(*  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

 begin

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

   moreover, dvd is defined whenever multiplication is*)

 axclass

   div < type

 instance  nat :: div ..

 consts

   div  :: "'a::div \<Rightarrow> 'a \<Rightarrow> 'a"          (infixl 70)

   mod  :: "'a::div \<Rightarrow> 'a \<Rightarrow> 'a"          (infixl 70)

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

 defs

   mod_def:   "m mod n == wfrec (trancl pred_nat)

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

   div_def:   "m div n == wfrec (trancl pred_nat) 

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

 (*The definition of dvd is polymorphic!*)

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

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

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

 constdefs

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

     "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 (trancl 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 (trancl 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 (case_tac "n=0", 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"

 apply (induct "m")

 apply (simp_all (no_asm_simp) add: mod_geq)

 done

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

 apply (case_tac "n=0")

 apply (simp_all add: mod_geq)

 done

 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)"

 apply (induct "k")

 apply (simp_all add: add_left_commute [of _ n])

 done

 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 (case_tac "n=0", simp)

 apply (case_tac "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 (case_tac "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 (case_tac "n=0", simp)

 apply (induct "m" rule: nat_less_induct)

 apply (subst mod_if)

 apply (simp_all (no_asm_simp) 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"

 apply(simp add: mod_div_equality)

 done

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

 apply(simp add: mod_div_equality2)

 done

 ML

 {*

 val div_mod_equality = thm "div_mod_equality";

 val div_mod_equality2 = thm "div_mod_equality2";

 structure CancelDivModData =

 struct

 val div_name = "Divides.op div";

 val mod_name = "Divides.op 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 [div_mod_equality,div_mod_equality2]

 val trans = trans

 val prove_eq_sums =

   let val simps = add_0 :: add_0_right :: 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"], 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 (case_tac "na<n", simp) 

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

 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))"

 by (auto simp add: quorem_def)

 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 (case_tac "m=0", simp_all)

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

 by (case_tac "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))"

 apply (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)

 done

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

 apply (case_tac "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 (case_tac "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 (no_asm) 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 (case_tac "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 (case_tac "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))"

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

 done

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

 apply (case_tac "b=0", simp)

 apply (case_tac "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 (case_tac "b=0", simp)

 apply (case_tac "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 (case_tac "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"

 apply (induct "m")

 apply (simp_all (no_asm_simp) add: div_geq)

 done

 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 (case_tac "n=0", auto)

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

 by (case_tac "n=0", auto)

 subsection{*The Divides Relation*}

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

 by (unfold dvd_def, blast)

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

 by (unfold dvd_def, blast)

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

 apply (unfold dvd_def)

 apply (blast intro: mult_0_right [symmetric])

 done

 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"

 by (unfold dvd_def, 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)"

 apply (unfold dvd_def)

 apply (blast intro: mult_1_right [symmetric])

 done

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

 apply (unfold dvd_def)

 apply (blast intro: mult_assoc)

 done

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

 apply (unfold dvd_def)

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

 done

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

 apply (unfold dvd_def)

 apply (blast intro: add_mult_distrib2 [symmetric])

 done

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

 apply (unfold dvd_def)

 apply (blast intro: diff_mult_distrib2 [symmetric])

 done

 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)"

 apply (unfold dvd_def)

 apply (blast intro: mult_left_commute)

 done

 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"

 apply (unfold 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"

 apply (unfold 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 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 = mod_eq_0_iff [THEN iffD1]

 declare mod_eq_0D [dest!]

 (*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

 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

 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";

 val div_mult_mult1 = thm "div_mult_mult1";

 val div_mult_mult2 = thm "div_mult_mult2";

 val div_1 = thm "div_1";

 val div_self = thm "div_self";

 val div_add_self2 = thm "div_add_self2";

 val div_add_self1 = thm "div_add_self1";

 val div_mult_self1 = thm "div_mult_self1";

 val div_mult_self2 = thm "div_mult_self2";

 val div_le_mono = thm "div_le_mono";

 val div_le_mono2 = thm "div_le_mono2";

 val div_le_dividend = thm "div_le_dividend";

 val div_less_dividend = thm "div_less_dividend";

 val mod_Suc = thm "mod_Suc";

 val dvdI = thm "dvdI";

 val dvdE = thm "dvdE";

 val dvd_0_right = thm "dvd_0_right";

 val dvd_0_left = thm "dvd_0_left";

 val dvd_0_left_iff = thm "dvd_0_left_iff";

 val dvd_1_left = thm "dvd_1_left";

 val dvd_1_iff_1 = thm "dvd_1_iff_1";

 val dvd_refl = thm "dvd_refl";

 val dvd_trans = thm "dvd_trans";

 val dvd_anti_sym = thm "dvd_anti_sym";

 val dvd_add = thm "dvd_add";

 val dvd_diff = thm "dvd_diff";

 val dvd_diffD = thm "dvd_diffD";

 val dvd_diffD1 = thm "dvd_diffD1";

 val dvd_mult = thm "dvd_mult";

 val dvd_mult2 = thm "dvd_mult2";

 val dvd_reduce = thm "dvd_reduce";

 val dvd_mod = thm "dvd_mod";

 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";

 val dvd_mod_iff = thm "dvd_mod_iff";

 val dvd_mult_cancel = thm "dvd_mult_cancel";

 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";

 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";

 val mult_dvd_mono = thm "mult_dvd_mono";

 val dvd_mult_left = thm "dvd_mult_left";

 val dvd_mult_right = thm "dvd_mult_right";

 val dvd_imp_le = thm "dvd_imp_le";

 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";

 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";

 val mod_eq_0_iff = thm "mod_eq_0_iff";

 val mod_eqD = thm "mod_eqD";

 *}

 (*

 lemma split_div:

 assumes m: "m \<noteq> 0"

 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"

        (is "?P = ?Q")

 proof

   assume P: ?P

   show ?Q

   proof (intro allI impI)

     fix i j

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

     show "P i"

     proof (cases)

       assume "i = 0"

       with n j P show "P i" by simp

     next

       assume "i \<noteq> 0"

       with n j P show "P i" by (simp add:add_ac div_mult_self1)

     qed

   qed

 next

   assume Q: ?Q

   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]

   show ?P by simp

 qed

 lemma split_mod:

 assumes m: "m \<noteq> 0"

 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"

        (is "?P = ?Q")

 proof

   assume P: ?P

   show ?Q

   proof (intro allI impI)

     fix i j

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

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

   qed

 next

   assume Q: ?Q

   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]

   show ?P by simp

 qed

 *)

 end