(* Title: HOL/IntDiv.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
*)
header{* The Division Operators div and mod *}
theory IntDiv
imports Int Divides FunDef
begin
definition divmod_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
--{*definition of quotient and remainder*}
[code]: "divmod_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
(if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
--{*for the division algorithm*}
[code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
else (2 * q, r))"
text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
"posDivAlg a b = (if a < b \<or> b \<le> 0 then (0, a)
else adjust b (posDivAlg a (2 * b)))"
by auto
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))") auto
text{*algorithm for the case @{text "a<0, b>0"}*}
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
"negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0 then (-1, a + b)
else adjust b (negDivAlg a (2 * b)))"
by auto
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))") auto
text{*algorithm for the general case @{term "b\<noteq>0"}*}
definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
[code inline]: "negateSnd = apsnd uminus"
definition divmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
--{*The full division algorithm considers all possible signs for a, b
including the special case @{text "a=0, b<0"} because
@{term negDivAlg} requires @{term "a<0"}.*}
"divmod a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
else if a = 0 then (0, 0)
else negateSnd (negDivAlg (-a) (-b))
else
if 0 < b then negDivAlg a b
else negateSnd (posDivAlg (-a) (-b)))"
instantiation int :: Divides.div
begin
definition
div_def: "a div b = fst (divmod a b)"
definition
mod_def: "a mod b = snd (divmod a b)"
instance ..
end
lemma divmod_mod_div:
"divmod p q = (p div q, p mod q)"
by (auto simp add: div_def mod_def)
text{*
Here is the division algorithm in ML:
\begin{verbatim}
fun posDivAlg (a,b) =
if a<b then (0,a)
else let val (q,r) = posDivAlg(a, 2*b)
in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
end
fun negDivAlg (a,b) =
if 0\<le>a+b then (~1,a+b)
else let val (q,r) = negDivAlg(a, 2*b)
in if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
end;
fun negateSnd (q,r:int) = (q,~r);
fun divmod (a,b) = if 0\<le>a then
if b>0 then posDivAlg (a,b)
else if a=0 then (0,0)
else negateSnd (negDivAlg (~a,~b))
else
if 0<b then negDivAlg (a,b)
else negateSnd (posDivAlg (~a,~b));
\end{verbatim}
*}
subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
lemma unique_quotient_lemma:
"[| b*q' + r' \<le> b*q + r; 0 \<le> r'; r' < b; r < b |]
==> q' \<le> (q::int)"
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
prefer 2 apply (simp add: right_diff_distrib)
apply (subgoal_tac "0 < b * (1 + q - q') ")
apply (erule_tac [2] order_le_less_trans)
prefer 2 apply (simp add: right_diff_distrib right_distrib)
apply (subgoal_tac "b * q' < b * (1 + q) ")
prefer 2 apply (simp add: right_diff_distrib right_distrib)
apply (simp add: mult_less_cancel_left)
done
lemma unique_quotient_lemma_neg:
"[| b*q' + r' \<le> b*q + r; r \<le> 0; b < r; b < r' |]
==> q \<le> (q'::int)"
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,
auto)
lemma unique_quotient:
"[| divmod_rel a b (q, r); divmod_rel a b (q', r'); b \<noteq> 0 |]
==> q = q'"
apply (simp add: divmod_rel_def linorder_neq_iff split: split_if_asm)
apply (blast intro: order_antisym
dest: order_eq_refl [THEN unique_quotient_lemma]
order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
done
lemma unique_remainder:
"[| divmod_rel a b (q, r); divmod_rel a b (q', r'); b \<noteq> 0 |]
==> r = r'"
apply (subgoal_tac "q = q'")
apply (simp add: divmod_rel_def)
apply (blast intro: unique_quotient)
done
subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
text{*And positive divisors*}
lemma adjust_eq [simp]:
"adjust b (q,r) =
(let diff = r-b in
if 0 \<le> diff then (2*q + 1, diff)
else (2*q, r))"
by (simp add: Let_def adjust_def)
declare posDivAlg.simps [simp del]
text{*use with a simproc to avoid repeatedly proving the premise*}
lemma posDivAlg_eqn:
"0 < b ==>
posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
by (rule posDivAlg.simps [THEN trans], simp)
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
theorem posDivAlg_correct:
assumes "0 \<le> a" and "0 < b"
shows "divmod_rel a b (posDivAlg a b)"
using prems apply (induct a b rule: posDivAlg.induct)
apply auto
apply (simp add: divmod_rel_def)
apply (subst posDivAlg_eqn, simp add: right_distrib)
apply (case_tac "a < b")
apply simp_all
apply (erule splitE)
apply (auto simp add: right_distrib Let_def)
done
subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
text{*And positive divisors*}
declare negDivAlg.simps [simp del]
text{*use with a simproc to avoid repeatedly proving the premise*}
lemma negDivAlg_eqn:
"0 < b ==>
negDivAlg a b =
(if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
by (rule negDivAlg.simps [THEN trans], simp)
(*Correctness of negDivAlg: it computes quotients correctly
It doesn't work if a=0 because the 0/b equals 0, not -1*)
lemma negDivAlg_correct:
assumes "a < 0" and "b > 0"
shows "divmod_rel a b (negDivAlg a b)"
using prems apply (induct a b rule: negDivAlg.induct)
apply (auto simp add: linorder_not_le)
apply (simp add: divmod_rel_def)
apply (subst negDivAlg_eqn, assumption)
apply (case_tac "a + b < (0\<Colon>int)")
apply simp_all
apply (erule splitE)
