src/HOL/Divides.thy

Adapted to changes in Transitive_Closure theory.

(*  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 =
  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)"

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

defs
  (*This definition helps prove the harder properties of div and mod.
    It is copied from IntDiv.thy; should it be overloaded?*)
  quorem_def: "quorem \<equiv> (%((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 (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.div";
val mod_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 [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"], 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 (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))"
  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 (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 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 = 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


subsection {* Code generation for div and mod *}

definition
  "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

code_modulename SML
  Divides Integer

code_modulename OCaml
  Divides Integer

hide (open) const divmod


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";
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