(* Title: HOL/IntDiv.thy ID: $Id: IntDiv.thy,v 1.21 2007/10/21 12:53:44 nipkow Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1999 University of Cambridge *) header{*The Division Operators div and mod; the Divides Relation dvd*} theory IntDiv imports IntArith Divides FunDef begin constdefs quorem :: "(int*int) * (int*int) => bool" --{*definition of quotient and remainder*} [code func]: "quorem == %((a,b), (q,r)). a = b*q + r & (if 0 < b then 0≤r & r<b else b<r & r ≤ 0)" adjust :: "[int, int*int] => int*int" --{*for the division algorithm*} [code func]: "adjust b == %(q,r). if 0 ≤ r-b then (2*q + 1, r-b) else (2*q, r)" text{*algorithm for the case @{text "a≥0, b>0"}*} function posDivAlg :: "int => int => int × int" where "posDivAlg a b = (if (a<b | b≤0) then (0,a) else adjust b (posDivAlg a (2*b)))" by auto termination by (relation "measure (%(a,b). nat(a - b + 1))") auto text{*algorithm for the case @{text "a<0, b>0"}*} function negDivAlg :: "int => int => int × int" where "negDivAlg a b = (if (0≤a+b | b≤0) then (-1,a+b) else adjust b (negDivAlg a (2*b)))" by auto termination by (relation "measure (%(a,b). nat(- a - b))") auto text{*algorithm for the general case @{term "b≠0"}*} constdefs negateSnd :: "int*int => int*int" [code func]: "negateSnd == %(q,r). (q,-r)" definition divAlg :: "int × int => int × int" --{*The full division algorithm considers all possible signs for a, b including the special case @{text "a=0, b<0"} because @{term negDivAlg} requires @{term "a<0"}.*} where "divAlg = (λ(a, b). (if 0≤a then if 0≤b then posDivAlg a b else if a=0 then (0, 0) else negateSnd (negDivAlg (-a) (-b)) else if 0<b then negDivAlg a b else negateSnd (posDivAlg (-a) (-b))))" instance int :: Divides.div div_def: "a div b == fst (divAlg (a, b))" mod_def: "a mod b == snd (divAlg (a, b))" .. lemma divAlg_mod_div: "divAlg (p, q) = (p div q, p mod q)" by (auto simp add: div_def mod_def) text{* Here is the division algorithm in ML: \begin{verbatim} fun posDivAlg (a,b) = if a<b then (0,a) else let val (q,r) = posDivAlg(a, 2*b) in if 0≤r-b then (2*q+1, r-b) else (2*q, r) end fun negDivAlg (a,b) = if 0≤a+b then (~1,a+b) else let val (q,r) = negDivAlg(a, 2*b) in if 0≤r-b then (2*q+1, r-b) else (2*q, r) end; fun negateSnd (q,r:int) = (q,~r); fun divAlg (a,b) = if 0≤a then if b>0 then posDivAlg (a,b) else if a=0 then (0,0) else negateSnd (negDivAlg (~a,~b)) else if 0<b then negDivAlg (a,b) else negateSnd (posDivAlg (~a,~b)); \end{verbatim} *} subsection{*Uniqueness and Monotonicity of Quotients and Remainders*} lemma unique_quotient_lemma: "[| b*q' + r' ≤ b*q + r; 0 ≤ r'; r' < b; r < b |] ==> q' ≤ (q::int)" apply (subgoal_tac "r' + b * (q'-q) ≤ r") prefer 2 apply (simp add: right_diff_distrib) apply (subgoal_tac "0 < b * (1 + q - q') ") apply (erule_tac [2] order_le_less_trans) prefer 2 apply (simp add: right_diff_distrib right_distrib) apply (subgoal_tac "b * q' < b * (1 + q) ") prefer 2 apply (simp add: right_diff_distrib right_distrib) apply (simp add: mult_less_cancel_left) done lemma unique_quotient_lemma_neg: "[| b*q' + r' ≤ b*q + r; r ≤ 0; b < r; b < r' |] ==> q ≤ (q'::int)" by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, auto) lemma unique_quotient: "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b ≠ 0 |] ==> q = q'" apply (simp add: quorem_def linorder_neq_iff split: split_if_asm) apply (blast intro: order_antisym dest: order_eq_refl [THEN unique_quotient_lemma] order_eq_refl [THEN unique_quotient_lemma_neg] sym)+ done lemma unique_remainder: "[| quorem ((a,b), (q,r)); quorem ((a,b), (q',r')); b ≠ 0 |] ==> r = r'" apply (subgoal_tac "q = q'") apply (simp add: quorem_def) apply (blast intro: unique_quotient) done subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*} text{*And positive divisors*} lemma adjust_eq [simp]: "adjust b (q,r) = (let diff = r-b in if 0 ≤ diff then (2*q + 1, diff) else (2*q, r))" by (simp add: Let_def adjust_def) declare posDivAlg.simps [simp del] text{*use with a simproc to avoid repeatedly proving the premise*} lemma posDivAlg_eqn: "0 < b ==> posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))" by (rule posDivAlg.simps [THEN trans], simp) text{*Correctness of @{term posDivAlg}: it computes quotients correctly*} theorem posDivAlg_correct: assumes "0 ≤ a" and "0 < b" shows "quorem ((a, b), posDivAlg a b)" using prems apply (induct a b rule: posDivAlg.induct) apply auto apply (simp add: quorem_def) apply (subst posDivAlg_eqn, simp add: right_distrib) apply (case_tac "a < b") apply simp_all apply (erule splitE) apply (auto simp add: right_distrib Let_def) done subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*} text{*And positive divisors*} declare negDivAlg.simps [simp del] text{*use with a simproc to avoid repeatedly proving the premise*} lemma negDivAlg_eqn: "0 < b ==> negDivAlg a b = (if 0≤a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))" by (rule negDivAlg.simps [THEN trans], simp) (*Correctness of negDivAlg: it computes quotients correctly It doesn't work if a=0 because the 0/b equals 0, not -1*) lemma negDivAlg_correct: assumes "a < 0" and "b > 0" shows "quorem ((a, b), negDivAlg a b)" using prems apply (induct a b rule: negDivAlg.induct) apply (auto simp add: linorder_not_le) apply (simp add: quorem_def) apply (subst negDivAlg_eqn, assumption) apply (case_tac "a + b < (0::int)") apply simp_all apply (erule splitE) apply (auto simp add: right_distrib Let_def) done subsection{*Existence Shown by Proving the Division Algorithm to be Correct*} (*the case a=0*) lemma quorem_0: "b ≠ 0 ==> quorem ((0,b), (0,0))" by (auto simp add: quorem_def linorder_neq_iff) lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)" by (subst posDivAlg.simps, auto) lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)" by (subst negDivAlg.simps, auto) lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)" by (simp add: negateSnd_def) lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)" by (auto simp add: split_ifs quorem_def) lemma divAlg_correct: "b ≠ 0 ==> quorem ((a,b), divAlg (a, b))" by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg posDivAlg_correct negDivAlg_correct) text{*Arbitrary definitions for division by zero. Useful to simplify certain equations.*} lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a" by (simp add: div_def mod_def divAlg_def posDivAlg.simps) text{*Basic laws about division and remainder*} lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)" apply (case_tac "b = 0", simp) apply (cut_tac a = a and b = b in divAlg_correct) apply (auto simp add: quorem_def div_def mod_def) done lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k" by(simp add: zmod_zdiv_equality[symmetric]) lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k" by(simp add: mult_commute zmod_zdiv_equality[symmetric]) text {* Tool setup *} ML_setup {* local structure CancelDivMod = CancelDivModFun( struct val div_name = @{const_name Divides.div}; val mod_name = @{const_name Divides.mod}; val mk_binop = HOLogic.mk_binop; val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT; val dest_sum = Int_Numeral_Simprocs.dest_sum; val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}]; val trans = trans; val prove_eq_sums = let val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac} in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end; end) in val cancel_zdiv_zmod_proc = NatArithUtils.prep_simproc ("cancel_zdiv_zmod", ["(m::int) + n"], K CancelDivMod.proc) end; Addsimprocs [cancel_zdiv_zmod_proc] *} lemma pos_mod_conj : "(0::int) < b ==> 0 ≤ a mod b & a mod b < b" apply (cut_tac a = a and b = b in divAlg_correct) apply (auto simp add: quorem_def mod_def) done lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1, standard] and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard] lemma neg_mod_conj : "b < (0::int) ==> a mod b ≤ 0 & b < a mod b" apply (cut_tac a = a and b = b in divAlg_correct) apply (auto simp add: quorem_def div_def mod_def) done lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1, standard] and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard] subsection{*General Properties of div and mod*} lemma quorem_div_mod: "b ≠ 0 ==> quorem ((a, b), (a div b, a mod b))" apply (cut_tac a = a and b = b in zmod_zdiv_equality) apply (force simp add: quorem_def linorder_neq_iff) done lemma quorem_div: "[| quorem((a,b),(q,r)); b ≠ 0 |] ==> a div b = q" by (simp add: quorem_div_mod [THEN unique_quotient]) lemma quorem_mod: "[| quorem((a,b),(q,r)); b ≠ 0 |] ==> a mod b = r" by (simp add: quorem_div_mod [THEN unique_remainder]) lemma div_pos_pos_trivial: "[| (0::int) ≤ a; a < b |] ==> a div b = 0" apply (rule quorem_div) apply (auto simp add: quorem_def) done lemma div_neg_neg_trivial: "[| a ≤ (0::int); b < a |] ==> a div b = 0" apply (rule quorem_div) apply (auto simp add: quorem_def) done lemma div_pos_neg_trivial: "[| (0::int) < a; a+b ≤ 0 |] ==> a div b = -1" apply (rule quorem_div) apply (auto simp add: quorem_def) done (*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*) lemma mod_pos_pos_trivial: "[| (0::int) ≤ a; a < b |] ==> a mod b = a" apply (rule_tac q = 0 in quorem_mod) apply (auto simp add: quorem_def) done lemma mod_neg_neg_trivial: "[| a ≤ (0::int); b < a |] ==> a mod b = a" apply (rule_tac q = 0 in quorem_mod) apply (auto simp add: quorem_def) done lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b ≤ 0 |] ==> a mod b = a+b" apply (rule_tac q = "-1" in quorem_mod) apply (auto simp add: quorem_def) done text{*There is no @{text mod_neg_pos_trivial}.*} (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*) lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)" apply (case_tac "b = 0", simp) apply (simp add: quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_div, THEN sym]) done (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*) lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))" apply (case_tac "b = 0", simp) apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod], auto) done subsection{*Laws for div and mod with Unary Minus*} lemma zminus1_lemma: "quorem((a,b),(q,r)) ==> quorem ((-a,b), (if r=0 then -q else -q - 1), (if r=0 then 0 else b-r))" by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib) lemma zdiv_zminus1_eq_if: "b ≠ (0::int) ==> (-a) div b = (if a mod b = 0 then - (a div b) else - (a div b) - 1)" by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div]) lemma zmod_zminus1_eq_if: "(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))" apply (case_tac "b = 0", simp) apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod]) done lemma zdiv_zminus2: "a div (-b) = (-a::int) div b" by (cut_tac a = "-a" in zdiv_zminus_zminus, auto) lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)" by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto) lemma zdiv_zminus2_eq_if: "b ≠ (0::int) ==> a div (-b) = (if a mod b = 0 then - (a div b) else - (a div b) - 1)" by (simp add: zdiv_zminus1_eq_if zdiv_zminus2) lemma zmod_zminus2_eq_if: "a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)" by (simp add: zmod_zminus1_eq_if zmod_zminus2) subsection{*Division of a Number by Itself*} lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 ≤ q" apply (subgoal_tac "0 < a*q") apply (simp add: zero_less_mult_iff, arith) done lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 ≤ r |] ==> q ≤ 1" apply (subgoal_tac "0 ≤ a* (1-q) ") apply (simp add: zero_le_mult_iff) apply (simp add: right_diff_distrib) done lemma self_quotient: "[| quorem((a,a),(q,r)); a ≠ (0::int) |] ==> q = 1" apply (simp add: split_ifs quorem_def linorder_neq_iff) apply (rule order_antisym, safe, simp_all) apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1) apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2) apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+ done lemma self_remainder: "[| quorem((a,a),(q,r)); a ≠ (0::int) |] ==> r = 0" apply (frule self_quotient, assumption) apply (simp add: quorem_def) done lemma zdiv_self [simp]: "a ≠ 0 ==> a div a = (1::int)" by (simp add: quorem_div_mod [THEN self_quotient]) (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *) lemma zmod_self [simp]: "a mod a = (0::int)" apply (case_tac "a = 0", simp) apply (simp add: quorem_div_mod [THEN self_remainder]) done subsection{*Computation of Division and Remainder*} lemma zdiv_zero [simp]: "(0::int) div b = 0" by (simp add: div_def divAlg_def) lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1" by (simp add: div_def divAlg_def) lemma zmod_zero [simp]: "(0::int) mod b = 0" by (simp add: mod_def divAlg_def) lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1" by (simp add: div_def divAlg_def) lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1" by (simp add: mod_def divAlg_def) text{*a positive, b positive *} lemma div_pos_pos: "[| 0 < a; 0 ≤ b |] ==> a div b = fst (posDivAlg a b)" by (simp add: div_def divAlg_def) lemma mod_pos_pos: "[| 0 < a; 0 ≤ b |] ==> a mod b = snd (posDivAlg a b)" by (simp add: mod_def divAlg_def) text{*a negative, b positive *} lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)" by (simp add: div_def divAlg_def) lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)" by (simp add: mod_def divAlg_def) text{*a positive, b negative *} lemma div_pos_neg: "[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))" by (simp add: div_def divAlg_def) lemma mod_pos_neg: "[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))" by (simp add: mod_def divAlg_def) text{*a negative, b negative *} lemma div_neg_neg: "[| a < 0; b ≤ 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))" by (simp add: div_def divAlg_def) lemma mod_neg_neg: "[| a < 0; b ≤ 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))" by (simp add: mod_def divAlg_def) text {*Simplify expresions in which div and mod combine numerical constants*} lemma quoremI: "[|a == b * q + r; if 0 < b then 0 ≤ r ∧ r < b else b < r ∧ r ≤ 0|] ==> quorem ((a, b), (q, r))" unfolding quorem_def by simp lemmas quorem_div_eq = quoremI [THEN quorem_div, THEN eq_reflection] lemmas quorem_mod_eq = quoremI [THEN quorem_mod, THEN eq_reflection] lemmas arithmetic_simps = arith_simps add_special OrderedGroup.add_0_left OrderedGroup.add_0_right mult_zero_left mult_zero_right mult_1_left mult_1_right (* simprocs adapted from HOL/ex/Binary.thy *) ML {* local infix ==; val op == = Logic.mk_equals; fun plus m n = @{term "plus :: int => int => int"} $ m $ n; fun mult m n = @{term "times :: int => int => int"} $ m $ n; val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps}; fun prove ctxt prop = Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss)); fun binary_proc proc ss ct = (case Thm.term_of ct of _ $ t $ u => (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of SOME args => proc (Simplifier.the_context ss) args | NONE => NONE) | _ => NONE); in fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) => if n = 0 then NONE else let val (k, l) = Integer.div_mod m n; fun mk_num x = HOLogic.mk_number HOLogic.intT x; in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))]) end); end; *} simproc_setup binary_int_div ("number_of m div number_of n :: int") = {* K (divmod_proc (@{thm quorem_div_eq})) *} simproc_setup binary_int_mod ("number_of m mod number_of n :: int") = {* K (divmod_proc (@{thm quorem_mod_eq})) *} (* The following 8 lemmas are made unnecessary by the above simprocs: *) lemmas div_pos_pos_number_of = div_pos_pos [of "number_of v" "number_of w", standard] lemmas div_neg_pos_number_of = div_neg_pos [of "number_of v" "number_of w", standard] lemmas div_pos_neg_number_of = div_pos_neg [of "number_of v" "number_of w", standard] lemmas div_neg_neg_number_of = div_neg_neg [of "number_of v" "number_of w", standard] lemmas mod_pos_pos_number_of = mod_pos_pos [of "number_of v" "number_of w", standard] lemmas mod_neg_pos_number_of = mod_neg_pos [of "number_of v" "number_of w", standard] lemmas mod_pos_neg_number_of = mod_pos_neg [of "number_of v" "number_of w", standard] lemmas mod_neg_neg_number_of = mod_neg_neg [of "number_of v" "number_of w", standard] lemmas posDivAlg_eqn_number_of [simp] = posDivAlg_eqn [of "number_of v" "number_of w", standard] lemmas negDivAlg_eqn_number_of [simp] = negDivAlg_eqn [of "number_of v" "number_of w", standard] text{*Special-case simplification *} lemma zmod_1 [simp]: "a mod (1::int) = 0" apply (cut_tac a = a and b = 1 in pos_mod_sign) apply (cut_tac [2] a = a and b = 1 in pos_mod_bound) apply (auto simp del:pos_mod_bound pos_mod_sign) done lemma zdiv_1 [simp]: "a div (1::int) = a" by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto) lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0" apply (cut_tac a = a and b = "-1" in neg_mod_sign) apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound) apply (auto simp del: neg_mod_sign neg_mod_bound) done lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a" by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto) (** The last remaining special cases for constant arithmetic: 1 div z and 1 mod z **) lemmas div_pos_pos_1_number_of [simp] = div_pos_pos [OF int_0_less_1, of "number_of w", standard] lemmas div_pos_neg_1_number_of [simp] = div_pos_neg [OF int_0_less_1, of "number_of w", standard] lemmas mod_pos_pos_1_number_of [simp] = mod_pos_pos [OF int_0_less_1, of "number_of w", standard] lemmas mod_pos_neg_1_number_of [simp] = mod_pos_neg [OF int_0_less_1, of "number_of w", standard] lemmas posDivAlg_eqn_1_number_of [simp] = posDivAlg_eqn [of concl: 1 "number_of w", standard] lemmas negDivAlg_eqn_1_number_of [simp] = negDivAlg_eqn [of concl: 1 "number_of w", standard] subsection{*Monotonicity in the First Argument (Dividend)*} lemma zdiv_mono1: "[| a ≤ a'; 0 < (b::int) |] ==> a div b ≤ a' div b" apply (cut_tac a = a and b = b in zmod_zdiv_equality) apply (cut_tac a = a' and b = b in zmod_zdiv_equality) apply (rule unique_quotient_lemma) apply (erule subst) apply (erule subst, simp_all) done lemma zdiv_mono1_neg: "[| a ≤ a'; (b::int) < 0 |] ==> a' div b ≤ a div b" apply (cut_tac a = a and b = b in zmod_zdiv_equality) apply (cut_tac a = a' and b = b in zmod_zdiv_equality) apply (rule unique_quotient_lemma_neg) apply (erule subst) apply (erule subst, simp_all) done subsection{*Monotonicity in the Second Argument (Divisor)*} lemma q_pos_lemma: "[| 0 ≤ b'*q' + r'; r' < b'; 0 < b' |] ==> 0 ≤ (q'::int)" apply (subgoal_tac "0 < b'* (q' + 1) ") apply (simp add: zero_less_mult_iff) apply (simp add: right_distrib) done lemma zdiv_mono2_lemma: "[| b*q + r = b'*q' + r'; 0 ≤ b'*q' + r'; r' < b'; 0 ≤ r; 0 < b'; b' ≤ b |] ==> q ≤ (q'::int)" apply (frule q_pos_lemma, assumption+) apply (subgoal_tac "b*q < b* (q' + 1) ") apply (simp add: mult_less_cancel_left) apply (subgoal_tac "b*q = r' - r + b'*q'") prefer 2 apply simp apply (simp (no_asm_simp) add: right_distrib) apply (subst add_commute, rule zadd_zless_mono, arith) apply (rule mult_right_mono, auto) done lemma zdiv_mono2: "[| (0::int) ≤ a; 0 < b'; b' ≤ b |] ==> a div b ≤ a div b'" apply (subgoal_tac "b ≠ 0") prefer 2 apply arith apply (cut_tac a = a and b = b in zmod_zdiv_equality) apply (cut_tac a = a and b = b' in zmod_zdiv_equality) apply (rule zdiv_mono2_lemma) apply (erule subst) apply (erule subst, simp_all) done lemma q_neg_lemma: "[| b'*q' + r' < 0; 0 ≤ r'; 0 < b' |] ==> q' ≤ (0::int)" apply (subgoal_tac "b'*q' < 0") apply (simp add: mult_less_0_iff, arith) done lemma zdiv_mono2_neg_lemma: "[| b*q + r = b'*q' + r'; b'*q' + r' < 0; r < b; 0 ≤ r'; 0 < b'; b' ≤ b |] ==> q' ≤ (q::int)" apply (frule q_neg_lemma, assumption+) apply (subgoal_tac "b*q' < b* (q + 1) ") apply (simp add: mult_less_cancel_left) apply (simp add: right_distrib) apply (subgoal_tac "b*q' ≤ b'*q'") prefer 2 apply (simp add: mult_right_mono_neg, arith) done lemma zdiv_mono2_neg: "[| a < (0::int); 0 < b'; b' ≤ b |] ==> a div b' ≤ a div b" apply (cut_tac a = a and b = b in zmod_zdiv_equality) apply (cut_tac a = a and b = b' in zmod_zdiv_equality) apply (rule zdiv_mono2_neg_lemma) apply (erule subst) apply (erule subst, simp_all) done subsection{*More Algebraic Laws for div and mod*} text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *} lemma zmult1_lemma: "[| quorem((b,c),(q,r)); c ≠ 0 |] ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))" by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib) lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)" apply (case_tac "c = 0", simp) apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div]) done lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)" apply (case_tac "c = 0", simp) apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod]) done lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c" apply (rule trans) apply (rule_tac s = "b*a mod c" in trans) apply (rule_tac [2] zmod_zmult1_eq) apply (simp_all add: mult_commute) done lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c" apply (rule zmod_zmult1_eq' [THEN trans]) apply (rule zmod_zmult1_eq) done lemma zdiv_zmult_self1 [simp]: "b ≠ (0::int) ==> (a*b) div b = a" by (simp add: zdiv_zmult1_eq) lemma zdiv_zmult_self2 [simp]: "b ≠ (0::int) ==> (b*a) div b = a" by (subst mult_commute, erule zdiv_zmult_self1) lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)" by (simp add: zmod_zmult1_eq) lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)" by (simp add: mult_commute zmod_zmult1_eq) lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)" proof assume "m mod d = 0" with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto next assume "EX q::int. m = d*q" thus "m mod d = 0" by auto qed lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1] text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *} lemma zadd1_lemma: "[| quorem((a,c),(aq,ar)); quorem((b,c),(bq,br)); c ≠ 0 |] ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))" by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib) (*NOT suitable for rewriting: the RHS has an instance of the LHS*) lemma zdiv_zadd1_eq: "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)" apply (case_tac "c = 0", simp) apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div) done lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c" apply (case_tac "c = 0", simp) apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod) done lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)" apply (case_tac "b = 0", simp) apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial) done lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)" apply (case_tac "b = 0", simp) apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial) done lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c" apply (rule trans [symmetric]) apply (rule zmod_zadd1_eq, simp) apply (rule zmod_zadd1_eq [symmetric]) done lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c" apply (rule trans [symmetric]) apply (rule zmod_zadd1_eq, simp) apply (rule zmod_zadd1_eq [symmetric]) done lemma zdiv_zadd_self1[simp]: "a ≠ (0::int) ==> (a+b) div a = b div a + 1" by (simp add: zdiv_zadd1_eq) lemma zdiv_zadd_self2[simp]: "a ≠ (0::int) ==> (b+a) div a = b div a + 1" by (simp add: zdiv_zadd1_eq) lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)" apply (case_tac "a = 0", simp) apply (simp add: zmod_zadd1_eq) done lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)" apply (case_tac "a = 0", simp) apply (simp add: zmod_zadd1_eq) done lemma zmod_zdiff1_eq: fixes a::int shows "(a - b) mod c = (a mod c - b mod c) mod c" (is "?l = ?r") proof - have "?l = (c + (a mod c - b mod c)) mod c" using zmod_zadd1_eq[of a "-b" c] by(simp add:ring_simps zmod_zminus1_eq_if) also have "… = ?r" by simp finally show ?thesis . qed subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *} (*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but 7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems to cause particular problems.