apply (auto simp add: right_distrib Let_def)
done
subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
(*the case a=0*)
lemma divmod_rel_0: "b \<noteq> 0 ==> divmod_rel 0 b (0, 0)"
by (auto simp add: divmod_rel_def linorder_neq_iff)
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
by (subst posDivAlg.simps, auto)
lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
by (subst negDivAlg.simps, auto)
lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
by (simp add: negateSnd_def)
lemma divmod_rel_neg: "divmod_rel (-a) (-b) qr ==> divmod_rel a b (negateSnd qr)"
by (auto simp add: split_ifs divmod_rel_def)
lemma divmod_correct: "b \<noteq> 0 ==> divmod_rel a b (divmod a b)"
by (force simp add: linorder_neq_iff divmod_rel_0 divmod_def divmod_rel_neg
posDivAlg_correct negDivAlg_correct)
text{*Arbitrary definitions for division by zero. Useful to simplify
certain equations.*}
lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
by (simp add: div_def mod_def divmod_def posDivAlg.simps)
text{*Basic laws about division and remainder*}
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
apply (case_tac "b = 0", simp)
apply (cut_tac a = a and b = b in divmod_correct)
apply (auto simp add: divmod_rel_def div_def mod_def)
done
lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
by(simp add: zmod_zdiv_equality[symmetric])
lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
by(simp add: mult_commute zmod_zdiv_equality[symmetric])
text {* Tool setup *}
ML {*
local
structure CancelDivMod = CancelDivModFun(
struct
val div_name = @{const_name Divides.div};
val mod_name = @{const_name Divides.mod};
val mk_binop = HOLogic.mk_binop;
val mk_sum = Int_Numeral_Simprocs.mk_sum HOLogic.intT;
val dest_sum = Int_Numeral_Simprocs.dest_sum;
val div_mod_eqs =
map mk_meta_eq [@{thm zdiv_zmod_equality},
@{thm zdiv_zmod_equality2}];
val trans = trans;
val prove_eq_sums =
let
val simps = @{thm diff_int_def} :: Int_Numeral_Simprocs.add_0s @ @{thms zadd_ac}
in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
end)
in
val cancel_zdiv_zmod_proc = Simplifier.simproc (the_context ())
"cancel_zdiv_zmod" ["(m::int) + n"] (K CancelDivMod.proc)
end;
Addsimprocs [cancel_zdiv_zmod_proc]
*}
lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
apply (cut_tac a = a and b = b in divmod_correct)
apply (auto simp add: divmod_rel_def mod_def)
done
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1, standard]
and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
apply (cut_tac a = a and b = b in divmod_correct)
apply (auto simp add: divmod_rel_def div_def mod_def)
done
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1, standard]
and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
subsection{*General Properties of div and mod*}
lemma divmod_rel_div_mod: "b \<noteq> 0 ==> divmod_rel a b (a div b, a mod b)"
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (force simp add: divmod_rel_def linorder_neq_iff)
done
lemma divmod_rel_div: "[| divmod_rel a b (q, r); b \<noteq> 0 |] ==> a div b = q"
by (simp add: divmod_rel_div_mod [THEN unique_quotient])
lemma divmod_rel_mod: "[| divmod_rel a b (q, r); b \<noteq> 0 |] ==> a mod b = r"
by (simp add: divmod_rel_div_mod [THEN unique_remainder])
lemma div_pos_pos_trivial: "[| (0::int) \<le> a; a < b |] ==> a div b = 0"
apply (rule divmod_rel_div)
apply (auto simp add: divmod_rel_def)
done
lemma div_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a div b = 0"
apply (rule divmod_rel_div)
apply (auto simp add: divmod_rel_def)
done
lemma div_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a div b = -1"
apply (rule divmod_rel_div)
apply (auto simp add: divmod_rel_def)
done
(*There is no div_neg_pos_trivial because 0 div b = 0 would supersede it*)
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a; a < b |] ==> a mod b = a"
apply (rule_tac q = 0 in divmod_rel_mod)
apply (auto simp add: divmod_rel_def)
done
lemma mod_neg_neg_trivial: "[| a \<le> (0::int); b < a |] ==> a mod b = a"
apply (rule_tac q = 0 in divmod_rel_mod)
apply (auto simp add: divmod_rel_def)
done
lemma mod_pos_neg_trivial: "[| (0::int) < a; a+b \<le> 0 |] ==> a mod b = a+b"
apply (rule_tac q = "-1" in divmod_rel_mod)
apply (auto simp add: divmod_rel_def)
done
text{*There is no @{text mod_neg_pos_trivial}.*}
(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
apply (case_tac "b = 0", simp)
apply (simp add: divmod_rel_div_mod [THEN divmod_rel_neg, simplified,
THEN divmod_rel_div, THEN sym])
done
(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
apply (case_tac "b = 0", simp)
apply (subst divmod_rel_div_mod [THEN divmod_rel_neg, simplified, THEN divmod_rel_mod],
auto)
done
subsection{*Laws for div and mod with Unary Minus*}
lemma zminus1_lemma:
"divmod_rel a b (q, r)
==> divmod_rel (-a) b (if r=0 then -q else -q - 1,
if r=0 then 0 else b-r)"
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_diff_distrib)
lemma zdiv_zminus1_eq_if:
"b \<noteq> (0::int)
==> (-a) div b =
(if a mod b = 0 then - (a div b) else - (a div b) - 1)"
by (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_div])
lemma zmod_zminus1_eq_if:
"(-a::int) mod b = (if a mod b = 0 then 0 else b - (a mod b))"
apply (case_tac "b = 0", simp)
apply (blast intro: divmod_rel_div_mod [THEN zminus1_lemma, THEN divmod_rel_mod])
done
lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
lemma zdiv_zminus2_eq_if:
"b \<noteq> (0::int)
