src/HOL/Divides.thy
author haftmann
Fri Oct 20 07:46:10 2017 +0200 (21 months ago)
changeset 66886 960509bfd47e
parent 66837 6ba663ff2b1c
child 67083 6b2c0681ef28
permissions -rw-r--r--
added lemmas and tuned proofs
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(*  Title:      HOL/Divides.thy
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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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section \<open>More on quotient and remainder\<close>
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theory Divides
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imports Parity
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begin
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subsection \<open>Numeral division with a pragmatic type class\<close>
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text \<open>
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  The following type class contains everything necessary to formulate
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  a division algorithm in ring structures with numerals, restricted
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  to its positive segments.  This is its primary motivation, and it
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  could surely be formulated using a more fine-grained, more algebraic
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  and less technical class hierarchy.
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\<close>
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class unique_euclidean_semiring_numeral = semiring_parity + linordered_semidom +
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  assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
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    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
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    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
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    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
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    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
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    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
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    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
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    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
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  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
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  fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
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    and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
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  assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
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    and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
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    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
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    else (2 * q, r))"
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    \<comment> \<open>These are conceptually definitions but force generated code
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    to be monomorphic wrt. particular instances of this class which
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    yields a significant speedup.\<close>
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begin
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lemma divmod_digit_1:
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  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
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  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
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    and "a mod (2 * b) - b = a mod b" (is "?Q")
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proof -
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  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
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    by (auto intro: trans)
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  with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
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  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
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  with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
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  define w where "w = a div b mod 2"
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  then have w_exhaust: "w = 0 \<or> w = 1" by auto
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  have mod_w: "a mod (2 * b) = a mod b + b * w"
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    by (simp add: w_def mod_mult2_eq ac_simps)
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  from assms w_exhaust have "w = 1"
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    by (auto simp add: mod_w) (insert mod_less, auto)
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  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
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  have "2 * (a div (2 * b)) = a div b - w"
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    by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
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  with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
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  then show ?P and ?Q
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    by (simp_all add: div mod add_implies_diff [symmetric])
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qed
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lemma divmod_digit_0:
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  assumes "0 < b" and "a mod (2 * b) < b"
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  shows "2 * (a div (2 * b)) = a div b" (is "?P")
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    and "a mod (2 * b) = a mod b" (is "?Q")
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proof -
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  define w where "w = a div b mod 2"
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  then have w_exhaust: "w = 0 \<or> w = 1" by auto
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  have mod_w: "a mod (2 * b) = a mod b + b * w"
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    by (simp add: w_def mod_mult2_eq ac_simps)
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  moreover have "b \<le> a mod b + b"
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  proof -
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    from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
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    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
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    then show ?thesis by simp
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  qed
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  moreover note assms w_exhaust
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  ultimately have "w = 0" by auto
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  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
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  have "2 * (a div (2 * b)) = a div b - w"
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    by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
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  with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
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  then show ?P and ?Q
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    by (simp_all add: div mod)
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qed
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lemma fst_divmod:
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  "fst (divmod m n) = numeral m div numeral n"
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  by (simp add: divmod_def)
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lemma snd_divmod:
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  "snd (divmod m n) = numeral m mod numeral n"
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  by (simp add: divmod_def)
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text \<open>
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  This is a formulation of one step (referring to one digit position)
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  in school-method division: compare the dividend at the current
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  digit position with the remainder from previous division steps
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  and evaluate accordingly.
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\<close>
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lemma divmod_step_eq [simp]:
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  "divmod_step l (q, r) = (if numeral l \<le> r
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    then (2 * q + 1, r - numeral l) else (2 * q, r))"
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  by (simp add: divmod_step_def)
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text \<open>
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  This is a formulation of school-method division.
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  If the divisor is smaller than the dividend, terminate.
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  If not, shift the dividend to the right until termination
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  occurs and then reiterate single division steps in the
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  opposite direction.
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\<close>
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lemma divmod_divmod_step:
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  "divmod m n = (if m < n then (0, numeral m)
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    else divmod_step n (divmod m (Num.Bit0 n)))"
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proof (cases "m < n")
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  case True then have "numeral m < numeral n" by simp
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  then show ?thesis
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    by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
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next
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  case False
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  have "divmod m n =
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    divmod_step n (numeral m div (2 * numeral n),
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      numeral m mod (2 * numeral n))"
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  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
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    case True
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    with divmod_step_eq
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      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
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        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