*) text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *} lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r ≤ 0 |] ==> b*c < b*(q mod c) + r" apply (subgoal_tac "b * (c - q mod c) < r * 1") apply (simp add: right_diff_distrib) apply (rule order_le_less_trans) apply (erule_tac [2] mult_strict_right_mono) apply (rule mult_left_mono_neg) apply (auto simp add: compare_rls add_commute [of 1] add1_zle_eq pos_mod_bound) done lemma zmult2_lemma_aux2: "[| (0::int) < c; b < r; r ≤ 0 |] ==> b * (q mod c) + r ≤ 0" apply (subgoal_tac "b * (q mod c) ≤ 0") apply arith apply (simp add: mult_le_0_iff) done lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 ≤ r; r < b |] ==> 0 ≤ b * (q mod c) + r" apply (subgoal_tac "0 ≤ b * (q mod c) ") apply arith apply (simp add: zero_le_mult_iff) done lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 ≤ r; r < b |] ==> b * (q mod c) + r < b * c" apply (subgoal_tac "r * 1 < b * (c - q mod c) ") apply (simp add: right_diff_distrib) apply (rule order_less_le_trans) apply (erule mult_strict_right_mono) apply (rule_tac [2] mult_left_mono) apply (auto simp add: compare_rls add_commute [of 1] add1_zle_eq pos_mod_bound) done lemma zmult2_lemma: "[| quorem ((a,b), (q,r)); b ≠ 0; 0 < c |] ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))" by (auto simp add: mult_ac quorem_def linorder_neq_iff zero_less_mult_iff right_distrib [symmetric] zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4) lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c" apply (case_tac "b = 0", simp) apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div]) done lemma zmod_zmult2_eq: "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b" apply (case_tac "b = 0", simp) apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod]) done subsection{*Cancellation of Common Factors in div*} lemma zdiv_zmult_zmult1_aux1: "[| (0::int) < b; c ≠ 0 |] ==> (c*a) div (c*b) = a div b" by (subst zdiv_zmult2_eq, auto) lemma zdiv_zmult_zmult1_aux2: "[| b < (0::int); c ≠ 0 |] ==> (c*a) div (c*b) = a div b" apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ") apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto) done lemma zdiv_zmult_zmult1: "c ≠ (0::int) ==> (c*a) div (c*b) = a div b" apply (case_tac "b = 0", simp) apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2) done lemma zdiv_zmult_zmult1_if[simp]: "(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)" by (simp add:zdiv_zmult_zmult1) (* lemma zdiv_zmult_zmult2: "c ≠ (0::int) ==> (a*c) div (b*c) = a div b" apply (drule zdiv_zmult_zmult1) apply (auto simp add: mult_commute) done *) subsection{*Distribution of Factors over mod*} lemma zmod_zmult_zmult1_aux1: "[| (0::int) < b; c ≠ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)" by (subst zmod_zmult2_eq, auto) lemma zmod_zmult_zmult1_aux2: "[| b < (0::int); c ≠ 0 |] ==> (c*a) mod (c*b) = c * (a mod b)" apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))") apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto) done lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)" apply (case_tac "b = 0", simp) apply (case_tac "c = 0", simp) apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2) done lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)" apply (cut_tac c = c in zmod_zmult_zmult1) apply (auto simp add: mult_commute) done lemma zmod_zmod_cancel: assumes "n dvd m" shows "(k::int) mod m mod n = k mod n" proof - from `n dvd m` obtain r where "m = n*r" by(auto simp:dvd_def) have "k mod n = (m * (k div m) + k mod m) mod n" using zmod_zdiv_equality[of k m] by simp also have "… = (m * (k div m) mod n + k mod m mod n) mod n" by(subst zmod_zadd1_eq, rule refl) also have "m * (k div m) mod n = 0" using `m = n*r` by(simp add:mult_ac) finally show ?thesis by simp qed subsection {*Splitting Rules for div and mod*} text{*The proofs of the two lemmas below are essentially identical*} lemma split_pos_lemma: "0<k ==> P(n div k :: int)(n mod k) = (∀i j. 0≤j & j<k & n = k*i + j --> P i j)" apply (rule iffI, clarify) apply (erule_tac P="P ?x ?y" in rev_mp) apply (subst zmod_zadd1_eq) apply (subst zdiv_zadd1_eq) apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial) txt{*converse direction*} apply (drule_tac x = "n div k" in spec) apply (drule_tac x = "n mod k" in spec, simp) done lemma split_neg_lemma: "k<0 ==> P(n div k :: int)(n mod k) = (∀i j. k<j & j≤0 & n = k*i + j --> P i j)" apply (rule iffI, clarify) apply (erule_tac P="P ?x ?y" in rev_mp) apply (subst zmod_zadd1_eq) apply (subst zdiv_zadd1_eq) apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial) txt{*converse direction*} apply (drule_tac x = "n div k" in spec) apply (drule_tac x = "n mod k" in spec, simp) done lemma split_zdiv: "P(n div k :: int) = ((k = 0 --> P 0) & (0<k --> (∀i j. 0≤j & j<k & n = k*i + j --> P i)) & (k<0 --> (∀i j. k<j & j≤0 & n = k*i + j --> P i)))" apply (case_tac "k=0", simp) apply (simp only: linorder_neq_iff) apply (erule disjE) apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] split_neg_lemma [of concl: "%x y. P x"]) done lemma split_zmod: "P(n mod k :: int) = ((k = 0 --> P n) & (0<k --> (∀i j. 0≤j & j<k & n = k*i + j --> P j)) & (k<0 --> (∀i j. k<j & j≤0 & n = k*i + j --> P j)))" apply (case_tac "k=0", simp) apply (simp only: linorder_neq_iff) apply (erule disjE) apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] split_neg_lemma [of concl: "%x y. P y"]) done (* Enable arith to deal with div 2 and mod 2: *) declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split] declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split] subsection{*Speeding up the Division Algorithm with Shifting*} text{*computing div by shifting *} lemma pos_zdiv_mult_2: "(0::int) ≤ a ==> (1 + 2*b) div (2*a) = b div a" proof cases assume "a=0" thus ?thesis by simp next assume "a≠0" and le_a: "0≤a" hence a_pos: "1 ≤ a" by arith hence one_less_a2: "1 < 2*a" by arith hence le_2a: "2 * (1 + b mod a) ≤ 2 * a" by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq) with a_pos have "0 ≤ b mod a" by simp hence le_addm: "0 ≤ 1 mod (2*a) + 2*(b mod a)" by (simp add: mod_pos_pos_trivial one_less_a2) with le_2a have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0" by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2 right_distrib) thus ?thesis by (subst zdiv_zadd1_eq, simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2 div_pos_pos_trivial) qed lemma neg_zdiv_mult_2: "a ≤ (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a" apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ") apply (rule_tac [2] pos_zdiv_mult_2) apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib) apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))") apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric], simp) done (*Not clear why this must be proved separately; probably number_of causes simplification problems*) lemma not_0_le_lemma: "~ 0 ≤ x ==> x ≤ (0::int)" by auto lemma zdiv_number_of_BIT[simp]: "number_of (v BIT