1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/IntDiv.thy Thu May 31 18:16:52 2007 +0200
1.3 @@ -0,0 +1,1406 @@
1.4 +(* Title: HOL/IntDiv.thy
1.5 + ID: $Id$
1.6 + Author: Lawrence C Paulson, Cambridge University Computer Laboratory
1.7 + Copyright 1999 University of Cambridge
1.8 +
1.9 +*)
1.10 +
1.11 +header{*The Division Operators div and mod; the Divides Relation dvd*}
1.12 +
1.13 +theory IntDiv
1.14 +imports IntArith Divides FunDef
1.15 +begin
1.16 +
1.17 +declare zless_nat_conj [simp]
1.18 +
1.19 +constdefs
1.20 + quorem :: "(int*int) * (int*int) => bool"
1.21 + --{*definition of quotient and remainder*}
1.22 + [code func]: "quorem == %((a,b), (q,r)).
1.23 + a = b*q + r &
1.24 + (if 0 < b then 0\<le>r & r<b else b<r & r \<le> 0)"
1.25 +
1.26 + adjust :: "[int, int*int] => int*int"
1.27 + --{*for the division algorithm*}
1.28 + [code func]: "adjust b == %(q,r). if 0 \<le> r-b then (2*q + 1, r-b)
1.29 + else (2*q, r)"
1.30 +
1.31 +text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
1.32 +function
1.33 + posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"
1.34 +where
1.35 + "posDivAlg a b =
1.36 + (if (a<b | b\<le>0) then (0,a)
1.37 + else adjust b (posDivAlg a (2*b)))"
1.38 +by auto
1.39 +termination by (relation "measure (%(a,b). nat(a - b + 1))") auto
1.40 +
1.41 +text{*algorithm for the case @{text "a<0, b>0"}*}
1.42 +function
1.43 + negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int"
1.44 +where
1.45 + "negDivAlg a b =
1.46 + (if (0\<le>a+b | b\<le>0) then (-1,a+b)
1.47 + else adjust b (negDivAlg a (2*b)))"
1.48 +by auto
1.49 +termination by (relation "measure (%(a,b). nat(- a - b))") auto
1.50 +
1.51 +text{*algorithm for the general case @{term "b\<noteq>0"}*}
1.52 +constdefs
1.53 + negateSnd :: "int*int => int*int"
1.54 + [code func]: "negateSnd == %(q,r). (q,-r)"
1.55 +
1.56 +definition
1.57 + divAlg :: "int \<times> int \<Rightarrow> int \<times> int"
1.58 + --{*The full division algorithm considers all possible signs for a, b
1.59 + including the special case @{text "a=0, b<0"} because
1.60 + @{term negDivAlg} requires @{term "a<0"}.*}
1.61 +where
1.62 + "divAlg = (\<lambda>(a, b). (if 0\<le>a then
1.63 + if 0\<le>b then posDivAlg a b
1.64 + else if a=0 then (0, 0)
1.65 + else negateSnd (negDivAlg (-a) (-b))
1.66 + else
1.67 + if 0<b then negDivAlg a b
1.68 + else negateSnd (posDivAlg (-a) (-b))))"
1.69 +
1.70 +instance int :: Divides.div
1.71 + div_def: "a div b == fst (divAlg (a, b))"
1.72 + mod_def: "a mod b == snd (divAlg (a, b))" ..
1.73 +
1.74 +lemma divAlg_mod_div:
1.75 + "divAlg (p, q) = (p div q, p mod q)"
1.76 + by (auto simp add: div_def mod_def)
1.77 +
1.78 +text{*
1.79 +Here is the division algorithm in ML:
1.80 +
1.81 +\begin{verbatim}
1.82 + fun posDivAlg (a,b) =
1.83 + if a<b then (0,a)
1.84 + else let val (q,r) = posDivAlg(a, 2*b)
1.85 + in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
1.86 + end
1.87 +
1.88 + fun negDivAlg (a,b) =
1.89 + if 0\<le>a+b then (~1,a+b)
1.90 + else let val (q,r) = negDivAlg(a, 2*b)
1.91 + in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
1.92 + end;
1.93 +
1.94 + fun negateSnd (q,r:int) = (q,~r);
1.95 +
1.96 + fun divAlg (a,b) = if 0\<le>a then
1.97 + if b>0 then posDivAlg (a,b)
1.98 + else if a=0 then (0,0)
1.99 + else negateSnd (negDivAlg (~a,~b))
1.100 + else
1.101 + if 0<b then negDivAlg (a,b)
1.102 + else negateSnd (posDivAlg (~a,~b));
1.103 +\end{verbatim}
1.104 +*}
1.105 +
1.106 +
1.107 +
1.108 +subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
1.109 +
1.110 +lemma unique_quotient_lemma:
1.111 + "[| b*q' + r' \<le> b*q + r; 0 \<le> r'; r' < b; r < b |]
1.112 + ==> q' \<le> (q::int)"
1.113 +apply (subgoal_tac "r' + b * (q'-q) \<le> r")
1.114 + prefer 2 apply (simp add: right_diff_distrib)
1.115 +apply (subgoal_tac "0 < b * (1 + q - q') ")
1.116 +apply (erule_tac [2] order_le_less_trans)
1.117 + prefer 2 apply (simp add: right_diff_distrib right_distrib)
1.118 +apply (subgoal_tac "b * q' < b * (1 + q) ")
1.119 + prefer 2 apply (simp add: right_diff_distrib right_distrib)
1.120 +apply (simp add: mult_less_cancel_left)
1.121 +done
1.122 +
1.123 +lemma unique_quotient_lemma_neg:
1.124 + "[| b*q' + r' \<le> b*q + r; r \<le> 0; b < r; b < r' |]
1.125 + ==> q \<le> (q'::int)"
1.126 +by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,
1.127 + auto)
1.128 +
1.129 +lemma unique_quotient:
1.130 + "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b \<noteq> 0 |]
1.131 + ==> q = q'"
1.132 +apply (simp add: quorem_def linorder_neq_iff split: split_if_asm)
1.133 +apply (blast intro: order_antisym
1.134 + dest: order_eq_refl [THEN unique_quotient_lemma]
1.135 + order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
1.136 +done
1.137 +
1.138 +
1.139 +lemma unique_remainder:
1.140 + "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b \<noteq> 0 |]
1.141 + ==> r = r'"
1.142 +apply (subgoal_tac "q = q'")
1.143 + apply (simp add: quorem_def)
1.144 +apply (blast intro: unique_quotient)
1.145 +done
1.146 +
1.147 +
1.148 +subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
1.149 +
1.150 +text{*And positive divisors*}
1.151 +
1.152 +lemma adjust_eq [simp]:
1.153 + "adjust b (q,r) =
1.154 + (let diff = r-b in
1.155 + if 0 \<le> diff then (2*q + 1, diff)
1.156 + else (2*q, r))"
1.157 +by (simp add: Let_def adjust_def)
1.158 +
1.159 +declare posDivAlg.simps [simp del]
1.160 +
1.161 +text{*use with a simproc to avoid repeatedly proving the premise*}
1.162 +lemma posDivAlg_eqn:
1.163 + "0 < b ==>
1.164 + posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
1.165 +by (rule posDivAlg.simps [THEN trans], simp)
1.166 +
1.167 +text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
1.168 +theorem posDivAlg_correct:
1.169 + assumes "0 \<le> a" and "0 < b"
1.170 + shows "quorem ((a, b), posDivAlg a b)"
1.171 +using prems apply (induct a b rule: posDivAlg.induct)
1.172 +apply auto
1.173 +apply (simp add: quorem_def)
1.174 +apply (subst posDivAlg_eqn, simp add: right_distrib)
1.175 +apply (case_tac "a < b")
1.176 +apply simp_all
1.177 +apply (erule splitE)
1.178 +apply (auto simp add: right_distrib Let_def)
1.179 +done
1.180 +
1.181 +
1.182 +subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
1.183 +
1.184 +text{*And positive divisors*}
1.185 +
1.186 +declare negDivAlg.simps [simp del]
1.187 +
1.188 +text{*use with a simproc to avoid repeatedly proving the premise*}
1.189 +lemma negDivAlg_eqn:
1.190 + "0 < b ==>
1.191 + negDivAlg a b =
