src/HOL/Divides.thy
author haftmann
Tue Jan 22 23:07:21 2008 +0100 (2008-01-22)
changeset 25942 a52309ac4a4d
parent 25571 c9e39eafc7a0
child 25947 1f2f4d941e9e
permissions -rw-r--r--
added class semiring_div
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(*  Title:      HOL/Divides.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div, mod and the divides relation "dvd" *}
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theory Divides
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imports Power
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uses "~~/src/Provers/Arith/cancel_div_mod.ML"
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begin
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subsection {* Syntactic division operations *}
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class div = times +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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begin
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definition
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  dvd  :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50)
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where
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  [code func del]: "m dvd n \<longleftrightarrow> (\<exists>k. n = m * k)"
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end
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subsection {* Abstract divisibility in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + div + 
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  assumes mod_div_equality: "a div b * b + a mod b = a"
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    and div_by_0: "a div 0 = 0"
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    and mult_div: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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begin
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lemma div_by_1: "a div 1 = a"
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  using mult_div [of one a] zero_neq_one by simp
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lemma mod_by_1: "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_by_0: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mult_mod: "a * b mod b = 0"
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proof (cases "b = 0")
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  case True then show ?thesis by (simp add: mod_by_0)
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next
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  case False with mult_div have abb: "a * b div b = a" .
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  from mod_div_equality have "a * b div b * b + a * b mod b = a * b" .
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  with abb have "a * b + a * b mod b = a * b + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self: "a mod a = 0"
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  using mult_mod [of one] by simp
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lemma div_self: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using mult_div [of _ one] by simp
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lemma div_0: "0 div a = 0"
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proof (cases "a = 0")
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  case True then show ?thesis by (simp add: div_by_0)
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next
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  case False with mult_div have "0 * a div a = 0" .
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  then show ?thesis by simp
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qed
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lemma mod_0: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp 
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lemma dvd_def_mod [code func]: "a dvd b \<longleftrightarrow> b mod a = 0"
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proof
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  assume "b mod a = 0"
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  with mod_div_equality [of b a] have "b div a * a = b" by simp
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  then have "b = a * (b div a)" unfolding mult_commute ..
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  then have "\<exists>c. b = a * c" ..
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  then show "a dvd b" unfolding dvd_def .
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next
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  assume "a dvd b"
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  then have "\<exists>c. b = a * c" unfolding dvd_def .
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  then obtain c where "b = a * c" ..
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  then have "b mod a = a * c mod a" by simp
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  then have "b mod a = c * a mod a" by (simp add: mult_commute)
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  then show "b mod a = 0" by (simp add: mult_mod)
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qed
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lemma dvd_refl: "a dvd a"
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  unfolding dvd_def_mod mod_self ..
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lemma dvd_trans:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" unfolding dvd_def by auto
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  moreover from assms obtain w where "c = b * w" unfolding dvd_def by auto
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  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
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  then show ?thesis unfolding dvd_def ..
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qed
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lemma one_dvd: "1 dvd a"
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  unfolding dvd_def by simp
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lemma dvd_0: "a dvd 0"
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unfolding dvd_def proof
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  show "0 = a * 0" by simp
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qed
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end
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subsection {* Division on the natural numbers *}
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instantiation nat :: semiring_div
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begin
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definition
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  div_def: "m div n == wfrec (pred_nat^+)
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                          (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m"
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lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+)
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               (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
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by (simp add: div_def)
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definition
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  mod_def: "m mod n == wfrec (pred_nat^+)
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                          (%f j. if j<n | n=0 then j else f (j-n)) m"
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lemma mod_eq: "(%m. m mod n) =
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              wfrec (pred_nat^+) (%f j. if j<n | n=0 then j else f (j-n))"
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by (simp add: mod_def)
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lemmas wf_less_trans = def_wfrec [THEN trans,
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  OF eq_reflection wf_pred_nat [THEN wf_trancl], standard]
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lemma div_less [simp]: "m < n \<Longrightarrow> m div n = (0\<Colon>nat)"
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  by (rule div_eq [THEN wf_less_trans]) simp
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lemma le_div_geq: "0 < n \<Longrightarrow> n \<le> m \<Longrightarrow> m div n = Suc ((m - n) div n)"
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  by (rule div_eq [THEN wf_less_trans]) (simp add: cut_apply less_eq)
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lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a\<Colon>nat)"
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  by (rule mod_eq [THEN wf_less_trans]) simp
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lemma mod_less [simp]: "m < n \<Longrightarrow> m mod n = (m\<Colon>nat)"
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  by (rule mod_eq [THEN wf_less_trans]) simp
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lemma le_mod_geq: "(n\<Colon>nat) \<le> m \<Longrightarrow> m mod n = (m - n) mod n"
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  by (cases "n = 0", simp, rule mod_eq [THEN wf_less_trans])
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    (simp add: cut_apply less_eq)
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lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
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  by (simp add: le_mod_geq)
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instance proof
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  fix n m :: nat
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  show "(m div n) * n + m mod n = m"
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    apply (cases "n = 0", simp)
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    apply (induct m rule: nat_less_induct [rule_format])