b) div number_of (w BIT bit.B0) = (if b=bit.B0 | (0::int) ≤ number_of w then number_of v div (number_of w) else (number_of v + (1::int)) div (number_of w))" apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac split: bit.split) done subsection{*Computing mod by Shifting (proofs resemble those for div)*} lemma pos_zmod_mult_2: "(0::int) ≤ a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)" apply (case_tac "a = 0", simp) apply (subgoal_tac "1 < a * 2") prefer 2 apply arith apply (subgoal_tac "2* (1 + b mod a) ≤ 2*a") apply (rule_tac [2] mult_left_mono) apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq pos_mod_bound) apply (subst zmod_zadd1_eq) apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial) apply (rule mod_pos_pos_trivial) apply (auto simp add: mod_pos_pos_trivial left_distrib) apply (subgoal_tac "0 ≤ b mod a", arith, simp) done lemma neg_zmod_mult_2: "a ≤ (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1" apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) = 1 + 2* ((-b - 1) mod (-a))") apply (rule_tac [2] pos_zmod_mult_2) apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib) apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))") prefer 2 apply simp apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric]) done lemma zmod_number_of_BIT [simp]: "number_of (v BIT b) mod number_of (w BIT bit.B0) = (case b of bit.B0 => 2 * (number_of v mod number_of w) | bit.B1 => if (0::int) ≤ number_of w then 2 * (number_of v mod number_of w) + 1 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)" apply (simp only: number_of_eq numeral_simps UNIV_I split: bit.split) apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2 not_0_le_lemma neg_zmod_mult_2 add_ac) done subsection{*Quotients of Signs*} lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0" apply (subgoal_tac "a div b ≤ -1", force) apply (rule order_trans) apply (rule_tac a' = "-1" in zdiv_mono1) apply (auto simp add: zdiv_minus1) done lemma div_nonneg_neg_le0: "[| (0::int) ≤ a; b < 0 |] ==> a div b ≤ 0" by (drule zdiv_mono1_neg, auto) lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 ≤ a div b) = (0 ≤ a)" apply auto apply (drule_tac [2] zdiv_mono1) apply (auto simp add: linorder_neq_iff) apply (simp (no_asm_use) add: linorder_not_less [symmetric]) apply (blast intro: div_neg_pos_less0) done lemma neg_imp_zdiv_nonneg_iff: "b < (0::int) ==> (0 ≤ a div b) = (a ≤ (0::int))" apply (subst zdiv_zminus_zminus [symmetric]) apply (subst pos_imp_zdiv_nonneg_iff, auto) done (*But not (a div b ≤ 0 iff a≤0); consider a=1, b=2 when a div b = 0.*) lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)" by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff) (*Again the law fails for ≤: consider a = -1, b = -2 when a div b = 0*) lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)" by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff) subsection {* The Divides Relation *} lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))" by (simp add: dvd_def zmod_eq_0_iff) instance int :: dvd_mod by default (simp add: zdvd_iff_zmod_eq_0) lemmas zdvd_iff_zmod_eq_0_number_of [simp] = zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard] lemma zdvd_0_right [iff]: "(m::int) dvd 0" by (simp add: dvd_def) lemma zdvd_0_left [iff,noatp]: "(0 dvd (m::int)) = (m = 0)" by (simp add: dvd_def) lemma zdvd_1_left [iff]: "1 dvd (m::int)" by (simp add: dvd_def) lemma zdvd_refl [simp]: "m dvd (m::int)" by (auto simp add: dvd_def intro: zmult_1_right [symmetric]) lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)" by (auto simp add: dvd_def intro: mult_assoc) lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))" apply (simp add: dvd_def, auto) apply (rule_tac [!] x = "-k" in exI, auto) done lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))" apply (simp add: dvd_def, auto) apply (rule_tac [!] x = "-k" in exI, auto) done lemma zdvd_abs1: "( ¦i::int¦ dvd j) = (i dvd j)" apply (cases "i > 0", simp) apply (simp add: dvd_def) apply (rule iffI) apply (erule exE) apply (rule_tac x="- k" in exI, simp) apply (erule exE) apply (rule_tac x="- k" in exI, simp) done lemma zdvd_abs2: "( (i::int) dvd ¦j¦) = (i dvd j)" apply (cases "j > 0", simp) apply (simp add: dvd_def) apply (rule iffI) apply (erule exE) apply (rule_tac x="- k" in exI, simp) apply (erule exE) apply (rule_tac x="- k" in exI, simp) done lemma zdvd_anti_sym: "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)" apply (simp add: dvd_def, auto) apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff) done lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)" apply (simp add: dvd_def) apply (blast intro: right_distrib [symmetric]) done lemma zdvd_dvd_eq: assumes anz:"a ≠ 0" and ab: "(a::int) dvd b" and ba:"b dvd a" shows "¦a¦ = ¦b¦" proof- from ab obtain k where k:"b = a*k" unfolding dvd_def by blast from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast from k k' have "a = a*k*k'" by simp with mult_cancel_left1[where c="a" and b="k*k'"] have kk':"k*k' = 1" using anz by (simp add: mult_assoc) hence "k = 1 ∧ k' = 1 ∨ k = -1 ∧ k' = -1" by (simp add: zmult_eq_1_iff) thus ?thesis using k k' by auto qed lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)" apply (simp add: dvd_def) apply (blast intro: right_diff_distrib [symmetric]) done lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)" apply (subgoal_tac "m = n + (m - n)") apply (erule ssubst) apply (blast intro: zdvd_zadd, simp) done lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n" apply (simp add: dvd_def) apply (blast intro: mult_left_commute) done lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n" apply (subst mult_commute) apply (erule zdvd_zmult) done lemma zdvd_triv_right [iff]: "(k::int) dvd m * k" apply (rule zdvd_zmult) apply (rule zdvd_refl) done lemma zdvd_triv_left [iff]: "(k::int) dvd k * m" apply (rule zdvd_zmult2) apply (rule zdvd_refl) done lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)" apply (simp add: dvd_def) apply (simp add: mult_assoc, blast) done lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)" apply (rule zdvd_zmultD2) apply (subst mult_commute, assumption) done lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n" apply (simp add: dvd_def, clarify) apply (rule_tac x = "k * ka" in exI) apply (simp add: mult_ac) done lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))" apply (rule iffI) apply (erule_tac [2] zdvd_zadd) apply (subgoal_tac "n = (n + k * m) - k * m") apply (erule ssubst) apply (erule zdvd_zdiff, simp_all) done lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n" apply (simp add: dvd_def) apply (auto simp add: zmod_zmult_zmult1) done lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)" apply (subgoal_tac "k dvd n * (m div n) + m mod n") apply (simp add: zmod_zdiv_equality [symmetric]) apply (simp only: zdvd_zadd zdvd_zmult2) done lemma zdvd_not_zless: "0 < m ==> m < n ==> ¬ n dvd (m::int)" apply (simp add: dvd_def, auto) apply (subgoal_tac "0 < n") prefer 2 apply (blast intro: order_less_trans) apply (simp add: zero_less_mult_iff) apply (subgoal_tac "n * k < n * 1") apply (drule mult_less_cancel_left [THEN iffD1], auto) done lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)" using zmod_zdiv_equality[where a="m" and b="n"] by (simp add: ring_simps) lemma zdvd_mult_div_cancel:"(n::int) dvd m ==> n * (m div n) = m" apply (subgoal_tac "m mod n = 0") apply (simp add: zmult_div_cancel) apply (simp only: zdvd_iff_zmod_eq_0) done lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k ≠ (0::int)" shows "m dvd n" proof- from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast {assume "n ≠ m*h" hence "k* n ≠ k* (m*h)" using kz by simp with h have False by (simp add: mult_assoc)} hence "n = m * h" by blast thus ?thesis by blast qed lemma zdvd_zmult_cancel_disj[simp]: "(k*m) dvd (k*n) = (k=0 | m dvd (n::int))" by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel) theorem ex_nat: "(∃x::nat. P x) = (∃x::int. 0 <= x ∧ P (nat x))" apply (simp split add: split_nat) apply (rule iffI) apply (erule exE) apply (rule_tac x = "int x" in exI) apply simp apply (erule exE) apply (rule_tac x = "nat x" in exI) apply (erule conjE) apply (erule_tac x = "nat x" in allE) apply simp done theorem zdvd_int: "(x dvd y) = (int x dvd int y)" apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric] nat_0_le cong add: conj_cong) apply (rule iffI) apply iprover apply (erule exE) apply (case_tac "x=0") apply (rule_tac x=0 in exI) apply simp apply (case_tac "0 ≤ k") apply iprover apply (simp add: neq0_conv linorder_not_le) apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]]) apply assumption apply (simp add: mult_ac) done lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( ¦x¦ = 1)" proof assume d: "x dvd 1" hence "int (nat ¦x¦) dvd int (nat 1)" by (simp add: zdvd_abs1) hence "nat ¦x¦ dvd 1" by (simp add: zdvd_int) hence "nat ¦x¦ = 1" by simp thus "¦x¦ = 1" by (cases "x < 0", auto) next assume "¦x¦=1" thus "x dvd 1" by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0) qed lemma zdvd_mult_cancel1: assumes mp:"m ≠(0::int)" shows "(m * n dvd m) = (¦n¦ = 1)" proof assume n1: "¦n¦ = 1" thus "m * n dvd m" by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff) next assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp from zdvd_mult_cancel[OF H2 mp] show "¦n¦ = 1" by (simp only: zdvd1_eq) qed lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))" apply (auto simp add: dvd_def nat_abs_mult_distrib) apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm) apply (rule_tac x = "-(int k)" in exI) apply (auto simp add: int_mult) done lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)" apply (auto simp add: dvd_def abs_if int_mult) apply (rule_tac [3] x = "nat k" in exI) apply (rule_tac [2] x = "-(int k)" in exI) apply (rule_tac x = "nat (-k)" in exI) apply (cut_tac [3] k = m in int_less_0_conv) apply (cut_tac k = m in int_less_0_conv) apply (auto simp add: zero_le_mult_iff mult_less_0_iff nat_mult_distrib [symmetric] nat_eq_iff2) done lemma nat_dvd_iff: "(nat z dvd m) = (if 0 ≤ z then (z dvd int m) else m = 0)" apply (auto simp add: dvd_def int_mult) apply (rule_tac x = "nat k" in exI) apply (cut_tac k = m in int_less_0_conv) apply (auto simp add: zero_le_mult_iff mult_less_0_iff nat_mult_distrib [symmetric] nat_eq_iff2) done lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))" apply (auto simp add: dvd_def) apply (rule_tac [!] x = "-k" in exI, auto) done lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))" apply (auto simp add: dvd_def) apply (drule minus_equation_iff [THEN iffD1]) apply (rule_tac [!] x = "-k" in exI, auto) done lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z ≤ (n::int)" apply (rule_tac z=n in int_cases) apply (auto simp add: dvd_int_iff) apply (rule_tac z=z in int_cases) apply (auto simp add: dvd_imp_le) done subsection{*Integer Powers*} instance int :: power .. primrec "p ^ 0 = 1" "p ^ (Suc n) = (p::int) * (p ^ n)" instance int :: recpower proof fix z :: int fix n :: nat show "z^0 = 1" by simp show "z^(Suc n) = z * (z^n)" by simp qed lemma of_int_power: "of_int (z ^ n) = (of_int z ^ n :: 'a::{recpower, ring_1})" by (induct n) (simp_all add: power_Suc) lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m" apply (induct "y", auto) apply (rule zmod_zmult1_eq [THEN trans]) apply (simp (no_asm_simp)) apply (rule zmod_zmult_distrib [symmetric]) done lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)" by (rule Power.power_add) lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)" by (rule Power.power_mult [symmetric]) lemma zero_less_zpower_abs_iff [simp]: "(0 < (abs x)^n) = (x ≠ (0::int) | n=0)" apply (induct "n") apply (auto simp add: zero_less_mult_iff) done lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n" apply (induct "n") apply (auto simp add: zero_le_mult_iff) done lemma int_power: "int (m^n) = (int m) ^ n" by (rule of_nat_power) text{*Compatibility binding*} lemmas zpower_int = int_power [symmetric] lemma zdiv_int: "int (a div b) = (int a) div (int b)" apply (subst split_div, auto) apply (subst split_zdiv, auto) apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient) apply (auto simp add: IntDiv.quorem_def of_nat_mult) done lemma zmod_int: "int (a mod b) = (int a) mod (int b)" apply (subst split_mod, auto) apply (subst split_zmod, auto) apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in unique_remainder) apply (auto simp add: IntDiv.quorem_def of_nat_mult) done text{*Suggested by Matthias Daum*} lemma int_power_div_base: "[|0 < m; 0 < k|] ==> k ^ m div k = (k::int) ^ (m - Suc 0)" apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)") apply (erule ssubst) apply (simp only: power_add) apply simp_all done text {* by Brian Huffman *} lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m" by (simp only: zmod_zminus1_eq_if mod_mod_trivial) lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)" by (simp only: diff_def zmod_zadd_left_eq [symmetric]) lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)" proof - have "(x + - (y mod m) mod m) mod m = (x + - y mod m) mod m" by (simp only: zminus_zmod) hence "(x + - (y mod m)) mod m = (x + - y) mod m" by (simp only: zmod_zadd_right_eq [symmetric]) thus "(x - y mod m) mod m = (x - y) mod m" by (simp only: diff_def) qed lemmas zmod_simps = IntDiv.zmod_zadd_left_eq [symmetric] IntDiv.zmod_zadd_right_eq [symmetric] IntDiv.zmod_zmult1_eq [symmetric] IntDiv.zmod_zmult1_eq' [symmetric] IntDiv.zpower_zmod zminus_zmod zdiff_zmod_left zdiff_zmod_right text {* code generator setup *} code_modulename SML IntDiv Integer code_modulename OCaml IntDiv Integer code_modulename Haskell IntDiv Integer end
lemma divAlg_mod_div:
divAlg (p, q) = (p div q, p mod q)
lemma unique_quotient_lemma:
[| b * q' + r' ≤ b * q + r; 0 ≤ r'; r' < b; r < b |] ==> q' ≤ q
lemma unique_quotient_lemma_neg:
[| b * q' + r' ≤ b * q + r; r ≤ 0; b < r; b < r' |] ==> q ≤ q'
lemma unique_quotient:
[| IntDiv.quorem ((a, b), q, r); IntDiv.quorem ((a, b), q', r'); b ≠ 0 |]
==> q = q'
lemma unique_remainder:
[| IntDiv.quorem ((a, b), q, r); IntDiv.quorem ((a, b), q', r'); b ≠ 0 |]
==> r = r'
lemma adjust_eq:
adjust b (q, r) =
(let diff = r - b in if 0 ≤ diff then (2 * q + 1, diff) else (2 * q, r))
lemma posDivAlg_eqn:
0 < b
==> posDivAlg a b = (if a < b then (0, a) else adjust b (posDivAlg a (2 * b)))
theorem posDivAlg_correct:
[| 0 ≤ a; 0 < b |] ==> IntDiv.quorem ((a, b), posDivAlg a b)
lemma negDivAlg_eqn:
0 < b
==> negDivAlg a b =
(if 0 ≤ a + b then (-1, a + b) else adjust b (negDivAlg a (2 * b)))
lemma negDivAlg_correct:
[| a < 0; 0 < b |] ==> IntDiv.quorem ((a, b), negDivAlg a b)
lemma quorem_0:
b ≠ 0 ==> IntDiv.quorem ((0, b), 0, 0)
lemma posDivAlg_0:
posDivAlg 0 b = (0, 0)
lemma negDivAlg_minus1:
negDivAlg -1 b = (-1, b - 1)
lemma negateSnd_eq:
negateSnd (q, r) = (q, - r)
lemma quorem_neg:
IntDiv.quorem ((- a, - b), qr) ==> IntDiv.quorem ((a, b), negateSnd qr)
lemma divAlg_correct:
b ≠ 0 ==> IntDiv.quorem ((a, b), divAlg (a, b))
lemma DIVISION_BY_ZERO:
a div 0 = 0 ∧ a mod 0 = a
lemma zmod_zdiv_equality:
a = b * (a div b) + a mod b
lemma zdiv_zmod_equality:
b * (a div b) + a mod b + k = a + k
lemma zdiv_zmod_equality2:
a div b * b + a mod b + k = a + k
lemma pos_mod_conj:
0 < b ==> 0 ≤ a mod b ∧ a mod b < b
lemma pos_mod_sign:
0 < b ==> 0 ≤ a mod b
and pos_mod_bound:
0 < b ==> a mod b < b
lemma neg_mod_conj:
b < 0 ==> a mod b ≤ 0 ∧ b < a mod b