1.192 + (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
1.193 +by (rule negDivAlg.simps [THEN trans], simp)
1.194 +
1.195 +(*Correctness of negDivAlg: it computes quotients correctly
1.196 + It doesn't work if a=0 because the 0/b equals 0, not -1*)
1.197 +lemma negDivAlg_correct:
1.198 + assumes "a < 0" and "b > 0"
1.199 + shows "quorem ((a, b), negDivAlg a b)"
1.200 +using prems apply (induct a b rule: negDivAlg.induct)
1.201 +apply (auto simp add: linorder_not_le)
1.202 +apply (simp add: quorem_def)
1.203 +apply (subst negDivAlg_eqn, assumption)
1.204 +apply (case_tac "a + b < (0\<Colon>int)")
1.205 +apply simp_all
1.206 +apply (erule splitE)
1.207 +apply (auto simp add: right_distrib Let_def)
1.208 +done
1.209 +
1.210 +
1.211 +subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
1.212 +
1.213 +(*the case a=0*)
1.214 +lemma quorem_0: "b \<noteq> 0 ==> quorem ((0,b), (0,0))"
1.215 +by (auto simp add: quorem_def linorder_neq_iff)
1.216 +
1.217 +lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
1.218 +by (subst posDivAlg.simps, auto)
1.219 +
1.220 +lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
1.221 +by (subst negDivAlg.simps, auto)
1.222 +
1.223 +lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
1.224 +by (simp add: negateSnd_def)
1.225 +
1.226 +lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)"
1.227 +by (auto simp add: split_ifs quorem_def)
1.228 +
1.229 +lemma divAlg_correct: "b \<noteq> 0 ==> quorem ((a,b), divAlg (a, b))"
1.230 +by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg
1.231 + posDivAlg_correct negDivAlg_correct)
1.232 +
1.233 +text{*Arbitrary definitions for division by zero. Useful to simplify
1.234 + certain equations.*}
1.235 +
1.236 +lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
1.237 +by (simp add: div_def mod_def divAlg_def posDivAlg.simps)
1.238 +
1.239 +
1.240 +text{*Basic laws about division and remainder*}
1.241 +
1.242 +lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
1.243 +apply (case_tac "b = 0", simp)
1.244 +apply (cut_tac a = a and b = b in divAlg_correct)
1.245 +apply (auto simp add: quorem_def div_def mod_def)
1.246 +done
1.247 +
1.248 +lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
1.249 +by(simp add: zmod_zdiv_equality[symmetric])
1.250 +
1.251 +lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
1.252 +by(simp add: mult_commute zmod_zdiv_equality[symmetric])
1.253 +
1.254 +text {* Tool setup *}
1.255 +
1.256 +ML_setup {*
1.257 +local
1.258 +
1.259 +structure CancelDivMod = CancelDivModFun(
1.260 +struct
1.261 + val div_name = @{const_name Divides.div};
1.262 + val mod_name = @{const_name Divides.mod};
1.263 + val mk_binop = HOLogic.mk_binop;
1.264 + val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
1.265 + val dest_sum = Int_Numeral_Simprocs.dest_sum;
1.266 + val div_mod_eqs =
1.267 + map mk_meta_eq [@{thm zdiv_zmod_equality},
1.268 + @{thm zdiv_zmod_equality2}];
1.269 + val trans = trans;
1.270 + val prove_eq_sums =
1.271 + let
1.272 + val simps = diff_int_def :: Int_Numeral_Simprocs.add_0s @ zadd_ac
1.273 + in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
1.274 +end)
1.275 +
1.276 +in
1.277 +
1.278 +val cancel_zdiv_zmod_proc = NatArithUtils.prep_simproc
1.279 + ("cancel_zdiv_zmod", ["(m::int) + n"], K CancelDivMod.proc)
1.280 +
1.281 +end;
1.282 +
1.283 +Addsimprocs [cancel_zdiv_zmod_proc]
1.284 +*}
1.285 +
1.286 +lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
1.287 +apply (cut_tac a = a and b = b in divAlg_correct)
1.288 +apply (auto simp add: quorem_def mod_def)
1.289 +done
1.290 +
1.291 +lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1, standard]
1.292 + and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
1.293 +
1.294 +lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
1.295 +apply (cut_tac a = a and b = b in divAlg_correct)
1.296 +apply (auto simp add: quorem_def div_def mod_def)
1.297 +done
1.298 +
1.299 +lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1, standard]
1.300 + and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
1.301 +
1.302 +
1.303 +
1.304 +subsection{*General Properties of div and mod*}
1.305 +
1.306 +lemma quorem_div_mod: "b \<noteq> 0 ==> quorem ((a, b), (a div b, a mod b))"
1.307 +apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1.308 +apply (force simp add: quorem_def linorder_neq_iff)
1.309 +done
1.310 +
1.311 +lemma quorem_div: "[| quorem((a,b),(q,r)); b \<noteq> 0 |] ==> a div b = q"
1.312 +by (simp add: quorem_div_mod [THEN unique_quotient])
1.313 +
1.314 +lemma quorem_mod: "[| quorem((a,b),(q,r)); b \<noteq> 0 |] ==> a mod b = r"
1.315 +by (simp add: quorem_div_mod [THEN unique_remainder])
1.316 +
1.317 +lemma div_pos_pos_trivial: "[| (0::int) \<le> a; a < b |] ==> a div b = 0"
1.318 +apply (rule quorem_div)
1.319 +apply (auto simp add: quorem_def)
1.320 +done
1.321 +
1.322 +lemma div_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a div b = 0"
1.323 +apply (rule quorem_div)
1.324 +apply (auto simp add: quorem_def)
1.325 +done
1.326 +
1.327 +lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a div b = -1"
1.328 +apply (rule quorem_div)
1.329 +apply (auto simp add: quorem_def)
1.330 +done
1.331 +
1.332 +(*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)
1.333 +
1.334 +lemma mod_pos_pos_trivial: "[| (0::int) \<le> a; a < b |] ==> a mod b = a"
1.335 +apply (rule_tac q = 0 in quorem_mod)
1.336 +apply (auto simp add: quorem_def)
1.337 +done
1.338 +
1.339 +lemma mod_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a mod b = a"
1.340 +apply (rule_tac q = 0 in quorem_mod)
1.341 +apply (auto simp add: quorem_def)
1.342 +done
1.343 +
1.344 +lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a mod b = a+b"
1.345 +apply (rule_tac q = "-1" in quorem_mod)
1.346 +apply (auto simp add: quorem_def)
1.347 +done
1.348 +
1.349 +text{*There is no @{text mod_neg_pos_trivial}.*}
1.350 +
1.351 +
1.352 +(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
1.353 +lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
1.354 +apply (case_tac "b = 0", simp)
1.355 +apply (simp add: quorem_div_mod [THEN quorem_neg, simplified,
1.356 + THEN quorem_div, THEN sym])
1.357 +
1.358 +done
1.359 +
1.360 +(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
1.361 +lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
1.362 +apply (case_tac "b = 0", simp)
1.363 +apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],
1.364 + auto)
1.365 +done
1.366 +
1.367 +
1.368 +subsection{*Laws for div and mod with Unary Minus*}