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    apply (subst mod_if)
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    apply (simp add: add_assoc add_diff_inverse le_div_geq)
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    done
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next
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  fix n :: nat
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  show "n div 0 = 0"
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    by (rule div_eq [THEN wf_less_trans], simp)
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next
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  fix n m :: nat
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  assume "n \<noteq> 0"
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  then show "m * n div n = m"
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    by (induct m) (simp_all add: le_div_geq)
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qed
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end
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subsubsection{*Simproc for Cancelling Div and Mod*}
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lemmas mod_div_equality = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
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lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
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  unfolding mult_commute [of n]
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  by (rule mod_div_equality)
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lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
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  by (simp add: mod_div_equality2)
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ML {*
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structure CancelDivModData =
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struct
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val div_name = @{const_name Divides.div};
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val mod_name = @{const_name Divides.mod};
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val mk_binop = HOLogic.mk_binop;
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val mk_sum = NatArithUtils.mk_sum;
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val dest_sum = NatArithUtils.dest_sum;
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(*logic*)
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val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
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val trans = trans
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val prove_eq_sums =
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  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
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  in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
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end;
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structure CancelDivMod = CancelDivModFun(CancelDivModData);
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val cancel_div_mod_proc = NatArithUtils.prep_simproc
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      ("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc);
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Addsimprocs[cancel_div_mod_proc];
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*}
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subsubsection {* Remainder *}
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lemmas DIVISION_BY_ZERO_MOD [simp] = mod_by_0 [of "a\<Colon>nat", standard]
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lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
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  by (induct m) (simp_all add: le_div_geq)
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lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
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  by (simp add: le_mod_geq linorder_not_less)
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lemma mod_1 [simp]: "m mod Suc 0 = 0"
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  by (induct m) (simp_all add: mod_geq)
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lemmas mod_self [simp] = semiring_div_class.mod_self [of "n\<Colon>nat", standard]
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lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
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  apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
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   apply (simp add: add_commute)
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  apply (subst le_mod_geq [symmetric], simp_all)
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  done
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lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
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  by (simp add: add_commute mod_add_self2)
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lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
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  by (induct k) (simp_all add: add_left_commute [of _ n])
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lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
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  by (simp add: mult_commute mod_mult_self1)
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lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
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  apply (cases "n = 0", simp)
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  apply (cases "k = 0", simp)
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  apply (induct m rule: nat_less_induct)
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  apply (subst mod_if, simp)
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  apply (simp add: mod_geq diff_mult_distrib)
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  done
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lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
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  by (simp add: mult_commute [of k] mod_mult_distrib)
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lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
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  apply (cases "n = 0", simp)
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  apply (induct m, simp)
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  apply (rename_tac k)
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  apply (cut_tac m = "k * n" and n = n in mod_add_self2)
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  apply (simp add: add_commute)
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  done
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lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
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  by (simp add: mult_commute mod_mult_self_is_0)
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subsubsection{*Quotient*}
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lemmas DIVISION_BY_ZERO_DIV [simp] = div_by_0 [of "a\<Colon>nat", standard]
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lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
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  by (simp add: le_div_geq linorder_not_less)
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lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
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  by (simp add: div_geq)
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(* a simple rearrangement of mod_div_equality: *)
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lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
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  by (cut_tac m = m and n = n in mod_div_equality2, arith)
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lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
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  apply (induct m rule: nat_less_induct)
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  apply (rename_tac m)
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  apply (case_tac "m<n", simp)
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  txt{*case @{term "n \<le> m"}*}
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  apply (simp add: mod_geq)
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  done
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lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
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  apply (drule mod_less_divisor [where m = m])
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  apply simp
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  done
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lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
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  by (simp add: mult_commute div_mult_self_is_m)
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(*mod_mult_distrib2 above is the counterpart for remainder*)
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subsubsection {* Proving advancedfacts about Quotient and Remainder *}
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definition
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  quorem :: "(nat*nat) * (nat*nat) => bool" where
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  (*This definition helps prove the harder properties of div and mod.
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    It is copied from IntDiv.thy; should it be overloaded?*)
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  "quorem = (%((a,b), (q,r)).
