1.1 --- a/src/HOL/IntDiv.thy Wed Mar 04 10:43:39 2009 +0100
1.2 +++ b/src/HOL/IntDiv.thy Wed Mar 04 10:45:52 2009 +0100
1.3 @@ -547,34 +547,6 @@
1.4 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
1.5 {* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}
1.6
1.7 -(* The following 8 lemmas are made unnecessary by the above simprocs: *)
1.8 -
1.9 -lemmas div_pos_pos_number_of =
1.10 - div_pos_pos [of "number_of v" "number_of w", standard]
1.11 -
1.12 -lemmas div_neg_pos_number_of =
1.13 - div_neg_pos [of "number_of v" "number_of w", standard]
1.14 -
1.15 -lemmas div_pos_neg_number_of =
1.16 - div_pos_neg [of "number_of v" "number_of w", standard]
1.17 -
1.18 -lemmas div_neg_neg_number_of =
1.19 - div_neg_neg [of "number_of v" "number_of w", standard]
1.20 -
1.21 -
1.22 -lemmas mod_pos_pos_number_of =
1.23 - mod_pos_pos [of "number_of v" "number_of w", standard]
1.24 -
1.25 -lemmas mod_neg_pos_number_of =
1.26 - mod_neg_pos [of "number_of v" "number_of w", standard]
1.27 -
1.28 -lemmas mod_pos_neg_number_of =
1.29 - mod_pos_neg [of "number_of v" "number_of w", standard]
1.30 -
1.31 -lemmas mod_neg_neg_number_of =
1.32 - mod_neg_neg [of "number_of v" "number_of w", standard]
1.33 -
1.34 -
1.35 lemmas posDivAlg_eqn_number_of [simp] =
1.36 posDivAlg_eqn [of "number_of v" "number_of w", standard]
1.37
1.38 @@ -584,15 +556,6 @@
1.39
1.40 text{*Special-case simplification *}
1.41
1.42 -lemma zmod_1 [simp]: "a mod (1::int) = 0"
1.43 -apply (cut_tac a = a and b = 1 in pos_mod_sign)
1.44 -apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
1.45 -apply (auto simp del:pos_mod_bound pos_mod_sign)
1.46 -done
1.47 -
1.48 -lemma zdiv_1 [simp]: "a div (1::int) = a"
1.49 -by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
1.50 -
1.51 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
1.52 apply (cut_tac a = a and b = "-1" in neg_mod_sign)
1.53 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
1.54 @@ -726,9 +689,6 @@
1.55 apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
1.56 done
1.57
1.58 -lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
1.59 -by (simp add: zdiv_zmult1_eq)
1.60 -
1.61 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
1.62 apply (case_tac "b = 0", simp)
1.63 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
1.64 @@ -754,7 +714,7 @@
1.65 assume not0: "b \<noteq> 0"
1.66 show "(a + c * b) div b = c + a div b"
1.67 unfolding zdiv_zadd1_eq [of a "c * b"] using not0
1.68 - by (simp add: zmod_zmult1_eq zmod_zdiv_trivial)
1.69 + by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
1.70 qed auto
1.71
1.72 lemma posDivAlg_div_mod:
1.73 @@ -784,41 +744,12 @@
1.74 show ?thesis by simp
1.75 qed
1.76
1.77 -lemma zdiv_zadd_self1: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"
1.78 -by (rule div_add_self1) (* already declared [simp] *)
1.79 -
1.80 -lemma zdiv_zadd_self2: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"
1.81 -by (rule div_add_self2) (* already declared [simp] *)
1.82 -
1.83 -lemma zdiv_zmult_self2: "b \<noteq> (0::int) ==> (b*a) div b = a"
1.84 -by (rule div_mult_self1_is_id) (* already declared [simp] *)
1.85 -
1.86 -lemma zmod_zmult_self1: "(a*b) mod b = (0::int)"
1.87 -by (rule mod_mult_self2_is_0) (* already declared [simp] *)
1.88 -
1.89 -lemma zmod_zmult_self2: "(b*a) mod b = (0::int)"
1.90 -by (rule mod_mult_self1_is_0) (* already declared [simp] *)
1.91 -
1.92 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
1.93 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
1.94
1.95 (* REVISIT: should this be generalized to all semiring_div types? *)
1.96 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
1.97
1.98 -lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
1.99 -by (rule mod_add_left_eq)
1.100 -
1.101 -lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
1.102 -by (rule mod_add_right_eq)
1.103 -
1.104 -lemma zmod_zadd_self1: "(a+b) mod a = b mod (a::int)"
1.105 -by (rule mod_add_self1) (* already declared [simp] *)
1.106 -
1.107 -lemma zmod_zadd_self2: "(b+a) mod a = b mod (a::int)"
1.108 -by (rule mod_add_self2) (* already declared [simp] *)
1.109 -
1.110 -lemma zmod_zdiff1_eq: "(a - b) mod c = (a mod c - b mod c) mod (c::int)"
1.111 -by (rule mod_diff_eq)
1.112
1.113 subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}
1.114
1.115 @@ -902,13 +833,6 @@
1.116 "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"
1.117 by (simp add:zdiv_zmult_zmult1)
1.118
1.119 -(*
1.120 -lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
1.121 -apply (drule zdiv_zmult_zmult1)
1.122 -apply (auto simp add: mult_commute)
1.123 -done
1.124 -*)
1.125 -
1.126
1.127 subsection{*Distribution of Factors over mod*}
1.128
1.129 @@ -933,9 +857,6 @@
1.130 apply (auto simp add: mult_commute)
1.131 done
1.132
1.133 -lemma zmod_zmod_cancel: "n dvd m \<Longrightarrow> (k::int) mod m mod n = k mod n"
1.134 -by (rule mod_mod_cancel)
1.135 -