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        by simp
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    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
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      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
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      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
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      by simp_all
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    ultimately show ?thesis by (simp only: divmod_def)
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  next
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    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
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      by (simp add: not_le)
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    with divmod_step_eq
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      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
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        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
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        by auto
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    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
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      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
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      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
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      by (simp_all only: zero_less_numeral)
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    ultimately show ?thesis by (simp only: divmod_def)
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  qed
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  then have "divmod m n =
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    divmod_step n (numeral m div numeral (Num.Bit0 n),
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      numeral m mod numeral (Num.Bit0 n))"
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    by (simp only: numeral.simps distrib mult_1)
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  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
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    by (simp add: divmod_def)
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  with False show ?thesis by simp
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qed
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text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
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lemma divmod_trivial [simp]:
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  "divmod Num.One Num.One = (numeral Num.One, 0)"
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  "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
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  "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
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  "divmod num.One (num.Bit0 n) = (0, Numeral1)"
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  "divmod num.One (num.Bit1 n) = (0, Numeral1)"
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  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
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text \<open>Division by an even number is a right-shift\<close>
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lemma divmod_cancel [simp]:
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  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
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  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
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proof -
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  have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
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    "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
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    by (simp_all only: numeral_mult numeral.simps distrib) simp_all
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  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
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  then show ?P and ?Q
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    by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
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      div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
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      add.commute del: numeral_times_numeral)
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qed
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text \<open>The really hard work\<close>
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lemma divmod_steps [simp]:
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  "divmod (num.Bit0 m) (num.Bit1 n) =
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      (if m \<le> n then (0, numeral (num.Bit0 m))
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       else divmod_step (num.Bit1 n)
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             (divmod (num.Bit0 m)
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               (num.Bit0 (num.Bit1 n))))"
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  "divmod (num.Bit1 m) (num.Bit1 n) =
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      (if m < n then (0, numeral (num.Bit1 m))
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       else divmod_step (num.Bit1 n)
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             (divmod (num.Bit1 m)
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               (num.Bit0 (num.Bit1 n))))"
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  by (simp_all add: divmod_divmod_step)
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lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps  
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text \<open>Special case: divisibility\<close>
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definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
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where
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  "divides_aux qr \<longleftrightarrow> snd qr = 0"
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lemma divides_aux_eq [simp]:
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  "divides_aux (q, r) \<longleftrightarrow> r = 0"
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  by (simp add: divides_aux_def)
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lemma dvd_numeral_simp [simp]:
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  "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
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  by (simp add: divmod_def mod_eq_0_iff_dvd)
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text \<open>Generic computation of quotient and remainder\<close>  
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lemma numeral_div_numeral [simp]: 
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  "numeral k div numeral l = fst (divmod k l)"
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  by (simp add: fst_divmod)
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lemma numeral_mod_numeral [simp]: 
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  "numeral k mod numeral l = snd (divmod k l)"
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  by (simp add: snd_divmod)
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lemma one_div_numeral [simp]:
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  "1 div numeral n = fst (divmod num.One n)"
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  by (simp add: fst_divmod)
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lemma one_mod_numeral [simp]:
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  "1 mod numeral n = snd (divmod num.One n)"
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  by (simp add: snd_divmod)
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text \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
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lemma cong_exp_iff_simps:
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  "numeral n mod numeral Num.One = 0
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    \<longleftrightarrow> True"
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  "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
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    \<longleftrightarrow> numeral n mod numeral q = 0"
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  "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
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    \<longleftrightarrow> False"
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  "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
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    \<longleftrightarrow> True"
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  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> True"
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  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> False"
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  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> (numeral n mod numeral q) = 0"
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  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> False"
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  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
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  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> False"
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  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> (numeral m mod numeral q) = 0"
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  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> False"
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  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
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    \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
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  by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
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end
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hide_fact (open) div_less mod_less mod_less_eq_dividend mod_mult2_eq div_mult2_eq
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subsection \<open>More on division\<close>
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instantiation nat :: unique_euclidean_semiring_numeral
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begin
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definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
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where
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   283
  divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
   284
haftmann@61275
   285
definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
haftmann@61275
   286
where
haftmann@61275
   287
  "divmod_step_nat l qr = (let (q, r) = qr
haftmann@61275
   288
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
   289
    else (2 * q, r))"
haftmann@61275
   290
haftmann@66808
   291
instance by standard
haftmann@66808
   292
  (auto simp add: divmod'_nat_def divmod_step_nat_def div_greater_zero_iff div_mult2_eq mod_mult2_eq)