==> a div (-b) =
(if a mod b = 0 then - (a div b) else - (a div b) - 1)"
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
lemma zmod_zminus2_eq_if:
"a mod (-b::int) = (if a mod b = 0 then 0 else (a mod b) - b)"
by (simp add: zmod_zminus1_eq_if zmod_zminus2)
subsection{*Division of a Number by Itself*}
lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
apply (subgoal_tac "0 < a*q")
apply (simp add: zero_less_mult_iff, arith)
done
lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
apply (subgoal_tac "0 \<le> a* (1-q) ")
apply (simp add: zero_le_mult_iff)
apply (simp add: right_diff_distrib)
done
lemma self_quotient: "[| divmod_rel a a (q, r); a \<noteq> (0::int) |] ==> q = 1"
apply (simp add: split_ifs divmod_rel_def linorder_neq_iff)
apply (rule order_antisym, safe, simp_all)
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
done
lemma self_remainder: "[| divmod_rel a a (q, r); a \<noteq> (0::int) |] ==> r = 0"
apply (frule self_quotient, assumption)
apply (simp add: divmod_rel_def)
done
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
by (simp add: divmod_rel_div_mod [THEN self_quotient])
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
lemma zmod_self [simp]: "a mod a = (0::int)"
apply (case_tac "a = 0", simp)
apply (simp add: divmod_rel_div_mod [THEN self_remainder])
done
subsection{*Computation of Division and Remainder*}
lemma zdiv_zero [simp]: "(0::int) div b = 0"
by (simp add: div_def divmod_def)
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
by (simp add: div_def divmod_def)
lemma zmod_zero [simp]: "(0::int) mod b = 0"
by (simp add: mod_def divmod_def)
lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
by (simp add: div_def divmod_def)
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
by (simp add: mod_def divmod_def)
text{*a positive, b positive *}
lemma div_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
by (simp add: div_def divmod_def)
lemma mod_pos_pos: "[| 0 < a; 0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
by (simp add: mod_def divmod_def)
text{*a negative, b positive *}
lemma div_neg_pos: "[| a < 0; 0 < b |] ==> a div b = fst (negDivAlg a b)"
by (simp add: div_def divmod_def)
lemma mod_neg_pos: "[| a < 0; 0 < b |] ==> a mod b = snd (negDivAlg a b)"
by (simp add: mod_def divmod_def)
text{*a positive, b negative *}
lemma div_pos_neg:
"[| 0 < a; b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
by (simp add: div_def divmod_def)
lemma mod_pos_neg:
"[| 0 < a; b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
by (simp add: mod_def divmod_def)
text{*a negative, b negative *}
lemma div_neg_neg:
"[| a < 0; b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
by (simp add: div_def divmod_def)
lemma mod_neg_neg:
"[| a < 0; b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
by (simp add: mod_def divmod_def)
text {*Simplify expresions in which div and mod combine numerical constants*}
lemma divmod_relI:
"\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
\<Longrightarrow> divmod_rel a b (q, r)"
unfolding divmod_rel_def by simp
lemmas divmod_rel_div_eq = divmod_relI [THEN divmod_rel_div, THEN eq_reflection]
lemmas divmod_rel_mod_eq = divmod_relI [THEN divmod_rel_mod, THEN eq_reflection]
lemmas arithmetic_simps =
arith_simps
add_special
OrderedGroup.add_0_left
OrderedGroup.add_0_right
mult_zero_left
mult_zero_right
mult_1_left
mult_1_right
(* simprocs adapted from HOL/ex/Binary.thy *)
ML {*
local
infix ==;
val op == = Logic.mk_equals;
fun plus m n = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n;
fun mult m n = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ m $ n;
val binary_ss = HOL_basic_ss addsimps @{thms arithmetic_simps};
fun prove ctxt prop =
Goal.prove ctxt [] [] prop (fn _ => ALLGOALS (full_simp_tac binary_ss));
fun binary_proc proc ss ct =
(case Thm.term_of ct of
_ $ t $ u =>
(case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
SOME args => proc (Simplifier.the_context ss) args
| NONE => NONE)
| _ => NONE);
in
fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
if n = 0 then NONE
else
let val (k, l) = Integer.div_mod m n;
fun mk_num x = HOLogic.mk_number HOLogic.intT x;
in SOME (rule OF [prove ctxt (t == plus (mult u (mk_num k)) (mk_num l))])
end);
end;
*}
simproc_setup binary_int_div ("number_of m div number_of n :: int") =
{* K (divmod_proc (@{thm divmod_rel_div_eq})) *}
simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
{* K (divmod_proc (@{thm divmod_rel_mod_eq})) *}
(* The following 8 lemmas are made unnecessary by the above simprocs: *)
lemmas div_pos_pos_number_of =
div_pos_pos [of "number_of v" "number_of w", standard]
lemmas div_neg_pos_number_of =
div_neg_pos [of "number_of v" "number_of w", standard]
lemmas div_pos_neg_number_of =
div_pos_neg [of "number_of v" "number_of w", standard]
lemmas div_neg_neg_number_of =
div_neg_neg [of "number_of v" "number_of w", standard]
lemmas mod_pos_pos_number_of =
mod_pos_pos [of "number_of v" "number_of w", standard]
lemmas mod_neg_pos_number_of =
mod_neg_pos [of "number_of v" "number_of w", standard]
lemmas mod_pos_neg_number_of =
mod_pos_neg [of "number_of v" "number_of w", standard]
lemmas mod_neg_neg_number_of =
mod_neg_neg [of "number_of v" "number_of w", standard]
lemmas posDivAlg_eqn_number_of [simp] =
posDivAlg_eqn [of "number_of v" "number_of w", standard]
lemmas negDivAlg_eqn_number_of [simp] =
negDivAlg_eqn [of "number_of v" "number_of w", standard]