1.369 +
1.370 +lemma zminus1_lemma:
1.371 + "quorem((a,b),(q,r))
1.372 + ==> quorem ((-a,b), (if r=0 then -q else -q - 1),
1.373 + (if r=0 then 0 else b-r))"
1.374 +by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)
1.375 +
1.376 +
1.377 +lemma zdiv_zminus1_eq_if:
1.378 + "b \<noteq> (0::int)
1.379 + ==> (-a) div b =
1.380 + (if a mod b = 0 then - (a div b) else - (a div b) - 1)"
1.381 +by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])
1.382 +
1.383 +lemma zmod_zminus1_eq_if:
1.384 + "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"
1.385 +apply (case_tac "b = 0", simp)
1.386 +apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])
1.387 +done
1.388 +
1.389 +lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
1.390 +by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
1.391 +
1.392 +lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
1.393 +by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
1.394 +
1.395 +lemma zdiv_zminus2_eq_if:
1.396 + "b \<noteq> (0::int)
1.397 + ==> a div (-b) =
1.398 + (if a mod b = 0 then - (a div b) else - (a div b) - 1)"
1.399 +by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
1.400 +
1.401 +lemma zmod_zminus2_eq_if:
1.402 + "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"
1.403 +by (simp add: zmod_zminus1_eq_if zmod_zminus2)
1.404 +
1.405 +
1.406 +subsection{*Division of a Number by Itself*}
1.407 +
1.408 +lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
1.409 +apply (subgoal_tac "0 < a*q")
1.410 + apply (simp add: zero_less_mult_iff, arith)
1.411 +done
1.412 +
1.413 +lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
1.414 +apply (subgoal_tac "0 \<le> a* (1-q) ")
1.415 + apply (simp add: zero_le_mult_iff)
1.416 +apply (simp add: right_diff_distrib)
1.417 +done
1.418 +
1.419 +lemma self_quotient: "[| quorem((a,a),(q,r)); a \<noteq> (0::int) |] ==> q = 1"
1.420 +apply (simp add: split_ifs quorem_def linorder_neq_iff)
1.421 +apply (rule order_antisym, safe, simp_all)
1.422 +apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
1.423 +apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
1.424 +apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
1.425 +done
1.426 +
1.427 +lemma self_remainder: "[| quorem((a,a),(q,r)); a \<noteq> (0::int) |] ==> r = 0"
1.428 +apply (frule self_quotient, assumption)
1.429 +apply (simp add: quorem_def)
1.430 +done
1.431 +
1.432 +lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
1.433 +by (simp add: quorem_div_mod [THEN self_quotient])
1.434 +
1.435 +(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
1.436 +lemma zmod_self [simp]: "a mod a = (0::int)"
1.437 +apply (case_tac "a = 0", simp)
1.438 +apply (simp add: quorem_div_mod [THEN self_remainder])
1.439 +done
1.440 +
1.441 +
1.442 +subsection{*Computation of Division and Remainder*}
1.443 +
1.444 +lemma zdiv_zero [simp]: "(0::int) div b = 0"
1.445 +by (simp add: div_def divAlg_def)
1.446 +
1.447 +lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
1.448 +by (simp add: div_def divAlg_def)
1.449 +
1.450 +lemma zmod_zero [simp]: "(0::int) mod b = 0"
1.451 +by (simp add: mod_def divAlg_def)
1.452 +
1.453 +lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
1.454 +by (simp add: div_def divAlg_def)
1.455 +
1.456 +lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
1.457 +by (simp add: mod_def divAlg_def)
1.458 +
1.459 +text{*a positive, b positive *}
1.460 +
1.461 +lemma div_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
1.462 +by (simp add: div_def divAlg_def)
1.463 +
1.464 +lemma mod_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
1.465 +by (simp add: mod_def divAlg_def)
1.466 +
1.467 +text{*a negative, b positive *}
1.468 +
1.469 +lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"
1.470 +by (simp add: div_def divAlg_def)
1.471 +
1.472 +lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"
1.473 +by (simp add: mod_def divAlg_def)
1.474 +
1.475 +text{*a positive, b negative *}
1.476 +
1.477 +lemma div_pos_neg:
1.478 + "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
1.479 +by (simp add: div_def divAlg_def)
1.480 +
1.481 +lemma mod_pos_neg:
1.482 + "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
1.483 +by (simp add: mod_def divAlg_def)
1.484 +
1.485 +text{*a negative, b negative *}
1.486 +
1.487 +lemma div_neg_neg:
1.488 + "[| a < 0; b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
1.489 +by (simp add: div_def divAlg_def)
1.490 +
1.491 +lemma mod_neg_neg:
1.492 + "[| a < 0; b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
1.493 +by (simp add: mod_def divAlg_def)
1.494 +
1.495 +text {*Simplify expresions in which div and mod combine numerical constants*}
1.496 +
1.497 +lemmas div_pos_pos_number_of [simp] =
1.498 + div_pos_pos [of "number_of v" "number_of w", standard]
1.499 +
1.500 +lemmas div_neg_pos_number_of [simp] =
1.501 + div_neg_pos [of "number_of v" "number_of w", standard]
1.502 +
1.503 +lemmas div_pos_neg_number_of [simp] =
1.504 + div_pos_neg [of "number_of v" "number_of w", standard]
1.505 +
1.506 +lemmas div_neg_neg_number_of [simp] =
1.507 + div_neg_neg [of "number_of v" "number_of w", standard]
1.508 +
1.509 +
1.510 +lemmas mod_pos_pos_number_of [simp] =
1.511 + mod_pos_pos [of "number_of v" "number_of w", standard]
1.512 +
1.513 +lemmas mod_neg_pos_number_of [simp] =
1.514 + mod_neg_pos [of "number_of v" "number_of w", standard]
1.515 +
1.516 +lemmas mod_pos_neg_number_of [simp] =
1.517 + mod_pos_neg [of "number_of v" "number_of w", standard]
1.518 +
1.519 +lemmas mod_neg_neg_number_of [simp] =
1.520 + mod_neg_neg [of "number_of v" "number_of w", standard]
1.521 +
1.522 +
1.523 +lemmas posDivAlg_eqn_number_of [simp] =
1.524 + posDivAlg_eqn [of "number_of v" "number_of w", standard]
1.525 +
1.526 +lemmas negDivAlg_eqn_number_of [simp] =
1.527 + negDivAlg_eqn [of "number_of v" "number_of w", standard]
1.528 +
1.529 +
1.530 +text{*Special-case simplification *}
1.531 +
1.532 +lemma zmod_1 [simp]: "a mod (1::int) = 0"
1.533 +apply (cut_tac a = a and b = 1 in pos_mod_sign)
1.534 +apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
1.535 +apply (auto simp del:pos_mod_bound pos_mod_sign)
1.536 +done
1.537 +
1.538 +lemma zdiv_1 [simp]: "a div (1::int) = a"
1.539 +by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
1.540 +
1.541 +lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
1.542 +apply (cut_tac a = a and b = "-1" in neg_mod_sign)
1.543 +apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
