src/HOL/Integ/IntDiv.thy
 author nipkow Wed Aug 18 11:09:40 2004 +0200 (2004-08-18) changeset 15140 322485b816ac parent 15131 c69542757a4d child 15221 8412cfdf3287 permissions -rw-r--r--
import -> imports
 paulson@6917 ` 1` ```(* Title: HOL/IntDiv.thy ``` paulson@6917 ` 2` ``` ID: \$Id\$ ``` paulson@6917 ` 3` ``` Author: Lawrence C Paulson, Cambridge University Computer Laboratory ``` paulson@6917 ` 4` ``` Copyright 1999 University of Cambridge ``` paulson@6917 ` 5` paulson@6917 ` 6` ```The division operators div, mod and the divides relation "dvd" ``` paulson@13183 ` 7` paulson@13183 ` 8` ```Here is the division algorithm in ML: ``` paulson@13183 ` 9` paulson@13183 ` 10` ``` fun posDivAlg (a,b) = ``` paulson@13183 ` 11` ``` if ar-b then (2*q+1, r-b) else (2*q, r) ``` paulson@13183 ` 14` ``` end ``` paulson@13183 ` 15` paulson@13183 ` 16` ``` fun negDivAlg (a,b) = ``` paulson@14288 ` 17` ``` if 0\a+b then (~1,a+b) ``` paulson@13183 ` 18` ``` else let val (q,r) = negDivAlg(a, 2*b) ``` paulson@14288 ` 19` ``` in if 0\r-b then (2*q+1, r-b) else (2*q, r) ``` paulson@13183 ` 20` ``` end; ``` paulson@13183 ` 21` paulson@13183 ` 22` ``` fun negateSnd (q,r:int) = (q,~r); ``` paulson@13183 ` 23` paulson@14288 ` 24` ``` fun divAlg (a,b) = if 0\a then ``` paulson@13183 ` 25` ``` if b>0 then posDivAlg (a,b) ``` paulson@13183 ` 26` ``` else if a=0 then (0,0) ``` paulson@13183 ` 27` ``` else negateSnd (negDivAlg (~a,~b)) ``` paulson@13183 ` 28` ``` else ``` paulson@13183 ` 29` ``` if 0 bool" ``` paulson@6917 ` 43` ``` "quorem == %((a,b), (q,r)). ``` paulson@6917 ` 44` ``` a = b*q + r & ``` paulson@14288 ` 45` ``` (if 0 < b then 0\r & r 0)" ``` paulson@6917 ` 46` paulson@11868 ` 47` ``` adjust :: "[int, int*int] => int*int" ``` paulson@14288 ` 48` ``` "adjust b == %(q,r). if 0 \ r-b then (2*q + 1, r-b) ``` paulson@11868 ` 49` ``` else (2*q, r)" ``` paulson@6917 ` 50` paulson@6917 ` 51` ```(** the division algorithm **) ``` paulson@6917 ` 52` paulson@6917 ` 53` ```(*for the case a>=0, b>0*) ``` paulson@6917 ` 54` ```consts posDivAlg :: "int*int => int*int" ``` paulson@11868 ` 55` ```recdef posDivAlg "inv_image less_than (%(a,b). nat(a - b + 1))" ``` paulson@6917 ` 56` ``` "posDivAlg (a,b) = ``` paulson@14288 ` 57` ``` (if (a0) then (0,a) ``` paulson@11868 ` 58` ``` else adjust b (posDivAlg(a, 2*b)))" ``` paulson@6917 ` 59` paulson@6917 ` 60` ```(*for the case a<0, b>0*) ``` paulson@6917 ` 61` ```consts negDivAlg :: "int*int => int*int" ``` paulson@6917 ` 62` ```recdef negDivAlg "inv_image less_than (%(a,b). nat(- a - b))" ``` paulson@6917 ` 63` ``` "negDivAlg (a,b) = ``` paulson@14288 ` 64` ``` (if (0\a+b | b\0) then (-1,a+b) ``` paulson@11868 ` 65` ``` else adjust b (negDivAlg(a, 2*b)))" ``` paulson@6917 ` 66` paulson@6917 ` 67` ```(*for the general case b~=0*) ``` paulson@6917 ` 68` paulson@6917 ` 69` ```constdefs ``` paulson@6917 ` 70` ``` negateSnd :: "int*int => int*int" ``` paulson@6917 ` 71` ``` "negateSnd == %(q,r). (q,-r)" ``` paulson@6917 ` 72` paulson@6917 ` 73` ``` (*The full division algorithm considers all possible signs for a, b ``` paulson@6917 ` 74` ``` including the special case a=0, b<0, because negDivAlg requires a<0*) ``` paulson@6917 ` 75` ``` divAlg :: "int*int => int*int" ``` paulson@6917 ` 76` ``` "divAlg == ``` paulson@14288 ` 77` ``` %(a,b). if 0\a then ``` paulson@14288 ` 78` ``` if 0\b then posDivAlg (a,b) ``` paulson@11868 ` 79` ``` else if a=0 then (0,0) ``` paulson@6917 ` 80` ``` else negateSnd (negDivAlg (-a,-b)) ``` paulson@6917 ` 81` ``` else ``` paulson@11868 ` 82` ``` if 0 b*q + r; 0 \ r'; 0 < b; r < b |] ``` paulson@14288 ` 98` ``` ==> q' \ (q::int)" ``` paulson@14288 ` 99` ```apply (subgoal_tac "r' + b * (q'-q) \ r") ``` paulson@14479 ` 100` ``` prefer 2 apply (simp add: right_diff_distrib) ``` paulson@13183 ` 101` ```apply (subgoal_tac "0 < b * (1 + q - q') ") ``` paulson@13183 ` 102` ```apply (erule_tac [2] order_le_less_trans) ``` paulson@14479 ` 103` ``` prefer 2 apply (simp add: right_diff_distrib right_distrib) ``` paulson@13183 ` 104` ```apply (subgoal_tac "b * q' < b * (1 + q) ") ``` paulson@14479 ` 105` ``` prefer 2 apply (simp add: right_diff_distrib right_distrib) ``` paulson@14387 ` 106` ```apply (simp add: mult_less_cancel_left) ``` paulson@13183 ` 107` ```done ``` paulson@13183 ` 108` paulson@13183 ` 109` ```lemma unique_quotient_lemma_neg: ``` paulson@14288 ` 110` ``` "[| b*q' + r' \ b*q + r; r \ 0; b < 0; b < r' |] ``` paulson@14288 ` 111` ``` ==> q \ (q'::int)" ``` paulson@13183 ` 112` ```by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, ``` paulson@13183 ` 113` ``` auto) ``` paulson@13183 ` 114` paulson@13183 ` 115` ```lemma unique_quotient: ``` paulson@13183 ` 116` ``` "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b ~= 0 |] ``` paulson@13183 ` 117` ``` ==> q = q'" ``` paulson@13183 ` 118` ```apply (simp add: quorem_def linorder_neq_iff split: split_if_asm) ``` paulson@13183 ` 119` ```apply (blast intro: order_antisym ``` paulson@13183 ` 120` ``` dest: order_eq_refl [THEN unique_quotient_lemma] ``` paulson@13183 ` 121` ``` order_eq_refl [THEN unique_quotient_lemma_neg] sym)+ ``` paulson@13183 ` 122` ```done ``` paulson@13183 ` 123` paulson@13183 ` 124` paulson@13183 ` 125` ```lemma unique_remainder: ``` paulson@13183 ` 126` ``` "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b ~= 0 |] ``` paulson@13183 ` 127` ``` ==> r = r'" ``` paulson@13183 ` 128` ```apply (subgoal_tac "q = q'") ``` paulson@13183 ` 129` ``` apply (simp add: quorem_def) ``` paulson@13183 ` 130` ```apply (blast intro: unique_quotient) ``` paulson@13183 ` 131` ```done ``` paulson@13183 ` 132` paulson@13183 ` 133` paulson@14271 ` 134` ```subsection{*Correctness of posDivAlg, the Algorithm for Non-Negative Dividends*} ``` paulson@14271 ` 135` paulson@14271 ` 136` ```text{*And positive divisors*} ``` paulson@13183 ` 137` paulson@13183 ` 138` ```lemma adjust_eq [simp]: ``` paulson@13183 ` 139` ``` "adjust b (q,r) = ``` paulson@13183 ` 140` ``` (let diff = r-b in ``` paulson@14288 ` 141` ``` if 0 \ diff then (2*q + 1, diff) ``` paulson@13183 ` 142` ``` else (2*q, r))" ``` paulson@13183 ` 143` ```by (simp add: Let_def adjust_def) ``` paulson@13183 ` 144` paulson@13183 ` 145` ```declare posDivAlg.simps [simp del] ``` paulson@13183 ` 146` paulson@13183 ` 147` ```(**use with a simproc to avoid repeatedly proving the premise*) ``` paulson@13183 ` 148` ```lemma posDivAlg_eqn: ``` paulson@13183 ` 149` ``` "0 < b ==> ``` paulson@13183 ` 150` ``` posDivAlg (a,b) = (if a a --> 0 < b --> quorem ((a, b), posDivAlg (a, b))" ``` paulson@13183 ` 156` ```apply (induct_tac a b rule: posDivAlg.induct, auto) ``` paulson@13183 ` 157` ``` apply (simp_all add: quorem_def) ``` paulson@13183 ` 158` ``` (*base case: a ``` paulson@13183 ` 176` ``` negDivAlg (a,b) = ``` paulson@14288 ` 177` ``` (if 0\a+b then (-1,a+b) else adjust b (negDivAlg(a, 2*b)))" ``` paulson@13183 ` 178` ```by (rule negDivAlg.simps [THEN trans], simp) ``` paulson@13183 ` 179` paulson@13183 ` 180` ```(*Correctness of negDivAlg: it computes quotients correctly ``` paulson@13183 ` 181` ``` It doesn't work if a=0 because the 0/b equals 0, not -1*) ``` paulson@13183 ` 182` ```lemma negDivAlg_correct [rule_format]: ``` paulson@13183 ` 183` ``` "a < 0 --> 0 < b --> quorem ((a, b), negDivAlg (a, b))" ``` paulson@13183 ` 184` ```apply (induct_tac a b rule: negDivAlg.induct, auto) ``` paulson@13183 ` 185` ``` apply (simp_all add: quorem_def) ``` paulson@14288 ` 186` ``` (*base case: 0\a+b*) ``` paulson@13183 ` 187` ``` apply (simp add: negDivAlg_eqn) ``` paulson@13183 ` 188` ```(*main argument*) ``` paulson@13183 ` 189` ```apply (subst negDivAlg_eqn, assumption) ``` paulson@13183 ` 190` ```apply (erule splitE) ``` paulson@14479 ` 191` ```apply (auto simp add: right_distrib Let_def) ``` paulson@13183 ` 192` ```done ``` paulson@13183 ` 193` paulson@13183 ` 194` paulson@14271 ` 195` ```subsection{*Existence Shown by Proving the Division Algorithm to be Correct*} ``` paulson@13183 ` 196` paulson@13183 ` 197` ```(*the case a=0*) ``` paulson@13183 ` 198` ```lemma quorem_0: "b ~= 0 ==> quorem ((0,b), (0,0))" ``` paulson@13183 ` 199` ```by (auto simp add: quorem_def linorder_neq_iff) ``` paulson@13183 ` 200` paulson@13183 ` 201` ```lemma posDivAlg_0 [simp]: "posDivAlg (0, b) = (0, 0)" ``` paulson@13183 ` 202` ```by (subst posDivAlg.simps, auto) ``` paulson@13183 ` 203` paulson@13183 ` 204` ```lemma negDivAlg_minus1 [simp]: "negDivAlg (-1, b) = (-1, b - 1)" ``` paulson@13183 ` 205` ```by (subst negDivAlg.simps, auto) ``` paulson@13183 ` 206` paulson@13183 ` 207` ```lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)" ``` paulson@13183 ` 208` ```by (unfold negateSnd_def, auto) ``` paulson@13183 ` 209` paulson@13183 ` 210` ```lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)" ``` paulson@13183 ` 211` ```by (auto simp add: split_ifs quorem_def) ``` paulson@13183 ` 212` paulson@13183 ` 213` ```lemma divAlg_correct: "b ~= 0 ==> quorem ((a,b), divAlg(a,b))" ``` paulson@13183 ` 214` ```by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg ``` paulson@13183 ` 215` ``` posDivAlg_correct negDivAlg_correct) ``` paulson@13183 ` 216` paulson@13183 ` 217` ```(** Arbitrary definitions for division by zero. Useful to simplify ``` paulson@13183 ` 