haftmann@61275
   293
haftmann@61275
   294
end
haftmann@61275
   295
haftmann@61275
   296
declare divmod_algorithm_code [where ?'a = nat, code]
paulson@14267
   297
haftmann@60868
   298
lemma Suc_0_div_numeral [simp]:
haftmann@60868
   299
  fixes k l :: num
haftmann@60868
   300
  shows "Suc 0 div numeral k = fst (divmod Num.One k)"
haftmann@60868
   301
  by (simp_all add: fst_divmod)
haftmann@60868
   302
haftmann@60868
   303
lemma Suc_0_mod_numeral [simp]:
haftmann@60868
   304
  fixes k l :: num
haftmann@60868
   305
  shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
haftmann@60868
   306
  by (simp_all add: snd_divmod)
haftmann@60868
   307
haftmann@66808
   308
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
haftmann@66808
   309
  where "divmod_nat m n = (m div n, m mod n)"
haftmann@66808
   310
haftmann@66808
   311
lemma fst_divmod_nat [simp]:
haftmann@66808
   312
  "fst (divmod_nat m n) = m div n"
haftmann@66808
   313
  by (simp add: divmod_nat_def)
haftmann@66808
   314
haftmann@66808
   315
lemma snd_divmod_nat [simp]:
haftmann@66808
   316
  "snd (divmod_nat m n) = m mod n"
haftmann@66808
   317
  by (simp add: divmod_nat_def)
haftmann@66808
   318
haftmann@66808
   319
lemma divmod_nat_if [code]:
haftmann@66808
   320
  "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@66808
   321
    let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
haftmann@66808
   322
  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
haftmann@66808
   323
haftmann@66808
   324
lemma [code]:
haftmann@66808
   325
  "m div n = fst (divmod_nat m n)"
haftmann@66808
   326
  "m mod n = snd (divmod_nat m n)"
haftmann@66808
   327
  by simp_all
haftmann@66808
   328
haftmann@64635
   329
inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
haftmann@64635
   330
  where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
haftmann@64635
   331
  | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
haftmann@64635
   332
  | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
haftmann@64635
   333
      \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
haftmann@64635
   334
haftmann@64635
   335
lemma eucl_rel_int_iff:    
haftmann@64635
   336
  "eucl_rel_int k l (q, r) \<longleftrightarrow> 
haftmann@64635
   337
    k = l * q + r \<and>
haftmann@64635
   338
     (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
haftmann@64635
   339
  by (cases "r = 0")
haftmann@64635
   340
    (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
haftmann@64635
   341
    simp add: ac_simps sgn_1_pos sgn_1_neg)
haftmann@33361
   342
haftmann@33361
   343
lemma unique_quotient_lemma:
haftmann@60868
   344
  "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
haftmann@33361
   345
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
   346
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
   347
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
   348
apply (erule_tac [2] order_le_less_trans)
webertj@49962
   349
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
   350
apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962
   351
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
   352
apply (simp add: mult_less_cancel_left)
haftmann@33361
   353
done
haftmann@33361
   354
haftmann@33361
   355
lemma unique_quotient_lemma_neg:
haftmann@60868
   356
  "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
haftmann@60868
   357
  by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
haftmann@33361
   358
haftmann@33361
   359
lemma unique_quotient:
haftmann@64635
   360
  "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> q = q'"
haftmann@64635
   361
  apply (simp add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
haftmann@64635
   362
  apply (blast intro: order_antisym
haftmann@64635
   363
    dest: order_eq_refl [THEN unique_quotient_lemma]
haftmann@64635
   364
    order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@64635
   365
  done
haftmann@33361
   366
haftmann@33361
   367
lemma unique_remainder:
haftmann@64635
   368
  "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> r = r'"
haftmann@33361
   369
apply (subgoal_tac "q = q'")
haftmann@64635
   370
 apply (simp add: eucl_rel_int_iff)
haftmann@33361
   371
apply (blast intro: unique_quotient)
haftmann@33361
   372
done
haftmann@33361
   373
haftmann@64635
   374
lemma eucl_rel_int:
haftmann@64635
   375
  "eucl_rel_int k l (k div l, k mod l)"
haftmann@64592
   376
proof (cases k rule: int_cases3)
haftmann@64592
   377
  case zero
haftmann@64592
   378
  then show ?thesis
haftmann@64635
   379
    by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
haftmann@64592
   380
next
haftmann@64592
   381
  case (pos n)
haftmann@64592
   382
  then show ?thesis
haftmann@64592
   383
    using div_mult_mod_eq [of n]
haftmann@64592
   384
    by (cases l rule: int_cases3)
haftmann@64592
   385
      (auto simp del: of_nat_mult of_nat_add
haftmann@64592
   386
        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
haftmann@64635
   387
        eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
haftmann@64592
   388
next
haftmann@64592
   389
  case (neg n)
haftmann@64592
   390
  then show ?thesis
haftmann@64592
   391
    using div_mult_mod_eq [of n]
haftmann@64592
   392
    by (cases l rule: int_cases3)
haftmann@64592
   393
      (auto simp del: of_nat_mult of_nat_add
haftmann@64592
   394
        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
haftmann@64635
   395
        eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
haftmann@64592
   396
qed
haftmann@33361
   397
huffman@47141
   398
lemma divmod_int_unique:
haftmann@64635
   399
  assumes "eucl_rel_int k l (q, r)"
haftmann@60868
   400
  shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
haftmann@64635
   401
  using assms eucl_rel_int [of k l]
haftmann@60868
   402
  using unique_quotient [of k l] unique_remainder [of k l]
haftmann@60868
   403
  by auto
haftmann@64592
   404
haftmann@64715
   405
lemma div_abs_eq_div_nat:
haftmann@64715
   406
  "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
haftmann@64715
   407
  by (simp add: divide_int_def)
haftmann@64715
   408
haftmann@64715
   409
lemma mod_abs_eq_div_nat:
haftmann@64715
   410
  "\<bar>k\<bar> mod \<bar>l\<bar> = int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)"
haftmann@66816
   411
  by (simp add: modulo_int_def)
haftmann@66816
   412
haftmann@66816
   413
lemma zdiv_int:
haftmann@66816
   414
  "int (a div b) = int a div int b"
haftmann@66816
   415
  by (simp add: divide_int_def sgn_1_pos)
haftmann@66816
   416
haftmann@66816
   417
lemma zmod_int:
haftmann@66816
   418
  "int (a mod b) = int a mod int b"
haftmann@66816
   419
  by (simp add: modulo_int_def sgn_1_pos)
haftmann@64715
   420
haftmann@64715
   421
lemma div_sgn_abs_cancel:
haftmann@64715
   422
  fixes k l v :: int
haftmann@64715
   423
  assumes "v \<noteq> 0"
haftmann@64715
   424
  shows "(sgn v * \<bar>k\<bar>) div (sgn v * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@64715
   425
proof -
haftmann@64715
   426
  from assms have "sgn v = - 1 \<or> sgn v = 1"
haftmann@64715
   427
    by (cases "v \<ge> 0") auto
haftmann@64715
   428
  then show ?thesis
blanchet@66630
   429
    using assms unfolding divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"]
blanchet@66630
   430
    by (fastforce simp add: not_less div_abs_eq_div_nat)
haftmann@64715
   431
qed
haftmann@64715
   432
haftmann@64715
   433
lemma div_eq_sgn_abs:
haftmann@64715
   434
  fixes k l v :: int
haftmann@64715
   435
  assumes "sgn k = sgn l"
haftmann@64715
   436
  shows "k div l = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@64715
   437
proof (cases "l = 0")
haftmann@64715
   438
  case True
haftmann@64715
   439
  then show ?thesis
haftmann@64715
   440
    by simp
haftmann@64715
   441
next
haftmann@64715
   442
  case False
haftmann@64715
   443
  with assms have "(sgn k * \<bar>k\<bar>) div (sgn l * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@66816
   444
    using div_sgn_abs_cancel [of l k l] by simp
haftmann@64715
   445
  then show ?thesis
haftmann@64715
   446
    by (simp add: sgn_mult_abs)
haftmann@64715
   447
qed
haftmann@64715
   448
haftmann@64715
   449
lemma div_dvd_sgn_abs:
haftmann@64715
   450
  fixes k l :: int
haftmann@64715
   451
  assumes "l dvd k"
haftmann@64715
   452
  shows "k div l = (sgn k * sgn l) * (\<bar>k\<bar> div \<bar>l\<bar>)"
haftmann@66816
   453
proof (cases "k = 0 \<or> l = 0")
haftmann@64715
   454
  case True
haftmann@64715
   455
  then show ?thesis
haftmann@66816
   456
    by auto
haftmann@64715
   457
next
haftmann@64715
   458
  case False
haftmann@66816
   459
  then have "k \<noteq> 0" and "l \<noteq> 0"
haftmann@66816
   460
    by auto
haftmann@64715
   461
  show ?thesis
haftmann@64715
   462
  proof (cases "sgn l = sgn k")
haftmann@64715
   463
    case True
haftmann@64715
   464
    then show ?thesis
haftmann@64715
   465
      by (simp add: div_eq_sgn_abs)
haftmann@64715
   466
  next
haftmann@64715
   467
    case False
haftmann@66816
   468
    with \<open>k \<noteq> 0\<close> \<open>l \<noteq> 0\<close>
haftmann@66816
   469
    have "sgn l * sgn k = - 1"
haftmann@66816
   470
      by (simp add: sgn_if split: if_splits)
haftmann@66816
   471
    with assms show ?thesis
haftmann@64715
   472
      unfolding divide_int_def [of k l]
haftmann@66816
   473
      by (auto simp add: zdiv_int ac_simps)
haftmann@64715
   474
  qed
haftmann@64715
   475
qed
haftmann@64715
   476
haftmann@64715
   477
lemma div_noneq_sgn_abs:
haftmann@64715
   478
  fixes k l :: int
haftmann@64715
   479
  assumes "l \<noteq> 0"
haftmann@64715
   480
  assumes "sgn k \<noteq> sgn l"
haftmann@64715
   481
  shows "k div l = - (\<bar>k\<bar> div \<bar>l\<bar>) - of_bool (\<not> l dvd k)"
haftmann@64715
   482
  using assms
haftmann@64715
   483
  by (simp only: divide_int_def [of k l], auto simp add: not_less zdiv_int)
haftmann@64715
   484
  
haftmann@64592
   485
text\<open>Basic laws about division and remainder\<close>
haftmann@64592
   486
huffman@47141
   487
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
haftmann@64635
   488
  using eucl_rel_int [of a b]
haftmann@64635
   489
  by (auto simp add: eucl_rel_int_iff prod_eq_iff)
haftmann@33361
   490
haftmann@66816
   491
lemmas pos_mod_sign = pos_mod_conj [THEN conjunct1]
haftmann@66816
   492
   and pos_mod_bound = pos_mod_conj [THEN conjunct2]
haftmann@33361
   493
huffman@47141
   494
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
haftmann@64635
   495
  using eucl_rel_int [of a b]
haftmann@64635
   496
  by (auto simp add: eucl_rel_int_iff prod_eq_iff)
haftmann@33361
   497
wenzelm@45607
   498
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
   499
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
   500
haftmann@33361
   501
wenzelm@60758
   502
subsubsection \<open>General Properties of div and mod\<close>
haftmann@33361
   503
haftmann@66886
   504
lemma div_pos_pos_trivial [simp]:
haftmann@66886
   505
  "k div l = 0" if "k \<ge> 0" and "k < l" for k l :: int
haftmann@66886
   506
  using that by (auto intro!: div_int_unique [of _ _ _ k] simp add: eucl_rel_int_iff)
haftmann@33361
   507
haftmann@66886
   508
lemma div_neg_neg_trivial [simp]:
haftmann@66886
   509
  "k div l = 0" if "k \<le> 0" and "l < k" for k l :: int
haftmann@66886
   510
  using that by (auto intro!: div_int_unique [of _ _ _ k] simp add: eucl_rel_int_iff)
haftmann@33361
   511
haftmann@33361
   512
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
   513
apply (rule div_int_unique)
haftmann@64635
   514
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