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                    a = b*q + r &
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                    (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"
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lemma unique_quotient_lemma:
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   325
     "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]
paulson@14267
   326
      ==> q' \<le> (q::nat)"
wenzelm@22718
   327
  apply (rule leI)
wenzelm@22718
   328
  apply (subst less_iff_Suc_add)
wenzelm@22718
   329
  apply (auto simp add: add_mult_distrib2)
wenzelm@22718
   330
  done
paulson@14267
   331
paulson@14267
   332
lemma unique_quotient:
wenzelm@22718
   333
     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
paulson@14267
   334
      ==> q = q'"
wenzelm@22718
   335
  apply (simp add: split_ifs quorem_def)
wenzelm@22718
   336
  apply (blast intro: order_antisym
wenzelm@22718
   337
    dest: order_eq_refl [THEN unique_quotient_lemma] sym)
wenzelm@22718
   338
  done
paulson@14267
   339
paulson@14267
   340
lemma unique_remainder:
wenzelm@22718
   341
     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
paulson@14267
   342
      ==> r = r'"
wenzelm@22718
   343
  apply (subgoal_tac "q = q'")
wenzelm@22718
   344
   prefer 2 apply (blast intro: unique_quotient)
wenzelm@22718
   345
  apply (simp add: quorem_def)
wenzelm@22718
   346
  done
paulson@14267
   347
nipkow@25162
   348
lemma quorem_div_mod: "b > 0 ==> quorem ((a, b), (a div b, a mod b))"
nipkow@25162
   349
unfolding quorem_def by simp
paulson@14267
   350
nipkow@25162
   351
lemma quorem_div: "[| quorem((a,b),(q,r));  b > 0 |] ==> a div b = q"
nipkow@25162
   352
by (simp add: quorem_div_mod [THEN unique_quotient])
paulson@14267
   353
nipkow@25162
   354
lemma quorem_mod: "[| quorem((a,b),(q,r));  b > 0 |] ==> a mod b = r"
nipkow@25162
   355
by (simp add: quorem_div_mod [THEN unique_remainder])
paulson@14267
   356
paulson@14267
   357
(** A dividend of zero **)
paulson@14267
   358
haftmann@25942
   359
lemmas div_0 [simp] = semiring_div_class.div_0 [of "n\<Colon>nat", standard]
paulson@14267
   360
haftmann@25942
   361
lemmas mod_0 [simp] = semiring_div_class.mod_0 [of "n\<Colon>nat", standard]
paulson@14267
   362
paulson@14267
   363
(** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
paulson@14267
   364
paulson@14267
   365
lemma quorem_mult1_eq:
nipkow@25162
   366
  "[| quorem((b,c),(q,r)); c > 0 |]
nipkow@25162
   367
   ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
nipkow@25162
   368
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
paulson@14267
   369
paulson@14267
   370
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
nipkow@25134
   371
apply (cases "c = 0", simp)
haftmann@25942
   372
thm DIVISION_BY_ZERO_DIV
nipkow@25134
   373
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
nipkow@25134
   374
done
paulson@14267
   375
paulson@14267
   376
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
nipkow@25134
   377
apply (cases "c = 0", simp)
nipkow@25134
   378
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
nipkow@25134
   379
done
paulson@14267
   380
paulson@14267
   381
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
wenzelm@22718
   382
  apply (rule trans)
wenzelm@22718
   383
   apply (rule_tac s = "b*a mod c" in trans)
wenzelm@22718
   384
    apply (rule_tac [2] mod_mult1_eq)
wenzelm@22718
   385
   apply (simp_all add: mult_commute)
wenzelm@22718
   386
  done
paulson@14267
   387
nipkow@25162
   388
lemma mod_mult_distrib_mod:
nipkow@25162
   389
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
nipkow@25162
   390
apply (rule mod_mult1_eq' [THEN trans])
nipkow@25162
   391
apply (rule mod_mult1_eq)
nipkow@25162
   392
done
paulson@14267
   393
paulson@14267
   394
(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
paulson@14267
   395
paulson@14267
   396
lemma quorem_add1_eq:
nipkow@25162
   397
  "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c > 0 |]
nipkow@25162
   398
   ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
nipkow@25162
   399
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
paulson@14267
   400
paulson@14267
   401
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   402
lemma div_add1_eq:
nipkow@25134
   403
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
nipkow@25134
   404
apply (cases "c = 0", simp)
nipkow@25134
   405
apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod)
nipkow@25134
   406
done
paulson@14267
   407
paulson@14267
   408
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
nipkow@25134
   409
apply (cases "c = 0", simp)
nipkow@25134
   410
apply (blast intro: quorem_div_mod quorem_add1_eq [THEN quorem_mod])
nipkow@25134
   411
done
paulson@14267
   412
paulson@14267
   413
haftmann@25942
   414
subsubsection {* Proving @{prop "a div (b*c) = (a div b) div c"} *}
paulson@14267
   415
paulson@14267
   416
(** first, a lemma to bound the remainder **)
paulson@14267
   417
paulson@14267
   418
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
   419
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
   420
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
   421
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
   422
  apply (simp add: add_mult_distrib2)
wenzelm@22718
   423
  done
paulson@10559
   424
wenzelm@22718
   425
lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
paulson@14267
   426
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
wenzelm@22718
   427
  by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
   428
paulson@14267
   429
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
wenzelm@22718
   430
  apply (cases "b = 0", simp)
wenzelm@22718
   431
  apply (cases "c = 0", simp)