218` ``` certain equations **) ``` paulson@13183 ` 219` paulson@14271 ` 220` ```lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a" ``` paulson@14271 ` 221` ```by (simp add: div_def mod_def divAlg_def posDivAlg.simps) ``` paulson@13183 ` 222` paulson@13183 ` 223` ```(** Basic laws about division and remainder **) ``` paulson@13183 ` 224` paulson@13183 ` 225` ```lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)" ``` paulson@15013 ` 226` ```apply (case_tac "b = 0", simp) ``` paulson@13183 ` 227` ```apply (cut_tac a = a and b = b in divAlg_correct) ``` paulson@13183 ` 228` ```apply (auto simp add: quorem_def div_def mod_def) ``` paulson@13183 ` 229` ```done ``` paulson@13183 ` 230` nipkow@13517 ` 231` ```lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k" ``` nipkow@13517 ` 232` ```by(simp add: zmod_zdiv_equality[symmetric]) ``` nipkow@13517 ` 233` nipkow@13517 ` 234` ```lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k" ``` nipkow@13517 ` 235` ```by(simp add: zmult_commute zmod_zdiv_equality[symmetric]) ``` nipkow@13517 ` 236` nipkow@13517 ` 237` ```use "IntDiv_setup.ML" ``` nipkow@13517 ` 238` paulson@14288 ` 239` ```lemma pos_mod_conj : "(0::int) < b ==> 0 \ a mod b & a mod b < b" ``` paulson@13183 ` 240` ```apply (cut_tac a = a and b = b in divAlg_correct) ``` paulson@13183 ` 241` ```apply (auto simp add: quorem_def mod_def) ``` paulson@13183 ` 242` ```done ``` paulson@13183 ` 243` nipkow@13788 ` 244` ```lemmas pos_mod_sign[simp] = pos_mod_conj [THEN conjunct1, standard] ``` nipkow@13788 ` 245` ``` and pos_mod_bound[simp] = pos_mod_conj [THEN conjunct2, standard] ``` paulson@13183 ` 246` paulson@14288 ` 247` ```lemma neg_mod_conj : "b < (0::int) ==> a mod b \ 0 & b < a mod b" ``` paulson@13183 ` 248` ```apply (cut_tac a = a and b = b in divAlg_correct) ``` paulson@13183 ` 249` ```apply (auto simp add: quorem_def div_def mod_def) ``` paulson@13183 ` 250` ```done ``` paulson@13183 ` 251` nipkow@13788 ` 252` ```lemmas neg_mod_sign[simp] = neg_mod_conj [THEN conjunct1, standard] ``` nipkow@13788 ` 253` ``` and neg_mod_bound[simp] = neg_mod_conj [THEN conjunct2, standard] ``` paulson@13183 ` 254` paulson@13183 ` 255` paulson@13260 ` 256` paulson@13183 ` 257` ```(** proving general properties of div and mod **) ``` paulson@13183 ` 258` paulson@13183 ` 259` ```lemma quorem_div_mod: "b ~= 0 ==> quorem ((a, b), (a div b, a mod b))" ``` paulson@13183 ` 260` ```apply (cut_tac a = a and b = b in zmod_zdiv_equality) ``` nipkow@13788 ` 261` ```apply (force simp add: quorem_def linorder_neq_iff) ``` paulson@13183 ` 262` ```done ``` paulson@13183 ` 263` paulson@13183 ` 264` ```lemma quorem_div: "[| quorem((a,b),(q,r)); b ~= 0 |] ==> a div b = q" ``` paulson@13183 ` 265` ```by (simp add: quorem_div_mod [THEN unique_quotient]) ``` paulson@13183 ` 266` paulson@13183 ` 267` ```lemma quorem_mod: "[| quorem((a,b),(q,r)); b ~= 0 |] ==> a mod b = r" ``` paulson@13183 ` 268` ```by (simp add: quorem_div_mod [THEN unique_remainder]) ``` paulson@13183 ` 269` paulson@14288 ` 270` ```lemma div_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a div b = 0" ``` paulson@13183 ` 271` ```apply (rule quorem_div) ``` paulson@13183 ` 272` ```apply (auto simp add: quorem_def) ``` paulson@13183 ` 273` ```done ``` paulson@13183 ` 274` paulson@14288 ` 275` ```lemma div_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a div b = 0" ``` paulson@13183 ` 276` ```apply (rule quorem_div) ``` paulson@13183 ` 277` ```apply (auto simp add: quorem_def) ``` paulson@13183 ` 278` ```done ``` paulson@13183 ` 279` paulson@14288 ` 280` ```lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a div b = -1" ``` paulson@13183 ` 281` ```apply (rule quorem_div) ``` paulson@13183 ` 282` ```apply (auto simp add: quorem_def) ``` paulson@13183 ` 283` ```done ``` paulson@13183 ` 284` paulson@13183 ` 285` ```(*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*) ``` paulson@13183 ` 286` paulson@14288 ` 287` ```lemma mod_pos_pos_trivial: "[| (0::int) \ a; a < b |] ==> a mod b = a" ``` paulson@13183 ` 288` ```apply (rule_tac q = 0 in quorem_mod) ``` paulson@13183 ` 289` ```apply (auto simp add: quorem_def) ``` paulson@13183 ` 290` ```done ``` paulson@13183 ` 291` paulson@14288 ` 292` ```lemma mod_neg_neg_trivial: "[| a \ (0::int); b < a |] ==> a mod b = a" ``` paulson@13183 ` 293` ```apply (rule_tac q = 0 in quorem_mod) ``` paulson@13183 ` 294` ```apply (auto simp add: quorem_def) ``` paulson@13183 ` 295` ```done ``` paulson@13183 ` 296` paulson@14288 ` 297` ```lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \ 0 |] ==> a mod b = a+b" ``` paulson@13183 ` 298` ```apply (rule_tac q = "-1" in quorem_mod) ``` paulson@13183 ` 299` ```apply (auto simp add: quorem_def) ``` paulson@13183 ` 300` ```done ``` paulson@13183 ` 301` paulson@13183 ` 302` ```(*There is no mod_neg_pos_trivial...*) ``` paulson@13183 ` 303` paulson@13183 ` 304` paulson@13183 ` 305` ```(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*) ``` paulson@13183 ` 306` ```lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)" ``` paulson@15013 ` 307` ```apply (case_tac "b = 0", simp) ``` paulson@13183 ` 308` ```apply (simp add: quorem_div_mod [THEN quorem_neg, simplified, ``` paulson@13183 ` 309` ``` THEN quorem_div, THEN sym]) ``` paulson@13183 ` 310` paulson@13183 ` 311` ```done ``` paulson@13183 ` 312` paulson@13183 ` 313` ```(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*) ``` paulson@13183 ` 314` ```lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))" ``` paulson@15013 ` 315` ```apply (case_tac "b = 0", simp) ``` paulson@13183 ` 316` ```apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod], ``` paulson@13183 ` 317` ``` auto) ``` paulson@13183 ` 318` ```done ``` paulson@13183 ` 319` paulson@14271 ` 320` ```subsection{*div, mod and unary minus*} ``` paulson@13183 ` 321` paulson@13183 ` 322` ```lemma zminus1_lemma: ``` paulson@13183 ` 323` ``` "quorem((a,b),(q,r)) ``` paulson@13183 ` 324` ``` ==> quorem ((-a,b), (if r=0 then -q else -q - 1), ``` paulson@13183 ` 325` ``` (if r=0 then 0 else b-r))" ``` paulson@14479 ` 326` ```by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib) ``` paulson@13183 ` 327` paulson@13183 ` 328` paulson@13183 ` 329` ```lemma zdiv_zminus1_eq_if: ``` paulson@13183 ` 330` ``` "b ~= (0::int) ``` paulson@13183 ` 331` ``` ==> (-a) div b = ``` paulson@13183 ` 332` ``` (if a mod b = 0 then - (a div b) else - (a div b) - 1)" ``` paulson@13183 ` 333` ```by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div]) ``` paulson@13183 ` 334` paulson@13183 ` 335` ```lemma zmod_zminus1_eq_if: ``` paulson@13183 ` 336` ``` "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))" ``` paulson@15013 ` 337` ```apply (case_tac "b = 0", simp) ``` paulson@13183 ` 338` ```apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod]) ``` paulson@13183 ` 339` ```done ``` paulson@13183 ` 340` paulson@13183 ` 341` ```lemma zdiv_zminus2: "a div (-b) = (-a::int) div b" ``` paulson@13183 ` 342` ```by (cut_tac a = "-a" in zdiv_zminus_zminus, auto) ``` paulson@13183 ` 343` paulson@13183 ` 344` ```lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)" ``` paulson@13183 ` 345` ```by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto) ``` paulson@13183 ` 346` paulson@13183 ` 347` ```lemma zdiv_zminus2_eq_if: ``` paulson@13183 ` 348` ``` "b ~= (0::int) ``` paulson@13183 ` 349` ``` ==> a div (-b) = ``` paulson@13183 ` 350` ``` (if a mod b = 0 then - (a div b) else - (a div b) - 1)" ``` paulson@13183 ` 351` ```by (simp add: zdiv_zminus1_eq_if zdiv_zminus2) ``` paulson@13183 ` 352` paulson@13183 ` 353` ```lemma zmod_zminus2_eq_if: ``` paulson@13183 ` 354` ``` "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)" ``` paulson@13183 ` 355` ```by (simp add: zmod_zminus1_eq_if zmod_zminus2) ``` paulson@13183 ` 356` paulson@13183 ` 357` paulson@14271 ` 358` ```subsection{*Division of a Number by Itself*} ``` paulson@13183 ` 359` paulson@14288 ` 360` ```lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \ q" ``` paulson@13183 ` 361` ```apply (subgoal_tac "0 < a*q") ``` paulson@14353 ` 362` ``` apply (simp add: zero_less_mult_iff, arith) ``` paulson@13183 ` 363` ```done ``` paulson@13183 ` 364` paulson@14288 ` 365` ```lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \ r |] ==> q \ 1" ``` paulson@14288 ` 366` ```apply (subgoal_tac "0 \ a* (1-q) ") ``` paulson@14353 ` 367` ``` apply (simp add: zero_le_mult_iff) ``` paulson@14479 ` 368` ```apply (simp add: right_diff_distrib) ``` paulson@13183 ` 369` ```done ``` paulson@13183 ` 370` paulson@13183 ` 371` ```lemma self_quotient: "[| quorem((a,a),(q,r)); a ~= (0::int) |] ==> q = 1" ``` paulson@13183 ` 372` ```apply (simp add: split_ifs quorem_def linorder_neq_iff) ``` berghofe@13601 ` 373` ```apply (rule order_antisym, safe, simp_all (no_asm_use)) ``` wenzelm@13524 ` 374` ```apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1) ``` wenzelm@13524 ` 375` ```apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2) ``` berghofe@13601 ` 376` ```apply (force intro: self_quotient_aux1 self_quotient_aux2 simp only: zadd_commute zmult_zminus)+ ``` paulson@13183 ` 377` ```done ``` paulson@13183 ` 378` paulson@13183 ` 379` ```lemma self_remainder: "[| quorem((a,a),(q,r)); a ~= (0::int) |] ==> r = 