text{*Special-case simplification *}
lemma zmod_1 [simp]: "a mod (1::int) = 0"
apply (cut_tac a = a and b = 1 in pos_mod_sign)
apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
apply (auto simp del:pos_mod_bound pos_mod_sign)
done
lemma zdiv_1 [simp]: "a div (1::int) = a"
by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
apply (auto simp del: neg_mod_sign neg_mod_bound)
done
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
(** The last remaining special cases for constant arithmetic:
1 div z and 1 mod z **)
lemmas div_pos_pos_1_number_of [simp] =
div_pos_pos [OF int_0_less_1, of "number_of w", standard]
lemmas div_pos_neg_1_number_of [simp] =
div_pos_neg [OF int_0_less_1, of "number_of w", standard]
lemmas mod_pos_pos_1_number_of [simp] =
mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
lemmas mod_pos_neg_1_number_of [simp] =
mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
lemmas posDivAlg_eqn_1_number_of [simp] =
posDivAlg_eqn [of concl: 1 "number_of w", standard]
lemmas negDivAlg_eqn_1_number_of [simp] =
negDivAlg_eqn [of concl: 1 "number_of w", standard]
subsection{*Monotonicity in the First Argument (Dividend)*}
lemma zdiv_mono1: "[| a \<le> a'; 0 < (b::int) |] ==> a div b \<le> a' div b"
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
apply (rule unique_quotient_lemma)
apply (erule subst)
apply (erule subst, simp_all)
done
lemma zdiv_mono1_neg: "[| a \<le> a'; (b::int) < 0 |] ==> a' div b \<le> a div b"
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
apply (rule unique_quotient_lemma_neg)
apply (erule subst)
apply (erule subst, simp_all)
done
subsection{*Monotonicity in the Second Argument (Divisor)*}
lemma q_pos_lemma:
"[| 0 \<le> b'*q' + r'; r' < b'; 0 < b' |] ==> 0 \<le> (q'::int)"
apply (subgoal_tac "0 < b'* (q' + 1) ")
apply (simp add: zero_less_mult_iff)
apply (simp add: right_distrib)
done
lemma zdiv_mono2_lemma:
"[| b*q + r = b'*q' + r'; 0 \<le> b'*q' + r';
r' < b'; 0 \<le> r; 0 < b'; b' \<le> b |]
==> q \<le> (q'::int)"
apply (frule q_pos_lemma, assumption+)
apply (subgoal_tac "b*q < b* (q' + 1) ")
apply (simp add: mult_less_cancel_left)
apply (subgoal_tac "b*q = r' - r + b'*q'")
prefer 2 apply simp
apply (simp (no_asm_simp) add: right_distrib)
apply (subst add_commute, rule zadd_zless_mono, arith)
apply (rule mult_right_mono, auto)
done
lemma zdiv_mono2:
"[| (0::int) \<le> a; 0 < b'; b' \<le> b |] ==> a div b \<le> a div b'"
apply (subgoal_tac "b \<noteq> 0")
prefer 2 apply arith
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
apply (rule zdiv_mono2_lemma)
apply (erule subst)
apply (erule subst, simp_all)
done
lemma q_neg_lemma:
"[| b'*q' + r' < 0; 0 \<le> r'; 0 < b' |] ==> q' \<le> (0::int)"
apply (subgoal_tac "b'*q' < 0")
apply (simp add: mult_less_0_iff, arith)
done
lemma zdiv_mono2_neg_lemma:
"[| b*q + r = b'*q' + r'; b'*q' + r' < 0;
r < b; 0 \<le> r'; 0 < b'; b' \<le> b |]
==> q' \<le> (q::int)"
apply (frule q_neg_lemma, assumption+)
apply (subgoal_tac "b*q' < b* (q + 1) ")
apply (simp add: mult_less_cancel_left)
apply (simp add: right_distrib)
apply (subgoal_tac "b*q' \<le> b'*q'")
prefer 2 apply (simp add: mult_right_mono_neg, arith)
done
lemma zdiv_mono2_neg:
"[| a < (0::int); 0 < b'; b' \<le> b |] ==> a div b' \<le> a div b"
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
apply (rule zdiv_mono2_neg_lemma)
apply (erule subst)
apply (erule subst, simp_all)
done
subsection{*More Algebraic Laws for div and mod*}
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
lemma zmult1_lemma:
"[| divmod_rel b c (q, r); c \<noteq> 0 |]
==> divmod_rel (a * b) c (a*q + a*r div c, a*r mod c)"
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
apply (case_tac "c = 0", simp)
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_div])
done
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
apply (case_tac "c = 0", simp)
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
done
lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
apply (rule trans)
apply (rule_tac s = "b*a mod c" in trans)
apply (rule_tac [2] zmod_zmult1_eq)
apply (simp_all add: mult_commute)
done
lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
apply (rule zmod_zmult1_eq' [THEN trans])
apply (rule zmod_zmult1_eq)
done
lemma zdiv_zmult_self1 [simp]: "b \<noteq> (0::int) ==> (a*b) div b = a"
by (simp add: zdiv_zmult1_eq)
lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
apply (case_tac "b = 0", simp)
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
done
lemma zmod_zmod_trivial: "(a mod b) mod b = a mod (b::int)"
apply (case_tac "b = 0", simp)
apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
done
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
lemma zadd1_lemma:
"[| divmod_rel a c (aq, ar); divmod_rel b c (bq, br); c \<noteq> 0 |]
==> divmod_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
by (force simp add: split_ifs divmod_rel_def linorder_neq_iff right_distrib)
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
lemma zdiv_zadd1_eq:
"(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
apply (case_tac "c = 0", simp)
apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_div)
done
lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
apply (case_tac "c = 0", simp)
apply (blast intro: zadd1_lemma [OF divmod_rel_div_mod divmod_rel_div_mod] divmod_rel_mod)
done
instance int :: ring_div
proof
fix a b c :: int