1.544 +apply (auto simp del: neg_mod_sign neg_mod_bound)
1.545 +done
1.546 +
1.547 +lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
1.548 +by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
1.549 +
1.550 +(** The last remaining special cases for constant arithmetic:
1.551 + 1 div z and 1 mod z **)
1.552 +
1.553 +lemmas div_pos_pos_1_number_of [simp] =
1.554 + div_pos_pos [OF int_0_less_1, of "number_of w", standard]
1.555 +
1.556 +lemmas div_pos_neg_1_number_of [simp] =
1.557 + div_pos_neg [OF int_0_less_1, of "number_of w", standard]
1.558 +
1.559 +lemmas mod_pos_pos_1_number_of [simp] =
1.560 + mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
1.561 +
1.562 +lemmas mod_pos_neg_1_number_of [simp] =
1.563 + mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
1.564 +
1.565 +
1.566 +lemmas posDivAlg_eqn_1_number_of [simp] =
1.567 + posDivAlg_eqn [of concl: 1 "number_of w", standard]
1.568 +
1.569 +lemmas negDivAlg_eqn_1_number_of [simp] =
1.570 + negDivAlg_eqn [of concl: 1 "number_of w", standard]
1.571 +
1.572 +
1.573 +
1.574 +subsection{*Monotonicity in the First Argument (Dividend)*}
1.575 +
1.576 +lemma zdiv_mono1: "[| a \<le> a'; 0 < (b::int) |] ==> a div b \<le> a' div b"
1.577 +apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1.578 +apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
1.579 +apply (rule unique_quotient_lemma)
1.580 +apply (erule subst)
1.581 +apply (erule subst, simp_all)
1.582 +done
1.583 +
1.584 +lemma zdiv_mono1_neg: "[| a \<le> a'; (b::int) < 0 |] ==> a' div b \<le> a div b"
1.585 +apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1.586 +apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
1.587 +apply (rule unique_quotient_lemma_neg)
1.588 +apply (erule subst)
1.589 +apply (erule subst, simp_all)
1.590 +done
1.591 +
1.592 +
1.593 +subsection{*Monotonicity in the Second Argument (Divisor)*}
1.594 +
1.595 +lemma q_pos_lemma:
1.596 + "[| 0 \<le> b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \<le> (q'::int)"
1.597 +apply (subgoal_tac "0 < b'* (q' + 1) ")
1.598 + apply (simp add: zero_less_mult_iff)
1.599 +apply (simp add: right_distrib)
1.600 +done
1.601 +
1.602 +lemma zdiv_mono2_lemma:
1.603 + "[| b*q + r = b'*q' + r'; 0 \<le> b'*q' + r';
1.604 + r' < b'; 0 \<le> r; 0 < b'; b' \<le> b |]
1.605 + ==> q \<le> (q'::int)"
1.606 +apply (frule q_pos_lemma, assumption+)
1.607 +apply (subgoal_tac "b*q < b* (q' + 1) ")
1.608 + apply (simp add: mult_less_cancel_left)
1.609 +apply (subgoal_tac "b*q = r' - r + b'*q'")
1.610 + prefer 2 apply simp
1.611 +apply (simp (no_asm_simp) add: right_distrib)
1.612 +apply (subst add_commute, rule zadd_zless_mono, arith)
1.613 +apply (rule mult_right_mono, auto)
1.614 +done
1.615 +
1.616 +lemma zdiv_mono2:
1.617 + "[| (0::int) \<le> a; 0 < b'; b' \<le> b |] ==> a div b \<le> a div b'"
1.618 +apply (subgoal_tac "b \<noteq> 0")
1.619 + prefer 2 apply arith
1.620 +apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1.621 +apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
1.622 +apply (rule zdiv_mono2_lemma)
1.623 +apply (erule subst)
1.624 +apply (erule subst, simp_all)
1.625 +done
1.626 +
1.627 +lemma q_neg_lemma:
1.628 + "[| b'*q' + r' < 0; 0 \<le> r'; 0 < b' |] ==> q' \<le> (0::int)"
1.629 +apply (subgoal_tac "b'*q' < 0")
1.630 + apply (simp add: mult_less_0_iff, arith)
1.631 +done
1.632 +
1.633 +lemma zdiv_mono2_neg_lemma:
1.634 + "[| b*q + r = b'*q' + r'; b'*q' + r' < 0;
1.635 + r < b; 0 \<le> r'; 0 < b'; b' \<le> b |]
1.636 + ==> q' \<le> (q::int)"
1.637 +apply (frule q_neg_lemma, assumption+)
1.638 +apply (subgoal_tac "b*q' < b* (q + 1) ")
1.639 + apply (simp add: mult_less_cancel_left)
1.640 +apply (simp add: right_distrib)
1.641 +apply (subgoal_tac "b*q' \<le> b'*q'")
1.642 + prefer 2 apply (simp add: mult_right_mono_neg, arith)
1.643 +done
1.644 +
1.645 +lemma zdiv_mono2_neg:
1.646 + "[| a < (0::int); 0 < b'; b' \<le> b |] ==> a div b' \<le> a div b"
1.647 +apply (cut_tac a = a and b = b in zmod_zdiv_equality)
1.648 +apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
1.649 +apply (rule zdiv_mono2_neg_lemma)
1.650 +apply (erule subst)
1.651 +apply (erule subst, simp_all)
1.652 +done
1.653 +
1.654 +subsection{*More Algebraic Laws for div and mod*}
1.655 +
1.656 +text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
1.657 +
1.658 +lemma zmult1_lemma:
1.659 + "[| quorem((b,c),(q,r)); c \<noteq> 0 |]
1.660 + ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
1.661 +by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
1.662 +
1.663 +lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
1.664 +apply (case_tac "c = 0", simp)
1.665 +apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
1.666 +done
1.667 +
1.668 +lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
1.669 +apply (case_tac "c = 0", simp)
1.670 +apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
1.671 +done
1.672 +
1.673 +lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
1.674 +apply (rule trans)
1.675 +apply (rule_tac s = "b*a mod c" in trans)
1.676 +apply (rule_tac [2] zmod_zmult1_eq)
1.677 +apply (simp_all add: mult_commute)
1.678 +done
1.679 +
1.680 +lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
1.681 +apply (rule zmod_zmult1_eq' [THEN trans])
1.682 +apply (rule zmod_zmult1_eq)
1.683 +done
1.684 +
1.685 +lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
1.686 +by (simp add: zdiv_zmult1_eq)
1.687 +
1.688 +lemma zdiv_zmult_self2 [simp]: "b \<noteq> (0::int) ==> (b*a) div b = a"
1.689 +by (subst mult_commute, erule zdiv_zmult_self1)
1.690 +
1.691 +lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"
1.692 +by (simp add: zmod_zmult1_eq)
1.693 +
1.694 +lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"
1.695 +by (simp add: mult_commute zmod_zmult1_eq)
1.696 +
1.697 +lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
1.698 +proof
1.699 + assume "m mod d = 0"
1.700 + with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto
1.701 +next
1.702 + assume "EX q::int. m = d*q"
1.703 + thus "m mod d = 0" by auto
1.704 +qed
1.705 +
1.706 +lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
1.707 +
1.708 +text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
1.709 +
1.710 +lemma zadd1_lemma:
1.711 + "[| quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); c \<noteq> 0 |]
1.712 + ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
1.713 +by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
1.714 +
1.715 +(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