0" ``` paulson@13183 ` 380` ```apply (frule self_quotient, assumption) ``` paulson@13183 ` 381` ```apply (simp add: quorem_def) ``` paulson@13183 ` 382` ```done ``` paulson@13183 ` 383` paulson@13183 ` 384` ```lemma zdiv_self [simp]: "a ~= 0 ==> a div a = (1::int)" ``` paulson@13183 ` 385` ```by (simp add: quorem_div_mod [THEN self_quotient]) ``` paulson@13183 ` 386` paulson@13183 ` 387` ```(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *) ``` paulson@13183 ` 388` ```lemma zmod_self [simp]: "a mod a = (0::int)" ``` paulson@15013 ` 389` ```apply (case_tac "a = 0", simp) ``` paulson@13183 ` 390` ```apply (simp add: quorem_div_mod [THEN self_remainder]) ``` paulson@13183 ` 391` ```done ``` paulson@13183 ` 392` paulson@13183 ` 393` paulson@14271 ` 394` ```subsection{*Computation of Division and Remainder*} ``` paulson@13183 ` 395` paulson@13183 ` 396` ```lemma zdiv_zero [simp]: "(0::int) div b = 0" ``` paulson@13183 ` 397` ```by (simp add: div_def divAlg_def) ``` paulson@13183 ` 398` paulson@13183 ` 399` ```lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1" ``` paulson@13183 ` 400` ```by (simp add: div_def divAlg_def) ``` paulson@13183 ` 401` paulson@13183 ` 402` ```lemma zmod_zero [simp]: "(0::int) mod b = 0" ``` paulson@13183 ` 403` ```by (simp add: mod_def divAlg_def) ``` paulson@13183 ` 404` paulson@13183 ` 405` ```lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1" ``` paulson@13183 ` 406` ```by (simp add: div_def divAlg_def) ``` paulson@13183 ` 407` paulson@13183 ` 408` ```lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1" ``` paulson@13183 ` 409` ```by (simp add: mod_def divAlg_def) ``` paulson@13183 ` 410` paulson@13183 ` 411` ```(** a positive, b positive **) ``` paulson@13183 ` 412` paulson@14288 ` 413` ```lemma div_pos_pos: "[| 0 < a; 0 \ b |] ==> a div b = fst (posDivAlg(a,b))" ``` paulson@13183 ` 414` ```by (simp add: div_def divAlg_def) ``` paulson@13183 ` 415` paulson@14288 ` 416` ```lemma mod_pos_pos: "[| 0 < a; 0 \ b |] ==> a mod b = snd (posDivAlg(a,b))" ``` paulson@13183 ` 417` ```by (simp add: mod_def divAlg_def) ``` paulson@13183 ` 418` paulson@13183 ` 419` ```(** a negative, b positive **) ``` paulson@13183 ` 420` paulson@13183 ` 421` ```lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg(a,b))" ``` paulson@13183 ` 422` ```by (simp add: div_def divAlg_def) ``` paulson@13183 ` 423` paulson@13183 ` 424` ```lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg(a,b))" ``` paulson@13183 ` 425` ```by (simp add: mod_def divAlg_def) ``` paulson@13183 ` 426` paulson@13183 ` 427` ```(** a positive, b negative **) ``` paulson@13183 ` 428` paulson@13183 ` 429` ```lemma div_pos_neg: ``` paulson@13183 ` 430` ``` "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd(negDivAlg(-a,-b)))" ``` paulson@13183 ` 431` ```by (simp add: div_def divAlg_def) ``` paulson@13183 ` 432` paulson@13183 ` 433` ```lemma mod_pos_neg: ``` paulson@13183 ` 434` ``` "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd(negDivAlg(-a,-b)))" ``` paulson@13183 ` 435` ```by (simp add: mod_def divAlg_def) ``` paulson@13183 ` 436` paulson@13183 ` 437` ```(** a negative, b negative **) ``` paulson@13183 ` 438` paulson@13183 ` 439` ```lemma div_neg_neg: ``` paulson@14288 ` 440` ``` "[| a < 0; b \ 0 |] ==> a div b = fst (negateSnd(posDivAlg(-a,-b)))" ``` paulson@13183 ` 441` ```by (simp add: div_def divAlg_def) ``` paulson@13183 ` 442` paulson@13183 ` 443` ```lemma mod_neg_neg: ``` paulson@14288 ` 444` ``` "[| a < 0; b \ 0 |] ==> a mod b = snd (negateSnd(posDivAlg(-a,-b)))" ``` paulson@13183 ` 445` ```by (simp add: mod_def divAlg_def) ``` paulson@13183 ` 446` paulson@13183 ` 447` ```text {*Simplify expresions in which div and mod combine numerical constants*} ``` paulson@13183 ` 448` paulson@13183 ` 449` ```declare div_pos_pos [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 450` ```declare div_neg_pos [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 451` ```declare div_pos_neg [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 452` ```declare div_neg_neg [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 453` paulson@13183 ` 454` ```declare mod_pos_pos [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 455` ```declare mod_neg_pos [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 456` ```declare mod_pos_neg [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 457` ```declare mod_neg_neg [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 458` paulson@13183 ` 459` ```declare posDivAlg_eqn [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 460` ```declare negDivAlg_eqn [of "number_of v" "number_of w", standard, simp] ``` paulson@13183 ` 461` paulson@13183 ` 462` paulson@13183 ` 463` ```(** Special-case simplification **) ``` paulson@13183 ` 464` paulson@13183 ` 465` ```lemma zmod_1 [simp]: "a mod (1::int) = 0" ``` paulson@13183 ` 466` ```apply (cut_tac a = a and b = 1 in pos_mod_sign) ``` nipkow@13788 ` 467` ```apply (cut_tac [2] a = a and b = 1 in pos_mod_bound) ``` nipkow@13788 ` 468` ```apply (auto simp del:pos_mod_bound pos_mod_sign) ``` nipkow@13788 ` 469` ```done ``` paulson@13183 ` 470` paulson@13183 ` 471` ```lemma zdiv_1 [simp]: "a div (1::int) = a" ``` paulson@13183 ` 472` ```by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto) ``` paulson@13183 ` 473` paulson@13183 ` 474` ```lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0" ``` paulson@13183 ` 475` ```apply (cut_tac a = a and b = "-1" in neg_mod_sign) ``` nipkow@13788 ` 476` ```apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound) ``` nipkow@13788 ` 477` ```apply (auto simp del: neg_mod_sign neg_mod_bound) ``` paulson@13183 ` 478` ```done ``` paulson@13183 ` 479` paulson@13183 ` 480` ```lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a" ``` paulson@13183 ` 481` ```by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto) ``` paulson@13183 ` 482` paulson@13183 ` 483` ```(** The last remaining special cases for constant arithmetic: ``` paulson@13183 ` 484` ``` 1 div z and 1 mod z **) ``` paulson@13183 ` 485` paulson@13183 ` 486` ```declare div_pos_pos [OF int_0_less_1, of "number_of w", standard, simp] ``` paulson@13183 ` 487` ```declare div_pos_neg [OF int_0_less_1, of "number_of w", standard, simp] ``` paulson@13183 ` 488` ```declare mod_pos_pos [OF int_0_less_1, of "number_of w", standard, simp] ``` paulson@13183 ` 489` ```declare mod_pos_neg [OF int_0_less_1, of "number_of w", standard, simp] ``` paulson@13183 ` 490` paulson@13183 ` 491` ```declare posDivAlg_eqn [of concl: 1 "number_of w", standard, simp] ``` paulson@13183 ` 492` ```declare negDivAlg_eqn [of concl: 1 "number_of w", standard, simp] ``` paulson@13183 ` 493` paulson@13183 ` 494` paulson@14271 ` 495` ```subsection{*Monotonicity in the First Argument (Dividend)*} ``` paulson@13183 ` 496` paulson@14288 ` 497` ```lemma zdiv_mono1: "[| a \ a'; 0 < (b::int) |] ==> a div b \ a' div b" ``` paulson@13183 ` 498` ```apply (cut_tac a = a and b = b in zmod_zdiv_equality) ``` paulson@13183 ` 499` ```apply (cut_tac a = a' and b = b in zmod_zdiv_equality) ``` paulson@13183 ` 500` ```apply (rule unique_quotient_lemma) ``` paulson@13183 ` 501` ```apply (erule subst) ``` paulson@13183 ` 502` ```apply (erule subst) ``` nipkow@13788 ` 503` ```apply (simp_all) ``` paulson@13183 ` 504` ```done ``` paulson@13183 ` 505` paulson@14288 ` 506` ```lemma zdiv_mono1_neg: "[| a \ a'; (b::int) < 0 |] ==> a' div b \ a div b" ``` paulson@13183 ` 507` ```apply (cut_tac a = a and b = b in zmod_zdiv_equality) ``` paulson@13183 ` 508` ```apply (cut_tac a = a' and b = b in zmod_zdiv_equality) ``` paulson@13183 ` 509` ```apply (rule unique_quotient_lemma_neg) ``` paulson@13183 ` 510` ```apply (erule subst) ``` paulson@13183 ` 511` ```apply (erule subst) ``` nipkow@13788 ` 512` ```apply (simp_all) ``` paulson@13183 ` 513` ```done ``` paulson@6917 ` 514` paulson@6917 ` 515` paulson@14271 ` 516` ```subsection{*Monotonicity in the Second Argument (Divisor)*} ``` paulson@13183 ` 517` paulson@13183 ` 518` ```lemma q_pos_lemma: ``` paulson@14288 ` 519` ``` "[| 0 \ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \ (q'::int)" ``` paulson@13183 ` 520` ```apply (subgoal_tac "0 < b'* (q' + 1) ") ``` paulson@14353 ` 521` ``` apply (simp add: zero_less_mult_iff) ``` paulson@14479 ` 522` ```apply (simp add: right_distrib) ``` paulson@13183 ` 523` ```done ``` paulson@13183 ` 524` paulson@13183 ` 525` ```lemma zdiv_mono2_lemma: ``` paulson@14288 ` 526` ``` "[| b*q + r = b'*q' + r'; 0 \ b'*q' + r'; ``` paulson@14288 ` 527` ``` r' < b'; 0 \ r; 0 < b'; b' \ b |] ``` paulson@14288 ` 528` ``` ==> q \ (q'::int)" ``` paulson@13183 ` 529` ```apply (frule q_pos_lemma, assumption+) ``` paulson@13183 ` 530` ```apply (subgoal_tac "b*q < b* (q' + 1) ") ``` paulson@14387 ` 531` ``` apply (simp add: mult_less_cancel_left) ``` paulson@13183 ` 532` ```apply (subgoal_tac "b*q = r' - r + b'*q'") ``` paulson@13183 ` 533` ``` prefer 2 apply simp ``` paulson@14479 ` 534` ```apply (simp (no_asm_simp) add: right_distrib) ``` paulson@13183 ` 535` ```apply (subst zadd_commute, rule zadd_zless_mono, arith) ``` paulson@14378 ` 536` ```apply (rule mult_right_mono, auto) ``` paulson@13183 ` 537` ```done ``` paulson@13183 ` 538` paulson@13183 ` 539` ```lemma zdiv_mono2: ``` paulson@14288 ` 