assume not0: "b \<noteq> 0"
show "(a + c * b) div b = c + a div b"
unfolding zdiv_zadd1_eq [of a "c * b"] using not0
by (simp add: zmod_zmult1_eq zmod_zdiv_trivial)
qed auto
lemma posDivAlg_div_mod:
assumes "k \<ge> 0"
and "l \<ge> 0"
shows "posDivAlg k l = (k div l, k mod l)"
proof (cases "l = 0")
case True then show ?thesis by (simp add: posDivAlg.simps)
next
case False with assms posDivAlg_correct
have "divmod_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
by simp
from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
show ?thesis by simp
qed
lemma negDivAlg_div_mod:
assumes "k < 0"
and "l > 0"
shows "negDivAlg k l = (k div l, k mod l)"
proof -
from assms have "l \<noteq> 0" by simp
from assms negDivAlg_correct
have "divmod_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
by simp
from divmod_rel_div [OF this `l \<noteq> 0`] divmod_rel_mod [OF this `l \<noteq> 0`]
show ?thesis by simp
qed
lemma zdiv_zadd_self1: "a \<noteq> (0::int) ==> (a+b) div a = b div a + 1"
by (rule div_add_self1) (* already declared [simp] *)
lemma zdiv_zadd_self2: "a \<noteq> (0::int) ==> (b+a) div a = b div a + 1"
by (rule div_add_self2) (* already declared [simp] *)
lemma zdiv_zmult_self2: "b \<noteq> (0::int) ==> (b*a) div b = a"
by (rule div_mult_self1_is_id) (* already declared [simp] *)
lemma zmod_zmult_self1: "(a*b) mod b = (0::int)"
by (rule mod_mult_self2_is_0) (* already declared [simp] *)
lemma zmod_zmult_self2: "(b*a) mod b = (0::int)"
by (rule mod_mult_self1_is_0) (* already declared [simp] *)
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
(* REVISIT: should this be generalized to all semiring_div types? *)
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
by (rule mod_add_left_eq)
lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
by (rule mod_add_right_eq)
lemma zmod_zadd_self1: "(a+b) mod a = b mod (a::int)"
by (rule mod_add_self1) (* already declared [simp] *)
lemma zmod_zadd_self2: "(b+a) mod a = b mod (a::int)"
by (rule mod_add_self2) (* already declared [simp] *)
lemma zmod_zdiff1_eq: "(a - b) mod c = (a mod c - b mod c) mod (c::int)"
by (rule mod_diff_eq)
subsection{*Proving @{term "a div (b*c) = (a div b) div c"} *}
(*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but
7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems
to cause particular problems.*)
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \<le> 0 |] ==> b*c < b*(q mod c) + r"
apply (subgoal_tac "b * (c - q mod c) < r * 1")
apply (simp add: algebra_simps)
apply (rule order_le_less_trans)
apply (erule_tac [2] mult_strict_right_mono)
apply (rule mult_left_mono_neg)
using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
apply (simp)
apply (simp)
done
lemma zmult2_lemma_aux2:
"[| (0::int) < c; b < r; r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
apply (subgoal_tac "b * (q mod c) \<le> 0")
apply arith
apply (simp add: mult_le_0_iff)
done
lemma zmult2_lemma_aux3: "[| (0::int) < c; 0 \<le> r; r < b |] ==> 0 \<le> b * (q mod c) + r"
apply (subgoal_tac "0 \<le> b * (q mod c) ")
apply arith
apply (simp add: zero_le_mult_iff)
done
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
apply (simp add: right_diff_distrib)
apply (rule order_less_le_trans)
apply (erule mult_strict_right_mono)
apply (rule_tac [2] mult_left_mono)
apply simp
using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
apply simp
done
lemma zmult2_lemma: "[| divmod_rel a b (q, r); b \<noteq> 0; 0 < c |]
==> divmod_rel a (b * c) (q div c, b*(q mod c) + r)"
by (auto simp add: mult_ac divmod_rel_def linorder_neq_iff
zero_less_mult_iff right_distrib [symmetric]
zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
apply (case_tac "b = 0", simp)
apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_div])
done
lemma zmod_zmult2_eq:
"(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
apply (case_tac "b = 0", simp)
apply (force simp add: divmod_rel_div_mod [THEN zmult2_lemma, THEN divmod_rel_mod])
done
subsection{*Cancellation of Common Factors in div*}
lemma zdiv_zmult_zmult1_aux1:
"[| (0::int) < b; c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
by (subst zdiv_zmult2_eq, auto)
lemma zdiv_zmult_zmult1_aux2:
"[| b < (0::int); c \<noteq> 0 |] ==> (c*a) div (c*b) = a div b"
apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
done
lemma zdiv_zmult_zmult1: "c \<noteq> (0::int) ==> (c*a) div (c*b) = a div b"
apply (case_tac "b = 0", simp)
apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
done
lemma zdiv_zmult_zmult1_if[simp]:
"(k*m) div (k*n) = (if k = (0::int) then 0 else m div n)"
by (simp add:zdiv_zmult_zmult1)
(*
lemma zdiv_zmult_zmult2: "c \<noteq> (0::int) ==> (a*c) div (b*c) = a div b"
apply (drule zdiv_zmult_zmult1)
apply (auto simp add: mult_commute)
done
*)
subsection{*Distribution of Factors over mod*}
lemma zmod_zmult_zmult1_aux1:
"[| (0::int) < b; c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
by (subst zmod_zmult2_eq, auto)
lemma zmod_zmult_zmult1_aux2:
"[| b < (0::int); c \<noteq> 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
done
lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
apply (case_tac "b = 0", simp)
apply (case_tac "c = 0", simp)
apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
done
lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
apply (cut_tac c = c in zmod_zmult_zmult1)
apply (auto simp add: mult_commute)
done
lemma zmod_zmod_cancel: "n dvd m \<Longrightarrow> (k::int) mod m mod n = k mod n"