1.716 +lemma zdiv_zadd1_eq:
1.717 + "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
1.718 +apply (case_tac "c = 0", simp)
1.719 +apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)
1.720 +done
1.721 +
1.722 +lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
1.723 +apply (case_tac "c = 0", simp)
1.724 +apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)
1.725 +done
1.726 +
1.727 +lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"
1.728 +apply (case_tac "b = 0", simp)
1.729 +apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
1.730 +done
1.731 +
1.732 +lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"
1.733 +apply (case_tac "b = 0", simp)
1.734 +apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
1.735 +done
1.736 +
1.737 +lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
1.738 +apply (rule trans [symmetric])
1.739 +apply (rule zmod_zadd1_eq, simp)
1.740 +apply (rule zmod_zadd1_eq [symmetric])
1.741 +done
1.742 +
1.743 +lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
1.744 +apply (rule trans [symmetric])
1.745 +apply (rule zmod_zadd1_eq, simp)
1.746 +apply (rule zmod_zadd1_eq [symmetric])
1.747 +done
1.748 +
1.749 +lemma zdiv_zadd_self1[simp]: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"
1.750 +by (simp add: zdiv_zadd1_eq)
1.751 +
1.752 +lemma zdiv_zadd_self2[simp]: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"
1.753 +by (simp add: zdiv_zadd1_eq)
1.754 +
1.755 +lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"
1.756 +apply (case_tac "a = 0", simp)
1.757 +apply (simp add: zmod_zadd1_eq)
1.758 +done
1.759 +
1.760 +lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"
1.761 +apply (case_tac "a = 0", simp)
1.762 +apply (simp add: zmod_zadd1_eq)
1.763 +done
1.764 +
1.765 +
1.766 +subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}
1.767 +
1.768 +(*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but
1.769 + 7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems
1.770 + to cause particular problems.*)
1.771 +
1.772 +text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
1.773 +
1.774 +lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \<le> 0 |] ==> b*c < b*(q mod c) + r"
1.775 +apply (subgoal_tac "b * (c - q mod c) < r * 1")
1.776 +apply (simp add: right_diff_distrib)
1.777 +apply (rule order_le_less_trans)
1.778 +apply (erule_tac [2] mult_strict_right_mono)
1.779 +apply (rule mult_left_mono_neg)
1.780 +apply (auto simp add: compare_rls add_commute [of 1]
1.781 + add1_zle_eq pos_mod_bound)
1.782 +done
1.783 +
1.784 +lemma zmult2_lemma_aux2:
1.785 + "[| (0::int) < c; b < r; r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
1.786 +apply (subgoal_tac "b * (q mod c) \<le> 0")
1.787 + apply arith
1.788 +apply (simp add: mult_le_0_iff)
1.789 +done
1.790 +
1.791 +lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \<le> r; r < b |] ==> 0 \<le> b * (q mod c) + r"
1.792 +apply (subgoal_tac "0 \<le> b * (q mod c) ")
1.793 +apply arith
1.794 +apply (simp add: zero_le_mult_iff)
1.795 +done
1.796 +
1.797 +lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
1.798 +apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
1.799 +apply (simp add: right_diff_distrib)
1.800 +apply (rule order_less_le_trans)
1.801 +apply (erule mult_strict_right_mono)
1.802 +apply (rule_tac [2] mult_left_mono)
1.803 +apply (auto simp add: compare_rls add_commute [of 1]
1.804 + add1_zle_eq pos_mod_bound)
1.805 +done
1.806 +
1.807 +lemma zmult2_lemma: "[| quorem ((a,b), (q,r)); b \<noteq> 0; 0 < c |]
1.808 + ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
1.809 +by (auto simp add: mult_ac quorem_def linorder_neq_iff
1.810 + zero_less_mult_iff right_distrib [symmetric]
1.811 + zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
1.812 +
1.813 +lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
1.814 +apply (case_tac "b = 0", simp)
1.815 +apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])
1.816 +done
1.817 +
1.818 +lemma zmod_zmult2_eq:
1.819 + "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
1.820 +apply (case_tac "b = 0", simp)
1.821 +apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])
1.822 +done
1.823 +
1.824 +
1.825 +subsection{*Cancellation of Common Factors in div*}
1.826 +
1.827 +lemma zdiv_zmult_zmult1_aux1:
1.828 + "[| (0::int) < b; c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
1.829 +by (subst zdiv_zmult2_eq, auto)
1.830 +
1.831 +lemma zdiv_zmult_zmult1_aux2:
1.832 + "[| b < (0::int); c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
1.833 +apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
1.834 +apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
1.835 +done
1.836 +
1.837 +lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
1.838 +apply (case_tac "b = 0", simp)
1.839 +apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
1.840 +done
1.841 +
1.842 +lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
1.843 +apply (drule zdiv_zmult_zmult1)
1.844 +apply (auto simp add: mult_commute)
1.845 +done
1.846 +
1.847 +
1.848 +
1.849 +subsection{*Distribution of Factors over mod*}
1.850 +
1.851 +lemma zmod_zmult_zmult1_aux1:
1.852 + "[| (0::int) < b; c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
1.853 +by (subst zmod_zmult2_eq, auto)
1.854 +
1.855 +lemma zmod_zmult_zmult1_aux2:
1.856 + "[| b < (0::int); c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
1.857 +apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
1.858 +apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
1.859 +done
1.860 +
1.861 +lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
1.862 +apply (case_tac "b = 0", simp)
1.863 +apply (case_tac "c = 0", simp)
1.864 +apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
1.865 +done
1.866 +
1.867 +lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
1.868 +apply (cut_tac c = c in zmod_zmult_zmult1)
1.869 +apply (auto simp add: mult_commute)
1.870 +done
1.871 +
1.872 +
1.873 +subsection {*Splitting Rules for div and mod*}
1.874 +
1.875 +text{*The proofs of the two lemmas below are essentially identical*}
1.876 +
1.877 +lemma split_pos_lemma:
1.878 + "0<k ==>
1.879 + P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
1.880 +apply (rule iffI, clarify)
1.881 + apply (erule_tac P="P ?x ?y" in rev_mp)
1.882 + apply (subst zmod_zadd1_eq)
1.883 + apply (subst zdiv_zadd1_eq)
1.884 + apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
1.885 +txt{*converse direction*}
1.886 +apply (drule_tac x = "n div k" in spec)