540` ``` "[| (0::int) \ a; 0 < b'; b' \ b |] ==> a div b \ a div b'" ``` paulson@13183 ` 541` ```apply (subgoal_tac "b ~= 0") ``` paulson@13183 ` 542` ``` prefer 2 apply arith ``` paulson@13183 ` 543` ```apply (cut_tac a = a and b = b in zmod_zdiv_equality) ``` paulson@13183 ` 544` ```apply (cut_tac a = a and b = b' in zmod_zdiv_equality) ``` paulson@13183 ` 545` ```apply (rule zdiv_mono2_lemma) ``` paulson@13183 ` 546` ```apply (erule subst) ``` paulson@13183 ` 547` ```apply (erule subst) ``` nipkow@13788 ` 548` ```apply (simp_all) ``` paulson@13183 ` 549` ```done ``` paulson@13183 ` 550` paulson@13183 ` 551` ```lemma q_neg_lemma: ``` paulson@14288 ` 552` ``` "[| b'*q' + r' < 0; 0 \ r'; 0 < b' |] ==> q' \ (0::int)" ``` paulson@13183 ` 553` ```apply (subgoal_tac "b'*q' < 0") ``` paulson@14353 ` 554` ``` apply (simp add: mult_less_0_iff, arith) ``` paulson@13183 ` 555` ```done ``` paulson@13183 ` 556` paulson@13183 ` 557` ```lemma zdiv_mono2_neg_lemma: ``` paulson@13183 ` 558` ``` "[| b*q + r = b'*q' + r'; b'*q' + r' < 0; ``` paulson@14288 ` 559` ``` r < b; 0 \ r'; 0 < b'; b' \ b |] ``` paulson@14288 ` 560` ``` ==> q' \ (q::int)" ``` paulson@13183 ` 561` ```apply (frule q_neg_lemma, assumption+) ``` paulson@13183 ` 562` ```apply (subgoal_tac "b*q' < b* (q + 1) ") ``` paulson@14387 ` 563` ``` apply (simp add: mult_less_cancel_left) ``` paulson@14479 ` 564` ```apply (simp add: right_distrib) ``` paulson@14288 ` 565` ```apply (subgoal_tac "b*q' \ b'*q'") ``` paulson@14378 ` 566` ``` prefer 2 apply (simp add: mult_right_mono_neg) ``` paulson@13183 ` 567` ```apply (subgoal_tac "b'*q' < b + b*q") ``` paulson@13183 ` 568` ``` apply arith ``` paulson@13183 ` 569` ```apply simp ``` paulson@13183 ` 570` ```done ``` paulson@13183 ` 571` paulson@13183 ` 572` ```lemma zdiv_mono2_neg: ``` paulson@14288 ` 573` ``` "[| a < (0::int); 0 < b'; b' \ b |] ==> a div b' \ a div b" ``` paulson@13183 ` 574` ```apply (cut_tac a = a and b = b in zmod_zdiv_equality) ``` paulson@13183 ` 575` ```apply (cut_tac a = a and b = b' in zmod_zdiv_equality) ``` paulson@13183 ` 576` ```apply (rule zdiv_mono2_neg_lemma) ``` paulson@13183 ` 577` ```apply (erule subst) ``` paulson@13183 ` 578` ```apply (erule subst) ``` nipkow@13788 ` 579` ```apply (simp_all) ``` paulson@13183 ` 580` ```done ``` paulson@13183 ` 581` paulson@13183 ` 582` paulson@14271 ` 583` ```subsection{*More Algebraic Laws for div and mod*} ``` paulson@13183 ` 584` paulson@13183 ` 585` ```(** proving (a*b) div c = a * (b div c) + a * (b mod c) **) ``` paulson@13183 ` 586` paulson@13183 ` 587` ```lemma zmult1_lemma: ``` paulson@13183 ` 588` ``` "[| quorem((b,c),(q,r)); c ~= 0 |] ``` paulson@13183 ` 589` ``` ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))" ``` paulson@14479 ` 590` ```by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib) ``` paulson@13183 ` 591` paulson@13183 ` 592` ```lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)" ``` paulson@15013 ` 593` ```apply (case_tac "c = 0", simp) ``` paulson@13183 ` 594` ```apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div]) ``` paulson@13183 ` 595` ```done ``` paulson@13183 ` 596` paulson@13183 ` 597` ```lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)" ``` paulson@15013 ` 598` ```apply (case_tac "c = 0", simp) ``` paulson@13183 ` 599` ```apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod]) ``` paulson@13183 ` 600` ```done ``` paulson@13183 ` 601` paulson@13183 ` 602` ```lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c" ``` paulson@13183 ` 603` ```apply (rule trans) ``` paulson@13183 ` 604` ```apply (rule_tac s = "b*a mod c" in trans) ``` paulson@13183 ` 605` ```apply (rule_tac [2] zmod_zmult1_eq) ``` paulson@13183 ` 606` ```apply (simp_all add: zmult_commute) ``` paulson@13183 ` 607` ```done ``` paulson@13183 ` 608` paulson@13183 ` 609` ```lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c" ``` paulson@13183 ` 610` ```apply (rule zmod_zmult1_eq' [THEN trans]) ``` paulson@13183 ` 611` ```apply (rule zmod_zmult1_eq) ``` paulson@13183 ` 612` ```done ``` paulson@13183 ` 613` paulson@13183 ` 614` ```lemma zdiv_zmult_self1 [simp]: "b ~= (0::int) ==> (a*b) div b = a" ``` paulson@13183 ` 615` ```by (simp add: zdiv_zmult1_eq) ``` paulson@13183 ` 616` paulson@13183 ` 617` ```lemma zdiv_zmult_self2 [simp]: "b ~= (0::int) ==> (b*a) div b = a" ``` paulson@13183 ` 618` ```by (subst zmult_commute, erule zdiv_zmult_self1) ``` paulson@13183 ` 619` paulson@13183 ` 620` ```lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)" ``` paulson@13183 ` 621` ```by (simp add: zmod_zmult1_eq) ``` paulson@13183 ` 622` paulson@13183 ` 623` ```lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)" ``` paulson@13183 ` 624` ```by (simp add: zmult_commute zmod_zmult1_eq) ``` paulson@13183 ` 625` paulson@13183 ` 626` ```lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)" ``` nipkow@13517 ` 627` ```proof ``` nipkow@13517 ` 628` ``` assume "m mod d = 0" ``` paulson@14473 ` 629` ``` with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto ``` nipkow@13517 ` 630` ```next ``` nipkow@13517 ` 631` ``` assume "EX q::int. m = d*q" ``` nipkow@13517 ` 632` ``` thus "m mod d = 0" by auto ``` nipkow@13517 ` 633` ```qed ``` paulson@13183 ` 634` paulson@13183 ` 635` ```declare zmod_eq_0_iff [THEN iffD1, dest!] ``` paulson@13183 ` 636` paulson@13183 ` 637` ```(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **) ``` paulson@13183 ` 638` paulson@13183 ` 639` ```lemma zadd1_lemma: ``` paulson@13183 ` 640` ``` "[| quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); c ~= 0 |] ``` paulson@13183 ` 641` ``` ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))" ``` paulson@14479 ` 642` ```by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib) ``` paulson@13183 ` 643` paulson@13183 ` 644` ```(*NOT suitable for rewriting: the RHS has an instance of the LHS*) ``` paulson@13183 ` 645` ```lemma zdiv_zadd1_eq: ``` paulson@13183 ` 646` ``` "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)" ``` paulson@15013 ` 647` ```apply (case_tac "c = 0", simp) ``` paulson@13183 ` 648` ```apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div) ``` paulson@13183 ` 649` ```done ``` paulson@13183 ` 650` paulson@13183 ` 651` ```lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c" ``` paulson@15013 ` 652` ```apply (case_tac "c = 0", simp) ``` paulson@13183 ` 653` ```apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod) ``` paulson@13183 ` 654` ```done ``` paulson@13183 ` 655` paulson@13183 ` 656` ```lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)" ``` paulson@15013 ` 657` ```apply (case_tac "b = 0", simp) ``` nipkow@13788 ` 658` ```apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial) ``` paulson@13183 ` 659` ```done ``` paulson@13183 ` 660` paulson@13183 ` 661` ```lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)" ``` paulson@15013 ` 662` ```apply (case_tac "b = 0", simp) ``` nipkow@13788 ` 663` ```apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial) ``` paulson@13183 ` 664` ```done ``` paulson@13183 ` 665` paulson@13183 ` 666` ```lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c" ``` paulson@13183 ` 667` ```apply (rule trans [symmetric]) ``` paulson@13183 ` 668` ```apply (rule zmod_zadd1_eq, simp) ``` paulson@13183 ` 669` ```apply (rule zmod_zadd1_eq [symmetric]) ``` paulson@13183 ` 670` ```done ``` paulson@13183 ` 671` paulson@13183 ` 672` ```lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c" ``` paulson@13183 ` 673` ```apply (rule trans [symmetric]) ``` paulson@13183 ` 674` ```apply (rule zmod_zadd1_eq, simp) ``` paulson@13183 ` 675` ```apply (rule zmod_zadd1_eq [symmetric]) ``` paulson@13183 ` 676` ```done ``` paulson@13183 ` 677` paulson@13183 ` 678` ```lemma zdiv_zadd_self1[simp]: "a ~= (0::int) ==> (a+b) div a = b div a + 1" ``` paulson@13183 ` 679` ```by (simp add: zdiv_zadd1_eq) ``` paulson@13183 ` 680` paulson@13183 ` 681` ```lemma zdiv_zadd_self2[simp]: "a ~= (0::int) ==> (b+a) div a = b div a + 1" ``` paulson@13183 ` 682` ```by (simp add: zdiv_zadd1_eq) ``` paulson@13183 ` 683` paulson@13183 ` 684` ```lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)" ``` paulson@15013 ` 685` ```apply (case_tac "a = 0", simp) ``` paulson@13183 ` 686` ```apply (simp add: zmod_zadd1_eq) ``` paulson@13183 ` 687` ```done ``` paulson@13183 ` 688` paulson@13183 ` 689` ```lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)" ``` paulson@15013 ` 690` ```apply (case_tac "a = 0", simp) ``` paulson@13183 ` 691` ```apply (simp add: zmod_zadd1_eq) ``` paulson@13183 ` 692` ```done ``` paulson@13183 ` 693` paulson@13183 ` 694` paulson@14271 ` 695` ```subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *} ``` paulson@13183 ` 696` paulson@13183 ` 697` ```(*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but ``` paulson@13183 ` 698` ``` 7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems ``` paulson@13183 ` 699` ``` to cause particular problems.