(*  Title:      HOL/Divides.thy

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1999  University of Cambridge

*)

header {* The division operators div, mod and the divides relation "dvd" *}

theory Divides

imports Datatype Power

begin

(*We use the same class for div and mod;

  moreover, dvd is defined whenever multiplication is*)

class div = type +

  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"

begin

notation

  div (infixl "\<^loc>div" 70)

notation

  mod (infixl "\<^loc>mod" 70)

end

notation

  div (infixl "div" 70)

notation

  mod (infixl "mod" 70)

instance nat :: "Divides.div"

  mod_def: "m mod n == wfrec (pred_nat^+)

                          (%f j. if j<n | n=0 then j else f (j-n)) m"

  div_def: "m div n == wfrec (pred_nat^+)

                          (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m" ..

definition

  (*The definition of dvd is polymorphic!*)

  dvd  :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where

  dvd_def: "m dvd n \<longleftrightarrow> (\<exists>k. n = m*k)"

definition

  quorem :: "(nat*nat) * (nat*nat) => bool" where

  (*This definition helps prove the harder properties of div and mod.

    It is copied from IntDiv.thy; should it be overloaded?*)

  "quorem = (%((a,b), (q,r)).

                    a = b*q + r &

                    (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"

subsection{*Initial Lemmas*}

lemmas wf_less_trans =

       def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],

                  standard]

lemma mod_eq: "(%m. m mod n) =

              wfrec (pred_nat^+) (%f j. if j<n | n=0 then j else f (j-n))"

by (simp add: mod_def)

lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+)

               (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"

by (simp add: div_def)

(** Aribtrary definitions for division by zero.  Useful to simplify

    certain equations **)

lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"

  by (rule div_eq [THEN wf_less_trans], simp)

lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"

  by (rule mod_eq [THEN wf_less_trans], simp)

subsection{*Remainder*}

lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"

  by (rule mod_eq [THEN wf_less_trans]) simp

lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"

  apply (cases "n=0")

   apply simp

  apply (rule mod_eq [THEN wf_less_trans])

  apply (simp add: cut_apply less_eq)

  done

(*Avoids the ugly ~m<n above*)

lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n"

  by (simp add: mod_geq linorder_not_less)

lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)"

  by (simp add: mod_geq)

lemma mod_1 [simp]: "m mod Suc 0 = 0"

  by (induct m) (simp_all add: mod_geq)

lemma mod_self [simp]: "n mod n = (0::nat)"

  by (cases "n = 0") (simp_all add: mod_geq)

lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"

  apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")

   apply (simp add: add_commute)

  apply (subst mod_geq [symmetric], simp_all)

  done

lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"

  by (simp add: add_commute mod_add_self2)

lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"

  by (induct k) (simp_all add: add_left_commute [of _ n])

lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"

  by (simp add: mult_commute mod_mult_self1)

lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"

  apply (cases "n = 0", simp)

  apply (cases "k = 0", simp)

  apply (induct m rule: nat_less_induct)

  apply (subst mod_if, simp)

  apply (simp add: mod_geq diff_mult_distrib)

  done

lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"

  by (simp add: mult_commute [of k] mod_mult_distrib)

lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"

  apply (cases "n = 0", simp)

  apply (induct m, simp)

  apply (rename_tac k)

  apply (cut_tac m = "k * n" and n = n in mod_add_self2)

  apply (simp add: add_commute)

  done

lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"

  by (simp add: mult_commute mod_mult_self_is_0)

subsection{*Quotient*}

lemma div_less [simp]: "m<n ==> m div n = (0::nat)"

  by (rule div_eq [THEN wf_less_trans], simp)

lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"

  apply (rule div_eq [THEN wf_less_trans])

  apply (simp add: cut_apply less_eq)

  done

(*Avoids the ugly ~m<n above*)

lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"

  by (simp add: div_geq linorder_not_less)

lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"

  by (simp add: div_geq)

(*Main Result about quotient and remainder.*)

lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"

  apply (cases "n = 0", simp)

  apply (induct m rule: nat_less_induct)

  apply (subst mod_if)

  apply (simp_all add: add_assoc div_geq add_diff_inverse)

  done

lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"

  apply (cut_tac m = m and n = n in mod_div_equality)

  apply (simp add: mult_commute)

  done

subsection{*Simproc for Cancelling Div and Mod*}

lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"

  by (simp add: mod_div_equality)

lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"

  by (simp add: mod_div_equality2)