1.887 +apply (drule_tac x = "n mod k" in spec, simp)
1.888 +done
1.889 +
1.890 +lemma split_neg_lemma:
1.891 + "k<0 ==>
1.892 + P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
1.893 +apply (rule iffI, clarify)
1.894 + apply (erule_tac P="P ?x ?y" in rev_mp)
1.895 + apply (subst zmod_zadd1_eq)
1.896 + apply (subst zdiv_zadd1_eq)
1.897 + apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
1.898 +txt{*converse direction*}
1.899 +apply (drule_tac x = "n div k" in spec)
1.900 +apply (drule_tac x = "n mod k" in spec, simp)
1.901 +done
1.902 +
1.903 +lemma split_zdiv:
1.904 + "P(n div k :: int) =
1.905 + ((k = 0 --> P 0) &
1.906 + (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
1.907 + (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
1.908 +apply (case_tac "k=0", simp)
1.909 +apply (simp only: linorder_neq_iff)
1.910 +apply (erule disjE)
1.911 + apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
1.912 + split_neg_lemma [of concl: "%x y. P x"])
1.913 +done
1.914 +
1.915 +lemma split_zmod:
1.916 + "P(n mod k :: int) =
1.917 + ((k = 0 --> P n) &
1.918 + (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
1.919 + (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
1.920 +apply (case_tac "k=0", simp)
1.921 +apply (simp only: linorder_neq_iff)
1.922 +apply (erule disjE)
1.923 + apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
1.924 + split_neg_lemma [of concl: "%x y. P y"])
1.925 +done
1.926 +
1.927 +(* Enable arith to deal with div 2 and mod 2: *)
1.928 +declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
1.929 +declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
1.930 +
1.931 +
1.932 +subsection{*Speeding up the Division Algorithm with Shifting*}
1.933 +
1.934 +text{*computing div by shifting *}
1.935 +
1.936 +lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
1.937 +proof cases
1.938 + assume "a=0"
1.939 + thus ?thesis by simp
1.940 +next
1.941 + assume "a\<noteq>0" and le_a: "0\<le>a"
1.942 + hence a_pos: "1 \<le> a" by arith
1.943 + hence one_less_a2: "1 < 2*a" by arith
1.944 + hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
1.945 + by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
1.946 + with a_pos have "0 \<le> b mod a" by simp
1.947 + hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
1.948 + by (simp add: mod_pos_pos_trivial one_less_a2)
1.949 + with le_2a
1.950 + have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
1.951 + by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
1.952 + right_distrib)
1.953 + thus ?thesis
1.954 + by (subst zdiv_zadd1_eq,
1.955 + simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
1.956 + div_pos_pos_trivial)
1.957 +qed
1.958 +
1.959 +lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
1.960 +apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
1.961 +apply (rule_tac [2] pos_zdiv_mult_2)
1.962 +apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
1.963 +apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
1.964 +apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
1.965 + simp)
1.966 +done
1.967 +
1.968 +
1.969 +(*Not clear why this must be proved separately; probably number_of causes
1.970 + simplification problems*)
1.971 +lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
1.972 +by auto
1.973 +
1.974 +lemma zdiv_number_of_BIT[simp]:
1.975 + "number_of (v BIT b) div number_of (w BIT bit.B0) =
1.976 + (if b=bit.B0 | (0::int) \<le> number_of w
1.977 + then number_of v div (number_of w)
1.978 + else (number_of v + (1::int)) div (number_of w))"
1.979 +apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)
1.980 +apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac
1.981 + split: bit.split)
1.982 +done
1.983 +
1.984 +
1.985 +subsection{*Computing mod by Shifting (proofs resemble those for div)*}
1.986 +
1.987 +lemma pos_zmod_mult_2:
1.988 + "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
1.989 +apply (case_tac "a = 0", simp)
1.990 +apply (subgoal_tac "1 < a * 2")
1.991 + prefer 2 apply arith
1.992 +apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
1.993 + apply (rule_tac [2] mult_left_mono)
1.994 +apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq
1.995 + pos_mod_bound)
1.996 +apply (subst zmod_zadd1_eq)
1.997 +apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
1.998 +apply (rule mod_pos_pos_trivial)
1.999 +apply (auto simp add: mod_pos_pos_trivial left_distrib)
1.1000 +apply (subgoal_tac "0 \<le> b mod a", arith, simp)
1.1001 +done
1.1002 +
1.1003 +lemma neg_zmod_mult_2:
1.1004 + "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
1.1005 +apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =
1.1006 + 1 + 2* ((-b - 1) mod (-a))")
1.1007 +apply (rule_tac [2] pos_zmod_mult_2)
1.1008 +apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
1.1009 +apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
1.1010 + prefer 2 apply simp
1.1011 +apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
1.1012 +done
1.1013 +
1.1014 +lemma zmod_number_of_BIT [simp]:
1.1015 + "number_of (v BIT b) mod number_of (w BIT bit.B0) =
1.1016 + (case b of
1.1017 + bit.B0 => 2 * (number_of v mod number_of w)
1.1018 + | bit.B1 => if (0::int) \<le> number_of w
1.1019 + then 2 * (number_of v mod number_of w) + 1
1.1020 + else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
1.1021 +apply (simp only: number_of_eq numeral_simps UNIV_I split: bit.split)
1.1022 +apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
1.1023 + not_0_le_lemma neg_zmod_mult_2 add_ac)
1.1024 +done
1.1025 +
1.1026 +
1.1027 +subsection{*Quotients of Signs*}
1.1028 +
1.1029 +lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"
1.1030 +apply (subgoal_tac "a div b \<le> -1", force)
1.1031 +apply (rule order_trans)
1.1032 +apply (rule_tac a' = "-1" in zdiv_mono1)
1.1033 +apply (auto simp add: zdiv_minus1)
1.1034 +done
1.1035 +
1.1036 +lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
1.1037 +by (drule zdiv_mono1_neg, auto)
1.1038 +
1.1039 +lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
1.1040 +apply auto
1.1041 +apply (drule_tac [2] zdiv_mono1)
1.1042 +apply (auto simp add: linorder_neq_iff)
1.1043 +apply (simp (no_asm_use) add: linorder_not_less [symmetric])
1.1044 +apply (blast intro: div_neg_pos_less0)
1.1045 +done
1.1046 +
1.1047 +lemma neg_imp_zdiv_nonneg_iff:
1.1048 + "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
1.1049 +apply (subst zdiv_zminus_zminus [symmetric])
1.1050 +apply (subst pos_imp_zdiv_nonneg_iff, auto)
1.1051 +done