   515
done
haftmann@33361
   516
haftmann@66801
   517
lemma div_positive_int:
haftmann@66801
   518
  "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
haftmann@66801
   519
  using that by (simp add: divide_int_def div_positive)
haftmann@66801
   520
haftmann@33361
   521
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
   522
haftmann@66886
   523
lemma mod_pos_pos_trivial [simp]:
haftmann@66886
   524
  "k mod l = k" if "k \<ge> 0" and "k < l" for k l :: int
haftmann@66886
   525
  using that by (auto intro!: mod_int_unique [of _ _ 0] simp add: eucl_rel_int_iff)
haftmann@33361
   526
haftmann@66886
   527
lemma mod_neg_neg_trivial [simp]:
haftmann@66886
   528
  "k mod l = k" if "k \<le> 0" and "l < k" for k l :: int
haftmann@66886
   529
  using that by (auto intro!: mod_int_unique [of _ _ 0] simp add: eucl_rel_int_iff)
haftmann@33361
   530
haftmann@33361
   531
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
   532
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@64635
   533
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
   534
done
haftmann@33361
   535
wenzelm@61799
   536
text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
wenzelm@60758
   537
wenzelm@60758
   538
wenzelm@60758
   539
subsubsection \<open>Laws for div and mod with Unary Minus\<close>
haftmann@33361
   540
haftmann@33361
   541
lemma zminus1_lemma:
haftmann@64635
   542
     "eucl_rel_int a b (q, r) ==> b \<noteq> 0
haftmann@64635
   543
      ==> eucl_rel_int (-a) b (if r=0 then -q else -q - 1,
haftmann@33361
   544
                          if r=0 then 0 else b-r)"
blanchet@66630
   545
by (force simp add: eucl_rel_int_iff right_diff_distrib)
haftmann@33361
   546
haftmann@33361
   547
haftmann@33361
   548
lemma zdiv_zminus1_eq_if:
lp15@60562
   549
     "b \<noteq> (0::int)
lp15@60562
   550
      ==> (-a) div b =
haftmann@33361
   551
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@64635
   552
by (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
   553
haftmann@33361
   554
lemma zmod_zminus1_eq_if:
haftmann@33361
   555
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
   556
apply (case_tac "b = 0", simp)
haftmann@64635
   557
apply (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
   558
done
haftmann@33361
   559
haftmann@64593
   560
lemma zmod_zminus1_not_zero:
haftmann@33361
   561
  fixes k l :: int
haftmann@33361
   562
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@64592
   563
  by (simp add: mod_eq_0_iff_dvd)
haftmann@64592
   564
haftmann@64593
   565
lemma zmod_zminus2_not_zero:
haftmann@64592
   566
  fixes k l :: int
haftmann@64592
   567
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@64592
   568
  by (simp add: mod_eq_0_iff_dvd)
haftmann@33361
   569
haftmann@33361
   570
lemma zdiv_zminus2_eq_if:
haftmann@66816
   571
  "b \<noteq> (0::int)
lp15@60562
   572
      ==> a div (-b) =
haftmann@33361
   573
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@66816
   574
  by (auto simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
   575
haftmann@33361
   576
lemma zmod_zminus2_eq_if:
haftmann@66816
   577
  "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
haftmann@66816
   578
  by (auto simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
   579
haftmann@33361
   580
wenzelm@60758
   581
subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
haftmann@33361
   582
haftmann@33361
   583
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@64246
   584
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
   585
using mult_div_mod_eq [symmetric, of a' b]
haftmann@64246
   586
apply -
haftmann@33361
   587
apply (rule unique_quotient_lemma)
haftmann@33361
   588
apply (erule subst)
haftmann@33361
   589
apply (erule subst, simp_all)
haftmann@33361
   590
done
haftmann@33361
   591
haftmann@33361
   592
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@64246
   593
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
   594
using mult_div_mod_eq [symmetric, of a' b]
haftmann@64246
   595
apply -
haftmann@33361
   596
apply (rule unique_quotient_lemma_neg)
haftmann@33361
   597
apply (erule subst)
haftmann@33361
   598
apply (erule subst, simp_all)
haftmann@33361
   599
done
haftmann@33361
   600
haftmann@33361
   601
wenzelm@60758
   602
subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
haftmann@33361
   603
haftmann@33361
   604
lemma q_pos_lemma:
haftmann@33361
   605
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
   606
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
   607
 apply (simp add: zero_less_mult_iff)
webertj@49962
   608
apply (simp add: distrib_left)
haftmann@33361
   609
done
haftmann@33361
   610
haftmann@33361
   611
lemma zdiv_mono2_lemma:
lp15@60562
   612
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
lp15@60562
   613
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
haftmann@33361
   614
      ==> q \<le> (q'::int)"
lp15@60562
   615
apply (frule q_pos_lemma, assumption+)
haftmann@33361
   616
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
   617
 apply (simp add: mult_less_cancel_left)
haftmann@33361
   618
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
   619
 prefer 2 apply simp
webertj@49962
   620
apply (simp (no_asm_simp) add: distrib_left)
haftmann@57512
   621
apply (subst add.commute, rule add_less_le_mono, arith)
haftmann@33361
   622
apply (rule mult_right_mono, auto)
haftmann@33361
   623
done
haftmann@33361
   624
haftmann@33361
   625
lemma zdiv_mono2:
haftmann@33361
   626
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
   627
apply (subgoal_tac "b \<noteq> 0")
haftmann@64246
   628
  prefer 2 apply arith
haftmann@64246
   629
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
   630
using mult_div_mod_eq [symmetric, of a b']
haftmann@64246
   631
apply -
haftmann@33361
   632
apply (rule zdiv_mono2_lemma)
haftmann@33361
   633
apply (erule subst)
haftmann@33361
   634
apply (erule subst, simp_all)
haftmann@33361
   635
done
haftmann@33361
   636
haftmann@33361
   637
lemma q_neg_lemma:
haftmann@33361
   638
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
   639
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
   640
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
   641
done
haftmann@33361
   642
haftmann@33361
   643
lemma zdiv_mono2_neg_lemma:
lp15@60562
   644
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
lp15@60562
   645
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
haftmann@33361
   646
      ==> q' \<le> (q::int)"
lp15@60562
   647
apply (frule q_neg_lemma, assumption+)
haftmann@33361
   648
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
   649
 apply (simp add: mult_less_cancel_left)
webertj@49962
   650
apply (simp add: distrib_left)
haftmann@33361
   651
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
   652
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
   653
done
haftmann@33361
   654
haftmann@33361
   655
lemma zdiv_mono2_neg:
haftmann@33361
   656
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@64246
   657
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
   658
using mult_div_mod_eq [symmetric, of a b']
haftmann@64246
   659
apply -
haftmann@33361
   660
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
   661
apply (erule subst)
haftmann@33361
   662
apply (erule subst, simp_all)
haftmann@33361
   663
done
haftmann@33361
   664
haftmann@33361
   665
wenzelm@60758
   666
subsubsection \<open>More Algebraic Laws for div and mod\<close>
wenzelm@60758
   667
haftmann@33361
   668
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@66814
   669
  by (fact div_mult1_eq)
haftmann@33361
   670
haftmann@33361
   671
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
   672
lemma zdiv_zadd1_eq:
haftmann@33361
   673
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@66814
   674
  by (fact div_add1_eq)
haftmann@33361
   675
haftmann@33361
   676
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
   677
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
   678
haftmann@33361
   679
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
   680
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
   681
haftmann@33361
   682
wenzelm@60758
   683
subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
haftmann@33361
   684
haftmann@33361
   685
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
   686
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
   687
  to cause particular problems.*)
haftmann@33361
   688
wenzelm@60758
   689
text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
haftmann@33361
   690
blanchet@55085
   691
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
haftmann@33361
   692
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
   693
 apply (simp add: algebra_simps)
haftmann@33361
   694
apply (rule order_le_less_trans)
haftmann@33361
   695
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
   696
 apply (rule mult_left_mono_neg)
huffman@35216
   697
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
   698
 apply (simp)
haftmann@33361
   699
apply (simp)
haftmann@33361
   700
done
haftmann@33361
   701
haftmann@33361
   702
lemma zmult2_lemma_aux2:
haftmann@33361
   703
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
   704
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
   705
 apply arith
haftmann@33361
   706
apply (simp add: mult_le_0_iff)
haftmann@33361
   707
done
haftmann@33361
   708
haftmann@33361
   709
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
   710
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
   711
apply arith
haftmann@33361
   712
apply (simp add: zero_le_mult_iff)
haftmann@33361
   713
done
haftmann@33361
   714
haftmann@33361
   715
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
   716
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
   717
 apply (simp add: right_diff_distrib)
haftmann@33361
   718
apply (rule order_less_le_trans)
haftmann@33361
   719
 apply (erule mult_strict_right_mono)
haftmann@33361
   720
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
   721
  apply simp
huffman@35216
   722
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
   723
apply simp
haftmann@33361
   724
done
haftmann@33361
   725
haftmann@64635
   726
lemma zmult2_lemma: "[| eucl_rel_int a b (q, r); 0 < c |]
haftmann@64635
   727
      ==> eucl_rel_int a (b * c) (q div c, b*(q mod c) + r)"
haftmann@64635
   728
by (auto simp add: mult.assoc eucl_rel_int_iff linorder_neq_iff
lp15@60562
   729
                   zero_less_mult_iff distrib_left [symmetric]
nipkow@62390
   730
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: if_split_asm)
haftmann@33361
   731
haftmann@53068
   732
lemma zdiv_zmult2_eq:
haftmann@53068
   733
  fixes a b c :: int
haftmann@53068
   734
  shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361
   735
apply (case_tac "b = 0", simp)
haftmann@64635
   736
apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
   737
done
haftmann@33361
   738
haftmann@33361
   739
lemma zmod_zmult2_eq:
haftmann@53068
   740
  fixes a b c :: int
haftmann@53068
   741
  shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361
   742
apply (case_tac "b = 0", simp)
haftmann@64635
   743
apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
   744
done
haftmann@33361
   745
huffman@47108
   746
lemma div_pos_geq:
huffman@47108
   747
  fixes k l :: int
huffman@47108
   748
  assumes "0 < l" and "l \<le> k"
huffman@47108
   749
  shows "k div l = (k - l) div l + 1"
huffman@47108
   750
proof -
huffman@47108
   751
  have "k = (k - l) + l" by simp
huffman@47108
   752
  then obtain j where k: "k = j + l" ..