wenzelm@22718
   432
  apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
wenzelm@22718
   433
  done
paulson@14267
   434
paulson@14267
   435
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
wenzelm@22718
   436
  apply (cases "b = 0", simp)
wenzelm@22718
   437
  apply (cases "c = 0", simp)
wenzelm@22718
   438
  apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
wenzelm@22718
   439
  done
paulson@14267
   440
paulson@14267
   441
haftmann@25942
   442
subsubsection{*Cancellation of Common Factors in Division*}
paulson@14267
   443
paulson@14267
   444
lemma div_mult_mult_lemma:
wenzelm@22718
   445
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   446
  by (auto simp add: div_mult2_eq)
paulson@14267
   447
paulson@14267
   448
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   449
  apply (cases "b = 0")
wenzelm@22718
   450
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
wenzelm@22718
   451
  done
paulson@14267
   452
paulson@14267
   453
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
wenzelm@22718
   454
  apply (drule div_mult_mult1)
wenzelm@22718
   455
  apply (auto simp add: mult_commute)
wenzelm@22718
   456
  done
paulson@14267
   457
paulson@14267
   458
haftmann@25942
   459
subsubsection{*Further Facts about Quotient and Remainder*}
paulson@14267
   460
paulson@14267
   461
lemma div_1 [simp]: "m div Suc 0 = m"
wenzelm@22718
   462
  by (induct m) (simp_all add: div_geq)
paulson@14267
   463
haftmann@25942
   464
lemmas div_self [simp] = semiring_div_class.div_self [of "n\<Colon>nat", standard]
paulson@14267
   465
paulson@14267
   466
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
wenzelm@22718
   467
  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
wenzelm@22718
   468
   apply (simp add: add_commute)
wenzelm@22718
   469
  apply (subst div_geq [symmetric], simp_all)
wenzelm@22718
   470
  done
paulson@14267
   471
paulson@14267
   472
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
wenzelm@22718
   473
  by (simp add: add_commute div_add_self2)
paulson@14267
   474
paulson@14267
   475
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
wenzelm@22718
   476
  apply (subst div_add1_eq)
wenzelm@22718
   477
  apply (subst div_mult1_eq, simp)
wenzelm@22718
   478
  done
paulson@14267
   479
paulson@14267
   480
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
wenzelm@22718
   481
  by (simp add: mult_commute div_mult_self1)
paulson@14267
   482
paulson@14267
   483
paulson@14267
   484
(* Monotonicity of div in first argument *)
paulson@14267
   485
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
   486
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
   487
apply (case_tac "k=0", simp)
paulson@15251
   488
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
   489
apply (case_tac "n<k")
paulson@14267
   490
(* 1  case n<k *)
paulson@14267
   491
apply simp
paulson@14267
   492
(* 2  case n >= k *)
paulson@14267
   493
apply (case_tac "m<k")
paulson@14267
   494
(* 2.1  case m<k *)
paulson@14267
   495
apply simp
paulson@14267
   496
(* 2.2  case m>=k *)
nipkow@15439
   497
apply (simp add: div_geq diff_le_mono)
paulson@14267
   498
done
paulson@14267
   499
paulson@14267
   500
(* Antimonotonicity of div in second argument *)
paulson@14267
   501
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   502
apply (subgoal_tac "0<n")
wenzelm@22718
   503
 prefer 2 apply simp
paulson@15251
   504
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   505
apply (rename_tac "k")
paulson@14267
   506
apply (case_tac "k<n", simp)
paulson@14267
   507
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
   508
 prefer 2 apply simp
paulson@14267
   509
apply (simp add: div_geq)
paulson@15251
   510
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   511
 prefer 2
paulson@14267
   512
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   513
apply (rule le_trans, simp)
nipkow@15439
   514
apply (simp)
paulson@14267
   515
done
paulson@14267
   516
paulson@14267
   517
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   518
apply (case_tac "n=0", simp)
paulson@14267
   519
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   520
apply (rule div_le_mono2)
paulson@14267
   521
apply (simp_all (no_asm_simp))
paulson@14267
   522
done
paulson@14267
   523
wenzelm@22718
   524
(* Similar for "less than" *)
paulson@17085
   525
lemma div_less_dividend [rule_format]:
paulson@14267
   526
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
paulson@15251
   527
apply (induct_tac m rule: nat_less_induct)
paulson@14267
   528
apply (rename_tac "m")
paulson@14267
   529
apply (case_tac "m<n", simp)
paulson@14267
   530
apply (subgoal_tac "0<n")
wenzelm@22718
   531
 prefer 2 apply simp
paulson@14267
   532
apply (simp add: div_geq)
paulson@14267
   533
apply (case_tac "n<m")
paulson@15251
   534
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   535
  apply (rule impI less_trans_Suc)+
paulson@14267
   536
apply assumption
nipkow@15439
   537
  apply (simp_all)
paulson@14267
   538
done
paulson@14267
   539
paulson@17085
   540
declare div_less_dividend [simp]
paulson@17085
   541
paulson@14267
   542
text{*A fact for the mutilated chess board*}
paulson@14267
   543
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   544
apply (case_tac "n=0", simp)
paulson@15251
   545
apply (induct "m" rule: nat_less_induct)
paulson@14267
   546
apply (case_tac "Suc (na) <n")
paulson@14267
   547
(* case Suc(na) < n *)
paulson@14267
   548
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   549
(* case n \<le> Suc(na) *)
paulson@16796
   550
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   551
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   552
done
paulson@14267
   553
paulson@14437
   554
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