1.1052 +
1.1053 +(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
1.1054 +lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
1.1055 +by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
1.1056 +
1.1057 +(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
1.1058 +lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
1.1059 +by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
1.1060 +
1.1061 +
1.1062 +subsection {* The Divides Relation *}
1.1063 +
1.1064 +lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
1.1065 +by(simp add:dvd_def zmod_eq_0_iff)
1.1066 +
1.1067 +lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
1.1068 + zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]
1.1069 +
1.1070 +lemma zdvd_0_right [iff]: "(m::int) dvd 0"
1.1071 +by (simp add: dvd_def)
1.1072 +
1.1073 +lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
1.1074 + by (simp add: dvd_def)
1.1075 +
1.1076 +lemma zdvd_1_left [iff]: "1 dvd (m::int)"
1.1077 + by (simp add: dvd_def)
1.1078 +
1.1079 +lemma zdvd_refl [simp]: "m dvd (m::int)"
1.1080 +by (auto simp add: dvd_def intro: zmult_1_right [symmetric])
1.1081 +
1.1082 +lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
1.1083 +by (auto simp add: dvd_def intro: mult_assoc)
1.1084 +
1.1085 +lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"
1.1086 + apply (simp add: dvd_def, auto)
1.1087 + apply (rule_tac [!] x = "-k" in exI, auto)
1.1088 + done
1.1089 +
1.1090 +lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"
1.1091 + apply (simp add: dvd_def, auto)
1.1092 + apply (rule_tac [!] x = "-k" in exI, auto)
1.1093 + done
1.1094 +lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)"
1.1095 + apply (cases "i > 0", simp)
1.1096 + apply (simp add: dvd_def)
1.1097 + apply (rule iffI)
1.1098 + apply (erule exE)
1.1099 + apply (rule_tac x="- k" in exI, simp)
1.1100 + apply (erule exE)
1.1101 + apply (rule_tac x="- k" in exI, simp)
1.1102 + done
1.1103 +lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)"
1.1104 + apply (cases "j > 0", simp)
1.1105 + apply (simp add: dvd_def)
1.1106 + apply (rule iffI)
1.1107 + apply (erule exE)
1.1108 + apply (rule_tac x="- k" in exI, simp)
1.1109 + apply (erule exE)
1.1110 + apply (rule_tac x="- k" in exI, simp)
1.1111 + done
1.1112 +
1.1113 +lemma zdvd_anti_sym:
1.1114 + "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
1.1115 + apply (simp add: dvd_def, auto)
1.1116 + apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
1.1117 + done
1.1118 +
1.1119 +lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
1.1120 + apply (simp add: dvd_def)
1.1121 + apply (blast intro: right_distrib [symmetric])
1.1122 + done
1.1123 +
1.1124 +lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a"
1.1125 + shows "\<bar>a\<bar> = \<bar>b\<bar>"
1.1126 +proof-
1.1127 + from ab obtain k where k:"b = a*k" unfolding dvd_def by blast
1.1128 + from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast
1.1129 + from k k' have "a = a*k*k'" by simp
1.1130 + with mult_cancel_left1[where c="a" and b="k*k'"]
1.1131 + have kk':"k*k' = 1" using anz by (simp add: mult_assoc)
1.1132 + hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
1.1133 + thus ?thesis using k k' by auto
1.1134 +qed
1.1135 +
1.1136 +lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
1.1137 + apply (simp add: dvd_def)
1.1138 + apply (blast intro: right_diff_distrib [symmetric])
1.1139 + done
1.1140 +
1.1141 +lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
1.1142 + apply (subgoal_tac "m = n + (m - n)")
1.1143 + apply (erule ssubst)
1.1144 + apply (blast intro: zdvd_zadd, simp)
1.1145 + done
1.1146 +
1.1147 +lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
1.1148 + apply (simp add: dvd_def)
1.1149 + apply (blast intro: mult_left_commute)
1.1150 + done
1.1151 +
1.1152 +lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
1.1153 + apply (subst mult_commute)
1.1154 + apply (erule zdvd_zmult)
1.1155 + done
1.1156 +
1.1157 +lemma zdvd_triv_right [iff]: "(k::int) dvd m * k"
1.1158 + apply (rule zdvd_zmult)
1.1159 + apply (rule zdvd_refl)
1.1160 + done
1.1161 +
1.1162 +lemma zdvd_triv_left [iff]: "(k::int) dvd k * m"
1.1163 + apply (rule zdvd_zmult2)
1.1164 + apply (rule zdvd_refl)
1.1165 + done
1.1166 +
1.1167 +lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
1.1168 + apply (simp add: dvd_def)
1.1169 + apply (simp add: mult_assoc, blast)
1.1170 + done
1.1171 +
1.1172 +lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
1.1173 + apply (rule zdvd_zmultD2)
1.1174 + apply (subst mult_commute, assumption)
1.1175 + done
1.1176 +
1.1177 +lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
1.1178 + apply (simp add: dvd_def, clarify)
1.1179 + apply (rule_tac x = "k * ka" in exI)
1.1180 + apply (simp add: mult_ac)
1.1181 + done
1.1182 +
1.1183 +lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
1.1184 + apply (rule iffI)
1.1185 + apply (erule_tac [2] zdvd_zadd)
1.1186 + apply (subgoal_tac "n = (n + k * m) - k * m")
1.1187 + apply (erule ssubst)
1.1188 + apply (erule zdvd_zdiff, simp_all)
1.1189 + done
1.1190 +
1.1191 +lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
1.1192 + apply (simp add: dvd_def)
1.1193 + apply (auto simp add: zmod_zmult_zmult1)
1.1194 + done
1.1195 +
1.1196 +lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
1.1197 + apply (subgoal_tac "k dvd n * (m div n) + m mod n")
1.1198 + apply (simp add: zmod_zdiv_equality [symmetric])
1.1199 + apply (simp only: zdvd_zadd zdvd_zmult2)
1.1200 + done
1.1201 +
1.1202 +lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
1.1203 + apply (simp add: dvd_def, auto)
1.1204 + apply (subgoal_tac "0 < n")
1.1205 + prefer 2
1.1206 + apply (blast intro: order_less_trans)
1.1207 + apply (simp add: zero_less_mult_iff)
1.1208 + apply (subgoal_tac "n * k < n * 1")
1.1209 + apply (drule mult_less_cancel_left [THEN iffD1], auto)
1.1210 + done
1.1211 +lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
1.1212 + using zmod_zdiv_equality[where a="m" and b="n"]
1.1213 + by (simp add: ring_eq_simps)
1.1214 +
1.1215 +lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
1.1216 +apply (subgoal_tac "m mod n = 0")
1.1217 + apply (simp add: zmult_div_cancel)
1.1218 +apply (simp only: zdvd_iff_zmod_eq_0)
1.1219 +done
1.1220 +
1.1221 +lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
1.1222 + shows "m dvd n"
1.1223 +proof-
1.1224 + from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
1.1225 + {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