by (rule mod_mod_cancel)
subsection {*Splitting Rules for div and mod*}
text{*The proofs of the two lemmas below are essentially identical*}
lemma split_pos_lemma:
"0<k ==>
P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
apply (rule iffI, clarify)
apply (erule_tac P="P ?x ?y" in rev_mp)
apply (subst zmod_zadd1_eq)
apply (subst zdiv_zadd1_eq)
apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
txt{*converse direction*}
apply (drule_tac x = "n div k" in spec)
apply (drule_tac x = "n mod k" in spec, simp)
done
lemma split_neg_lemma:
"k<0 ==>
P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
apply (rule iffI, clarify)
apply (erule_tac P="P ?x ?y" in rev_mp)
apply (subst zmod_zadd1_eq)
apply (subst zdiv_zadd1_eq)
apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
txt{*converse direction*}
apply (drule_tac x = "n div k" in spec)
apply (drule_tac x = "n mod k" in spec, simp)
done
lemma split_zdiv:
"P(n div k :: int) =
((k = 0 --> P 0) &
(0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
(k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
apply (case_tac "k=0", simp)
apply (simp only: linorder_neq_iff)
apply (erule disjE)
apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
split_neg_lemma [of concl: "%x y. P x"])
done
lemma split_zmod:
"P(n mod k :: int) =
((k = 0 --> P n) &
(0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
(k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
apply (case_tac "k=0", simp)
apply (simp only: linorder_neq_iff)
apply (erule disjE)
apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
split_neg_lemma [of concl: "%x y. P y"])
done
(* Enable arith to deal with div 2 and mod 2: *)
declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
subsection{*Speeding up the Division Algorithm with Shifting*}
text{*computing div by shifting *}
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
proof cases
assume "a=0"
thus ?thesis by simp
next
assume "a\<noteq>0" and le_a: "0\<le>a"
hence a_pos: "1 \<le> a" by arith
hence one_less_a2: "1 < 2*a" by arith
hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
by (simp add: mult_le_cancel_left add_commute [of 1] add1_zle_eq)
with a_pos have "0 \<le> b mod a" by simp
hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
by (simp add: mod_pos_pos_trivial one_less_a2)
with le_2a
have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
right_distrib)
thus ?thesis
by (subst zdiv_zadd1_eq,
simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
div_pos_pos_trivial)
qed
lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
apply (rule_tac [2] pos_zdiv_mult_2)
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
simp)
done
(*Not clear why this must be proved separately; probably number_of causes
simplification problems*)
lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
by auto
lemma zdiv_number_of_Bit0 [simp]:
"number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =
number_of v div (number_of w :: int)"
by (simp only: number_of_eq numeral_simps) simp
lemma zdiv_number_of_Bit1 [simp]:
"number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =
(if (0::int) \<le> number_of w
then number_of v div (number_of w)
else (number_of v + (1::int)) div (number_of w))"
apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)
apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
done
subsection{*Computing mod by Shifting (proofs resemble those for div)*}
lemma pos_zmod_mult_2:
"(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
apply (case_tac "a = 0", simp)
apply (subgoal_tac "1 < a * 2")
prefer 2 apply arith
apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
apply (rule_tac [2] mult_left_mono)
apply (auto simp add: add_commute [of 1] mult_commute add1_zle_eq
pos_mod_bound)
apply (subst zmod_zadd1_eq)
apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
apply (rule mod_pos_pos_trivial)
apply (auto simp add: mod_pos_pos_trivial ring_distribs)
apply (subgoal_tac "0 \<le> b mod a", arith, simp)
done
lemma neg_zmod_mult_2:
"a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =
1 + 2* ((-b - 1) mod (-a))")
apply (rule_tac [2] pos_zmod_mult_2)
apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
prefer 2 apply simp
apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
done
lemma zmod_number_of_Bit0 [simp]:
"number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =
(2::int) * (number_of v mod number_of w)"
apply (simp only: number_of_eq numeral_simps)
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
not_0_le_lemma neg_zmod_mult_2 add_ac)
done
lemma zmod_number_of_Bit1 [simp]:
"number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =
(if (0::int) \<le> number_of w
then 2 * (number_of v mod number_of w) + 1
else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
apply (simp only: number_of_eq numeral_simps)
apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
not_0_le_lemma neg_zmod_mult_2 add_ac)
done
subsection{*Quotients of Signs*}
lemma div_neg_pos_less0: "[| a < (0::int); 0 < b |] ==> a div b < 0"
apply (subgoal_tac "a div b \<le> -1", force)
apply (rule order_trans)
apply (rule_tac a' = "-1" in zdiv_mono1)
apply (auto simp add: zdiv_minus1)
done
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
by (drule zdiv_mono1_neg, auto)
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
apply auto
apply (drule_tac [2] zdiv_mono1)