lemma neg_mod_sign:
b < 0 ==> a mod b ≤ 0
and neg_mod_bound:
b < 0 ==> b < a mod b
lemma quorem_div_mod:
b ≠ 0 ==> IntDiv.quorem ((a, b), a div b, a mod b)
lemma quorem_div:
[| IntDiv.quorem ((a, b), q, r); b ≠ 0 |] ==> a div b = q
lemma quorem_mod:
[| IntDiv.quorem ((a, b), q, r); b ≠ 0 |] ==> a mod b = r
lemma div_pos_pos_trivial:
[| 0 ≤ a; a < b |] ==> a div b = 0
lemma div_neg_neg_trivial:
[| a ≤ 0; b < a |] ==> a div b = 0
lemma div_pos_neg_trivial:
[| 0 < a; a + b ≤ 0 |] ==> a div b = -1
lemma mod_pos_pos_trivial:
[| 0 ≤ a; a < b |] ==> a mod b = a
lemma mod_neg_neg_trivial:
[| a ≤ 0; b < a |] ==> a mod b = a
lemma mod_pos_neg_trivial:
[| 0 < a; a + b ≤ 0 |] ==> a mod b = a + b
lemma zdiv_zminus_zminus:
- a div - b = a div b
lemma zmod_zminus_zminus:
- a mod - b = - (a mod b)
lemma zminus1_lemma:
IntDiv.quorem ((a, b), q, r)
==> IntDiv.quorem
((- a, b), if r = 0 then - q else - q - 1, if r = 0 then 0 else b - r)
lemma zdiv_zminus1_eq_if:
b ≠ 0 ==> - a div b = (if a mod b = 0 then - (a div b) else - (a div b) - 1)
lemma zmod_zminus1_eq_if:
- a mod b = (if a mod b = 0 then 0 else b - a mod b)
lemma zdiv_zminus2:
a div - b = - a div b
lemma zmod_zminus2:
a mod - b = - (- a mod b)
lemma zdiv_zminus2_eq_if:
b ≠ 0 ==> a div - b = (if a mod b = 0 then - (a div b) else - (a div b) - 1)
lemma zmod_zminus2_eq_if:
a mod - b = (if a mod b = 0 then 0 else a mod b - b)
lemma self_quotient_aux1:
[| 0 < a; a = r + a * q; r < a |] ==> 1 ≤ q
lemma self_quotient_aux2:
[| 0 < a; a = r + a * q; 0 ≤ r |] ==> q ≤ 1
lemma self_quotient:
[| IntDiv.quorem ((a, a), q, r); a ≠ 0 |] ==> q = 1
lemma self_remainder:
[| IntDiv.quorem ((a, a), q, r); a ≠ 0 |] ==> r = 0
lemma zdiv_self:
a ≠ 0 ==> a div a = 1
lemma zmod_self:
a mod a = 0
lemma zdiv_zero:
0 div b = 0
lemma div_eq_minus1:
0 < b ==> -1 div b = -1
lemma zmod_zero:
0 mod b = 0
lemma zdiv_minus1:
0 < b ==> -1 div b = -1
lemma zmod_minus1:
0 < b ==> -1 mod b = b - 1
lemma div_pos_pos:
[| 0 < a; 0 ≤ b |] ==> a div b = fst (posDivAlg a b)
lemma mod_pos_pos:
[| 0 < a; 0 ≤ b |] ==> a mod b = snd (posDivAlg a b)
lemma div_neg_pos:
[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)
lemma mod_neg_pos:
[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)
lemma div_pos_neg:
[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (- a) (- b)))
lemma mod_pos_neg:
[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (- a) (- b)))
lemma div_neg_neg:
[| a < 0; b ≤ 0 |] ==> a div b = fst (negateSnd (posDivAlg (- a) (- b)))
lemma mod_neg_neg:
[| a < 0; b ≤ 0 |] ==> a mod b = snd (negateSnd (posDivAlg (- a) (- b)))
lemma quoremI:
[| a == b * q + r; if 0 < b then 0 ≤ r ∧ r < b else b < r ∧ r ≤ 0 |]
==> IntDiv.quorem ((a, b), q, r)
lemma quorem_div_eq:
[| a1 == b1 * y + r1; if 0 < b1 then 0 ≤ r1 ∧ r1 < b1 else b1 < r1 ∧ r1 ≤ 0;
b1 ≠ 0 |]
==> a1 div b1 == y
lemma quorem_mod_eq:
[| a1 == b1 * q1 + y; if 0 < b1 then 0 ≤ y ∧ y < b1 else b1 < y ∧ y ≤ 0;
b1 ≠ 0 |]
==> a1 mod b1 == y
lemma arithmetic_simps:
bit.B0 ≠ bit.B1
bit.B1 ≠ bit.B0
Numeral.Pls BIT bit.B0 = Numeral.Pls
Numeral.Min BIT bit.B1 = Numeral.Min
Numeral.pred Numeral.Pls = Numeral.Min
Numeral.pred Numeral.Min = Numeral.Min BIT bit.B0
Numeral.pred (k BIT bit.B1) = k BIT bit.B0
Numeral.pred (k BIT bit.B0) = Numeral.pred k BIT bit.B1
Numeral.succ Numeral.Pls = Numeral.Pls BIT bit.B1
Numeral.succ Numeral.Min = Numeral.Pls
Numeral.succ (k BIT bit.B1) = Numeral.succ k BIT bit.B0
Numeral.succ (k BIT bit.B0) = k BIT bit.B1
Numeral.Pls + k = k
Numeral.Min + k = Numeral.pred k
k BIT bit.B0 + l BIT b = (k + l) BIT b
k BIT bit.B1 + l BIT bit.B0 = (k + l) BIT bit.B1
k BIT bit.B1 + l BIT bit.B1 = (k + Numeral.succ l) BIT bit.B0
- Numeral.Pls = Numeral.Pls
- Numeral.Min = Numeral.Pls BIT bit.B1
- k BIT bit.B1 = Numeral.pred (- k) BIT bit.B1
- k BIT bit.B0 = (- k) BIT bit.B0
Numeral.Pls * w = Numeral.Pls
Numeral.Min * k = - k
k BIT bit.B1 * l = (k * l) BIT bit.B0 + l
k BIT bit.B0 * l = (k * l) BIT bit.B0
k + Numeral.Pls = k
k + Numeral.Min = Numeral.pred k
¦0::'a¦ = (0::'a)
¦1::'a¦ = (1::'a)
number_of v + number_of w = number_of (v + w)
- number_of w = number_of (- w)
- (1::'a) = (-1::'a)
number_of v * number_of w = number_of (v * w)
number_of v - number_of w = number_of (v + - w)
¦number_of x¦ = (if number_of x < (0::'a) then - number_of x else number_of x)
(1::'a) + (1::'a) = (2::'a)
(1::'a) + number_of w = number_of (Numeral.Pls BIT bit.B1 + w)
number_of v + (1::'a) = number_of (v + Numeral.Pls BIT bit.B1)
(0::'a) + a = a
a + (0::'a) = a
(0::'a) * a = (0::'a)
a * (0::'a) = (0::'a)
(1::'a) * a = a
a * (1::'a) = a
lemma div_pos_pos_number_of:
[| 0 < number_of v; 0 ≤ number_of w |]
==> number_of v div number_of w = fst (posDivAlg (number_of v) (number_of w))
lemma div_neg_pos_number_of:
[| number_of v < 0; 0 < number_of w |]
==> number_of v div number_of w = fst (negDivAlg (number_of v) (number_of w))
lemma div_pos_neg_number_of:
[| 0 < number_of v; number_of w < 0 |]
==> number_of v div number_of w =
fst (negateSnd (negDivAlg (- number_of v) (- number_of w)))
lemma div_neg_neg_number_of:
[| number_of v < 0; number_of w ≤ 0 |]
==> number_of v div number_of w =
fst (negateSnd (posDivAlg (- number_of v) (- number_of w)))
lemma mod_pos_pos_number_of:
[| 0 < number_of v; 0 ≤ number_of w |]
==> number_of v mod number_of w = snd (posDivAlg (number_of v) (number_of w))
lemma mod_neg_pos_number_of:
[| number_of v < 0; 0 < number_of w |]
==> number_of v mod number_of w = snd (negDivAlg (number_of v) (number_of w))
lemma mod_pos_neg_number_of:
[| 0 < number_of v; number_of w < 0 |]
==> number_of v mod number_of w =
snd (negateSnd (negDivAlg (- number_of v) (- number_of w)))
lemma mod_neg_neg_number_of:
[| number_of v < 0; number_of w ≤ 0 |]
==> number_of v mod number_of w =
snd (negateSnd (posDivAlg (- number_of v) (- number_of w)))
lemma posDivAlg_eqn_number_of:
0 < number_of v
==> posDivAlg (number_of w) (number_of v) =
(if number_of w < number_of v then (0, number_of w)
else adjust (number_of v) (posDivAlg (number_of w) (2 * number_of v)))
lemma negDivAlg_eqn_number_of:
0 < number_of v
==> negDivAlg (number_of w) (number_of v) =
(if 0 ≤ number_of w + number_of v then (-1, number_of w + number_of v)
else adjust (number_of v) (negDivAlg (number_of w) (2 * number_of v)))
lemma zmod_1:
a mod 1 = 0
lemma zdiv_1:
a div 1 = a
lemma zmod_minus1_right:
a mod -1 = 0
lemma zdiv_minus1_right:
a div -1 = - a
lemma div_pos_pos_1_number_of:
0 ≤ number_of w ==> 1 div number_of w = fst (posDivAlg 1 (number_of w))
lemma div_pos_neg_1_number_of:
number_of w < 0
==> 1 div number_of w = fst (negateSnd (negDivAlg (- 1) (- number_of w)))
lemma mod_pos_pos_1_number_of:
0 ≤ number_of w ==> 1 mod number_of w = snd (posDivAlg 1 (number_of w))
lemma mod_pos_neg_1_number_of:
number_of w < 0
==> 1 mod number_of w = snd (negateSnd (negDivAlg (- 1) (- number_of w)))
lemma posDivAlg_eqn_1_number_of:
0 < number_of w
==> posDivAlg 1 (number_of w) =
(if 1 < number_of w then (0, 1)
else adjust (number_of w) (posDivAlg 1 (2 * number_of w)))
lemma negDivAlg_eqn_1_number_of:
0 < number_of w
==> negDivAlg 1 (number_of w) =
(if 0 ≤ 1 + number_of w then (-1, 1 + number_of w)
else adjust (number_of w) (negDivAlg 1 (2 * number_of w)))
lemma zdiv_mono1:
[| a ≤ a'; 0 < b |] ==> a div b ≤ a' div b
lemma zdiv_mono1_neg:
[| a ≤ a'; b < 0 |] ==> a' div b ≤ a div b
lemma q_pos_lemma:
[| 0 ≤ b' * q' + r'; r' < b'; 0 < b' |] ==> 0 ≤ q'
lemma zdiv_mono2_lemma:
[| b * q + r = b' * q' + r'; 0 ≤ b' * q' + r'; r' < b'; 0 ≤ r; 0 < b'; b' ≤ b |]
==> q ≤ q'
lemma zdiv_mono2:
[| 0 ≤ a; 0 < b'; b' ≤ b |] ==> a div b ≤ a div b'
lemma q_neg_lemma:
[| b' * q' + r' < 0; 0 ≤ r'; 0 < b' |] ==> q' ≤ 0
lemma zdiv_mono2_neg_lemma:
[| b * q + r = b' * q' + r'; b' * q' + r' < 0; r < b; 0 ≤ r'; 0 < b'; b' ≤ b |]
==> q' ≤ q
lemma zdiv_mono2_neg:
[| a < 0; 0 < b'; b' ≤ b |] ==> a div b' ≤ a div b
lemma zmult1_lemma:
[| IntDiv.quorem ((b, c), q, r); c ≠ 0 |]
==> IntDiv.quorem ((a * b, c), a * q + a * r div c, a * r mod c)
lemma zdiv_zmult1_eq:
a * b div c = a * (b div c) + a * (b mod c) div c
lemma zmod_zmult1_eq:
a * b mod c = a * (b mod c) mod c
lemma zmod_zmult1_eq':
a * b mod c = a mod c * b mod c
lemma zmod_zmult_distrib:
a * b mod c = a mod c * (b mod c) mod c
lemma zdiv_zmult_self1:
b ≠ 0 ==> a * b div b = a
lemma zdiv_zmult_self2:
b ≠ 0 ==> b * a div b = a
lemma zmod_zmult_self1:
a * b mod b = 0
lemma zmod_zmult_self2:
b * a mod b = 0
lemma zmod_eq_0_iff:
(m mod d = 0) = (∃q. m = d * q)
lemma zmod_eq_0D:
m1 mod d1 = 0 ==> ∃q. m1 = d1 * q
lemma zadd1_lemma:
[| IntDiv.quorem ((a, c), aq, ar); IntDiv.quorem ((b, c), bq, br); c ≠ 0 |]
==> IntDiv.quorem ((a + b, c), aq + bq + (ar + br) div c, (ar + br) mod c)
lemma zdiv_zadd1_eq:
(a + b) div c = a div c + b div c + (a mod c + b mod c) div c
lemma zmod_zadd1_eq:
(a + b) mod c = (a mod c + b mod c) mod c
lemma mod_div_trivial:
a mod b div b = 0
lemma mod_mod_trivial:
a mod b mod b = a mod b
lemma zmod_zadd_left_eq:
(a + b) mod c = (a mod c + b) mod c
lemma zmod_zadd_right_eq:
(a + b) mod c = (a + b mod c) mod c
lemma zdiv_zadd_self1:
a ≠ 0 ==> (a + b) div a = b div a + 1
lemma zdiv_zadd_self2:
a ≠ 0 ==> (b + a) div a = b div a + 1
lemma zmod_zadd_self1:
(a + b) mod a = b mod a
lemma zmod_zadd_self2:
(b + a) mod a = b mod a
lemma zmod_zdiff1_eq:
(a - b) mod c = (a mod c - b mod c) mod c
lemma zmult2_lemma_aux1:
[| 0 < c; b < r; r ≤ 0 |] ==> b * c < b * (q mod c) + r
lemma zmult2_lemma_aux2:
[| 0 < c; b < r; r ≤ 0 |] ==> b * (q mod c) + r ≤ 0
lemma zmult2_lemma_aux3:
[| 0 < c; 0 ≤ r; r < b |] ==> 0 ≤ b * (q mod c) + r
lemma zmult2_lemma_aux4:
[| 0 < c; 0 ≤ r; r < b |] ==> b * (q mod c) + r < b * c
lemma zmult2_lemma:
[| IntDiv.quorem ((a, b), q, r); b ≠ 0; 0 < c |]
==> IntDiv.quorem ((a, b * c), q div c, b * (q mod c) + r)
lemma zdiv_zmult2_eq:
0 < c ==> a div (b * c) = a div b div c
lemma zmod_zmult2_eq:
0 < c ==> a mod (b * c) = b * (a div b mod c) + a mod b
lemma zdiv_zmult_zmult1_aux1:
[| 0 < b; c ≠ 0 |] ==> c * a div (c * b) = a div b
lemma zdiv_zmult_zmult1_aux2:
[| b < 0; c ≠ 0 |] ==> c * a div (c * b) = a div b
lemma zdiv_zmult_zmult1:
c ≠ 0 ==> c * a div (c * b) = a div b
lemma zdiv_zmult_zmult1_if:
k * m div (k * n) = (if k = 0 then 0 else m div n)
lemma zmod_zmult_zmult1_aux1:
[| 0 < b; c ≠ 0 |] ==> c * a mod (c * b) = c * (a mod b)
lemma zmod_zmult_zmult1_aux2:
[| b < 0; c ≠ 0 |] ==> c * a mod (c * b) = c * (a mod b)
lemma zmod_zmult_zmult1:
c * a mod (c * b) = c * (a mod b)
lemma zmod_zmult_zmult2:
a * c mod (b * c) = a mod b * c
lemma zmod_zmod_cancel:
n dvd m ==> k mod m mod n = k mod n
lemma split_pos_lemma:
0 < k
==> P (n div k) (n mod k) = (∀i j. 0 ≤ j ∧ j < k ∧ n = k * i + j --> P i j)
lemma split_neg_lemma:
k < 0
==> P (n div k) (n mod k) = (∀i j. k < j ∧ j ≤ 0 ∧ n = k * i + j --> P i j)
lemma split_zdiv:
P (n div k) =
((k = 0 --> P 0) ∧
(0 < k --> (∀i j. 0 ≤ j ∧ j < k ∧ n = k * i + j --> P i)) ∧
(k < 0 --> (∀i j. k < j ∧ j ≤ 0 ∧ n = k * i + j --> P i)))
lemma split_zmod:
P (n mod k) =
((k = 0 --> P n) ∧
(0 < k --> (∀i j. 0 ≤ j ∧ j < k ∧ n = k * i + j --> P j)) ∧
(k < 0 --> (∀i j. k < j ∧ j ≤ 0 ∧ n = k * i + j --> P j)))
lemma pos_zdiv_mult_2:
0 ≤ a ==> (1 + 2 * b) div (2 * a) = b div a
lemma neg_zdiv_mult_2:
a ≤ 0 ==> (1 + 2 * b) div (2 * a) = (b + 1) div a
lemma not_0_le_lemma:
¬ 0 ≤ x ==> x ≤ 0
lemma zdiv_number_of_BIT:
number_of (v BIT b) div number_of (w BIT bit.B0) =
(if b = bit.B0 ∨ 0 ≤ number_of w then number_of v div number_of w
else (number_of v + 1) div number_of w)
lemma pos_zmod_mult_2:
0 ≤ a ==> (1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)
lemma neg_zmod_mult_2:
a ≤ 0 ==> (1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1
lemma zmod_number_of_BIT:
number_of (v BIT b) mod number_of (w BIT bit.B0) =
(case b of bit.B0 => 2 * (number_of v mod number_of w)
| bit.B1 =>
if 0 ≤ number_of w then 2 * (number_of v mod number_of w) + 1
else 2 * ((number_of v + 1) mod number_of w) - 1)
lemma div_neg_pos_less0:
[| a < 0; 0 < b |] ==> a div b < 0
lemma div_nonneg_neg_le0:
[| 0 ≤ a; b < 0 |] ==> a div b ≤ 0
lemma pos_imp_zdiv_nonneg_iff:
0 < b ==> (0 ≤ a div b) = (0 ≤ a)
lemma neg_imp_zdiv_nonneg_iff:
b < 0 ==> (0 ≤ a div b) = (a ≤ 0)
lemma pos_imp_zdiv_neg_iff:
0 < b ==> (a div b < 0) = (a < 0)
lemma neg_imp_zdiv_neg_iff:
b < 0 ==> (a div b < 0) = (0 < a)
lemma zdvd_iff_zmod_eq_0:
(m dvd n) = (n mod m = 0)
lemma zdvd_iff_zmod_eq_0_number_of:
(number_of x dvd number_of y) = (number_of y mod number_of x = 0)
lemma zdvd_0_right:
m dvd 0
lemma zdvd_0_left:
(0 dvd m) = (m = 0)
lemma zdvd_1_left:
1 dvd m
lemma zdvd_refl:
m dvd m
lemma zdvd_trans:
[| m dvd n; n dvd k |] ==> m dvd k
lemma zdvd_zminus_iff:
(m dvd - n) = (m dvd n)
lemma zdvd_zminus2_iff:
(- m dvd n) = (m dvd n)
lemma zdvd_abs1:
(¦i¦ dvd j) = (i dvd j)
lemma zdvd_abs2:
(i dvd ¦j¦) = (i dvd j)
lemma zdvd_anti_sym:
[| 0 < m; 0 < n; m dvd n; n dvd m |] ==> m = n
lemma zdvd_zadd:
[| k dvd m; k dvd n |] ==> k dvd m + n
lemma zdvd_dvd_eq:
[| a ≠ 0; a dvd b; b dvd a |] ==> ¦a¦ = ¦b¦
lemma zdvd_zdiff:
[| k dvd m; k dvd n |] ==> k dvd m - n
lemma zdvd_zdiffD:
[| k dvd m - n; k dvd n |] ==> k dvd m
lemma zdvd_zmult:
k dvd n ==> k dvd m * n
lemma zdvd_zmult2:
k dvd m ==> k dvd m * n
lemma zdvd_triv_right:
k dvd m * k
lemma zdvd_triv_left:
k dvd k * m
lemma zdvd_zmultD2:
j * k dvd n ==> j dvd n
lemma zdvd_zmultD:
j * k dvd n ==> k dvd n
lemma zdvd_zmult_mono:
[| i dvd m; j dvd n |] ==> i * j dvd m * n
lemma zdvd_reduce:
(k dvd n + k * m) = (k dvd n)
lemma zdvd_zmod:
[| f dvd m; f dvd n |] ==> f dvd m mod n
lemma zdvd_zmod_imp_zdvd:
[| k dvd m mod n; k dvd n |] ==> k dvd m
lemma zdvd_not_zless:
[| 0 < m; m < n |] ==> ¬ n dvd m
lemma zmult_div_cancel:
n * (m div n) = m - m mod n
lemma zdvd_mult_div_cancel:
n dvd m ==> n * (m div n) = m
lemma zdvd_mult_cancel:
[| k * m dvd k * n; k ≠ 0 |] ==> m dvd n
lemma zdvd_zmult_cancel_disj:
(k * m dvd k * n) = (k = 0 ∨ m dvd n)
theorem ex_nat:
(∃x. P x) = (∃x≥0. P (nat x))
theorem zdvd_int:
(x dvd y) = (int x dvd int y)
lemma zdvd1_eq:
(x dvd 1) = (¦x¦ = 1)
lemma zdvd_mult_cancel1:
m ≠ 0 ==> (m * n dvd m) = (¦n¦ = 1)
lemma int_dvd_iff:
(int m dvd z) = (m dvd nat ¦z¦)
lemma dvd_int_iff:
(z dvd int m) = (nat ¦z¦ dvd m)
lemma nat_dvd_iff:
(nat z dvd m) = (if 0 ≤ z then z dvd int m else m = 0)
lemma zminus_dvd_iff:
(- z dvd w) = (z dvd w)
lemma dvd_zminus_iff:
(z dvd - w) = (z dvd w)
lemma zdvd_imp_le:
[| z dvd n; 0 < n |] ==> z ≤ n
lemma of_int_power:
of_int (z ^ n) = of_int z ^ n
lemma zpower_zmod:
(x mod m) ^ y mod m = x ^ y mod m
lemma zpower_zadd_distrib:
x ^ (y + z) = x ^ y * x ^ z
lemma zpower_zpower:
(x ^ y) ^ z = x ^ (y * z)
lemma zero_less_zpower_abs_iff:
(0 < ¦x¦ ^ n) = (x ≠ 0 ∨ n = 0)
lemma zero_le_zpower_abs:
0 ≤ ¦x¦ ^ n
lemma int_power:
int (m ^ n) = int m ^ n
lemma zpower_int:
int m ^ n = int (m ^ n)
lemma zdiv_int:
int (a div b) = int a div int b
lemma zmod_int:
int (a mod b) = int a mod int b
lemma int_power_div_base:
[| 0 < m; 0 < k |] ==> k ^ m div k = k ^ (m - Suc 0)
lemma zminus_zmod:
- (x mod m) mod m = - x mod m
lemma zdiff_zmod_left:
(x mod m - y) mod m = (x - y) mod m
lemma zdiff_zmod_right:
(x - y mod m) mod m = (x - y) mod m
lemma zmod_simps:
(a mod c + b) mod c = (a + b) mod c
(a + b mod c) mod c = (a + b) mod c
a * (b mod c) mod c = a * b mod c
a mod c * b mod c = a * b mod c
(x mod m) ^ y mod m = x ^ y mod m
- (x mod m) mod m = - x mod m
(x mod m - y) mod m = (x - y) mod m
(x - y mod m) mod m = (x - y) mod m