1.136
1.137 subsection {*Splitting Rules for div and mod*}
1.138
1.139 @@ -1070,7 +991,7 @@
1.140 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =
1.141 1 + 2* ((-b - 1) mod (-a))")
1.142 apply (rule_tac [2] pos_zmod_mult_2)
1.143 -apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
1.144 +apply (auto simp add: right_diff_distrib)
1.145 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
1.146 prefer 2 apply simp
1.147 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
1.148 @@ -1132,38 +1053,8 @@
1.149
1.150 subsection {* The Divides Relation *}
1.151
1.152 -lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
1.153 - by (rule dvd_eq_mod_eq_0)
1.154 -
1.155 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
1.156 - zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]
1.157 -
1.158 -lemma zdvd_0_right: "(m::int) dvd 0"
1.159 - by (rule dvd_0_right) (* already declared [iff] *)
1.160 -
1.161 -lemma zdvd_0_left: "(0 dvd (m::int)) = (m = 0)"
1.162 - by (rule dvd_0_left_iff) (* already declared [noatp,simp] *)
1.163 -
1.164 -lemma zdvd_1_left: "1 dvd (m::int)"
1.165 - by (rule one_dvd) (* already declared [simp] *)
1.166 -
1.167 -lemma zdvd_refl: "m dvd (m::int)"
1.168 - by (rule dvd_refl) (* already declared [simp] *)
1.169 -
1.170 -lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
1.171 - by (rule dvd_trans)
1.172 -
1.173 -lemma zdvd_zminus_iff: "m dvd -n \<longleftrightarrow> m dvd (n::int)"
1.174 - by (rule dvd_minus_iff) (* already declared [simp] *)
1.175 -
1.176 -lemma zdvd_zminus2_iff: "-m dvd n \<longleftrightarrow> m dvd (n::int)"
1.177 - by (rule minus_dvd_iff) (* already declared [simp] *)
1.178 -
1.179 -lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)"
1.180 - by (rule abs_dvd_iff) (* already declared [simp] *)
1.181 -
1.182 -lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)"
1.183 - by (rule dvd_abs_iff) (* already declared [simp] *)
1.184 + dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
1.185
1.186 lemma zdvd_anti_sym:
1.187 "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
1.188 @@ -1171,58 +1062,32 @@
1.189 apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
1.190 done
1.191
1.192 -lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
1.193 - by (rule dvd_add)
1.194 -
1.195 -lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a"
1.196 +lemma zdvd_dvd_eq: assumes "a \<noteq> 0" and "(a::int) dvd b" and "b dvd a"
1.197 shows "\<bar>a\<bar> = \<bar>b\<bar>"
1.198 proof-
1.199 - from ab obtain k where k:"b = a*k" unfolding dvd_def by blast
1.200 - from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast
1.201 + from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast
1.202 + from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast
1.203 from k k' have "a = a*k*k'" by simp
1.204 with mult_cancel_left1[where c="a" and b="k*k'"]
1.205 - have kk':"k*k' = 1" using anz by (simp add: mult_assoc)
1.206 + have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
1.207 hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
1.208 thus ?thesis using k k' by auto
1.209 qed
1.210
1.211 -lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
1.212 - by (rule Ring_and_Field.dvd_diff)
1.213 -
1.214 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
1.215 apply (subgoal_tac "m = n + (m - n)")
1.216 apply (erule ssubst)
1.217 - apply (blast intro: zdvd_zadd, simp)
1.218 + apply (blast intro: dvd_add, simp)
1.219 done
1.220
1.221 -lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
1.222 - by (rule dvd_mult)
1.223 -
1.224 -lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
1.225 - by (rule dvd_mult2)
1.226 -
1.227 -lemma zdvd_triv_right: "(k::int) dvd m * k"
1.228 - by (rule dvd_triv_right) (* already declared [simp] *)
1.229 -
1.230 -lemma zdvd_triv_left: "(k::int) dvd k * m"
1.231 - by (rule dvd_triv_left) (* already declared [simp] *)
1.232 -
1.233 -lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
1.234 - by (rule dvd_mult_left)
1.235 -
1.236 -lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
1.237 - by (rule dvd_mult_right)
1.238 -
1.239 -lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
1.240 - by (rule mult_dvd_mono)
1.241 -
1.242 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
1.243 - apply (rule iffI)
1.244 - apply (erule_tac [2] zdvd_zadd)
1.245 - apply (subgoal_tac "n = (n + k * m) - k * m")
1.246 - apply (erule ssubst)
1.247 - apply (erule zdvd_zdiff, simp_all)
1.248 - done
1.249 +apply (rule iffI)
1.250 + apply (erule_tac [2] dvd_add)
1.251 + apply (subgoal_tac "n = (n + k * m) - k * m")
1.252 + apply (erule ssubst)
1.253 + apply (erule dvd_diff)
1.254 + apply(simp_all)
1.255 +done
1.256
1.257 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
1.258 apply (simp add: dvd_def)
1.259 @@ -1232,7 +1097,7 @@
1.260 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
1.261 apply (subgoal_tac "k dvd n * (m div n) + m mod n")