ML

{*

structure CancelDivModData =

struct

val div_name = @{const_name Divides.div};

val mod_name = @{const_name Divides.mod};

val mk_binop = HOLogic.mk_binop;

val mk_sum = NatArithUtils.mk_sum;

val dest_sum = NatArithUtils.dest_sum;

(*logic*)

val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]

val trans = trans

val prove_eq_sums =

  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}

  in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;

end;

structure CancelDivMod = CancelDivModFun(CancelDivModData);

val cancel_div_mod_proc = NatArithUtils.prep_simproc

      ("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc);

Addsimprocs[cancel_div_mod_proc];

*}

(* a simple rearrangement of mod_div_equality: *)

lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"

  by (cut_tac m = m and n = n in mod_div_equality2, arith)

lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"

  apply (induct m rule: nat_less_induct)

  apply (rename_tac m)

  apply (case_tac "m<n", simp)

  txt{*case @{term "n \<le> m"}*}

  apply (simp add: mod_geq)

  done

lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"

  apply (drule mod_less_divisor [where m = m])

  apply simp

  done

lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"

  by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)

lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"

  by (simp add: mult_commute div_mult_self_is_m)

(*mod_mult_distrib2 above is the counterpart for remainder*)

subsection{*Proving facts about Quotient and Remainder*}

lemma unique_quotient_lemma:

     "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]

      ==> q' \<le> (q::nat)"

  apply (rule leI)

  apply (subst less_iff_Suc_add)

  apply (auto simp add: add_mult_distrib2)

  done

lemma unique_quotient:

     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]

      ==> q = q'"

  apply (simp add: split_ifs quorem_def)

  apply (blast intro: order_antisym

    dest: order_eq_refl [THEN unique_quotient_lemma] sym)

  done

lemma unique_remainder:

     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]

      ==> r = r'"

  apply (subgoal_tac "q = q'")

   prefer 2 apply (blast intro: unique_quotient)

  apply (simp add: quorem_def)

  done

lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"

  unfolding quorem_def by simp

lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"

  by (simp add: quorem_div_mod [THEN unique_quotient])

lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"

  by (simp add: quorem_div_mod [THEN unique_remainder])

(** A dividend of zero **)

lemma div_0 [simp]: "0 div m = (0::nat)"

  by (cases "m = 0") simp_all

lemma mod_0 [simp]: "0 mod m = (0::nat)"

  by (cases "m = 0") simp_all

(** proving (a*b) div c = a * (b div c) + a * (b mod c) **)

lemma quorem_mult1_eq:

     "[| quorem((b,c),(q,r));  0 < c |]

      ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"

  by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)

lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"

  apply (cases "c = 0", simp)

  apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])

  done

lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"

  apply (cases "c = 0", simp)

  apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])

  done

lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"

  apply (rule trans)

   apply (rule_tac s = "b*a mod c" in trans)

    apply (rule_tac [2] mod_mult1_eq)

   apply (simp_all add: mult_commute)

  done

lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"

  apply (rule mod_mult1_eq' [THEN trans])

  apply (rule mod_mult1_eq)

  done

(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)

lemma quorem_add1_eq:

     "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]

      ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"

  by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)

(*NOT suitable for rewriting: the RHS has an instance of the LHS*)

lemma div_add1_eq:

     "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"

  apply (cases "c = 0", simp)

  apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)

  done

lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"

  apply (cases "c = 0", simp)

  apply (blast intro: quorem_div_mod quorem_div_mod quorem_add1_eq [THEN quorem_mod])

  done

subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}

(** first, a lemma to bound the remainder **)

lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"

  apply (cut_tac m = q and n = c in mod_less_divisor)

  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)

  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)

  apply (simp add: add_mult_distrib2)

  done

lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]

      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"

  by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)

lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"

  apply (cases "b = 0", simp)

  apply (cases "c = 0", simp)

  apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])

  done

lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"

  apply (cases "b = 0", simp)

  apply (cases "c = 0", simp)

  apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])

  done

subsection{*Cancellation of Common Factors in Division*}

lemma div_mult_mult_lemma:

    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"

  by (auto simp add: div_mult2_eq)

lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"

  apply (cases "b = 0")

  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)

  done

lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"

  apply (drule div_mult_mult1)

  apply (auto simp add: mult_commute)

  done

(*Distribution of Factors over Remainders:

Could prove these as in Integ/IntDiv.ML, but we already have

mod_mult_distrib and mod_mult_distrib2 above!

Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"

qed "mod_mult_mult1";

Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";

qed "mod_mult_mult2";

 ***)

subsection{*Further Facts about Quotient and Remainder*}

lemma div_1 [simp]: "m div Suc 0 = m"

  by (induct m) (simp_all add: div_geq)

lemma div_self [simp]: "0<n ==> n div n = (1::nat)"

  by (simp add: div_geq)

lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"

  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")

   apply (simp add: add_commute)

  apply (subst div_geq [symmetric], simp_all)

  done

lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"

  by (simp add: add_commute div_add_self2)

lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"

  apply (subst div_add1_eq)

  apply (subst div_mult1_eq, simp)

  done

lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"

  by (simp add: mult_commute div_mult_self1)

(* Monotonicity of div in first argument *)

lemma div_le_mono [rule_format (no_asm)]:

    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"

apply (case_tac "k=0", simp)

apply (induct "n" rule: nat_less_induct, clarify)

apply (case_tac "n<k")

(* 1  case n<k *)

apply simp

(* 2  case n >= k *)

apply (case_tac "m<k")

(* 2.1  case m<k *)

apply simp

(* 2.2  case m>=k *)

apply (simp add: div_geq diff_le_mono)

done

(* Antimonotonicity of div in second argument *)

lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"

apply (subgoal_tac "0<n")

 prefer 2 apply simp

apply (induct_tac k rule: nat_less_induct)

apply (rename_tac "k")

apply (case_tac "k<n", simp)

apply (subgoal_tac "~ (k<m) ")

 prefer 2 apply simp

apply (simp add: div_geq)

apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")

 prefer 2

 apply (blast intro: div_le_mono diff_le_mono2)

apply (rule le_trans, simp)

apply (simp)

done

lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"

apply (case_tac "n=0", simp)

apply (subgoal_tac "m div n \<le> m div 1", simp)

apply (rule div_le_mono2)

apply (simp_all (no_asm_simp))

done

(* Similar for "less than" *)

lemma div_less_dividend [rule_format]:

     "!!n::nat. 1<n ==> 0 < m --> m div n < m"

apply (induct_tac m rule: nat_less_induct)

apply (rename_tac "m")

apply (case_tac "m<n", simp)

apply (subgoal_tac "0<n")

 prefer 2 apply simp

apply (simp add: div_geq)

apply (case_tac "n<m")

 apply (subgoal_tac "(m-n) div n < (m-n) ")

  apply (rule impI less_trans_Suc)+

apply assumption

  apply (simp_all)

done

declare div_less_dividend [simp]

text{*A fact for the mutilated chess board*}

lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"

apply (case_tac "n=0", simp)

apply (induct "m" rule: nat_less_induct)

apply (case_tac "Suc (na) <n")

(* case Suc(na) < n *)

apply (frule lessI [THEN less_trans], simp add: less_not_refl3)

(* case n \<le> Suc(na) *)

apply (simp add: linorder_not_less le_Suc_eq mod_geq)

apply (auto simp add: Suc_diff_le le_mod_geq)

done

lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"

  by (cases "n = 0") auto

lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"

  by (cases "n = 0") auto

subsection{*The Divides Relation*}

lemma dvdI [intro?]: "n = m * k ==> m dvd n"

  unfolding dvd_def by blast

lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"

  unfolding dvd_def by blast

lemma dvd_0_right [iff]: "m dvd (0::nat)"

  unfolding dvd_def by (blast intro: mult_0_right [symmetric])

lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"

  by (force simp add: dvd_def)

lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"

  by (blast intro: dvd_0_left)

lemma dvd_1_left [iff]: "Suc 0 dvd k"

  unfolding dvd_def by simp

lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"

  by (simp add: dvd_def)

lemma dvd_refl [simp]: "m dvd (m::nat)"

  unfolding dvd_def by (blast intro: mult_1_right [symmetric])

lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"

  unfolding dvd_def by (blast intro: mult_assoc)

lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"

  unfolding dvd_def

  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)

lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"

  unfolding dvd_def

  by (blast intro: add_mult_distrib2 [symmetric])

lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"

  unfolding dvd_def

  by (blast intro: diff_mult_distrib2 [symmetric])

lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"