apply (auto simp add: linorder_neq_iff)
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
apply (blast intro: div_neg_pos_less0)
done
lemma neg_imp_zdiv_nonneg_iff:
"b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
apply (subst zdiv_zminus_zminus [symmetric])
apply (subst pos_imp_zdiv_nonneg_iff, auto)
done
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
subsection {* The Divides Relation *}
lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
by (rule dvd_eq_mod_eq_0)
lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
zdvd_iff_zmod_eq_0 [of "number_of x" "number_of y", standard]
lemma zdvd_0_right: "(m::int) dvd 0"
by (rule dvd_0_right) (* already declared [iff] *)
lemma zdvd_0_left: "(0 dvd (m::int)) = (m = 0)"
by (rule dvd_0_left_iff) (* already declared [noatp,simp] *)
lemma zdvd_1_left: "1 dvd (m::int)"
by (rule one_dvd) (* already declared [simp] *)
lemma zdvd_refl [simp]: "m dvd (m::int)"
by (rule dvd_refl) (* TODO: declare generic dvd_refl [simp] *)
lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
by (rule dvd_trans)
lemma zdvd_zminus_iff: "m dvd -n \<longleftrightarrow> m dvd (n::int)"
by (rule dvd_minus_iff)
lemma zdvd_zminus2_iff: "-m dvd n \<longleftrightarrow> m dvd (n::int)"
by (rule minus_dvd_iff)
lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)"
by (cases "i > 0") (simp_all add: zdvd_zminus2_iff)
lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)"
by (cases "j > 0") (simp_all add: zdvd_zminus_iff)
lemma zdvd_anti_sym:
"0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
apply (simp add: dvd_def, auto)
apply (simp add: mult_assoc zero_less_mult_iff zmult_eq_1_iff)
done
lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
by (rule dvd_add)
lemma zdvd_dvd_eq: assumes anz:"a \<noteq> 0" and ab: "(a::int) dvd b" and ba:"b dvd a"
shows "\<bar>a\<bar> = \<bar>b\<bar>"
proof-
from ab obtain k where k:"b = a*k" unfolding dvd_def by blast
from ba obtain k' where k':"a = b*k'" unfolding dvd_def by blast
from k k' have "a = a*k*k'" by simp
with mult_cancel_left1[where c="a" and b="k*k'"]
have kk':"k*k' = 1" using anz by (simp add: mult_assoc)
hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
thus ?thesis using k k' by auto
qed
lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
by (rule Ring_and_Field.dvd_diff)
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
apply (subgoal_tac "m = n + (m - n)")
apply (erule ssubst)
apply (blast intro: zdvd_zadd, simp)
done
lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
by (rule dvd_mult)
lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
by (rule dvd_mult2)
lemma zdvd_triv_right: "(k::int) dvd m * k"
by (rule dvd_triv_right) (* already declared [simp] *)
lemma zdvd_triv_left: "(k::int) dvd k * m"
by (rule dvd_triv_left) (* already declared [simp] *)
lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
by (rule dvd_mult_left)
lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
by (rule dvd_mult_right)
lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
by (rule mult_dvd_mono)
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
apply (rule iffI)
apply (erule_tac [2] zdvd_zadd)
apply (subgoal_tac "n = (n + k * m) - k * m")
apply (erule ssubst)
apply (erule zdvd_zdiff, simp_all)
done
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
apply (simp add: dvd_def)
apply (auto simp add: zmod_zmult_zmult1)
done
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
apply (subgoal_tac "k dvd n * (m div n) + m mod n")
apply (simp add: zmod_zdiv_equality [symmetric])
apply (simp only: zdvd_zadd zdvd_zmult2)
done
lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
apply (auto elim!: dvdE)
apply (subgoal_tac "0 < n")
prefer 2
apply (blast intro: order_less_trans)
apply (simp add: zero_less_mult_iff)
apply (subgoal_tac "n * k < n * 1")
apply (drule mult_less_cancel_left [THEN iffD1], auto)
done
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
using zmod_zdiv_equality[where a="m" and b="n"]
by (simp add: algebra_simps)
lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
apply (subgoal_tac "m mod n = 0")
apply (simp add: zmult_div_cancel)
apply (simp only: zdvd_iff_zmod_eq_0)
done
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
shows "m dvd n"
proof-
from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
{assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
with h have False by (simp add: mult_assoc)}
hence "n = m * h" by blast
thus ?thesis by simp
qed
lemma zdvd_zmult_cancel_disj[simp]:
"(k*m) dvd (k*n) = (k=0 | m dvd (n::int))"
by (auto simp: zdvd_zmult_mono dest: zdvd_mult_cancel)
theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
apply (simp split add: split_nat)
apply (rule iffI)
apply (erule exE)
apply (rule_tac x = "int x" in exI)
apply simp
apply (erule exE)
apply (rule_tac x = "nat x" in exI)
apply (erule conjE)
apply (erule_tac x = "nat x" in allE)
apply simp
done
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
proof -
have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
proof -
fix k
assume A: "int y = int x * k"
then show "x dvd y" proof (cases k)
case (1 n) with A have "y = x * n" by (simp add: zmult_int)
then show ?thesis ..
next
case (2 n) with A have "int y = int x * (- int (Suc n))" by simp
also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
also have "\<dots> = - int (x * Suc n)" by (simp only: zmult_int)
finally have "- int (x * Suc n) = int y" ..