wenzelm@22718
   555
  by (cases "n = 0") auto
paulson@14437
   556
paulson@14437
   557
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
wenzelm@22718
   558
  by (cases "n = 0") auto
paulson@14437
   559
paulson@14267
   560
haftmann@25942
   561
subsubsection{*The Divides Relation*}
paulson@14267
   562
paulson@14267
   563
lemma dvdI [intro?]: "n = m * k ==> m dvd n"
wenzelm@22718
   564
  unfolding dvd_def by blast
paulson@14267
   565
paulson@14267
   566
lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
wenzelm@22718
   567
  unfolding dvd_def by blast
nipkow@13152
   568
paulson@14267
   569
lemma dvd_0_right [iff]: "m dvd (0::nat)"
wenzelm@22718
   570
  unfolding dvd_def by (blast intro: mult_0_right [symmetric])
paulson@14267
   571
paulson@14267
   572
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
wenzelm@22718
   573
  by (force simp add: dvd_def)
paulson@14267
   574
paulson@14267
   575
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
wenzelm@22718
   576
  by (blast intro: dvd_0_left)
paulson@14267
   577
paulson@24286
   578
declare dvd_0_left_iff [noatp]
paulson@24286
   579
paulson@14267
   580
lemma dvd_1_left [iff]: "Suc 0 dvd k"
wenzelm@22718
   581
  unfolding dvd_def by simp
paulson@14267
   582
paulson@14267
   583
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
wenzelm@22718
   584
  by (simp add: dvd_def)
paulson@14267
   585
haftmann@25942
   586
lemmas dvd_refl [simp] = semiring_div_class.dvd_refl [of "m\<Colon>nat", standard]
haftmann@25942
   587
lemmas dvd_trans [trans] = semiring_div_class.dvd_trans [of "m\<Colon>nat" n p, standard]
paulson@14267
   588
paulson@14267
   589
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
wenzelm@22718
   590
  unfolding dvd_def
wenzelm@22718
   591
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
paulson@14267
   592
haftmann@23684
   593
text {* @{term "op dvd"} is a partial order *}
haftmann@23684
   594
haftmann@25942
   595
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"]
haftmann@23684
   596
  by unfold_locales (auto intro: dvd_trans dvd_anti_sym)
haftmann@23684
   597
paulson@14267
   598
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
wenzelm@22718
   599
  unfolding dvd_def
wenzelm@22718
   600
  by (blast intro: add_mult_distrib2 [symmetric])
paulson@14267
   601
paulson@14267
   602
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
wenzelm@22718
   603
  unfolding dvd_def
wenzelm@22718
   604
  by (blast intro: diff_mult_distrib2 [symmetric])
paulson@14267
   605
paulson@14267
   606
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
wenzelm@22718
   607
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   608
  apply (blast intro: dvd_add)
wenzelm@22718
   609
  done
paulson@14267
   610
paulson@14267
   611
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
wenzelm@22718
   612
  by (drule_tac m = m in dvd_diff, auto)
paulson@14267
   613
paulson@14267
   614
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
wenzelm@22718
   615
  unfolding dvd_def by (blast intro: mult_left_commute)
paulson@14267
   616
paulson@14267
   617
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
wenzelm@22718
   618
  apply (subst mult_commute)
wenzelm@22718
   619
  apply (erule dvd_mult)
wenzelm@22718
   620
  done
paulson@14267
   621
paulson@17084
   622
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
wenzelm@22718
   623
  by (rule dvd_refl [THEN dvd_mult])
paulson@17084
   624
paulson@17084
   625
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
wenzelm@22718
   626
  by (rule dvd_refl [THEN dvd_mult2])
paulson@14267
   627
paulson@14267
   628
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
wenzelm@22718
   629
  apply (rule iffI)
wenzelm@22718
   630
   apply (erule_tac [2] dvd_add)
wenzelm@22718
   631
   apply (rule_tac [2] dvd_refl)
wenzelm@22718
   632
  apply (subgoal_tac "n = (n+k) -k")
wenzelm@22718
   633
   prefer 2 apply simp
wenzelm@22718
   634
  apply (erule ssubst)
wenzelm@22718
   635
  apply (erule dvd_diff)
wenzelm@22718
   636
  apply (rule dvd_refl)
wenzelm@22718
   637
  done
paulson@14267
   638
paulson@14267
   639
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
wenzelm@22718
   640
  unfolding dvd_def
wenzelm@22718
   641
  apply (case_tac "n = 0", auto)
wenzelm@22718
   642
  apply (blast intro: mod_mult_distrib2 [symmetric])
wenzelm@22718
   643
  done
paulson@14267
   644
paulson@14267
   645
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
wenzelm@22718
   646
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
wenzelm@22718
   647
   apply (simp add: mod_div_equality)
wenzelm@22718
   648
  apply (simp only: dvd_add dvd_mult)
wenzelm@22718
   649
  done
paulson@14267
   650
paulson@14267
   651
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
wenzelm@22718
   652
  by (blast intro: dvd_mod_imp_dvd dvd_mod)
paulson@14267
   653
paulson@14267
   654
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
wenzelm@22718
   655
  unfolding dvd_def
wenzelm@22718
   656
  apply (erule exE)
wenzelm@22718
   657
  apply (simp add: mult_ac)
wenzelm@22718
   658
  done
paulson@14267
   659
paulson@14267
   660
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
wenzelm@22718
   661
  apply auto
wenzelm@22718
   662
   apply (subgoal_tac "m*n dvd m*1")
wenzelm@22718
   663
   apply (drule dvd_mult_cancel, auto)
wenzelm@22718
   664
  done
paulson@14267
   665
paulson@14267
   666
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
wenzelm@22718
   667
  apply (subst mult_commute)
wenzelm@22718
   668
  apply (erule dvd_mult_cancel1)
wenzelm@22718
   669
  done
paulson@14267
   670
paulson@14267
   671
lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
wenzelm@22718
   672
  apply (unfold dvd_def, clarify)
