(*  Title:      HOL/Divides.thy

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1999  University of Cambridge

The division operators div, mod and the divides relation "dvd"

*)

theory Divides = NatArith:

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

  moreover, dvd is defined whenever multiplication is*)

axclass

  div < type

instance  nat :: div ..

instance  nat :: plus_ac0

proof qed (rule add_commute add_assoc add_0)+

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: diff_less 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 not_less_iff_le)

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_tac "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_tac "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_tac "m" rule: nat_less_induct)

apply (subst mod_if, simp)

apply (simp add: mod_geq diff_less 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_tac "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: diff_less 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 not_less_iff_le)

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_tac "m" rule: nat_less_induct)

apply (subst mod_if)

apply (simp_all (no_asm_simp) add: add_assoc div_geq add_diff_inverse diff_less)

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 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_tac "m" rule: nat_less_induct)

apply (case_tac "na<n", simp) 

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

apply (simp add: mod_geq diff_less)

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;  0 < 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_tac "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_tac "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_less 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 add: diff_less)

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, simp]:

     "!!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 add: diff_less)

done

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_tac "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: not_less_iff_le le_Suc_eq mod_geq)

apply (auto simp add: Suc_diff_le diff_less le_mod_geq)

done

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 not_less_iff_le [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

(* k dvd (m*k) *)

declare dvd_refl [THEN dvd_mult, iff] dvd_refl [THEN dvd_mult2, iff]

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)

declare mod_eq_0_iff [THEN iffD1, 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)

  apply (simp_all add: quorem_def, 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

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_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