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1987 |
apply (erule disjE) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1988 |
apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1989 |
split_neg_lemma [of concl: "%x y. P y"]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1990 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1991 |
|
33730
1755ca4ec022
Fixed splitting of div and mod on integers (split theorem differed from implementation).
webertj
parents:
33728
diff
changeset
|
1992 |
text {* Enable (lin)arith to deal with @{const div} and @{const mod} |
1755ca4ec022
Fixed splitting of div and mod on integers (split theorem differed from implementation).
webertj
parents:
33728
diff
changeset
|
1993 |
when these are applied to some constant that is of the form |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1994 |
@{term "numeral k"}: *} |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1995 |
declare split_zdiv [of _ _ "numeral k", arith_split] for k |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
1996 |
declare split_zmod [of _ _ "numeral k", arith_split] for k |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1997 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
1998 |
|
47166 | 1999 |
subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *} |
2000 |
||
2001 |
lemma pos_divmod_int_rel_mult_2: |
|
2002 |
assumes "0 \<le> b" |
|
2003 |
assumes "divmod_int_rel a b (q, r)" |
|
2004 |
shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)" |
|
2005 |
using assms unfolding divmod_int_rel_def by auto |
|
2006 |
||
2007 |
lemma neg_divmod_int_rel_mult_2: |
|
2008 |
assumes "b \<le> 0" |
|
2009 |
assumes "divmod_int_rel (a + 1) b (q, r)" |
|
2010 |
shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)" |
|
2011 |
using assms unfolding divmod_int_rel_def by auto |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2012 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2013 |
text{*computing div by shifting *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2014 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2015 |
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a" |
47166 | 2016 |
using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod] |
2017 |
by (rule div_int_unique) |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2018 |
|
35815
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset
|
2019 |
lemma neg_zdiv_mult_2: |
10e723e54076
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
boehmes
parents:
35673
diff
changeset
|
2020 |
assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a" |
47166 | 2021 |
using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod] |
2022 |
by (rule div_int_unique) |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2023 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2024 |
(* FIXME: add rules for negative numerals *) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2025 |
lemma zdiv_numeral_Bit0 [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2026 |
"numeral (Num.Bit0 v) div numeral (Num.Bit0 w) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2027 |
numeral v div (numeral w :: int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2028 |
unfolding numeral.simps unfolding mult_2 [symmetric] |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2029 |
by (rule div_mult_mult1, simp) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2030 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2031 |
lemma zdiv_numeral_Bit1 [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2032 |
"numeral (Num.Bit1 v) div numeral (Num.Bit0 w) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2033 |
(numeral v div (numeral w :: int))" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2034 |
unfolding numeral.simps |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2035 |
unfolding mult_2 [symmetric] add_commute [of _ 1] |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2036 |
by (rule pos_zdiv_mult_2, simp) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2037 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2038 |
lemma pos_zmod_mult_2: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2039 |
fixes a b :: int |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2040 |
assumes "0 \<le> a" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2041 |
shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)" |
47166 | 2042 |
using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod] |
2043 |
by (rule mod_int_unique) |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2044 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2045 |
lemma neg_zmod_mult_2: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2046 |
fixes a b :: int |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2047 |
assumes "a \<le> 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2048 |
shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1" |
47166 | 2049 |
using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod] |
2050 |
by (rule mod_int_unique) |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2051 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2052 |
(* FIXME: add rules for negative numerals *) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2053 |
lemma zmod_numeral_Bit0 [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2054 |
"numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2055 |
(2::int) * (numeral v mod numeral w)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2056 |
unfolding numeral_Bit0 [of v] numeral_Bit0 [of w] |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2057 |
unfolding mult_2 [symmetric] by (rule mod_mult_mult1) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2058 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2059 |
lemma zmod_numeral_Bit1 [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2060 |
"numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2061 |
2 * (numeral v mod numeral w) + (1::int)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2062 |
unfolding numeral_Bit1 [of v] numeral_Bit0 [of w] |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2063 |
unfolding mult_2 [symmetric] add_commute [of _ 1] |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2064 |
by (rule pos_zmod_mult_2, simp) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2065 |
|
39489 | 2066 |
lemma zdiv_eq_0_iff: |
2067 |
"(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") |
|
2068 |
proof |
|
2069 |
assume ?L |
|
2070 |
have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp |
|
2071 |
with `?L` show ?R by blast |
|
2072 |
next |
|
2073 |
assume ?R thus ?L |
|
2074 |
by(auto simp: div_pos_pos_trivial div_neg_neg_trivial) |
|
2075 |
qed |
|
2076 |
||
2077 |
||
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
2078 |
subsubsection {* Quotients of Signs *} |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2079 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2080 |
lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2081 |
apply (subgoal_tac "a div b \<le> -1", force) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2082 |