wenzelm@22718
   673
  apply (rule_tac x = "k*ka" in exI)
wenzelm@22718
   674
  apply (simp add: mult_ac)
wenzelm@22718
   675
  done
paulson@14267
   676
paulson@14267
   677
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
wenzelm@22718
   678
  by (simp add: dvd_def mult_assoc, blast)
paulson@14267
   679
paulson@14267
   680
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
wenzelm@22718
   681
  apply (unfold dvd_def, clarify)
wenzelm@22718
   682
  apply (rule_tac x = "i*k" in exI)
wenzelm@22718
   683
  apply (simp add: mult_ac)
wenzelm@22718
   684
  done
paulson@14267
   685
paulson@14267
   686
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
wenzelm@22718
   687
  apply (unfold dvd_def, clarify)
wenzelm@22718
   688
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
wenzelm@22718
   689
  apply (erule conjE)
wenzelm@22718
   690
  apply (rule le_trans)
wenzelm@22718
   691
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
wenzelm@22718
   692
   apply (erule_tac [2] Suc_leI, simp)
wenzelm@22718
   693
  done
paulson@14267
   694
haftmann@25942
   695
lemmas dvd_eq_mod_eq_0 = dvd_def_mod [of "k\<Colon>nat" n, standard]
paulson@14267
   696
paulson@14267
   697
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
wenzelm@22718
   698
  apply (subgoal_tac "m mod n = 0")
wenzelm@22718
   699
   apply (simp add: mult_div_cancel)
wenzelm@22718
   700
  apply (simp only: dvd_eq_mod_eq_0)
wenzelm@22718
   701
  done
paulson@14267
   702
haftmann@21408
   703
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
wenzelm@22718
   704
  apply (unfold dvd_def)
wenzelm@22718
   705
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   706
  apply (simp add: power_add)
wenzelm@22718
   707
  done
haftmann@21408
   708
nipkow@25162
   709
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
wenzelm@22718
   710
  by (induct n) auto
haftmann@21408
   711
haftmann@21408
   712
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
wenzelm@22718
   713
  apply (induct j)
wenzelm@22718
   714
   apply (simp_all add: le_Suc_eq)
wenzelm@22718
   715
  apply (blast dest!: dvd_mult_right)
wenzelm@22718
   716
  done
haftmann@21408
   717
haftmann@21408
   718
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
wenzelm@22718
   719
  apply (rule power_le_imp_le_exp, assumption)
wenzelm@22718
   720
  apply (erule dvd_imp_le, simp)
wenzelm@22718
   721
  done
haftmann@21408
   722
paulson@14267
   723
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
wenzelm@22718
   724
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
   725
wenzelm@22718
   726
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
   727
paulson@14267
   728
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
   729
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
wenzelm@22718
   730
  apply (cut_tac m = m in mod_div_equality)
wenzelm@22718
   731
  apply (simp only: add_ac)
wenzelm@22718
   732
  apply (blast intro: sym)
wenzelm@22718
   733
  done
paulson@14267
   734
paulson@14131
   735
nipkow@13152
   736
lemma split_div:
nipkow@13189
   737
 "P(n div k :: nat) =
nipkow@13189
   738
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
   739
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   740
proof
nipkow@13189
   741
  assume P: ?P
nipkow@13189
   742
  show ?Q
nipkow@13189
   743
  proof (cases)
nipkow@13189
   744
    assume "k = 0"
nipkow@13189
   745
    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   746
  next
nipkow@13189
   747
    assume not0: "k \<noteq> 0"
nipkow@13189
   748
    thus ?Q
nipkow@13189
   749
    proof (simp, intro allI impI)
nipkow@13189
   750
      fix i j
nipkow@13189
   751
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
   752
      show "P i"
nipkow@13189
   753
      proof (cases)
wenzelm@22718
   754
        assume "i = 0"
wenzelm@22718
   755
        with n j P show "P i" by simp
nipkow@13189
   756
      next
wenzelm@22718
   757
        assume "i \<noteq> 0"
wenzelm@22718
   758
        with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
   759
      qed
nipkow@13189
   760
    qed
nipkow@13189
   761
  qed
nipkow@13189
   762
next
nipkow@13189
   763
  assume Q: ?Q
nipkow@13189
   764
  show ?P
nipkow@13189
   765
  proof (cases)
nipkow@13189
   766
    assume "k = 0"
nipkow@13189
   767
    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   768
  next
nipkow@13189
   769
    assume not0: "k \<noteq> 0"
nipkow@13189
   770
    with Q have R: ?R by simp
nipkow@13189
   771
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   772
    show ?P by simp
nipkow@13189
   773
  qed
nipkow@13189
   774
qed
nipkow@13189
   775
berghofe@13882
   776
lemma split_div_lemma:
paulson@14267
   777
  "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
nipkow@25162
   778
apply (rule iffI)
nipkow@25162
   779
 apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
nipkow@25162
   780
   prefer 3; apply assumption
nipkow@25162
   781
  apply (simp_all add: quorem_def)
nipkow@25162
   782
 apply arith
nipkow@25162
   783
apply (rule conjI)
nipkow@25162
   784
 apply (rule_tac P="%x. n * (m div n) \<le> x" in
berghofe@13882
   785
    subst [OF mod_div_equality [of _ n]])
nipkow@25162
   786
 apply (simp only: add: mult_ac)
nipkow@25162
   787
 apply (rule_tac P="%x. x < n + n * (m div n)" in
berghofe@13882
   788
    subst [OF mod_div_equality [of _ n]])
nipkow@25162
   789
apply (simp only: add: mult_ac add_ac)
nipkow@25162
   790
apply (rule add_less_mono1, simp)
nipkow@25162
   791
done
berghofe@13882
   792
berghofe@13882
   793
theorem split_div':
berghofe@13882
   794
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   795
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
   796
  apply (case_tac "0 < n")
berghofe@13882
   797