  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])

  apply (blast intro: dvd_add)

  done

lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"

  by (drule_tac m = m in dvd_diff, auto)

lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"

  unfolding dvd_def by (blast intro: mult_left_commute)

lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"

  apply (subst mult_commute)

  apply (erule dvd_mult)

  done

lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"

  by (rule dvd_refl [THEN dvd_mult])

lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"

  by (rule dvd_refl [THEN dvd_mult2])

lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"

  apply (rule iffI)

   apply (erule_tac [2] dvd_add)

   apply (rule_tac [2] dvd_refl)

  apply (subgoal_tac "n = (n+k) -k")

   prefer 2 apply simp

  apply (erule ssubst)

  apply (erule dvd_diff)

  apply (rule dvd_refl)

  done

lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"

  unfolding dvd_def

  apply (case_tac "n = 0", auto)

  apply (blast intro: mod_mult_distrib2 [symmetric])

  done

lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"

  apply (subgoal_tac "k dvd (m div n) *n + m mod n")

   apply (simp add: mod_div_equality)

  apply (simp only: dvd_add dvd_mult)

  done

lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"

  by (blast intro: dvd_mod_imp_dvd dvd_mod)

lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"

  unfolding dvd_def

  apply (erule exE)

  apply (simp add: mult_ac)

  done

lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"

  apply auto

   apply (subgoal_tac "m*n dvd m*1")

   apply (drule dvd_mult_cancel, auto)

  done

lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"

  apply (subst mult_commute)

  apply (erule dvd_mult_cancel1)

  done

lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"

  apply (unfold dvd_def, clarify)

  apply (rule_tac x = "k*ka" in exI)

  apply (simp add: mult_ac)

  done

lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"

  by (simp add: dvd_def mult_assoc, blast)

lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"

  apply (unfold dvd_def, clarify)

  apply (rule_tac x = "i*k" in exI)

  apply (simp add: mult_ac)

  done

lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"

  apply (unfold dvd_def, clarify)

  apply (simp_all (no_asm_use) add: zero_less_mult_iff)

  apply (erule conjE)

  apply (rule le_trans)

   apply (rule_tac [2] le_refl [THEN mult_le_mono])

   apply (erule_tac [2] Suc_leI, simp)

  done

lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"

  apply (unfold dvd_def)

  apply (case_tac "k=0", simp, safe)

   apply (simp add: mult_commute)

  apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])

  apply (subst mult_commute, simp)

  done

lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"

  apply (subgoal_tac "m mod n = 0")

   apply (simp add: mult_div_cancel)

  apply (simp only: dvd_eq_mod_eq_0)

  done

lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"

  apply (unfold dvd_def)

  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])

  apply (simp add: power_add)

  done

lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"

  by (induct n) auto

lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"

  apply (induct j)

   apply (simp_all add: le_Suc_eq)

  apply (blast dest!: dvd_mult_right)

  done

lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"

  apply (rule power_le_imp_le_exp, assumption)

  apply (erule dvd_imp_le, simp)

  done

lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"

  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]

(*Loses information, namely we also have r<d provided d is nonzero*)

lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"

  apply (cut_tac m = m in mod_div_equality)

  apply (simp only: add_ac)

  apply (blast intro: sym)

  done

lemma split_div:

 "P(n div k :: nat) =

 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"

 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")

proof

  assume P: ?P

  show ?Q

  proof (cases)

    assume "k = 0"

    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)

  next

    assume not0: "k \<noteq> 0"

    thus ?Q

    proof (simp, intro allI impI)

      fix i j

      assume n: "n = k*i + j" and j: "j < k"

      show "P i"

      proof (cases)

        assume "i = 0"

        with n j P show "P i" by simp

      next

        assume "i \<noteq> 0"

        with not0 n j P show "P i" by(simp add:add_ac)

      qed

    qed

  qed

next

  assume Q: ?Q

  show ?P

  proof (cases)

    assume "k = 0"

    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)

  next

    assume not0: "k \<noteq> 0"

    with Q have R: ?R by simp

    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]

    show ?P by simp

  qed

qed

lemma split_div_lemma:

  "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"

  apply (rule iffI)

  apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)

prefer 3; apply assumption

  apply (simp_all add: quorem_def) apply arith

  apply (rule conjI)

  apply (rule_tac P="%x. n * (m div n) \<le> x" in

    subst [OF mod_div_equality [of _ n]])

  apply (simp only: add: mult_ac)

  apply (rule_tac P="%x. x < n + n * (m div n)" in

    subst [OF mod_div_equality [of _ n]])

  apply (simp only: add: mult_ac add_ac)

  apply (rule add_less_mono1, simp)

  done

theorem split_div':

  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>

   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"

  apply (case_tac "0 < n")

  apply (simp only: add: split_div_lemma)

  apply (simp_all add: DIVISION_BY_ZERO_DIV)

  done

lemma split_mod:

 "P(n mod k :: nat) =

 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"

 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")

proof

  assume P: ?P

  show ?Q

  proof (cases)

    assume "k = 0"

    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)

  next

    assume not0: "k \<noteq> 0"

    thus ?Q

    proof (simp, intro allI impI)

      fix i j

      assume "n = k*i + j" "j < k"

      thus "P j" using not0 P by(simp add:add_ac mult_ac)

    qed

  qed

next

  assume Q: ?Q

  show ?P

  proof (cases)

    assume "k = 0"

    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)

  next

    assume not0: "k \<noteq> 0"

    with Q have R: ?R by simp

    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]

    show ?P by simp

  qed

qed

theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"

  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in

    subst [OF mod_div_equality [of _ n]])

  apply arith

  done

lemma div_mod_equality':

  fixes m n :: nat

  shows "m div n * n = m - m mod n"

proof -

  have "m mod n \<le> m mod n" ..

  from div_mod_equality have 

    "m div n * n + m mod n - m mod n = m - m mod n" by simp

  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have

    "m div n * n + (m mod n - m mod n) = m - m mod n"

    by simp

  then show ?thesis by simp

qed

subsection {*An ``induction'' law for modulus arithmetic.*}

lemma mod_induct_0:

  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"

  and base: "P i" and i: "i<p"

  shows "P 0"

proof (rule ccontr)

  assume contra: "\<not>(P 0)"

  from i have p: "0<p" by simp

  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")

  proof

    fix k

    show "?A k"

    proof (induct k)

      show "?A 0" by simp  -- "by contradiction"

    next

      fix n

      assume ih: "?A n"

      show "?A (Suc n)"

      proof (clarsimp)

        assume y: "P (p - Suc n)"

        have n: "Suc n < p"

        proof (rule ccontr)

          assume "\<not>(Suc n < p)"

          hence "p - Suc n = 0"

            by simp

          with y contra show "False"

            by simp

        qed

        hence n2: "Suc (p - Suc n) = p-n" by arith

        from p have "p - Suc n < p" by arith

        with y step have z: "P ((Suc (p - Suc n)) mod p)"

          by blast

        show "False"

        proof (cases "n=0")

          case True

          with z n2 contra show ?thesis by simp

        next

          case False

          with p have "p-n < p" by arith

          with z n2 False ih show ?thesis by simp

        qed

      qed

    qed

  qed

  moreover

  from i obtain k where "0<k \<and> i+k=p"

    by (blast dest: less_imp_add_positive)

  hence "0<k \<and> i=p-k" by auto

  moreover

  note base

  ultimately

  show "False" by blast

qed

lemma mod_induct:

  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"

  and base: "P i" and i: "i<p" and j: "j<p"

  shows "P j"

proof -

  have "\<forall>j<p. P j"

  proof

    fix j

    show "j<p \<longrightarrow> P j" (is "?A j")

    proof (induct j)

      from step base i show "?A 0"

        by (auto elim: mod_induct_0)

    next

      fix k

      assume ih: "?A k"

      show "?A (Suc k)"

      proof

        assume suc: "Suc k < p"

        hence k: "k<p" by simp

        with ih have "P k" ..

        with step k have "P (Suc k mod p)"

          by blast

        moreover

        from suc have "Suc k mod p = Suc k"

          by simp

        ultimately

        show "P (Suc k)" by simp

      qed

    qed

  qed

  with j show ?thesis by blast

qed

lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"

  apply (rule trans [symmetric])

   apply (rule mod_add1_eq, simp)

  apply (rule mod_add1_eq [symmetric])

  done

lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"

  apply (rule trans [symmetric])

   apply (rule mod_add1_eq, simp)

  apply (rule mod_add1_eq [symmetric])

  done

lemma mod_div_decomp:

  fixes n k :: nat

  obtains m q where "m = n div k" and "q = n mod k"

    and "n = m * k + q"

proof -

  from mod_div_equality have "n = n div k * k + n mod k" by auto

  moreover have "n div k = n div k" ..