eberlm@63499
   753
  with assms show ?thesis by (simp add: div_add_self2)
huffman@47108
   754
qed
huffman@47108
   755
huffman@47108
   756
lemma mod_pos_geq:
huffman@47108
   757
  fixes k l :: int
huffman@47108
   758
  assumes "0 < l" and "l \<le> k"
huffman@47108
   759
  shows "k mod l = (k - l) mod l"
huffman@47108
   760
proof -
huffman@47108
   761
  have "k = (k - l) + l" by simp
huffman@47108
   762
  then obtain j where k: "k = j + l" ..
huffman@47108
   763
  with assms show ?thesis by simp
huffman@47108
   764
qed
huffman@47108
   765
haftmann@33361
   766
wenzelm@60758
   767
subsubsection \<open>Splitting Rules for div and mod\<close>
wenzelm@60758
   768
wenzelm@60758
   769
text\<open>The proofs of the two lemmas below are essentially identical\<close>
haftmann@33361
   770
haftmann@33361
   771
lemma split_pos_lemma:
lp15@60562
   772
 "0<k ==>
haftmann@33361
   773
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
haftmann@66886
   774
  by auto
haftmann@33361
   775
haftmann@33361
   776
lemma split_neg_lemma:
haftmann@33361
   777
 "k<0 ==>
haftmann@33361
   778
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
haftmann@66886
   779
  by auto
haftmann@33361
   780
haftmann@33361
   781
lemma split_zdiv:
haftmann@33361
   782
 "P(n div k :: int) =
lp15@60562
   783
  ((k = 0 --> P 0) &
lp15@60562
   784
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
haftmann@33361
   785
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@66886
   786
  apply (cases "k = 0")
haftmann@66886
   787
  apply simp
haftmann@33361
   788
apply (simp only: linorder_neq_iff)
haftmann@66886
   789
 apply (auto simp add: split_pos_lemma [of concl: "%x y. P x"]
haftmann@33361
   790
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
   791
done
haftmann@33361
   792
haftmann@33361
   793
lemma split_zmod:
haftmann@33361
   794
 "P(n mod k :: int) =
lp15@60562
   795
  ((k = 0 --> P n) &
lp15@60562
   796
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
haftmann@33361
   797
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
   798
apply (case_tac "k=0", simp)
haftmann@33361
   799
apply (simp only: linorder_neq_iff)
lp15@60562
   800
apply (erule disjE)
lp15@60562
   801
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
haftmann@33361
   802
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
   803
done
haftmann@33361
   804
haftmann@63950
   805
text \<open>Enable (lin)arith to deal with @{const divide} and @{const modulo}
webertj@33730
   806
  when these are applied to some constant that is of the form
wenzelm@60758
   807
  @{term "numeral k"}:\<close>
huffman@47108
   808
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
   809
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
   810
haftmann@33361
   811
wenzelm@61799
   812
subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
huffman@47166
   813
haftmann@64635
   814
lemma pos_eucl_rel_int_mult_2:
huffman@47166
   815
  assumes "0 \<le> b"
haftmann@64635
   816
  assumes "eucl_rel_int a b (q, r)"
haftmann@64635
   817
  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
haftmann@64635
   818
  using assms unfolding eucl_rel_int_iff by auto
haftmann@64635
   819
haftmann@64635
   820
lemma neg_eucl_rel_int_mult_2:
huffman@47166
   821
  assumes "b \<le> 0"
haftmann@64635
   822
  assumes "eucl_rel_int (a + 1) b (q, r)"
haftmann@64635
   823
  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
haftmann@64635
   824
  using assms unfolding eucl_rel_int_iff by auto
haftmann@33361
   825
wenzelm@60758
   826
text\<open>computing div by shifting\<close>
haftmann@33361
   827
haftmann@33361
   828
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
haftmann@64635
   829
  using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
huffman@47166
   830
  by (rule div_int_unique)
haftmann@33361
   831
lp15@60562
   832
lemma neg_zdiv_mult_2:
boehmes@35815
   833
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
haftmann@64635
   834
  using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
huffman@47166
   835
  by (rule div_int_unique)
haftmann@33361
   836
huffman@47108
   837
(* FIXME: add rules for negative numerals *)
huffman@47108
   838
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
   839
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
   840
    numeral v div (numeral w :: int)"
huffman@47108
   841
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
   842
  by (rule div_mult_mult1, simp)
huffman@47108
   843
huffman@47108
   844
lemma zdiv_numeral_Bit1 [simp]:
lp15@60562
   845
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
huffman@47108
   846
    (numeral v div (numeral w :: int))"
huffman@47108
   847
  unfolding numeral.simps
haftmann@57512
   848
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
   849
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
   850
haftmann@33361
   851
lemma pos_zmod_mult_2:
haftmann@33361
   852
  fixes a b :: int
haftmann@33361
   853
  assumes "0 \<le> a"
haftmann@33361
   854
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
haftmann@64635
   855
  using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
huffman@47166
   856
  by (rule mod_int_unique)
haftmann@33361
   857
haftmann@33361
   858
lemma neg_zmod_mult_2:
haftmann@33361
   859
  fixes a b :: int
haftmann@33361
   860
  assumes "a \<le> 0"
haftmann@33361
   861
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
haftmann@64635
   862
  using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
huffman@47166
   863
  by (rule mod_int_unique)
haftmann@33361
   864
huffman@47108
   865
(* FIXME: add rules for negative numerals *)
huffman@47108
   866
lemma zmod_numeral_Bit0 [simp]:
lp15@60562
   867
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
huffman@47108
   868
    (2::int) * (numeral v mod numeral w)"
huffman@47108
   869
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
   870
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
   871
huffman@47108
   872
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
   873
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
   874
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
   875
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
haftmann@57512
   876
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
   877
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
   878
nipkow@39489
   879
lemma zdiv_eq_0_iff:
haftmann@66886
   880
  "i div k = 0 \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i" (is "?L = ?R")
haftmann@66886
   881
  for i k :: int
nipkow@39489
   882
proof
nipkow@39489
   883
  assume ?L
haftmann@66886
   884
  moreover have "?L \<longrightarrow> ?R"
haftmann@66886
   885
    by (rule split_zdiv [THEN iffD2]) simp
haftmann@66886
   886
  ultimately show ?R
haftmann@66886
   887
    by blast
nipkow@39489
   888
next
haftmann@66886
   889
  assume ?R then show ?L
haftmann@66886
   890
    by auto
nipkow@39489
   891
qed
nipkow@39489
   892
haftmann@63947
   893
lemma zmod_trival_iff:
haftmann@63947
   894
  fixes i k :: int
haftmann@63947
   895
  shows "i mod k = i \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i"
haftmann@63947
   896
proof -
haftmann@63947
   897
  have "i mod k = i \<longleftrightarrow> i div k = 0"
haftmann@64242
   898
    by safe (insert div_mult_mod_eq [of i k], auto)
haftmann@63947
   899
  with zdiv_eq_0_iff
haftmann@63947
   900
  show ?thesis
haftmann@63947
   901
    by simp
haftmann@63947
   902
qed
nipkow@39489
   903
haftmann@64785
   904
  
wenzelm@60758
   905
subsubsection \<open>Quotients of Signs\<close>
haftmann@33361
   906
haftmann@60868
   907