1.262 apply (simp add: zmod_zdiv_equality [symmetric])
1.263 - apply (simp only: zdvd_zadd zdvd_zmult2)
1.264 + apply (simp only: dvd_add dvd_mult2)
1.265 done
1.266
1.267 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
1.268 @@ -1252,7 +1117,7 @@
1.269 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
1.270 apply (subgoal_tac "m mod n = 0")
1.271 apply (simp add: zmult_div_cancel)
1.272 -apply (simp only: zdvd_iff_zmod_eq_0)
1.273 +apply (simp only: dvd_eq_mod_eq_0)
1.274 done
1.275
1.276 lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
1.277 @@ -1265,10 +1130,6 @@
1.278 thus ?thesis by simp
1.279 qed
1.280
1.281 -lemma zdvd_zmult_cancel_disj[simp]:
1.282 - "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"
1.283 -by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel)
1.284 -
1.285
1.286 theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
1.287 apply (simp split add: split_nat)
1.288 @@ -1300,44 +1161,38 @@
1.289 then show ?thesis by (simp only: negative_eq_positive) auto
1.290 qed
1.291 qed
1.292 - then show ?thesis by (auto elim!: dvdE simp only: zdvd_triv_left int_mult)
1.293 + then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left int_mult)
1.294 qed
1.295
1.296 lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
1.297 proof
1.298 - assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)
1.299 + assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
1.300 hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
1.301 hence "nat \<bar>x\<bar> = 1" by simp
1.302 thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
1.303 next
1.304 assume "\<bar>x\<bar>=1" thus "x dvd 1"
1.305 - by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)
1.306 + by(cases "x < 0",simp_all add: minus_equation_iff dvd_eq_mod_eq_0)
1.307 qed
1.308 lemma zdvd_mult_cancel1:
1.309 assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
1.310 proof
1.311 assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
1.312 - by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)
1.313 + by (cases "n >0", auto simp add: minus_dvd_iff minus_equation_iff)
1.314 next
1.315 assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
1.316 from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
1.317 qed
1.318
1.319 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
1.320 - unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus_iff)
1.321 + unfolding zdvd_int by (cases "z \<ge> 0") simp_all
1.322
1.323 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
1.324 - unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus2_iff)
1.325 + unfolding zdvd_int by (cases "z \<ge> 0") simp_all
1.326
1.327 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
1.328 by (auto simp add: dvd_int_iff)
1.329
1.330 -lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
1.331 - by (rule minus_dvd_iff)
1.332 -
1.333 -lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
1.334 - by (rule dvd_minus_iff)
1.335 -
1.336 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
1.337 apply (rule_tac z=n in int_cases)
1.338 apply (auto simp add: dvd_int_iff)
1.339 @@ -1367,10 +1222,13 @@
1.340 apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
1.341 done
1.342
1.343 +lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
1.344 +by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
1.345 +
1.346 text{*Suggested by Matthias Daum*}
1.347 lemma int_power_div_base:
1.348 "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
1.349 -apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")
1.350 +apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
1.351 apply (erule ssubst)
1.352 apply (simp only: power_add)
1.353 apply simp_all
1.354 @@ -1387,8 +1245,8 @@
1.355 by (rule mod_diff_right_eq [symmetric])
1.356
1.357 lemmas zmod_simps =
1.358 - IntDiv.zmod_zadd_left_eq [symmetric]
1.359 - IntDiv.zmod_zadd_right_eq [symmetric]
1.360 + mod_add_left_eq [symmetric]
1.361 + mod_add_right_eq [symmetric]
1.362 IntDiv.zmod_zmult1_eq [symmetric]
1.363 mod_mult_left_eq [symmetric]
1.364 IntDiv.zpower_zmod
1.365 @@ -1463,14 +1321,14 @@
1.366 assume H: "x mod n = y mod n"
1.367 hence "x mod n - y mod n = 0" by simp
1.368 hence "(x mod n - y mod n) mod n = 0" by simp
1.369 - hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])
1.370 - thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)
1.371 + hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
1.372 + thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
1.373 next
1.374 assume H: "n dvd x - y"
1.375 then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
1.376 hence "x = n*k + y" by simp
1.377 hence "x mod n = (n*k + y) mod n" by simp
1.378 - thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)
1.379 + thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
1.380 qed
1.381
1.382 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"