then show ?thesis by (simp only: negative_eq_positive) auto
qed
qed
then show ?thesis by (auto elim!: dvdE simp only: zdvd_triv_left int_mult)
qed
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = ( \<bar>x\<bar> = 1)"
proof
assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by (simp add: zdvd_abs1)
hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
hence "nat \<bar>x\<bar> = 1" by simp
thus "\<bar>x\<bar> = 1" by (cases "x < 0", auto)
next
assume "\<bar>x\<bar>=1" thus "x dvd 1"
by(cases "x < 0",simp_all add: minus_equation_iff zdvd_iff_zmod_eq_0)
qed
lemma zdvd_mult_cancel1:
assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
proof
assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
by (cases "n >0", auto simp add: zdvd_zminus2_iff minus_equation_iff)
next
assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
qed
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus_iff)
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
unfolding zdvd_int by (cases "z \<ge> 0") (simp_all add: zdvd_zminus2_iff)
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
by (auto simp add: dvd_int_iff)
lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
by (rule minus_dvd_iff)
lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
by (rule dvd_minus_iff)
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
apply (rule_tac z=n in int_cases)
apply (auto simp add: dvd_int_iff)
apply (rule_tac z=z in int_cases)
apply (auto simp add: dvd_imp_le)
done
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
apply (induct "y", auto)
apply (rule zmod_zmult1_eq [THEN trans])
apply (simp (no_asm_simp))
apply (rule zmod_zmult_distrib [symmetric])
done
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
apply (subst split_div, auto)
apply (subst split_zdiv, auto)
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
done
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
apply (subst split_mod, auto)
apply (subst split_zmod, auto)
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia
in unique_remainder)
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
done
text{*Suggested by Matthias Daum*}
lemma int_power_div_base:
"\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
apply (subgoal_tac "k ^ m = k ^ ((m - 1) + 1)")
apply (erule ssubst)
apply (simp only: power_add)
apply simp_all
done
text {* by Brian Huffman *}
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
by (rule mod_minus_eq [symmetric])
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
by (rule mod_diff_left_eq [symmetric])
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
by (rule mod_diff_right_eq [symmetric])
lemmas zmod_simps =
IntDiv.zmod_zadd_left_eq [symmetric]
IntDiv.zmod_zadd_right_eq [symmetric]
IntDiv.zmod_zmult1_eq [symmetric]
IntDiv.zmod_zmult1_eq' [symmetric]
IntDiv.zpower_zmod
zminus_zmod zdiff_zmod_left zdiff_zmod_right
text {* Distributive laws for function @{text nat}. *}
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
apply (rule linorder_cases [of y 0])
apply (simp add: div_nonneg_neg_le0)
apply simp
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
done
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
lemma nat_mod_distrib:
"\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)
apply (simp add: nat_eq_iff zmod_int)
done
text{*Suggested by Matthias Daum*}
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
apply (subgoal_tac "nat x div nat k < nat x")
apply (simp (asm_lr) add: nat_div_distrib [symmetric])
apply (rule Divides.div_less_dividend, simp_all)
done
text {* code generator setup *}
context ring_1
begin
lemma of_int_num [code]:
"of_int k = (if k = 0 then 0 else if k < 0 then
- of_int (- k) else let
(l, m) = divmod k 2;
l' = of_int l
in if m = 0 then l' + l' else l' + l' + 1)"
proof -
have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>
of_int k = of_int (k div 2 * 2 + 1)"
proof -
have "k mod 2 < 2" by (auto intro: pos_mod_bound)
moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
moreover assume "k mod 2 \<noteq> 0"
ultimately have "k mod 2 = 1" by arith
moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
ultimately show ?thesis by auto
qed
have aux2: "\<And>x. of_int 2 * x = x + x"
proof -
fix x
have int2: "(2::int) = 1 + 1" by arith
show "of_int 2 * x = x + x"
unfolding int2 of_int_add left_distrib by simp
qed
have aux3: "\<And>x. x * of_int 2 = x + x"
proof -
fix x
have int2: "(2::int) = 1 + 1" by arith
show "x * of_int 2 = x + x"
unfolding int2 of_int_add right_distrib by simp
qed
from aux1 show ?thesis by (auto simp add: divmod_mod_div Let_def aux2 aux3)
qed
end
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
proof
assume H: "x mod n = y mod n"
hence "x mod n - y mod n = 0" by simp
hence "(x mod n - y mod n) mod n = 0" by simp
hence "(x - y) mod n = 0" by (simp add: zmod_zdiff1_eq[symmetric])
thus "n dvd x - y" by (simp add: zdvd_iff_zmod_eq_0)
next
assume H: "n dvd x - y"
then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
hence "x = n*k + y" by simp
hence "x mod n = (n*k + y) mod n" by simp
thus "x mod n = y mod n" by (simp add: zmod_zadd_left_eq)
qed
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"
shows "\<exists>q. x = y + n * q"
proof-
from xy have th: "int x - int y = int (x - y)" by simp
from xyn have "int x mod int n = int y mod int n"
by (simp add: zmod_int[symmetric])
hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])
hence "n dvd x - y" by (simp add: th zdvd_int)
then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
qed
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
(is "?lhs = ?rhs")
proof
assume H: "x mod n = y mod n"
{assume xy: "x \<le> y"
from H have th: "y mod n = x mod n" by simp
from nat_mod_eq_lemma[OF th xy] have ?rhs
apply clarify apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
moreover
{assume xy: "y \<le> x"
from nat_mod_eq_lemma[OF H xy] have ?rhs
apply clarify apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
ultimately show ?rhs using linear[of x y] by blast
next
assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
thus ?lhs by simp
qed
code_modulename SML
IntDiv Integer
code_modulename OCaml
IntDiv Integer
code_modulename Haskell
IntDiv Integer
end