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@60868
   908
by (simp add: divide_int_def)
haftmann@60868
   909
haftmann@60868
   910
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@63950
   911
by (simp add: modulo_int_def)
haftmann@60868
   912
haftmann@33361
   913
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
   914
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
   915
apply (rule order_trans)
haftmann@33361
   916
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
   917
apply (auto simp add: div_eq_minus1)
haftmann@33361
   918
done
haftmann@33361
   919
haftmann@33361
   920
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
   921
by (drule zdiv_mono1_neg, auto)
haftmann@33361
   922
haftmann@33361
   923
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
   924
by (drule zdiv_mono1, auto)
haftmann@33361
   925
wenzelm@61799
   926
text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
wenzelm@61799
   927
conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
wenzelm@60758
   928
They should all be simp rules unless that causes too much search.\<close>
nipkow@33804
   929
haftmann@33361
   930
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
   931
apply auto
haftmann@33361
   932
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
   933
apply (auto simp add: linorder_neq_iff)
haftmann@33361
   934
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
   935
apply (blast intro: div_neg_pos_less0)
haftmann@33361
   936
done
haftmann@33361
   937
haftmann@60868
   938
lemma pos_imp_zdiv_pos_iff:
haftmann@60868
   939
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
haftmann@60868
   940
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
haftmann@60868
   941
by arith
haftmann@60868
   942
haftmann@33361
   943
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
   944
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
   945
apply (subst div_minus_minus [symmetric])
haftmann@33361
   946
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
   947
done
haftmann@33361
   948
haftmann@33361
   949
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
   950
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
   951
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
   952
haftmann@33361
   953
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
   954
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
   955
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
   956
nipkow@33804
   957
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
   958
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
   959
apply rule
nipkow@33804
   960
 apply rule
nipkow@33804
   961
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
   962
 apply(cases "b=0")apply simp
nipkow@33804
   963
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
   964
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
   965
done
nipkow@33804
   966
nipkow@39489
   967
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
   968
apply (rule split_zmod[THEN iffD2])
nipkow@44890
   969
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
   970
done
nipkow@39489
   971
haftmann@60868
   972
haftmann@60868
   973
subsubsection \<open>Computation of Division and Remainder\<close>
haftmann@60868
   974
haftmann@66806
   975
instantiation int :: unique_euclidean_semiring_numeral
haftmann@61275
   976
begin
haftmann@61275
   977
haftmann@61275
   978
definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
haftmann@61275
   979
where
haftmann@61275
   980
  "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
   981
haftmann@61275
   982
definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
haftmann@61275
   983
where
haftmann@61275
   984
  "divmod_step_int l qr = (let (q, r) = qr
haftmann@61275
   985
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
   986
    else (2 * q, r))"
haftmann@61275
   987
haftmann@61275
   988
instance
haftmann@61275
   989
  by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
haftmann@66886
   990
    pos_imp_zdiv_pos_iff zmod_zmult2_eq zdiv_zmult2_eq)
haftmann@61275
   991
haftmann@61275
   992
end
haftmann@61275
   993
haftmann@61275
   994
declare divmod_algorithm_code [where ?'a = int, code]
lp15@60562
   995
haftmann@60930
   996
context
haftmann@60930
   997
begin
haftmann@60930
   998
  
haftmann@60930
   999
qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
haftmann@60868
  1000
where
haftmann@60868
  1001
  "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
haftmann@60868
  1002
haftmann@60930
  1003
qualified lemma adjust_div_eq [simp, code]:
haftmann@60868
  1004
  "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
haftmann@60868
  1005
  by (simp add: adjust_div_def)
haftmann@60868
  1006
haftmann@60930
  1007
qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
  1008
where
haftmann@60868
  1009
  [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
haftmann@60868
  1010
haftmann@60868
  1011
lemma minus_numeral_div_numeral [simp]:
haftmann@60868
  1012
  "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  1013
proof -
haftmann@60868
  1014
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  1015
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  1016
  then show ?thesis
haftmann@60868
  1017
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  1018
qed
haftmann@60868
  1019
haftmann@60868
  1020
lemma minus_numeral_mod_numeral [simp]:
haftmann@60868
  1021
  "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@66816
  1022
proof (cases "snd (divmod m n) = (0::int)")
haftmann@66816
  1023
  case True
haftmann@60868
  1024
  then show ?thesis
haftmann@66816
  1025
    by (simp add: mod_eq_0_iff_dvd divides_aux_def)
haftmann@66816
  1026
next
haftmann@66816
  1027
  case False
haftmann@66816
  1028
  then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@66816
  1029
    by (simp only: snd_divmod modulo_int_def) auto
haftmann@66816
  1030
  then show ?thesis
haftmann@66816
  1031
    by (simp add: divides_aux_def adjust_div_def)
haftmann@66816
  1032
      (simp add: divides_aux_def modulo_int_def)
haftmann@60868
  1033
qed
haftmann@60868
  1034
haftmann@60868
  1035
lemma numeral_div_minus_numeral [simp]:
haftmann@60868
  1036
  "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  1037
proof -
haftmann@60868
  1038
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  1039
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  1040
  then show ?thesis
haftmann@60868
  1041
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  1042
qed
haftmann@60868
  1043
  
haftmann@60868
  1044
lemma numeral_mod_minus_numeral [simp]:
haftmann@60868
  1045
  "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@66816
  1046
proof (cases "snd (divmod m n) = (0::int)")
haftmann@66816
  1047
  case True
haftmann@60868
  1048
  then show ?thesis
haftmann@66816
  1049
    by (simp add: mod_eq_0_iff_dvd divides_aux_def)
haftmann@66816
  1050
next
haftmann@66816
  1051
  case False
haftmann@66816
  1052
  then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@66816
  1053
    by (simp only: snd_divmod modulo_int_def) auto
haftmann@66816
  1054
  then show ?thesis
haftmann@66816
  1055
    by (simp add: divides_aux_def adjust_div_def)
haftmann@66816
  1056
      (simp add: divides_aux_def modulo_int_def)
haftmann@60868
  1057
qed
haftmann@60868
  1058
haftmann@60868
  1059
lemma minus_one_div_numeral [simp]:
haftmann@60868
  1060
  "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  1061
  using minus_numeral_div_numeral [of Num.One n] by simp  
haftmann@60868
  1062
haftmann@60868
  1063
lemma minus_one_mod_numeral [simp]:
haftmann@60868
  1064
  "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  1065
  using minus_numeral_mod_numeral [of Num.One n] by simp
haftmann@60868
  1066
haftmann@60868
  1067
lemma one_div_minus_numeral [simp]:
haftmann@60868
  1068
  "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  1069
  using numeral_div_minus_numeral [of Num.One n] by simp
haftmann@60868
  1070
  
haftmann@60868
  1071
lemma one_mod_minus_numeral [simp]:
haftmann@60868
  1072
  "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  1073
  using numeral_mod_minus_numeral [of Num.One n] by simp
haftmann@60868
  1074
haftmann@60930
  1075
end
haftmann@60930
  1076
haftmann@60868
  1077
haftmann@60868
  1078
subsubsection \<open>Further properties\<close>
haftmann@60868
  1079
haftmann@66817
  1080
lemma div_int_pos_iff:
haftmann@66817
  1081
  "k div l \<ge> 0 \<longleftrightarrow> k = 0 \<or> l = 0 \<or> k \<ge> 0 \<and> l \<ge> 0
haftmann@66817
  1082
    \<or> k < 0 \<and> l < 0"
haftmann@66817
  1083
  for k l :: int
haftmann@66817
  1084
  apply (cases "k = 0 \<or> l = 0")
haftmann@66817
  1085
   apply (auto simp add: pos_imp_zdiv_nonneg_iff neg_imp_zdiv_nonneg_iff)
haftmann@66817
  1086
  apply (rule ccontr)
haftmann@66817
  1087
  apply (simp add: neg_imp_zdiv_nonneg_iff)
haftmann@66817
  1088
  done
haftmann@66817
  1089
haftmann@66817
  1090
lemma mod_int_pos_iff:
haftmann@66817
  1091
  "k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0"
haftmann@66817
  1092
  for k l :: int
haftmann@66817
  1093
  apply (cases "l > 0")
haftmann@66817
  1094
   apply (simp_all add: dvd_eq_mod_eq_0)
haftmann@66817
  1095
  apply (use neg_mod_conj [of l k] in \<open>auto simp add: le_less not_less\<close>)
haftmann@66817
  1096
  done
haftmann@66817
  1097
haftmann@60868
  1098
text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
haftmann@60868
  1099
haftmann@60868
  1100
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@64635
  1101
  by (rule div_int_unique [of a b q r]) (simp add: eucl_rel_int_iff)
haftmann@60868
  1102
haftmann@60868
  1103
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@60868
  1104
  by (rule div_int_unique [of a b q r],
haftmann@64635
  1105
    simp add: eucl_rel_int_iff)
haftmann@60868
  1106
haftmann@60868
  1107
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  1108
  by (rule mod_int_unique [of a b q r],
haftmann@64635
  1109
    simp add: eucl_rel_int_iff)
haftmann@60868
  1110
haftmann@60868
  1111
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  1112
  by (rule mod_int_unique [of a b q r],
haftmann@64635
  1113
    simp add: eucl_rel_int_iff)
haftmann@33361
  1114
wenzelm@61944
  1115
lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
haftmann@33361
  1116
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  1117
wenzelm@60758
  1118
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  1119
lemma int_power_div_base:
haftmann@33361
  1120
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  1121
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  1122
 apply (erule ssubst)
haftmann@33361
  1123
 apply (simp only: power_add)
haftmann@33361
  1124
 apply simp_all
haftmann@33361
  1125
done
haftmann@33361
  1126
wenzelm@61799
  1127
text \<open>Distributive laws for function \<open>nat\<close>.\<close>
haftmann@33361
  1128
haftmann@33361
  1129
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  1130
apply (rule linorder_cases [of y 0])
haftmann@33361
  1131
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  1132
apply simp
haftmann@33361
  1133
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  1134
done
haftmann@33361
  1135
haftmann@33361
  1136
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  1137
lemma nat_mod_distrib:
haftmann@33361
  1138
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  1139
apply (case_tac "y = 0", simp)
haftmann@33361
  1140
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  1141
done
haftmann@33361
  1142
wenzelm@60758
  1143
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  1144
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
haftmann@33361
  1145
apply (subgoal_tac "nat x div nat k < nat x")
nipkow@34225
  1146
 apply (simp add: nat_div_distrib [symmetric])
haftmann@66808
  1147
apply (rule div_less_dividend, simp_all)
haftmann@33361
  1148
done
haftmann@33361
  1149
haftmann@66837
  1150
lemma mod_eq_dvd_iff_nat:
haftmann@66837
  1151
  "m mod q = n mod q \<longleftrightarrow> q dvd m - n" if "m \<ge> n" for m n q :: nat
haftmann@66837
  1152
proof -
haftmann@66837
  1153
  have "int m mod int q = int n mod int q \<longleftrightarrow> int q dvd int m - int n"
haftmann@66837
  1154
    by (simp add: mod_eq_dvd_iff)
haftmann@66837
  1155
  with that have "int (m mod q) = int (n mod q) \<longleftrightarrow> int q dvd int (m - n)"
haftmann@66837
  1156
    by (simp only: of_nat_mod of_nat_diff)
haftmann@66837
  1157
  then show ?thesis
haftmann@66837
  1158
    by (simp add: zdvd_int)
haftmann@66837
  1159
qed
haftmann@66837
  1160
haftmann@66837
  1161
lemma mod_eq_nat1E:
haftmann@66837
  1162
  fixes m n q :: nat
haftmann@66837
  1163
  assumes "m mod q = n mod q" and "m \<ge> n"
haftmann@66837
  1164
  obtains s where "m = n + q * s"
haftmann@66837
  1165
proof -
haftmann@66837
  1166
  from assms have "q dvd m - n"
haftmann@66837
  1167
    by (simp add: mod_eq_dvd_iff_nat)
haftmann@66837
  1168
  then obtain s where "m - n = q * s" ..
haftmann@66837
  1169
  with \<open>m \<ge> n\<close> have "m = n + q * s"
haftmann@66837
  1170
    by simp
haftmann@66837
  1171
  with that show thesis .
haftmann@66837
  1172
qed
haftmann@66837
  1173
haftmann@66837
  1174
lemma mod_eq_nat2E:
haftmann@66837
  1175
  fixes m n q :: nat
haftmann@66837
  1176
  assumes "m mod q = n mod q" and "n \<ge> m"
haftmann@66837
  1177
  obtains s where "n = m + q * s"
haftmann@66837
  1178
  using assms mod_eq_nat1E [of n q m] by (auto simp add: ac_simps)
haftmann@66837
  1179
haftmann@66837
  1180
lemma nat_mod_eq_lemma:
haftmann@66837
  1181
  assumes "(x::nat) mod n = y mod n" and "y \<le> x"
haftmann@33361
  1182
  shows "\<exists>q. x = y + n * q"
haftmann@66837
  1183
  using assms by (rule mod_eq_nat1E) rule
haftmann@33361
  1184
lp15@60562
  1185
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
haftmann@33361
  1186
  (is "?lhs = ?rhs")
haftmann@33361
  1187
proof
haftmann@33361
  1188
  assume H: "x mod n = y mod n"
haftmann@33361
  1189
  {assume xy: "x \<le> y"
haftmann@33361
  1190
    from H have th: "y mod n = x mod n" by simp
lp15@60562
  1191
    from nat_mod_eq_lemma[OF th xy] have ?rhs
haftmann@33361
  1192
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
haftmann@33361
  1193
  moreover
haftmann@33361
  1194
  {assume xy: "y \<le> x"
lp15@60562
  1195
    from nat_mod_eq_lemma[OF H xy] have ?rhs
haftmann@33361
  1196
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
lp15@60562
  1197
  ultimately  show ?rhs using linear[of x y] by blast
haftmann@33361
  1198
next
haftmann@33361
  1199
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
haftmann@33361
  1200
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
haftmann@33361
  1201
  thus  ?lhs by simp
haftmann@33361
  1202
qed
haftmann@33361
  1203
haftmann@66808
  1204
haftmann@60868
  1205
subsubsection \<open>Dedicated simproc for calculation\<close>
haftmann@60868
  1206
wenzelm@60758
  1207
text \<open>
haftmann@60868
  1208
  There is space for improvement here: the calculation itself
haftmann@66808
  1209
  could be carried out outside the logic, and a generic simproc
haftmann@60868
  1210
  (simplifier setup) for generic calculation would be helpful. 