  apply (simp only: add: split_div_lemma)
berghofe@13882
   798
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
berghofe@13882
   799
  done
berghofe@13882
   800
nipkow@13189
   801
lemma split_mod:
nipkow@13189
   802
 "P(n mod k :: nat) =
nipkow@13189
   803
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   804
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   805
proof
nipkow@13189
   806
  assume P: ?P
nipkow@13189
   807
  show ?Q
nipkow@13189
   808
  proof (cases)
nipkow@13189
   809
    assume "k = 0"
nipkow@13189
   810
    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   811
  next
nipkow@13189
   812
    assume not0: "k \<noteq> 0"
nipkow@13189
   813
    thus ?Q
nipkow@13189
   814
    proof (simp, intro allI impI)
nipkow@13189
   815
      fix i j
nipkow@13189
   816
      assume "n = k*i + j" "j < k"
nipkow@13189
   817
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
   818
    qed
nipkow@13189
   819
  qed
nipkow@13189
   820
next
nipkow@13189
   821
  assume Q: ?Q
nipkow@13189
   822
  show ?P
nipkow@13189
   823
  proof (cases)
nipkow@13189
   824
    assume "k = 0"
nipkow@13189
   825
    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   826
  next
nipkow@13189
   827
    assume not0: "k \<noteq> 0"
nipkow@13189
   828
    with Q have R: ?R by simp
nipkow@13189
   829
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   830
    show ?P by simp
nipkow@13189
   831
  qed
nipkow@13189
   832
qed
nipkow@13189
   833
berghofe@13882
   834
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
berghofe@13882
   835
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
berghofe@13882
   836
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
   837
  apply arith
berghofe@13882
   838
  done
berghofe@13882
   839
haftmann@22800
   840
lemma div_mod_equality':
haftmann@22800
   841
  fixes m n :: nat
haftmann@22800
   842
  shows "m div n * n = m - m mod n"
haftmann@22800
   843
proof -
haftmann@22800
   844
  have "m mod n \<le> m mod n" ..
haftmann@22800
   845
  from div_mod_equality have 
haftmann@22800
   846
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
haftmann@22800
   847
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
haftmann@22800
   848
    "m div n * n + (m mod n - m mod n) = m - m mod n"
haftmann@22800
   849
    by simp
haftmann@22800
   850
  then show ?thesis by simp
haftmann@22800
   851
qed
haftmann@22800
   852
haftmann@22800
   853
haftmann@25942
   854
subsubsection {*An ``induction'' law for modulus arithmetic.*}
paulson@14640
   855
paulson@14640
   856
lemma mod_induct_0:
paulson@14640
   857
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
   858
  and base: "P i" and i: "i<p"
paulson@14640
   859
  shows "P 0"
paulson@14640
   860
proof (rule ccontr)
paulson@14640
   861
  assume contra: "\<not>(P 0)"
paulson@14640
   862
  from i have p: "0<p" by simp
paulson@14640
   863
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
   864
  proof
paulson@14640
   865
    fix k
paulson@14640
   866
    show "?A k"
paulson@14640
   867
    proof (induct k)
paulson@14640
   868
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
   869
    next
paulson@14640
   870
      fix n
paulson@14640
   871
      assume ih: "?A n"
paulson@14640
   872
      show "?A (Suc n)"
paulson@14640
   873
      proof (clarsimp)
wenzelm@22718
   874
        assume y: "P (p - Suc n)"
wenzelm@22718
   875
        have n: "Suc n < p"
wenzelm@22718
   876
        proof (rule ccontr)
wenzelm@22718
   877
          assume "\<not>(Suc n < p)"
wenzelm@22718
   878
          hence "p - Suc n = 0"
wenzelm@22718
   879
            by simp
wenzelm@22718
   880
          with y contra show "False"
wenzelm@22718
   881
            by simp
wenzelm@22718
   882
        qed
wenzelm@22718
   883
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
   884
        from p have "p - Suc n < p" by arith
wenzelm@22718
   885
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
   886
          by blast
wenzelm@22718
   887
        show "False"
wenzelm@22718
   888
        proof (cases "n=0")
wenzelm@22718
   889
          case True
wenzelm@22718
   890
          with z n2 contra show ?thesis by simp
wenzelm@22718
   891
        next
wenzelm@22718
   892
          case False
wenzelm@22718
   893
          with p have "p-n < p" by arith
wenzelm@22718
   894
          with z n2 False ih show ?thesis by simp
wenzelm@22718
   895
        qed
paulson@14640
   896
      qed
paulson@14640
   897
    qed
paulson@14640
   898
  qed
paulson@14640
   899
  moreover
paulson@14640
   900
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
   901
    by (blast dest: less_imp_add_positive)
paulson@14640
   902
  hence "0<k \<and> i=p-k" by auto
paulson@14640
   903
  moreover
paulson@14640
   904
  note base
paulson@14640
   905
  ultimately
paulson@14640
   906
  show "False" by blast
paulson@14640
   907
qed
paulson@14640
   908
paulson@14640
   909
lemma mod_induct:
paulson@14640
   910
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
   911
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
   912
  shows "P j"
paulson@14640
   913
proof -
paulson@14640
   914
  have "\<forall>j<p. P j"
paulson@14640
   915
  proof
paulson@14640
   916
    fix j
paulson@14640
   917
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
   918
    proof (induct j)
paulson@14640
   919
      from step base i show "?A 0"
wenzelm@22718
   920
        by (auto elim: mod_induct_0)
paulson@14640
   921
    next
paulson@14640
   922
      fix k
paulson@14640
   923
      assume ih: "?A k"
paulson@14640
   924
      show "?A (Suc k)"
paulson@14640
   925
      proof
wenzelm@22718
   926
        assume suc: "Suc k < p"
wenzelm@22718
   927
        hence k: "k<p" by simp
wenzelm@22718
   928
        with ih have "P k" ..
