--- a/src/HOL/IntDiv.thy Wed Mar 04 10:43:39 2009 +0100
+++ b/src/HOL/IntDiv.thy Wed Mar 04 10:45:52 2009 +0100
@@ -547,34 +547,6 @@
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]
@@ -584,15 +556,6 @@
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)
@@ -726,9 +689,6 @@
apply (blast intro: divmod_rel_div_mod [THEN zmult1_lemma, THEN divmod_rel_mod])
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)
@@ -754,7 +714,7 @@
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)
+ by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
qed auto
lemma posDivAlg_div_mod:
@@ -784,41 +744,12 @@
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"} *}
@@ -902,13 +833,6 @@
"(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*}
@@ -933,9 +857,6 @@
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*}
@@ -1070,7 +991,7 @@
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 (auto simp add: 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])
@@ -1132,38 +1053,8 @@
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: "m dvd (m::int)"
- by (rule dvd_refl) (* already declared [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) (* already declared [simp] *)
-
-lemma zdvd_zminus2_iff: "-m dvd n \<longleftrightarrow> m dvd (n::int)"
- by (rule minus_dvd_iff) (* already declared [simp] *)
-
-lemma zdvd_abs1: "( \<bar>i::int\<bar> dvd j) = (i dvd j)"
- by (rule abs_dvd_iff) (* already declared [simp] *)
-
-lemma zdvd_abs2: "( (i::int) dvd \<bar>j\<bar>) = (i dvd j)"
- by (rule dvd_abs_iff) (* already declared [simp] *)
+ dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
lemma zdvd_anti_sym:
"0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
@@ -1171,58 +1062,32 @@
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"
+lemma zdvd_dvd_eq: assumes "a \<noteq> 0" and "(a::int) dvd b" and "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 `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast
+ from `b dvd a` 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)
+ have kk':"k*k' = 1" using `a\<noteq>0` 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)
+ apply (blast intro: dvd_add, 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
+apply (rule iffI)
+ apply (erule_tac [2] dvd_add)
+ apply (subgoal_tac "n = (n + k * m) - k * m")
+ apply (erule ssubst)
+ apply (erule dvd_diff)
+ apply(simp_all)
+done
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
apply (simp add: dvd_def)
@@ -1232,7 +1097,7 @@
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)
+ apply (simp only: dvd_add dvd_mult2)
done
lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
@@ -1252,7 +1117,7 @@
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)
+apply (simp only: dvd_eq_mod_eq_0)
done
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
@@ -1265,10 +1130,6 @@
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)
@@ -1300,44 +1161,38 @@
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)
+ then show ?thesis by (auto elim!: dvdE simp only: dvd_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)
+ assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
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)
+ by(cases "x < 0",simp_all add: minus_equation_iff dvd_eq_mod_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)
+ by (cases "n >0", auto simp add: minus_dvd_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)
+ unfolding zdvd_int by (cases "z \<ge> 0") simp_all
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)
+ unfolding zdvd_int by (cases "z \<ge> 0") simp_all
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)
@@ -1367,10 +1222,13 @@
apply (auto simp add: IntDiv.divmod_rel_def of_nat_mult)
done
+lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
+by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
+
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 (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
apply (erule ssubst)
apply (simp only: power_add)
apply simp_all
@@ -1387,8 +1245,8 @@
by (rule mod_diff_right_eq [symmetric])
lemmas zmod_simps =
- IntDiv.zmod_zadd_left_eq [symmetric]
- IntDiv.zmod_zadd_right_eq [symmetric]
+ mod_add_left_eq [symmetric]
+ mod_add_right_eq [symmetric]
IntDiv.zmod_zmult1_eq [symmetric]
mod_mult_left_eq [symmetric]
IntDiv.zpower_zmod
@@ -1463,14 +1321,14 @@
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)
+ hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
+ thus "n dvd x - y" by (simp add: dvd_eq_mod_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)
+ thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
qed
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"