*) ``` paulson@13183 ` 700` paulson@13183 ` 701` ```(** first, four lemmas to bound the remainder for the cases b<0 and b>0 **) ``` paulson@13183 ` 702` paulson@14288 ` 703` ```lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \ 0 |] ==> b*c < b*(q mod c) + r" ``` paulson@13183 ` 704` ```apply (subgoal_tac "b * (c - q mod c) < r * 1") ``` paulson@14479 ` 705` ```apply (simp add: right_diff_distrib) ``` paulson@13183 ` 706` ```apply (rule order_le_less_trans) ``` paulson@14378 ` 707` ```apply (erule_tac [2] mult_strict_right_mono) ``` paulson@14378 ` 708` ```apply (rule mult_left_mono_neg) ``` paulson@14271 ` 709` ```apply (auto simp add: compare_rls zadd_commute [of 1] ``` paulson@13183 ` 710` ``` add1_zle_eq pos_mod_bound) ``` paulson@13183 ` 711` ```done ``` paulson@13183 ` 712` paulson@14288 ` 713` ```lemma zmult2_lemma_aux2: "[| (0::int) < c; b < r; r \ 0 |] ==> b * (q mod c) + r \ 0" ``` paulson@14288 ` 714` ```apply (subgoal_tac "b * (q mod c) \ 0") ``` paulson@13183 ` 715` ``` apply arith ``` paulson@14353 ` 716` ```apply (simp add: mult_le_0_iff) ``` paulson@13183 ` 717` ```done ``` paulson@13183 ` 718` paulson@14288 ` 719` ```lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \ r; r < b |] ==> 0 \ b * (q mod c) + r" ``` paulson@14288 ` 720` ```apply (subgoal_tac "0 \ b * (q mod c) ") ``` paulson@13183 ` 721` ```apply arith ``` paulson@14353 ` 722` ```apply (simp add: zero_le_mult_iff) ``` paulson@13183 ` 723` ```done ``` paulson@13183 ` 724` paulson@14288 ` 725` ```lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \ r; r < b |] ==> b * (q mod c) + r < b * c" ``` paulson@13183 ` 726` ```apply (subgoal_tac "r * 1 < b * (c - q mod c) ") ``` paulson@14479 ` 727` ```apply (simp add: right_diff_distrib) ``` paulson@13183 ` 728` ```apply (rule order_less_le_trans) ``` paulson@14378 ` 729` ```apply (erule mult_strict_right_mono) ``` paulson@14387 ` 730` ```apply (rule_tac [2] mult_left_mono) ``` paulson@14271 ` 731` ```apply (auto simp add: compare_rls zadd_commute [of 1] ``` paulson@13183 ` 732` ``` add1_zle_eq pos_mod_bound) ``` paulson@13183 ` 733` ```done ``` paulson@13183 ` 734` paulson@13183 ` 735` ```lemma zmult2_lemma: "[| quorem ((a,b), (q,r)); b ~= 0; 0 < c |] ``` paulson@13183 ` 736` ``` ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))" ``` paulson@14271 ` 737` ```by (auto simp add: mult_ac quorem_def linorder_neq_iff ``` paulson@14479 ` 738` ``` zero_less_mult_iff right_distrib [symmetric] ``` wenzelm@13524 ` 739` ``` zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4) ``` paulson@13183 ` 740` paulson@13183 ` 741` ```lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c" ``` paulson@15013 ` 742` ```apply (case_tac "b = 0", simp) ``` paulson@13183 ` 743` ```apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div]) ``` paulson@13183 ` 744` ```done ``` paulson@13183 ` 745` paulson@13183 ` 746` ```lemma zmod_zmult2_eq: ``` paulson@13183 ` 747` ``` "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b" ``` paulson@15013 ` 748` ```apply (case_tac "b = 0", simp) ``` paulson@13183 ` 749` ```apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod]) ``` paulson@13183 ` 750` ```done ``` paulson@13183 ` 751` paulson@13183 ` 752` paulson@14271 ` 753` ```subsection{*Cancellation of Common Factors in div*} ``` paulson@13183 ` 754` wenzelm@13524 ` 755` ```lemma zdiv_zmult_zmult1_aux1: "[| (0::int) < b; c ~= 0 |] ==> (c*a) div (c*b) = a div b" ``` paulson@13183 ` 756` ```by (subst zdiv_zmult2_eq, auto) ``` paulson@13183 ` 757` wenzelm@13524 ` 758` ```lemma zdiv_zmult_zmult1_aux2: "[| b < (0::int); c ~= 0 |] ==> (c*a) div (c*b) = a div b" ``` paulson@13183 ` 759` ```apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ") ``` wenzelm@13524 ` 760` ```apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto) ``` paulson@13183 ` 761` ```done ``` paulson@13183 ` 762` paulson@13183 ` 763` ```lemma zdiv_zmult_zmult1: "c ~= (0::int) ==> (c*a) div (c*b) = a div b" ``` paulson@15013 ` 764` ```apply (case_tac "b = 0", simp) ``` wenzelm@13524 ` 765` ```apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2) ``` paulson@13183 ` 766` ```done ``` paulson@13183 ` 767` paulson@13183 ` 768` ```lemma zdiv_zmult_zmult2: "c ~= (0::int) ==> (a*c) div (b*c) = a div b" ``` paulson@13183 ` 769` ```apply (drule zdiv_zmult_zmult1) ``` paulson@13183 ` 770` ```apply (auto simp add: zmult_commute) ``` paulson@13183 ` 771` ```done ``` paulson@13183 ` 772` paulson@13183 ` 773` paulson@13183 ` 774` paulson@14271 ` 775` ```subsection{*Distribution of Factors over mod*} ``` paulson@13183 ` 776` wenzelm@13524 ` 777` ```lemma zmod_zmult_zmult1_aux1: "[| (0::int) < b; c ~= 0 |] ==> (c*a) mod (c*b) = c * (a mod b)" ``` paulson@13183 ` 778` ```by (subst zmod_zmult2_eq, auto) ``` paulson@13183 ` 779` wenzelm@13524 ` 780` ```lemma zmod_zmult_zmult1_aux2: "[| b < (0::int); c ~= 0 |] ==> (c*a) mod (c*b) = c * (a mod b)" ``` paulson@13183 ` 781` ```apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))") ``` wenzelm@13524 ` 782` ```apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto) ``` paulson@13183 ` 783` ```done ``` paulson@13183 ` 784` paulson@13183 ` 785` ```lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)" ``` paulson@15013 ` 786` ```apply (case_tac "b = 0", simp) ``` paulson@15013 ` 787` ```apply (case_tac "c = 0", simp) ``` wenzelm@13524 ` 788` ```apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2) ``` paulson@13183 ` 789` ```done ``` paulson@13183 ` 790` paulson@13183 ` 791` ```lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)" ``` paulson@13183 ` 792` ```apply (cut_tac c = c in zmod_zmult_zmult1) ``` paulson@13183 ` 793` ```apply (auto simp add: zmult_commute) ``` paulson@13183 ` 794` ```done ``` paulson@13183 ` 795` paulson@13183 ` 796` paulson@14271 ` 797` ```subsection {*Splitting Rules for div and mod*} ``` paulson@13260 ` 798` paulson@13260 ` 799` ```text{*The proofs of the two lemmas below are essentially identical*} ``` paulson@13260 ` 800` paulson@13260 ` 801` ```lemma split_pos_lemma: ``` paulson@13260 ` 802` ``` "0 ``` paulson@14288 ` 803` ``` P(n div k :: int)(n mod k) = (\i j. 0\j & j P i j)" ``` paulson@13260 ` 804` ```apply (rule iffI) ``` paulson@13260 ` 805` ``` apply clarify ``` paulson@13260 ` 806` ``` apply (erule_tac P="P ?x ?y" in rev_mp) ``` paulson@13260 ` 807` ``` apply (subst zmod_zadd1_eq) ``` paulson@13260 ` 808` ``` apply (subst zdiv_zadd1_eq) ``` paulson@13260 ` 809` ``` apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial) ``` paulson@13260 ` 810` ```txt{*converse direction*} ``` paulson@13260 ` 811` ```apply (drule_tac x = "n div k" in spec) ``` paulson@13260 ` 812` ```apply (drule_tac x = "n mod k" in spec) ``` nipkow@13788 ` 813` ```apply (simp) ``` paulson@13260 ` 814` ```done ``` paulson@13260 ` 815` paulson@13260 ` 816` ```lemma split_neg_lemma: ``` paulson@13260 ` 817` ``` "k<0 ==> ``` paulson@14288 ` 818` ``` P(n div k :: int)(n mod k) = (\i j. k0 & n = k*i + j --> P i j)" ``` paulson@13260 ` 819` ```apply (rule iffI) ``` paulson@13260 ` 820` ``` apply clarify ``` paulson@13260 ` 821` ``` apply (erule_tac P="P ?x ?y" in rev_mp) ``` paulson@13260 ` 822` ``` apply (subst zmod_zadd1_eq) ``` paulson@13260 ` 823` ``` apply (subst zdiv_zadd1_eq) ``` paulson@13260 ` 824` ``` apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial) ``` paulson@13260 ` 825` ```txt{*converse direction*} ``` paulson@13260 ` 826` ```apply (drule_tac x = "n div k" in spec) ``` paulson@13260 ` 827` ```apply (drule_tac x = "n mod k" in spec) ``` nipkow@13788 ` 828` ```apply (simp) ``` paulson@13260 ` 829` ```done ``` paulson@13260 ` 830` paulson@13260 ` 831` ```lemma split_zdiv: ``` paulson@13260 ` 832` ``` "P(n div k :: int) = ``` paulson@13260 ` 833` ``` ((k = 0 --> P 0) & ``` paulson@14288 ` 834` ``` (0 (\i j. 0\j & j P i)) & ``` paulson@14288 ` 835` ``` (k<0 --> (\i j. k0 & n = k*i + j --> P i)))" ``` paulson@13260 ` 836` ```apply (case_tac "k=0") ``` paulson@15013 ` 837` ``` apply (simp) ``` paulson@13260 ` 838` ```apply (simp only: linorder_neq_iff) ``` paulson@13260 ` 839` ```apply (erule disjE) ``` paulson@13260 ` 840` ``` apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] ``` paulson@13260 ` 841` ``` split_neg_lemma [of concl: "%x y. P x"]) ``` paulson@13260 ` 842` ```done ``` paulson@13260 ` 843` paulson@13260 ` 844` ```lemma split_zmod: ``` paulson@13260 ` 845` ``` "P(n mod k :: int) = ``` paulson@13260 ` 846` ``` ((k = 0 --> P n) & ``` paulson@14288 ` 847` ``` (0 (\i j. 0\j & j P j)) & ``` paulson@14288 ` 848` ``` (k<0 --> (\i j. k0 & n = k*i + j --> P j)))" ``` paulson@13260 ` 849` ```apply (case_tac "k=0") ``` paulson@15013 ` 850` ``` apply (simp) ``` paulson@13260 ` 851` ```apply (simp only: linorder_neq_iff) ``` paulson@13260 ` 852` ```apply (erule disjE) ``` paulson@13260 ` 853` ``` apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] ``` paulson@13260 ` 854` ``` split_neg_lemma [of concl: "%x y. P y"]) ``` paulson@13260 ` 855` ```done ``` paulson@13260 ` 856` paulson@13260 ` 857` ```(* Enable arith to deal with div 2 and mod 2: *) ``` nipkow@13266 ` 858` ```declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split] ``` nipkow@13266 ` 859` ```declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split] ``` paulson@13260 ` 860` paulson@13260 ` 861` paulson@14271 ` 862` ```subsection{*Speeding up the Division Algorithm with Shifting*} ``` paulson@13183 ` 863` paulson@13183 ` 864` ```(** computing div by shifting **) ``` paulson@13183 ` 865` paulson@14288 ` 866` ```lemma pos_zdiv_mult_2: "(0::int) \ a ==> (1 + 2*b) div (2*a) = b div a" ``` paulson@14288 ` 867` ```proof cases ``` paulson@14288 ` 868` ``` assume "a=0" ``` paulson@14288 ` 869` ``` thus ?thesis by simp ``` paulson@14288 ` 870` ```next ``` paulson@14288 ` 871` ``` assume "a\0" and le_a: "0\a" ``` paulson@14288 ` 872` ``` hence a_pos: "1 \ a" by arith ``` paulson@14288 ` 873` ``` hence one_less_a2: "1 < 2*a" by arith ``` paulson@14288 ` 874` ``` hence le_2a: "2 * (1 + b mod a) \ 2 * a" ``` paulson@14288 ` 875` ``` by (simp add: mult_le_cancel_left zadd_commute [of 1] add1_zle_eq) ``` paulson@14288 ` 876` ``` with a_pos have "0 \ b mod a" by simp ``` paulson@14288 ` 877` ``` hence le_addm: "0 \ 1 mod (2*a) + 2*(b mod a)" ``` paulson@14288 ` 878` ``` by (simp add: mod_pos_pos_trivial one_less_a2) ``` paulson@14288 ` 879` ``` with le_2a ``` paulson@14288 ` 880` ``` have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0" ``` paulson@14288 ` 881` ``` by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2 ``` paulson@14288 ` 882` ``` right_distrib) ``` paulson@14288 ` 883` ``` thus ?thesis ``` paulson@14288 ` 884` ``` by (subst zdiv_zadd1_eq, ``` paulson@14288 ` 885` ``` simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2 ``` paulson@14288 ` 886` ``` div_pos_pos_trivial) ``` paulson@14288 ` 887` ```qed ``` paulson@13183 ` 888` paulson@14288 ` 889` ```lemma neg_zdiv_mult_2: "a \ (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a" ``` paulson@13183 ` 890` ```apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ") ``` paulson@13183 ` 891` ```apply (rule_tac [2] pos_zdiv_mult_2) ``` paulson@14479 ` 892` ```apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib) ``` paulson@13183 ` 893` ```apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))") ``` paulson@14479 ` 894` ```apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric], ``` paulson@13183 ` 895` ``` simp) ``` paulson@13183 ` 896` ```done ``` paulson@13183 ` 897` paulson@13183 ` 898` paulson@13183 ` 899` ```(*Not clear why this must be proved separately; probably number_of causes ``` paulson@13183 ` 900` ``` simplification problems*) ``` paulson@14288 ` 901` ```lemma not_0_le_lemma: "~ 0 \ x ==> x \ (0::int)" ``` paulson@13183 ` 902` ```by auto ``` paulson@13183 ` 903` paulson@13183 ` 904` ```lemma zdiv_number_of_BIT[simp]: ``` paulson@13183 ` 905` ``` "number_of (v BIT b) div number_of (w BIT False) = ``` paulson@14288 ` 906` ``` (if ~b | (0::int) \ number_of w ``` paulson@13183 ` 907` ``` then number_of v div (number_of w) ``` paulson@13183 ` 908` ``` else (number_of v + (1::int)) div (number_of w))" ``` paulson@15013 ` 909` ```apply (simp only: number_of_eq Bin_simps UNIV_I split: split_if) ``` paulson@15013 ` 910` ```apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac) ``` paulson@13183 ` 911` ```done ``` paulson@13183 ` 912` paulson@13183 ` 913` paulson@15013 ` 914` ```subsection{*Computing mod by Shifting (proofs resemble those for div)*} ``` paulson@13183 ` 915` paulson@13183 ` 916` ```lemma pos_zmod_mult_2: ``` paulson@14288 ` 917` ``` "(0::int) \ a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)" ``` paulson@15013 ` 918` ```apply (case_tac "a = 0", simp) ``` paulson@14288 ` 919` ```apply (subgoal_tac "1 \ a") ``` paulson@13183 ` 920` ``` prefer 2 apply arith ``` paulson@13183 ` 921` ```apply (subgoal_tac "1 < a * 2") ``` paulson@13183 ` 922` ``` prefer 2 apply arith ``` paulson@14288 ` 923` ```apply (subgoal_tac "2* (1 + b mod a) \ 2*a") ``` paulson@14387 ` 924` ``` apply (rule_tac [2] mult_left_mono) ``` paulson@13183 ` 925` ```apply (auto simp add: zadd_commute [of 1] zmult_commute add1_zle_eq ``` paulson@13183 ` 926` ``` pos_mod_bound) ``` paulson@13183 ` 927` ```apply (subst zmod_zadd1_eq) ``` paulson@13183 ` 928` ```apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial) ``` paulson@13183 ` 929` ```apply (rule mod_pos_pos_trivial) ``` paulson@14288 ` 930` ```apply (auto simp add: mod_pos_pos_trivial left_distrib) ``` paulson@14288 ` 931` ```apply (subgoal_tac "0 \ b mod a", arith) ``` nipkow@13788 ` 932` ```apply (simp) ``` paulson@13183 ` 933` ```done ``` paulson@13183 ` 934` paulson@13183 ` 935` ```lemma neg_zmod_mult_2: ``` paulson@14288 ` 936` ``` "a \ (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1" ``` paulson@13183 ` 937` ```apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = ``` paulson@13183 ` 938` ``` 1 + 2* ((-b - 1) mod (-a))") ``` paulson@13183 ` 939` ```apply (rule_tac [2] pos_zmod_mult_2) ``` paulson@14479 ` 940` ```apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib) ``` paulson@13183 ` 941` ```apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))") ``` paulson@13183 ` 942` ``` prefer 2 apply simp ``` paulson@14479 ` 943` ```apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric]) ``` paulson@13183 ` 944` ```done ``` paulson@13183 ` 945` paulson@13183 ` 946` ```lemma zmod_number_of_BIT [simp]: ``` paulson@13183 ` 947` ``` "number_of (v BIT b) mod number_of (w BIT False) = ``` paulson@13183 ` 948` ``` (if b then ``` paulson@14288 ` 949` ``` if (0::int) \ number_of w ``` paulson@13183 ` 950` ``` then 2 * (number_of v mod number_of w) + 1 ``` paulson@13183 ` 951` ``` else 2 * ((number_of v + (1::int)) mod number_of w) - 1 ``` paulson@13183 ` 952` ``` else 2 * (number_of v mod number_of w))" ``` paulson@15013 ` 953` ```apply (simp only: number_of_eq Bin_simps UNIV_I split: split_if) ``` paulson@15013 ` 954` ```apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 ``` paulson@15013 ` 955` ``` not_0_le_lemma neg_zmod_mult_2 add_ac) ``` paulson@13183 ` 956` ```done ``` paulson@13183 ` 957` paulson@13183 ` 958` paulson@13183 ` 959` paulson@15013 ` 960` ```subsection{*Quotients of Signs*} ``` paulson@13183 ` 961` paulson@13183 ` 962` ```lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0" ``` paulson@14288 ` 963` ```apply (subgoal_tac "a div b \ -1", force) ``` paulson@13183 ` 964` ```apply (rule order_trans) ``` paulson@13183 ` 965` ```apply (rule_tac a' = "-1" in zdiv_mono1) ``` paulson@13183 ` 966` ```apply (auto simp add: zdiv_minus1) ``` paulson@13183 ` 967` ```done ``` paulson@13183 ` 968` paulson@14288 ` 969` ```lemma div_nonneg_neg_le0: "[| (0::int) \ a; b < 0 |] ==> a div b \ 0" ``` paulson@13183 ` 970` ```by (drule zdiv_mono1_neg, auto) ``` paulson@13183 ` 971` paulson@14288 ` 972` ```lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \ a div b) = (0 \ a)" ``` paulson@13183 ` 973` ```apply auto ``` paulson@13183 ` 974` ```apply (drule_tac [2] zdiv_mono1) ``` paulson@13183 ` 975` ```apply (auto simp add: linorder_neq_iff) ``` paulson@13183 ` 976` ```apply (simp (no_asm_use) add: linorder_not_less [symmetric]) ``` paulson@13183 ` 977` ```apply (blast intro: div_neg_pos_less0) ``` paulson@13183 ` 978` ```done ``` paulson@13183 ` 979` paulson@13183 ` 980` ```lemma neg_imp_zdiv_nonneg_iff: ``` paulson@14288 ` 981` ``` "b < (0::int) ==> (0 \ a div b) = (a \ (0::int))" ``` paulson@13183 ` 982` ```apply (subst zdiv_zminus_zminus [symmetric]) ``` paulson@13183 ` 983` ```apply (subst pos_imp_zdiv_nonneg_iff, auto) ``` paulson@13183 ` 984` ```done ``` paulson@13183 ` 985` paulson@14288 ` 986` ```(*But not (a div b \ 0 iff a\0); consider a=1, b=2 when a div b = 0.*) ``` paulson@13183 ` 987` ```lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)" ``` paulson@13183 ` 988` ```by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff) ``` paulson@13183 ` 989` paulson@14288 ` 990` ```(*Again the law fails for \: consider a = -1, b = -2 when a div b = 0*) ``` paulson@13183 ` 991` ```lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)" ``` paulson@13183 ` 992` ```by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff) ``` paulson@13183 ` 993` paulson@13837 ` 994` paulson@14271 ` 995` ```subsection {* The Divides Relation *} ``` paulson@13837 ` 996` paulson@13837 ` 997` ```lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))" ``` paulson@13837 ` 998` ```by(simp add:dvd_def zmod_eq_0_iff) ``` paulson@13837 ` 999` paulson@13837 ` 1000` ```lemma zdvd_0_right [iff]: "(m::int) dvd 0" ``` paulson@13837 ` 1001` ``` apply (unfold dvd_def) ``` paulson@14479 ` 1002` ``` apply (blast intro: mult_zero_right [symmetric]) ``` paulson@13837 ` 1003` ``` done ``` paulson@13837 ` 1004` paulson@13837 ` 1005` ```lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)" ``` paulson@13837 ` 1006` ``` by (unfold dvd_def, auto) ``` paulson@13837 ` 1007` paulson@13837 ` 1008` ```lemma zdvd_1_left [iff]: "1 dvd (m::int)" ``` paulson@13837 ` 1009` ``` by (unfold dvd_def, simp) ``` paulson@13837 ` 1010` paulson@13837 ` 1011` ```lemma zdvd_refl [simp]: "m dvd (m::int)" ``` paulson@13837 ` 1012` ``` apply (unfold dvd_def) ``` paulson@13837 ` 1013` ``` apply (blast intro: zmult_1_right [symmetric]) ``` paulson@13837 ` 1014` ``` done ``` paulson@13837 ` 1015` paulson@13837 ` 1016` ```lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)" ``` paulson@13837 ` 1017` ``` apply (unfold dvd_def) ``` paulson@13837 ` 1018` ``` apply (blast intro: zmult_assoc) ``` paulson@13837 ` 1019` ``` done ``` paulson@13837 ` 1020` paulson@13837 ` 1021` ```lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))" ``` paulson@13837 ` 1022` ``` apply (unfold dvd_def, auto) ``` paulson@13837 ` 1023` ``` apply (rule_tac [!] x = "-k" in exI, auto) ``` paulson@13837 ` 1024` ``` done ``` paulson@13837 ` 1025` paulson@13837 ` 1026` ```lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))" ``` paulson@13837 ` 1027` ``` apply (unfold dvd_def, auto) ``` paulson@13837 ` 1028` ``` apply (rule_tac [!] x = "-k" in exI, auto) ``` paulson@13837 ` 1029` ``` done ``` paulson@13837 ` 1030` paulson@13837 ` 1031` ```lemma zdvd_anti_sym: ``` paulson@13837 ` 1032` ``` "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)" ``` paulson@13837 ` 1033` ``` apply (unfold dvd_def, auto) ``` paulson@14353 ` 1034` ``` apply (simp add: zmult_assoc zmult_eq_self_iff zero_less_mult_iff zmult_eq_1_iff) ``` paulson@13837 ` 1035` ``` done ``` paulson@13837 ` 1036` paulson@13837 ` 1037` ```lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)" ``` paulson@13837 ` 1038` ``` apply (unfold dvd_def) ``` paulson@14479 ` 1039` ``` apply (blast intro: right_distrib [symmetric]) ``` paulson@13837 ` 1040` ``` done ``` paulson@13837 ` 1041` paulson@13837 ` 1042` ```lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)" ``` paulson@13837 ` 1043` ``` apply (unfold dvd_def) ``` paulson@14479 ` 1044` ``` apply (blast intro: right_diff_distrib [symmetric]) ``` paulson@13837 ` 1045` ``` done ``` paulson@13837 ` 1046` paulson@13837 ` 1047` ```lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)" ``` paulson@13837 ` 1048` ``` apply (subgoal_tac "m = n + (m - n)") ``` paulson@13837 ` 1049` ``` apply (erule ssubst) ``` paulson@13837 ` 1050` ``` apply (blast intro: zdvd_zadd, simp) ``` paulson@13837 ` 1051` ``` done ``` paulson@13837 ` 1052` paulson@13837 ` 1053` ```lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n" ``` paulson@13837 ` 1054` ``` apply (unfold dvd_def) ``` paulson@14271 ` 1055` ``` apply (blast intro: mult_left_commute) ``` paulson@13837 ` 1056` ``` done ``` paulson@13837 ` 1057` paulson@13837 ` 1058` ```lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n" ``` paulson@13837 ` 1059` ``` apply (subst zmult_commute) ``` paulson@13837 ` 1060` ``` apply (erule zdvd_zmult) ``` paulson@13837 ` 1061` ``` done ``` paulson@13837 ` 1062` paulson@13837 ` 1063` ```lemma [iff]: "(k::int) dvd m * k" ``` paulson@13837 ` 1064` ``` apply (rule zdvd_zmult) ``` paulson@13837 ` 1065` ``` apply (rule zdvd_refl) ``` paulson@13837 ` 1066` ``` done ``` paulson@13837 ` 1067` paulson@13837 ` 1068` ```lemma [iff]: "(k::int) dvd k * m" ``` paulson@13837 ` 1069` ``` apply (rule zdvd_zmult2) ``` paulson@13837 ` 1070` ``` apply (rule zdvd_refl) ``` paulson@13837 ` 1071` ``` done ``` paulson@13837 ` 1072` paulson@13837 ` 1073` ```lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)" ``` paulson@13837 ` 1074` ``` apply (unfold dvd_def) ``` paulson@13837 ` 1075` ``` apply (simp add: zmult_assoc, blast) ``` paulson@13837 ` 1076` ``` done ``` paulson@13837 ` 1077` paulson@13837 ` 1078` ```lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)" ``` paulson@13837 ` 1079` ``` apply (rule zdvd_zmultD2) ``` paulson@13837 ` 1080` ``` apply (subst zmult_commute, assumption) ``` paulson@13837 ` 1081` ``` done ``` paulson@13837 ` 1082` paulson@13837 ` 1083` ```lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n" ``` paulson@13837 ` 1084` ``` apply (unfold dvd_def, clarify) ``` paulson@13837 ` 1085` ``` apply (rule_tac x = "k * ka" in exI) ``` paulson@14271 ` 1086` ``` apply (simp add: mult_ac) ``` paulson@13837 ` 1087` ``` done ``` paulson@13837 ` 1088` paulson@13837 ` 1089` ```lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))" ``` paulson@13837 ` 1090` ``` apply (rule iffI) ``` paulson@13837 ` 1091` ``` apply (erule_tac [2] zdvd_zadd) ``` paulson@13837 ` 1092` ``` apply (subgoal_tac "n = (n + k * m) - k * m") ``` paulson@13837 ` 1093` ``` apply (erule ssubst) ``` paulson@13837 ` 1094` ``` apply (erule zdvd_zdiff, simp_all) ``` paulson@13837 ` 1095` ``` done ``` paulson@13837 ` 1096` paulson@13837 ` 1097` ```lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n" ``` paulson@13837 ` 1098` ``` apply (unfold dvd_def) ``` paulson@13837 ` 1099` ``` apply (auto simp add: zmod_zmult_zmult1) ``` paulson@13837 ` 1100` ``` done ``` paulson@13837 ` 1101` paulson@13837 ` 1102` ```lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)" ``` paulson@13837 ` 1103` ``` apply (subgoal_tac "k dvd n * (m div n) + m mod n") ``` paulson@13837 ` 1104` ``` apply (simp add: zmod_zdiv_equality [symmetric]) ``` paulson@13837 ` 1105` ``` apply (simp only: zdvd_zadd zdvd_zmult2) ``` paulson@13837 ` 1106` ``` done ``` paulson@13837 ` 1107` paulson@13837 ` 1108` ```lemma zdvd_not_zless: "0 < m ==> m < n ==> \ n dvd (m::int)" ``` paulson@13837 ` 1109` ``` apply (unfold dvd_def, auto) ``` paulson@13837 ` 1110` ``` apply (subgoal_tac "0 < n") ``` paulson@13837 ` 1111` ``` prefer 2 ``` paulson@14378 ` 1112` ``` apply (blast intro: order_less_trans) ``` paulson@14353 ` 1113` ``` apply (simp add: zero_less_mult_iff) ``` paulson@13837 ` 1114` ``` apply (subgoal_tac "n * k < n * 1") ``` paulson@14387 ` 1115` ``` apply (drule mult_less_cancel_left [THEN iffD1], auto) ``` paulson@13837 ` 1116` ``` done ``` paulson@13837 ` 1117` paulson@13837 ` 1118` ```lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))" ``` paulson@13837 ` 1119` ``` apply (auto simp add: dvd_def nat_abs_mult_distrib) ``` paulson@14353 ` 1120` ``` apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm) ``` paulson@14353 ` 1121` ``` apply (rule_tac x = "-(int k)" in exI) ``` paulson@13837 ` 1122` ``` apply (auto simp add: zmult_int [symmetric]) ``` paulson@13837 ` 1123` ``` done ``` paulson@13837 ` 1124` paulson@13837 ` 1125` ```lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)" ``` paulson@15003 ` 1126` ``` apply (auto simp add: dvd_def abs_if zmult_int [symmetric]) ``` paulson@13837 ` 1127` ``` apply (rule_tac [3] x = "nat k" in exI) ``` paulson@13837 ` 1128` ``` apply (rule_tac [2] x = "-(int k)" in exI) ``` paulson@13837 ` 1129` ``` apply (rule_tac x = "nat (-k)" in exI) ``` paulson@13837 ` 1130` ``` apply (cut_tac [3] k = m in int_less_0_conv) ``` paulson@13837 ` 1131` ``` apply (cut_tac k = m in int_less_0_conv) ``` paulson@14353 ` 1132` ``` apply (auto simp add: zero_le_mult_iff mult_less_0_iff ``` paulson@13837 ` 1133` ``` nat_mult_distrib [symmetric] nat_eq_iff2) ``` paulson@13837 ` 1134` ``` done ``` paulson@13837 ` 1135` paulson@13837 ` 1136` ```lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \ z then (z dvd int m) else m = 0)" ``` paulson@13837 ` 1137` ``` apply (auto simp add: dvd_def zmult_int [symmetric]) ``` paulson@13837 ` 1138` ``` apply (rule_tac x = "nat k" in exI) ``` paulson@13837 ` 1139` ``` apply (cut_tac k = m in int_less_0_conv) ``` paulson@14353 ` 1140` ``` apply (auto simp add: zero_le_mult_iff mult_less_0_iff ``` paulson@13837 ` 1141` ``` nat_mult_distrib [symmetric] nat_eq_iff2) ``` paulson@13837 ` 1142` ``` done ``` paulson@13837 ` 1143` paulson@13837 ` 1144` ```lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))" ``` paulson@13837 ` 1145` ``` apply (auto simp add: dvd_def) ``` paulson@13837 ` 1146` ``` apply (rule_tac [!] x = "-k" in exI, auto) ``` paulson@13837 ` 1147` ``` done ``` paulson@13837 ` 1148` paulson@13837 ` 1149` ```lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))" ``` paulson@13837 ` 1150` ``` apply (auto simp add: dvd_def) ``` paulson@14378 ` 1151` ``` apply (drule minus_equation_iff [THEN iffD1]) ``` paulson@13837 ` 1152` ``` apply (rule_tac [!] x = "-k" in exI, auto) ``` paulson@13837 ` 1153` ``` done ``` paulson@13837 ` 1154` paulson@14288 ` 1155` ```lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \ (n::int)" ``` paulson@13837 ` 1156` ``` apply (rule_tac z=n in int_cases) ``` paulson@13837 ` 1157` ``` apply (auto simp add: dvd_int_iff) ``` paulson@13837 ` 1158` ``` apply (rule_tac z=z in int_cases) ``` paulson@13837 ` 1159` ``` apply (auto simp add: dvd_imp_le) ``` paulson@13837 ` 1160` ``` done ``` paulson@13837 ` 1161` paulson@13837 ` 1162` paulson@14353 ` 1163` ```subsection{*Integer Powers*} ``` paulson@14353 ` 1164` paulson@14353 ` 1165` ```instance int :: power .. ``` paulson@14353 ` 1166` paulson@14353 ` 1167` ```primrec ``` paulson@14353 ` 1168` ``` "p ^ 0 = 1" ``` paulson@14353 ` 1169` ``` "p ^ (Suc n) = (p::int) * (p ^ n)" ``` paulson@14353 ` 1170` paulson@14353 ` 1171` paulson@15003 ` 1172` ```instance int :: recpower ``` paulson@14353 ` 1173` ```proof ``` paulson@14353 ` 1174` ``` fix z :: int ``` paulson@14353 ` 1175` ``` fix n :: nat ``` paulson@14353 ` 1176` ``` show "z^0 = 1" by simp ``` paulson@14353 ` 1177` ``` show "z^(Suc n) = z * (z^n)" by simp ``` paulson@14353 ` 1178` ```qed ``` paulson@14353 ` 1179` paulson@14353 ` 1180` paulson@14353 ` 1181` ```lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m" ``` paulson@14353 ` 1182` ```apply (induct_tac "y", auto) ``` paulson@14353 ` 1183` ```apply (rule zmod_zmult1_eq [THEN trans]) ``` paulson@14353 ` 1184` ```apply (simp (no_asm_simp)) ``` paulson@14353 ` 1185` ```apply (rule zmod_zmult_distrib [symmetric]) ``` paulson@14353 ` 1186` ```done ``` paulson@14353 ` 1187` paulson@14353 ` 1188` ```lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)" ``` paulson@14353 ` 1189` ``` by (rule Power.power_add) ``` paulson@14353 ` 1190` paulson@14353 ` 1191` ```lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)" ``` paulson@14353 ` 