wenzelm@60758
  1211
\<close>
haftmann@53067
  1212
haftmann@60868
  1213
simproc_setup numeral_divmod
haftmann@66806
  1214
  ("0 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  1215
   "0 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1216
   "0 div - 1 :: int" | "0 mod - 1 :: int" |
haftmann@66806
  1217
   "0 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1218
   "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
haftmann@66806
  1219
   "1 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  1220
   "1 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1221
   "1 div - 1 :: int" | "1 mod - 1 :: int" |
haftmann@66806
  1222
   "1 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1223
   "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
haftmann@60868
  1224
   "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
haftmann@60868
  1225
   "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
haftmann@60868
  1226
   "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
haftmann@66806
  1227
   "numeral a div 0 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  1228
   "numeral a div 1 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1229
   "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
haftmann@66806
  1230
   "numeral a div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1231
   "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
haftmann@60868
  1232
   "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
haftmann@60868
  1233
   "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
haftmann@60868
  1234
   "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
haftmann@60868
  1235
   "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
haftmann@60868
  1236
   "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
haftmann@60868
  1237
\<open> let
haftmann@60868
  1238
    val if_cong = the (Code.get_case_cong @{theory} @{const_name If});
haftmann@60868
  1239
    fun successful_rewrite ctxt ct =
haftmann@60868
  1240
      let
haftmann@60868
  1241
        val thm = Simplifier.rewrite ctxt ct
haftmann@60868
  1242
      in if Thm.is_reflexive thm then NONE else SOME thm end;
haftmann@60868
  1243
  in fn phi =>
haftmann@60868
  1244
    let
haftmann@60868
  1245
      val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
haftmann@60868
  1246
        one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
haftmann@60868
  1247
        one_div_minus_numeral one_mod_minus_numeral
haftmann@60868
  1248
        numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
haftmann@60868
  1249
        numeral_div_minus_numeral numeral_mod_minus_numeral
haftmann@60930
  1250
        div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
haftmann@60868
  1251
        numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
haftmann@60868
  1252
        divmod_cancel divmod_steps divmod_step_eq fst_conv snd_conv numeral_One
haftmann@60930
  1253
        case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
haftmann@60868
  1254
        minus_minus numeral_times_numeral mult_zero_right mult_1_right}
haftmann@60868
  1255
        @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
haftmann@60868
  1256
      fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
haftmann@60868
  1257
        (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
haftmann@60868
  1258
    in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
haftmann@60868
  1259
  end;
haftmann@60868
  1260
\<close>
blanchet@34126
  1261
haftmann@35673
  1262
wenzelm@60758
  1263
subsubsection \<open>Code generation\<close>
haftmann@33361
  1264
haftmann@60868
  1265
lemma [code]:
haftmann@60868
  1266
  fixes k :: int
haftmann@60868
  1267
  shows 
haftmann@60868
  1268
    "k div 0 = 0"
haftmann@60868
  1269
    "k mod 0 = k"
haftmann@60868
  1270
    "0 div k = 0"
haftmann@60868
  1271
    "0 mod k = 0"
haftmann@60868
  1272
    "k div Int.Pos Num.One = k"
haftmann@60868
  1273
    "k mod Int.Pos Num.One = 0"
haftmann@60868
  1274
    "k div Int.Neg Num.One = - k"
haftmann@60868
  1275
    "k mod Int.Neg Num.One = 0"
haftmann@60868
  1276
    "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
haftmann@60868
  1277
    "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
haftmann@60930
  1278
    "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
haftmann@60930
  1279
    "Int.Neg m mod Int.Pos n = Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
haftmann@60930
  1280
    "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
haftmann@60930
  1281
    "Int.Pos m mod Int.Neg n = - Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
haftmann@60868
  1282
    "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
haftmann@60868
  1283
    "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
haftmann@60868
  1284
  by simp_all
haftmann@53069
  1285
haftmann@52435
  1286
code_identifier
haftmann@52435
  1287
  code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1288
haftmann@60868
  1289
lemma dvd_eq_mod_eq_0_numeral:
haftmann@66806
  1290
  "numeral x dvd (numeral y :: 'a) \<longleftrightarrow> numeral y mod numeral x = (0 :: 'a::semidom_modulo)"
haftmann@60868
  1291
  by (fact dvd_eq_mod_eq_0)
haftmann@60868
  1292
haftmann@64246
  1293
declare minus_div_mult_eq_mod [symmetric, nitpick_unfold]
haftmann@64246
  1294
haftmann@66808
  1295
haftmann@66808
  1296
subsubsection \<open>Lemmas of doubtful value\<close>
haftmann@66808
  1297
haftmann@66808
  1298
lemma mod_mult_self3':
haftmann@66808
  1299
  "Suc (k * n + m) mod n = Suc m mod n"
haftmann@66808
  1300
  by (fact Suc_mod_mult_self3)
haftmann@66808
  1301
haftmann@66808
  1302
lemma mod_Suc_eq_Suc_mod:
haftmann@66808
  1303
  "Suc m mod n = Suc (m mod n) mod n"
haftmann@66808
  1304
  by (simp add: mod_simps)
haftmann@66808
  1305
haftmann@66808
  1306
lemma div_geq:
haftmann@66808
  1307
  "m div n = Suc ((m - n) div n)" if "0 < n" and " \<not> m < n" for m n :: nat
haftmann@66808
  1308
  by (rule le_div_geq) (use that in \<open>simp_all add: not_less\<close>)
haftmann@66808
  1309
haftmann@66808
  1310
lemma mod_geq:
haftmann@66808
  1311
  "m mod n = (m - n) mod n" if "\<not> m < n" for m n :: nat
haftmann@66808
  1312
  by (rule le_mod_geq) (use that in \<open>simp add: not_less\<close>)
haftmann@66808
  1313
haftmann@66808
  1314
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
haftmann@66808
  1315
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@66808
  1316
haftmann@66808
  1317
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
haftmann@66808
  1318
haftmann@66808
  1319
(*Loses information, namely we also have r<d provided d is nonzero*)
haftmann@66808
  1320
lemma mod_eqD:
haftmann@66808
  1321
  fixes m d r q :: nat
haftmann@66808
  1322
  assumes "m mod d = r"
haftmann@66808
  1323
  shows "\<exists>q. m = r + q * d"
haftmann@66808
  1324
proof -
haftmann@66808
  1325
  from div_mult_mod_eq obtain q where "q * d + m mod d = m" by blast
haftmann@66808
  1326
  with assms have "m = r + q * d" by simp
haftmann@66808
  1327
  then show ?thesis ..
haftmann@66808
  1328
qed
haftmann@66808
  1329
haftmann@66815
  1330
lemmas even_times_iff = even_mult_iff -- \<open>FIXME duplicate\<close>
haftmann@66815
  1331
haftmann@66815
  1332
lemma mod_2_not_eq_zero_eq_one_nat:
haftmann@66815
  1333
  fixes n :: nat
haftmann@66815
  1334
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
haftmann@66815
  1335
  by (fact not_mod_2_eq_0_eq_1)
haftmann@66815
  1336
haftmann@66815
  1337
lemma even_int_iff [simp]: "even (int n) \<longleftrightarrow> even n"
haftmann@66815
  1338
  by (fact even_of_nat)
haftmann@66815
  1339
haftmann@66816
  1340
lemma is_unit_int:
haftmann@66816
  1341
  "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
haftmann@66816
  1342
  by auto
haftmann@66816
  1343
haftmann@33361
  1344
end