apply (rule order_trans) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2083 |
apply (rule_tac a' = "-1" in zdiv_mono1) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2084 |
apply (auto simp add: div_eq_minus1) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2085 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2086 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2087 |
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2088 |
by (drule zdiv_mono1_neg, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2089 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2090 |
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2091 |
by (drule zdiv_mono1, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2092 |
|
33804 | 2093 |
text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"} |
2094 |
conditional upon the sign of @{text a} or @{text b}. There are many more. |
|
2095 |
They should all be simp rules unless that causes too much search. *} |
|
2096 |
||
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2097 |
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2098 |
apply auto |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2099 |
apply (drule_tac [2] zdiv_mono1) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2100 |
apply (auto simp add: linorder_neq_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2101 |
apply (simp (no_asm_use) add: linorder_not_less [symmetric]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2102 |
apply (blast intro: div_neg_pos_less0) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2103 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2104 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2105 |
lemma neg_imp_zdiv_nonneg_iff: |
33804 | 2106 |
"b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))" |
47159 | 2107 |
apply (subst div_minus_minus [symmetric]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2108 |
apply (subst pos_imp_zdiv_nonneg_iff, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2109 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2110 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2111 |
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2112 |
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2113 |
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2114 |
|
39489 | 2115 |
lemma pos_imp_zdiv_pos_iff: |
2116 |
"0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i" |
|
2117 |
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k] |
|
2118 |
by arith |
|
2119 |
||
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2120 |
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2121 |
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2122 |
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2123 |
|
33804 | 2124 |
lemma nonneg1_imp_zdiv_pos_iff: |
2125 |
"(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)" |
|
2126 |
apply rule |
|
2127 |
apply rule |
|
2128 |
using div_pos_pos_trivial[of a b]apply arith |
|
2129 |
apply(cases "b=0")apply simp |
|
2130 |
using div_nonneg_neg_le0[of a b]apply arith |
|
2131 |
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp |
|
2132 |
done |
|
2133 |
||
39489 | 2134 |
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m" |
2135 |
apply (rule split_zmod[THEN iffD2]) |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44766
diff
changeset
|
2136 |
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le) |
39489 | 2137 |
done |
2138 |
||
2139 |
||
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2140 |
subsubsection {* The Divides Relation *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2141 |
|
47268 | 2142 |
lemma dvd_neg_numeral_left [simp]: |
2143 |
fixes y :: "'a::comm_ring_1" |
|
2144 |
shows "(neg_numeral k) dvd y \<longleftrightarrow> (numeral k) dvd y" |
|
2145 |
unfolding neg_numeral_def minus_dvd_iff .. |
|
2146 |
||
2147 |
lemma dvd_neg_numeral_right [simp]: |
|
2148 |
fixes x :: "'a::comm_ring_1" |
|
2149 |
shows "x dvd (neg_numeral k) \<longleftrightarrow> x dvd (numeral k)" |
|
2150 |
unfolding neg_numeral_def dvd_minus_iff .. |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2151 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2152 |
lemmas dvd_eq_mod_eq_0_numeral [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2153 |
dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2154 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2155 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2156 |
subsubsection {* Further properties *} |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2157 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2158 |
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2159 |
using zmod_zdiv_equality[where a="m" and b="n"] |
47142 | 2160 |
by (simp add: algebra_simps) (* FIXME: generalize *) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2161 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2162 |
lemma zdiv_int: "int (a div b) = (int a) div (int b)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2163 |
apply (subst split_div, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2164 |
apply (subst split_zdiv, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2165 |
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2166 |
apply (auto simp add: divmod_int_rel_def of_nat_mult) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2167 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2168 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2169 |
lemma zmod_int: "int (a mod b) = (int a) mod (int b)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2170 |
apply (subst split_mod, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2171 |
apply (subst split_zmod, auto) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2172 |
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2173 |
in unique_remainder) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2174 |
apply (auto simp add: divmod_int_rel_def of_nat_mult) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2175 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2176 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2177 |
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2178 |
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2179 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2180 |
text{*Suggested by Matthias Daum*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2181 |
lemma int_power_div_base: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2182 |
"\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2183 |
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2184 |
apply (erule ssubst) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2185 |
apply (simp only: power_add) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2186 |
apply simp_all |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2187 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2188 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2189 |
text {* by Brian Huffman *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2190 |
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2191 |