  moreover have "n mod k = n mod k" ..

  note that ultimately show thesis by blast

qed

subsection {* Code generation for div, mod and dvd on nat *}

definition [code func del]:

  "divmod (m\<Colon>nat) n = (m div n, m mod n)"

lemma divmod_zero [code]: "divmod m 0 = (0, m)"

  unfolding divmod_def by simp

lemma divmod_succ [code]:

  "divmod m (Suc k) = (if m < Suc k then (0, m) else

    let

      (p, q) = divmod (m - Suc k) (Suc k)

    in (Suc p, q))"

  unfolding divmod_def Let_def split_def

  by (auto intro: div_geq mod_geq)

lemma div_divmod [code]: "m div n = fst (divmod m n)"

  unfolding divmod_def by simp

lemma mod_divmod [code]: "m mod n = snd (divmod m n)"

  unfolding divmod_def by simp

definition

  dvd_nat :: "nat \<Rightarrow> nat \<Rightarrow> bool"

where

  "dvd_nat m n \<longleftrightarrow> n mod m = (0 \<Colon> nat)"

lemma [code inline]:

  "op dvd = dvd_nat"

  by (auto simp add: dvd_nat_def dvd_eq_mod_eq_0 expand_fun_eq)

code_modulename SML

  Divides Integer

code_modulename OCaml

  Divides Integer

hide (open) const divmod dvd_nat

subsection {* Legacy bindings *}

ML

{*

val div_def = thm "div_def"

val mod_def = thm "mod_def"

val dvd_def = thm "dvd_def"

val quorem_def = thm "quorem_def"

val wf_less_trans = thm "wf_less_trans";

val mod_eq = thm "mod_eq";

val div_eq = thm "div_eq";

val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";

val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";

val mod_less = thm "mod_less";

val mod_geq = thm "mod_geq";

val le_mod_geq = thm "le_mod_geq";

val mod_if = thm "mod_if";

val mod_1 = thm "mod_1";

val mod_self = thm "mod_self";

val mod_add_self2 = thm "mod_add_self2";

val mod_add_self1 = thm "mod_add_self1";

val mod_mult_self1 = thm "mod_mult_self1";

val mod_mult_self2 = thm "mod_mult_self2";

val mod_mult_distrib = thm "mod_mult_distrib";

val mod_mult_distrib2 = thm "mod_mult_distrib2";

val mod_mult_self_is_0 = thm "mod_mult_self_is_0";

val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";

val div_less = thm "div_less";

val div_geq = thm "div_geq";

val le_div_geq = thm "le_div_geq";

val div_if = thm "div_if";

val mod_div_equality = thm "mod_div_equality";

val mod_div_equality2 = thm "mod_div_equality2";

val div_mod_equality = thm "div_mod_equality";

val div_mod_equality2 = thm "div_mod_equality2";

val mult_div_cancel = thm "mult_div_cancel";

val mod_less_divisor = thm "mod_less_divisor";

val div_mult_self_is_m = thm "div_mult_self_is_m";

val div_mult_self1_is_m = thm "div_mult_self1_is_m";

val unique_quotient_lemma = thm "unique_quotient_lemma";

val unique_quotient = thm "unique_quotient";

val unique_remainder = thm "unique_remainder";

val div_0 = thm "div_0";

val mod_0 = thm "mod_0";

val div_mult1_eq = thm "div_mult1_eq";

val mod_mult1_eq = thm "mod_mult1_eq";

val mod_mult1_eq' = thm "mod_mult1_eq'";

val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";

val div_add1_eq = thm "div_add1_eq";

val mod_add1_eq = thm "mod_add1_eq";

val mod_add_left_eq = thm "mod_add_left_eq";

 val mod_add_right_eq = thm "mod_add_right_eq";

val mod_lemma = thm "mod_lemma";

val div_mult2_eq = thm "div_mult2_eq";

val mod_mult2_eq = thm "mod_mult2_eq";

val div_mult_mult_lemma = thm "div_mult_mult_lemma";