(* 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
uses "~~/src/Provers/Arith/cancel_div_mod.ML"
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" (infixl "\<^loc>div" 70)
fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<^loc>mod" 70)
instance nat :: Divides.div
div_def: "m div n == wfrec (pred_nat^+)
(%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m"
mod_def: "m mod n == wfrec (pred_nat^+)
(%f j. if j<n | n=0 then j else f (j-n)) m" ..
definition (in times)
dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<^loc>dvd" 50)
where
"m \<^loc>dvd n \<longleftrightarrow> (\<exists>k. n = m \<^loc>* k)"
lemmas dvd_def = dvd_def [folded times_class.dvd]
class dvd_mod = times + div + zero + -- {* for code generation *}
assumes dvd_def_mod: "times.dvd (op \<^loc>*) x y \<longleftrightarrow> y \<^loc>mod x = \<^loc>0"
lemmas dvd_def_mod [code func] = dvd_def_mod [folded times_class.dvd]
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
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)
text {* @{term "op dvd"} is a partial order *}
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> m \<noteq> n"]
by unfold_locales (auto intro: dvd_trans dvd_anti_sym)
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
instance nat :: dvd_mod
by default (simp add: times_class.dvd [symmetric] dvd_eq_mod_eq_0)
code_modulename SML
Divides Nat
code_modulename OCaml
Divides Nat
code_modulename Haskell
Divides Nat
hide (open) const divmod
end