by (rule mod_minus_eq [symmetric]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2192 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2193 |
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2194 |
by (rule mod_diff_left_eq [symmetric]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2195 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2196 |
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2197 |
by (rule mod_diff_right_eq [symmetric]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2198 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2199 |
lemmas zmod_simps = |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2200 |
mod_add_left_eq [symmetric] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2201 |
mod_add_right_eq [symmetric] |
47142 | 2202 |
mod_mult_right_eq[symmetric] |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2203 |
mod_mult_left_eq [symmetric] |
47164 | 2204 |
power_mod |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2205 |
zminus_zmod zdiff_zmod_left zdiff_zmod_right |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2206 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2207 |
text {* Distributive laws for function @{text nat}. *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2208 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2209 |
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2210 |
apply (rule linorder_cases [of y 0]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2211 |
apply (simp add: div_nonneg_neg_le0) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2212 |
apply simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2213 |
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2214 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2215 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2216 |
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2217 |
lemma nat_mod_distrib: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2218 |
"\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2219 |
apply (case_tac "y = 0", simp) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2220 |
apply (simp add: nat_eq_iff zmod_int) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2221 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2222 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2223 |
text {* transfer setup *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2224 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2225 |
lemma transfer_nat_int_functions: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2226 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2227 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2228 |
by (auto simp add: nat_div_distrib nat_mod_distrib) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2229 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2230 |
lemma transfer_nat_int_function_closures: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2231 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2232 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2233 |
apply (cases "y = 0") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2234 |
apply (auto simp add: pos_imp_zdiv_nonneg_iff) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2235 |
apply (cases "y = 0") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2236 |
apply auto |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2237 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2238 |
|
35644 | 2239 |
declare transfer_morphism_nat_int [transfer add return: |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2240 |
transfer_nat_int_functions |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2241 |
transfer_nat_int_function_closures |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2242 |
] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2243 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2244 |
lemma transfer_int_nat_functions: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2245 |
"(int x) div (int y) = int (x div y)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2246 |
"(int x) mod (int y) = int (x mod y)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2247 |
by (auto simp add: zdiv_int zmod_int) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2248 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2249 |
lemma transfer_int_nat_function_closures: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2250 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2251 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2252 |
by (simp_all only: is_nat_def transfer_nat_int_function_closures) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2253 |
|
35644 | 2254 |
declare transfer_morphism_int_nat [transfer add return: |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2255 |
transfer_int_nat_functions |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2256 |
transfer_int_nat_function_closures |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2257 |
] |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2258 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2259 |
text{*Suggested by Matthias Daum*} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2260 |
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2261 |
apply (subgoal_tac "nat x div nat k < nat x") |
34225 | 2262 |
apply (simp add: nat_div_distrib [symmetric]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2263 |
apply (rule Divides.div_less_dividend, simp_all) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2264 |
done |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2265 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2266 |
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2267 |
proof |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2268 |
assume H: "x mod n = y mod n" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2269 |
hence "x mod n - y mod n = 0" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2270 |
hence "(x mod n - y mod n) mod n = 0" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2271 |
hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2272 |
thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2273 |
next |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2274 |
assume H: "n dvd x - y" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2275 |
then obtain k where k: "x-y = n*k" unfolding dvd_def by blast |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2276 |
hence "x = n*k + y" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2277 |
hence "x mod n = (n*k + y) mod n" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2278 |
thus "x mod n = y mod n" by (simp add: mod_add_left_eq) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2279 |
qed |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2280 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2281 |
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2282 |
shows "\<exists>q. x = y + n * q" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2283 |
proof- |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2284 |
from xy have th: "int x - int y = int (x - y)" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2285 |
from xyn have "int x mod int n = int y mod int n" |