wenzelm@22718
   929
        with step k have "P (Suc k mod p)"
wenzelm@22718
   930
          by blast
wenzelm@22718
   931
        moreover
wenzelm@22718
   932
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
   933
          by simp
wenzelm@22718
   934
        ultimately
wenzelm@22718
   935
        show "P (Suc k)" by simp
paulson@14640
   936
      qed
paulson@14640
   937
    qed
paulson@14640
   938
  qed
paulson@14640
   939
  with j show ?thesis by blast
paulson@14640
   940
qed
paulson@14640
   941
paulson@14640
   942
chaieb@18202
   943
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
chaieb@18202
   944
  apply (rule trans [symmetric])
wenzelm@22718
   945
   apply (rule mod_add1_eq, simp)
chaieb@18202
   946
  apply (rule mod_add1_eq [symmetric])
chaieb@18202
   947
  done
chaieb@18202
   948
chaieb@18202
   949
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
wenzelm@22718
   950
  apply (rule trans [symmetric])
wenzelm@22718
   951
   apply (rule mod_add1_eq, simp)
wenzelm@22718
   952
  apply (rule mod_add1_eq [symmetric])
wenzelm@22718
   953
  done
chaieb@18202
   954
haftmann@22800
   955
lemma mod_div_decomp:
haftmann@22800
   956
  fixes n k :: nat
haftmann@22800
   957
  obtains m q where "m = n div k" and "q = n mod k"
haftmann@22800
   958
    and "n = m * k + q"
haftmann@22800
   959
proof -
haftmann@22800
   960
  from mod_div_equality have "n = n div k * k + n mod k" by auto
haftmann@22800
   961
  moreover have "n div k = n div k" ..
haftmann@22800
   962
  moreover have "n mod k = n mod k" ..
haftmann@22800
   963
  note that ultimately show thesis by blast
haftmann@22800
   964
qed
haftmann@22800
   965
haftmann@20589
   966
haftmann@25942
   967
subsubsection {* Code generation for div, mod and dvd on nat *}
haftmann@20589
   968
haftmann@22845
   969
definition [code func del]:
haftmann@20589
   970
  "divmod (m\<Colon>nat) n = (m div n, m mod n)"
haftmann@20589
   971
wenzelm@22718
   972
lemma divmod_zero [code]: "divmod m 0 = (0, m)"
haftmann@20589
   973
  unfolding divmod_def by simp
haftmann@20589
   974
haftmann@20589
   975
lemma divmod_succ [code]:
haftmann@20589
   976
  "divmod m (Suc k) = (if m < Suc k then (0, m) else
haftmann@20589
   977
    let
haftmann@20589
   978
      (p, q) = divmod (m - Suc k) (Suc k)
wenzelm@22718
   979
    in (Suc p, q))"
haftmann@20589
   980
  unfolding divmod_def Let_def split_def
haftmann@20589
   981
  by (auto intro: div_geq mod_geq)
haftmann@20589
   982
wenzelm@22718
   983
lemma div_divmod [code]: "m div n = fst (divmod m n)"
haftmann@20589
   984
  unfolding divmod_def by simp
haftmann@20589
   985
wenzelm@22718
   986
lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
haftmann@20589
   987
  unfolding divmod_def by simp
haftmann@20589
   988
haftmann@21191
   989
code_modulename SML
haftmann@23017
   990
  Divides Nat
haftmann@20640
   991
haftmann@21911
   992
code_modulename OCaml
haftmann@23017
   993
  Divides Nat
haftmann@23017
   994
haftmann@23017
   995
code_modulename Haskell
haftmann@23017
   996
  Divides Nat
haftmann@21911
   997
haftmann@23684
   998
hide (open) const divmod
paulson@14267
   999
paulson@3366
  1000
end