1.1226 + with h have False by (simp add: mult_assoc)}
1.1227 + hence "n = m * h" by blast
1.1228 + thus ?thesis by blast
1.1229 +qed
1.1230 +
1.1231 +theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
1.1232 + apply (simp split add: split_nat)
1.1233 + apply (rule iffI)
1.1234 + apply (erule exE)
1.1235 + apply (rule_tac x = "int x" in exI)
1.1236 + apply simp
1.1237 + apply (erule exE)
1.1238 + apply (rule_tac x = "nat x" in exI)
1.1239 + apply (erule conjE)
1.1240 + apply (erule_tac x = "nat x" in allE)
1.1241 + apply simp
1.1242 + done
1.1243 +
1.1244 +theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
1.1245 + apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric]
1.1246 + nat_0_le cong add: conj_cong)
1.1247 + apply (rule iffI)
1.1248 + apply iprover
1.1249 + apply (erule exE)
1.1250 + apply (case_tac "x=0")
1.1251 + apply (rule_tac x=0 in exI)
1.1252 + apply simp
1.1253 + apply (case_tac "0 \<le> k")
1.1254 + apply iprover
1.1255 + apply (simp add: linorder_not_le)
1.1256 + apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]])
1.1257 + apply assumption
1.1258 + apply (simp add: mult_ac)
1.1259 + done
1.1260 +
1.1261 +lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
1.1262 +proof
1.1263 + assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)
1.1264 + hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
1.1265 + hence "nat \<bar>x\<bar> = 1" by simp
1.1266 + thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
1.1267 +next
1.1268 + assume "\<bar>x\<bar>=1" thus "x dvd 1"
1.1269 + by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)
1.1270 +qed
1.1271 +lemma zdvd_mult_cancel1:
1.1272 + assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
1.1273 +proof
1.1274 + assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
1.1275 + by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)
1.1276 +next
1.1277 + assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
1.1278 + from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
1.1279 +qed
1.1280 +
1.1281 +lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
1.1282 + apply (auto simp add: dvd_def nat_abs_mult_distrib)
1.1283 + apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)
1.1284 + apply (rule_tac x = "-(int k)" in exI)
1.1285 + apply (auto simp add: int_mult)
1.1286 + done
1.1287 +
1.1288 +lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
1.1289 + apply (auto simp add: dvd_def abs_if int_mult)
1.1290 + apply (rule_tac [3] x = "nat k" in exI)
1.1291 + apply (rule_tac [2] x = "-(int k)" in exI)
1.1292 + apply (rule_tac x = "nat (-k)" in exI)
1.1293 + apply (cut_tac [3] k = m in int_less_0_conv)
1.1294 + apply (cut_tac k = m in int_less_0_conv)
1.1295 + apply (auto simp add: zero_le_mult_iff mult_less_0_iff
1.1296 + nat_mult_distrib [symmetric] nat_eq_iff2)
1.1297 + done
1.1298 +
1.1299 +lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
1.1300 + apply (auto simp add: dvd_def int_mult)
1.1301 + apply (rule_tac x = "nat k" in exI)
1.1302 + apply (cut_tac k = m in int_less_0_conv)
1.1303 + apply (auto simp add: zero_le_mult_iff mult_less_0_iff
1.1304 + nat_mult_distrib [symmetric] nat_eq_iff2)
1.1305 + done
1.1306 +
1.1307 +lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
1.1308 + apply (auto simp add: dvd_def)
1.1309 + apply (rule_tac [!] x = "-k" in exI, auto)
1.1310 + done
1.1311 +
1.1312 +lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
1.1313 + apply (auto simp add: dvd_def)
1.1314 + apply (drule minus_equation_iff [THEN iffD1])
1.1315 + apply (rule_tac [!] x = "-k" in exI, auto)
1.1316 + done
1.1317 +
1.1318 +lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
1.1319 + apply (rule_tac z=n in int_cases)
1.1320 + apply (auto simp add: dvd_int_iff)
1.1321 + apply (rule_tac z=z in int_cases)
1.1322 + apply (auto simp add: dvd_imp_le)
1.1323 + done
1.1324 +
1.1325 +
1.1326 +subsection{*Integer Powers*}
1.1327 +
1.1328 +instance int :: power ..
1.1329 +
1.1330 +primrec
1.1331 + "p ^ 0 = 1"
1.1332 + "p ^ (Suc n) = (p::int) * (p ^ n)"
1.1333 +
1.1334 +
1.1335 +instance int :: recpower
1.1336 +proof
1.1337 + fix z :: int
1.1338 + fix n :: nat
1.1339 + show "z^0 = 1" by simp
1.1340 + show "z^(Suc n) = z * (z^n)" by simp
1.1341 +qed
1.1342 +
1.1343 +
1.1344 +lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
1.1345 +apply (induct "y", auto)
1.1346 +apply (rule zmod_zmult1_eq [THEN trans])
1.1347 +apply (simp (no_asm_simp))
1.1348 +apply (rule zmod_zmult_distrib [symmetric])
1.1349 +done
1.1350 +
1.1351 +lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"
1.1352 + by (rule Power.power_add)
1.1353 +
1.1354 +lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"
1.1355 + by (rule Power.power_mult [symmetric])
1.1356 +
1.1357 +lemma zero_less_zpower_abs_iff [simp]:
1.1358 + "(0 < (abs x)^n) = (x \<noteq> (0::int) | n=0)"
1.1359 +apply (induct "n")
1.1360 +apply (auto simp add: zero_less_mult_iff)
1.1361 +done
1.1362 +
1.1363 +lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"
1.1364 +apply (induct "n")
1.1365 +apply (auto simp add: zero_le_mult_iff)
1.1366 +done
1.1367 +
1.1368 +lemma int_power: "int (m^n) = (int m) ^ n"
1.1369 + by (induct n, simp_all add: int_mult)
1.1370 +
1.1371 +text{*Compatibility binding*}
1.1372 +lemmas zpower_int = int_power [symmetric]
1.1373 +
1.1374 +lemma zdiv_int: "int (a div b) = (int a) div (int b)"
1.1375 +apply (subst split_div, auto)
1.1376 +apply (subst split_zdiv, auto)
1.1377 +apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
1.1378 +apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
1.1379 +done
1.1380 +
1.1381 +lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
1.1382 +apply (subst split_mod, auto)
1.1383 +apply (subst split_zmod, auto)
1.1384 +apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia
1.1385 + in unique_remainder)
1.1386 +apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
1.1387 +done
1.1388 +
1.1389 +text{*Suggested by Matthias Daum*}
1.1390 +lemma int_power_div_base:
1.1391 + "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
1.1392 +apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")
1.1393 + apply (erule ssubst)
1.1394 + apply (simp only: power_add)
1.1395 + apply simp_all
1.1396 +done
1.1397 +
1.1398 +text {* code serializer setup *}
1.1399 +
1.1400 +code_modulename SML
1.1401 + IntDiv Integer
1.1402 +
1.1403 +code_modulename OCaml
1.1404 + IntDiv Integer
1.1405 +
1.1406 +code_modulename Haskell
1.1407 + IntDiv Divides
1.1408 +
1.1409 +end