46551
866bce5442a3
simplify projections on simultaneous computations of div and mod; tuned structure (from Florian Haftmann)
huffman
parents:
46026
diff
changeset
|
2286 |
by (simp add: zmod_int [symmetric]) |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2287 |
hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2288 |
hence "n dvd x - y" by (simp add: th zdvd_int) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2289 |
then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2290 |
qed |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2291 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2292 |
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2293 |
(is "?lhs = ?rhs") |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2294 |
proof |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2295 |
assume H: "x mod n = y mod n" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2296 |
{assume xy: "x \<le> y" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2297 |
from H have th: "y mod n = x mod n" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2298 |
from nat_mod_eq_lemma[OF th xy] have ?rhs |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2299 |
apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2300 |
moreover |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2301 |
{assume xy: "y \<le> x" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2302 |
from nat_mod_eq_lemma[OF H xy] have ?rhs |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2303 |
apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2304 |
ultimately show ?rhs using linear[of x y] by blast |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2305 |
next |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2306 |
assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2307 |
hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2308 |
thus ?lhs by simp |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2309 |
qed |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2310 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2311 |
lemma div_nat_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2312 |
"(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2313 |
by (simp add: nat_div_distrib) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2314 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2315 |
lemma one_div_nat_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2316 |
"Suc 0 div numeral v' = nat (1 div numeral v')" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2317 |
by (subst nat_div_distrib, simp_all) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2318 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2319 |
lemma mod_nat_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2320 |
"(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2321 |
by (simp add: nat_mod_distrib) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2322 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2323 |
lemma one_mod_nat_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2324 |
"Suc 0 mod numeral v' = nat (1 mod numeral v')" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2325 |
by (subst nat_mod_distrib) simp_all |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2326 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2327 |
lemma mod_2_not_eq_zero_eq_one_int: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2328 |
fixes k :: int |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2329 |
shows "k mod 2 \<noteq> 0 \<longleftrightarrow> k mod 2 = 1" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2330 |
by auto |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2331 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2332 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2333 |
subsubsection {* Tools setup *} |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2334 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2335 |
text {* Nitpick *} |
34126 | 2336 |
|
41792
ff3cb0c418b7
renamed "nitpick\_def" to "nitpick_unfold" to reflect its new semantics
blanchet
parents:
41550
diff
changeset
|
2337 |
lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality' |
34126 | 2338 |
|
35673 | 2339 |
|
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2340 |
subsubsection {* Code generation *} |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2341 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2342 |
definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2343 |
"pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2344 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2345 |
lemma pdivmod_posDivAlg [code]: |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2346 |
"pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2347 |
by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2348 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2349 |
lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2350 |
apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2351 |
then pdivmod k l |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2352 |
else (let (r, s) = pdivmod k l in |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46560
diff
changeset
|
2353 |
if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))" |
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2354 |
proof - |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2355 |
have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2356 |
show ?thesis |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2357 |
by (simp add: divmod_int_mod_div pdivmod_def) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2358 |
(auto simp add: aux not_less not_le zdiv_zminus1_eq_if |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2359 |
zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2360 |
qed |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2361 |
|
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2362 |
lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2363 |
apsnd ((op *) (sgn l)) (if sgn k = sgn l |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2364 |
then pdivmod k l |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2365 |
else (let (r, s) = pdivmod k l in |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2366 |
if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2367 |
proof - |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2368 |
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" |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2369 |
by (auto simp add: not_less sgn_if) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2370 |
then show ?thesis by (simp add: divmod_int_pdivmod) |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2371 |
qed |
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2372 |
|
33364 | 2373 |
code_modulename SML |
2374 |
Divides Arith |
|
2375 |
||
2376 |
code_modulename OCaml |
|
2377 |
Divides Arith |
|
2378 |
||
2379 |
code_modulename Haskell |
|
2380 |
Divides Arith |
|
2381 |
||
33361
1f18de40b43f
combined former theories Divides and IntDiv to one theory Divides
haftmann
parents:
33340
diff
changeset
|
2382 |
end |