1192` ``` by (rule Power.power_mult [symmetric]) ``` paulson@14353 ` 1193` paulson@14353 ` 1194` ```lemma zero_less_zpower_abs_iff [simp]: ``` paulson@14353 ` 1195` ``` "(0 < (abs x)^n) = (x \ (0::int) | n=0)" ``` paulson@14353 ` 1196` ```apply (induct_tac "n") ``` paulson@14353 ` 1197` ```apply (auto simp add: zero_less_mult_iff) ``` paulson@14353 ` 1198` ```done ``` paulson@14353 ` 1199` paulson@14353 ` 1200` ```lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n" ``` paulson@14353 ` 1201` ```apply (induct_tac "n") ``` paulson@14353 ` 1202` ```apply (auto simp add: zero_le_mult_iff) ``` paulson@14353 ` 1203` ```done ``` paulson@14353 ` 1204` obua@15101 ` 1205` ```lemma zdiv_int: "int (a div b) = (int a) div (int b)" ``` obua@15101 ` 1206` ```apply (subst split_div, auto) ``` obua@15101 ` 1207` ```apply (subst split_zdiv, auto) ``` obua@15101 ` 1208` ```apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient) ``` obua@15101 ` 1209` ```apply (auto simp add: IntDiv.quorem_def int_eq_of_nat) ``` obua@15101 ` 1210` ```done ``` obua@15101 ` 1211` obua@15101 ` 1212` ```lemma zmod_int: "int (a mod b) = (int a) mod (int b)" ``` obua@15101 ` 1213` ```apply (subst split_mod, auto) ``` obua@15101 ` 1214` ```apply (subst split_zmod, auto) ``` obua@15101 ` 1215` ```apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder) ``` obua@15101 ` 1216` ```apply (auto simp add: IntDiv.quorem_def int_eq_of_nat) ``` obua@15101 ` 1217` ```done ``` paulson@14353 ` 1218` paulson@13183 ` 1219` ```ML ``` paulson@13183 ` 1220` ```{* ``` paulson@13183 ` 1221` ```val quorem_def = thm "quorem_def"; ``` paulson@13183 ` 1222` paulson@13183 ` 1223` ```val unique_quotient = thm "unique_quotient"; ``` paulson@13183 ` 1224` ```val unique_remainder = thm "unique_remainder"; ``` paulson@13183 ` 1225` ```val adjust_eq = thm "adjust_eq"; ``` paulson@13183 ` 1226` ```val posDivAlg_eqn = thm "posDivAlg_eqn"; ``` paulson@13183 ` 1227` ```val posDivAlg_correct = thm "posDivAlg_correct"; ``` paulson@13183 ` 1228` ```val negDivAlg_eqn = thm "negDivAlg_eqn"; ``` paulson@13183 ` 1229` ```val negDivAlg_correct = thm "negDivAlg_correct"; ``` paulson@13183 ` 1230` ```val quorem_0 = thm "quorem_0"; ``` paulson@13183 ` 1231` ```val posDivAlg_0 = thm "posDivAlg_0"; ``` paulson@13183 ` 1232` ```val negDivAlg_minus1 = thm "negDivAlg_minus1"; ``` paulson@13183 ` 1233` ```val negateSnd_eq = thm "negateSnd_eq"; ``` paulson@13183 ` 1234` ```val quorem_neg = thm "quorem_neg"; ``` paulson@13183 ` 1235` ```val divAlg_correct = thm "divAlg_correct"; ``` paulson@13183 ` 1236` ```val DIVISION_BY_ZERO = thm "DIVISION_BY_ZERO"; ``` paulson@13183 ` 1237` ```val zmod_zdiv_equality = thm "zmod_zdiv_equality"; ``` paulson@13183 ` 1238` ```val pos_mod_conj = thm "pos_mod_conj"; ``` paulson@13183 ` 1239` ```val pos_mod_sign = thm "pos_mod_sign"; ``` paulson@13183 ` 1240` ```val neg_mod_conj = thm "neg_mod_conj"; ``` paulson@13183 ` 1241` ```val neg_mod_sign = thm "neg_mod_sign"; ``` paulson@13183 ` 1242` ```val quorem_div_mod = thm "quorem_div_mod"; ``` paulson@13183 ` 1243` ```val quorem_div = thm "quorem_div"; ``` paulson@13183 ` 1244` ```val quorem_mod = thm "quorem_mod"; ``` paulson@13183 ` 1245` ```val div_pos_pos_trivial = thm "div_pos_pos_trivial"; ``` paulson@13183 ` 1246` ```val div_neg_neg_trivial = thm "div_neg_neg_trivial"; ``` paulson@13183 ` 1247` ```val div_pos_neg_trivial = thm "div_pos_neg_trivial"; ``` paulson@13183 ` 1248` ```val mod_pos_pos_trivial = thm "mod_pos_pos_trivial"; ``` paulson@13183 ` 1249` ```val mod_neg_neg_trivial = thm "mod_neg_neg_trivial"; ``` paulson@13183 ` 1250` ```val mod_pos_neg_trivial = thm "mod_pos_neg_trivial"; ``` paulson@13183 ` 1251` ```val zdiv_zminus_zminus = thm "zdiv_zminus_zminus"; ``` paulson@13183 ` 1252` ```val zmod_zminus_zminus = thm "zmod_zminus_zminus"; ``` paulson@13183 ` 1253` ```val zdiv_zminus1_eq_if = thm "zdiv_zminus1_eq_if"; ``` paulson@13183 ` 1254` ```val zmod_zminus1_eq_if = thm "zmod_zminus1_eq_if"; ``` paulson@13183 ` 1255` ```val zdiv_zminus2 = thm "zdiv_zminus2"; ``` paulson@13183 ` 1256` ```val zmod_zminus2 = thm "zmod_zminus2"; ``` paulson@13183 ` 1257` ```val zdiv_zminus2_eq_if = thm "zdiv_zminus2_eq_if"; ``` paulson@13183 ` 1258` ```val zmod_zminus2_eq_if = thm "zmod_zminus2_eq_if"; ``` paulson@13183 ` 1259` ```val self_quotient = thm "self_quotient"; ``` paulson@13183 ` 1260` ```val self_remainder = thm "self_remainder"; ``` paulson@13183 ` 1261` ```val zdiv_self = thm "zdiv_self"; ``` paulson@13183 ` 1262` ```val zmod_self = thm "zmod_self"; ``` paulson@13183 ` 1263` ```val zdiv_zero = thm "zdiv_zero"; ``` paulson@13183 ` 1264` ```val div_eq_minus1 = thm "div_eq_minus1"; ``` paulson@13183 ` 1265` ```val zmod_zero = thm "zmod_zero"; ``` paulson@13183 ` 1266` ```val zdiv_minus1 = thm "zdiv_minus1"; ``` paulson@13183 ` 1267` ```val zmod_minus1 = thm "zmod_minus1"; ``` paulson@13183 ` 1268` ```val div_pos_pos = thm "div_pos_pos"; ``` paulson@13183 ` 1269` ```val mod_pos_pos = thm "mod_pos_pos"; ``` paulson@13183 ` 1270` ```val div_neg_pos = thm "div_neg_pos"; ``` paulson@13183 ` 1271` ```val mod_neg_pos = thm "mod_neg_pos"; ``` paulson@13183 ` 1272` ```val div_pos_neg = thm "div_pos_neg"; ``` paulson@13183 ` 1273` ```val mod_pos_neg = thm "mod_pos_neg"; ``` paulson@13183 ` 1274` ```val div_neg_neg = thm "div_neg_neg"; ``` paulson@13183 ` 1275` ```val mod_neg_neg = thm "mod_neg_neg"; ``` paulson@13183 ` 1276` ```val zmod_1 = thm "zmod_1"; ``` paulson@13183 ` 1277` ```val zdiv_1 = thm "zdiv_1"; ``` paulson@13183 ` 1278` ```val zmod_minus1_right = thm "zmod_minus1_right"; ``` paulson@13183 ` 1279` ```val zdiv_minus1_right = thm "zdiv_minus1_right"; ``` paulson@13183 ` 1280` ```val zdiv_mono1 = thm "zdiv_mono1"; ``` paulson@13183 ` 1281` ```val zdiv_mono1_neg = thm "zdiv_mono1_neg"; ``` paulson@13183 ` 1282` ```val zdiv_mono2 = thm "zdiv_mono2"; ``` paulson@13183 ` 1283` ```val zdiv_mono2_neg = thm "zdiv_mono2_neg"; ``` paulson@13183 ` 1284` ```val zdiv_zmult1_eq = thm "zdiv_zmult1_eq"; ``` paulson@13183 ` 1285` ```val zmod_zmult1_eq = thm "zmod_zmult1_eq"; ``` paulson@13183 ` 1286` ```val zmod_zmult1_eq' = thm "zmod_zmult1_eq'"; ``` paulson@13183 ` 1287` ```val zmod_zmult_distrib = thm "zmod_zmult_distrib"; ``` paulson@13183 ` 1288` ```val zdiv_zmult_self1 = thm "zdiv_zmult_self1"; ``` paulson@13183 ` 1289` ```val zdiv_zmult_self2 = thm "zdiv_zmult_self2"; ``` paulson@13183 ` 1290` ```val zmod_zmult_self1 = thm "zmod_zmult_self1"; ``` paulson@13183 ` 1291` ```val zmod_zmult_self2 = thm "zmod_zmult_self2"; ``` paulson@13183 ` 1292` ```val zmod_eq_0_iff = thm "zmod_eq_0_iff"; ``` paulson@13183 ` 1293` ```val zdiv_zadd1_eq = thm "zdiv_zadd1_eq"; ``` paulson@13183 ` 1294` ```val zmod_zadd1_eq = thm "zmod_zadd1_eq"; ``` paulson@13183 ` 1295` ```val mod_div_trivial = thm "mod_div_trivial"; ``` paulson@13183 ` 1296` ```val mod_mod_trivial = thm "mod_mod_trivial"; ``` paulson@13183 ` 1297` ```val zmod_zadd_left_eq = thm "zmod_zadd_left_eq"; ``` paulson@13183 ` 1298` ```val zmod_zadd_right_eq = thm "zmod_zadd_right_eq"; ``` paulson@13183 ` 1299` ```val zdiv_zadd_self1 = thm "zdiv_zadd_self1"; ``` paulson@13183 ` 1300` ```val zdiv_zadd_self2 = thm "zdiv_zadd_self2"; ``` paulson@13183 ` 1301` ```val zmod_zadd_self1 = thm "zmod_zadd_self1"; ``` paulson@13183 ` 1302` ```val zmod_zadd_self2 = thm "zmod_zadd_self2"; ``` paulson@13183 ` 1303` ```val zdiv_zmult2_eq = thm "zdiv_zmult2_eq"; ``` paulson@13183 ` 1304` ```val zmod_zmult2_eq = thm "zmod_zmult2_eq"; ``` paulson@13183 ` 1305` ```val zdiv_zmult_zmult1 = thm "zdiv_zmult_zmult1"; ``` paulson@13183 ` 1306` ```val zdiv_zmult_zmult2 = thm "zdiv_zmult_zmult2"; ``` paulson@13183 ` 1307` ```val zmod_zmult_zmult1 = thm "zmod_zmult_zmult1"; ``` paulson@13183 ` 1308` ```val zmod_zmult_zmult2 = thm "zmod_zmult_zmult2"; ``` paulson@13183 ` 1309` ```val pos_zdiv_mult_2 = thm "pos_zdiv_mult_2"; ``` paulson@13183 ` 1310` ```val neg_zdiv_mult_2 = thm "neg_zdiv_mult_2"; ``` paulson@13183 ` 1311` ```val zdiv_number_of_BIT = thm "zdiv_number_of_BIT"; ``` paulson@13183 ` 1312` ```val pos_zmod_mult_2 = thm "pos_zmod_mult_2"; ``` paulson@13183 ` 1313` ```val neg_zmod_mult_2 = thm "neg_zmod_mult_2"; ``` paulson@13183 ` 1314` ```val zmod_number_of_BIT = thm "zmod_number_of_BIT"; ``` paulson@13183 ` 1315` ```val div_neg_pos_less0 = thm "div_neg_pos_less0"; ``` paulson@13183 ` 1316` ```val div_nonneg_neg_le0 = thm "div_nonneg_neg_le0"; ``` paulson@13183 ` 1317` ```val pos_imp_zdiv_nonneg_iff = thm "pos_imp_zdiv_nonneg_iff"; ``` paulson@13183 ` 1318` ```val neg_imp_zdiv_nonneg_iff = thm "neg_imp_zdiv_nonneg_iff"; ``` paulson@13183 ` 1319` ```val pos_imp_zdiv_neg_iff = thm "pos_imp_zdiv_neg_iff"; ``` paulson@13183 ` 1320` ```val neg_imp_zdiv_neg_iff = thm "neg_imp_zdiv_neg_iff"; ``` paulson@14353 ` 1321` paulson@14353 ` 1322` ```val zpower_zmod = thm "zpower_zmod"; ``` paulson@14353 ` 1323` ```val zpower_zadd_distrib = thm "zpower_zadd_distrib"; ``` paulson@14353 ` 1324` ```val zpower_zpower = thm "zpower_zpower"; ``` paulson@14353 ` 1325` ```val zero_less_zpower_abs_iff = thm "zero_less_zpower_abs_iff"; ``` paulson@14353 ` 1326` ```val zero_le_zpower_abs = thm "zero_le_zpower_abs"; ``` paulson@13183 ` 1327` ```*} ``` paulson@13183 ` 1328` paulson@6917 ` 1329` ```end ```