header {* Greatest common divisor and least common multiple *}
theory GCD
imports Fact Parity
begin
declare One_nat_def [simp del]
subsection {* GCD and LCM definitions *}
class gcd = zero + one + dvd +
fixes
gcd :: "'a => 'a => 'a" and
lcm :: "'a => 'a => 'a"
begin
abbreviation
coprime :: "'a => 'a => bool"
where
"coprime x y == (gcd x y = 1)"
end
instantiation nat :: gcd
begin
fun
gcd_nat :: "nat => nat => nat"
where
"gcd_nat x y =
(if y = 0 then x else gcd y (x mod y))"
definition
lcm_nat :: "nat => nat => nat"
where
"lcm_nat x y = x * y div (gcd x y)"
instance proof qed
end
instantiation int :: gcd
begin
definition
gcd_int :: "int => int => int"
where
"gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
definition
lcm_int :: "int => int => int"
where
"lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
instance proof qed
end
subsection {* Transfer setup *}
lemma transfer_nat_int_gcd:
"(x::int) >= 0 ==> y >= 0 ==> gcd (nat x) (nat y) = nat (gcd x y)"
"(x::int) >= 0 ==> y >= 0 ==> lcm (nat x) (nat y) = nat (lcm x y)"
unfolding gcd_int_def lcm_int_def
by auto
lemma transfer_nat_int_gcd_closures:
"x >= (0::int) ==> y >= 0 ==> gcd x y >= 0"
"x >= (0::int) ==> y >= 0 ==> lcm x y >= 0"
by (auto simp add: gcd_int_def lcm_int_def)
declare transfer_morphism_nat_int[transfer add return:
transfer_nat_int_gcd transfer_nat_int_gcd_closures]
lemma transfer_int_nat_gcd:
"gcd (int x) (int y) = int (gcd x y)"
"lcm (int x) (int y) = int (lcm x y)"
by (unfold gcd_int_def lcm_int_def, auto)
lemma transfer_int_nat_gcd_closures:
"is_nat x ==> is_nat y ==> gcd x y >= 0"
"is_nat x ==> is_nat y ==> lcm x y >= 0"
by (auto simp add: gcd_int_def lcm_int_def)
declare transfer_morphism_int_nat[transfer add return:
transfer_int_nat_gcd transfer_int_nat_gcd_closures]
subsection {* GCD properties *}
lemma gcd_nat_induct:
fixes m n :: nat
assumes "!!m. P m 0"
and "!!m n. 0 < n ==> P n (m mod n) ==> P m n"
shows "P m n"
apply (rule gcd_nat.induct)
apply (case_tac "y = 0")
using assms apply simp_all
done
lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
by (simp add: gcd_int_def)
lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
by (simp add: gcd_int_def)
lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"
by(simp add: gcd_int_def)
lemma gcd_abs_int: "gcd (x::int) y = gcd (abs x) (abs y)"
by (simp add: gcd_int_def)
lemma gcd_abs1_int[simp]: "gcd (abs x) (y::int) = gcd x y"
by (metis abs_idempotent gcd_abs_int)
lemma gcd_abs2_int[simp]: "gcd x (abs y::int) = gcd x y"
by (metis abs_idempotent gcd_abs_int)
lemma gcd_cases_int:
fixes x :: int and y
assumes "x >= 0 ==> y >= 0 ==> P (gcd x y)"
and "x >= 0 ==> y <= 0 ==> P (gcd x (-y))"
and "x <= 0 ==> y >= 0 ==> P (gcd (-x) y)"
and "x <= 0 ==> y <= 0 ==> P (gcd (-x) (-y))"
shows "P (gcd x y)"
by (insert assms, auto, arith)
lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
by (simp add: gcd_int_def)
lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
by (simp add: lcm_int_def)
lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
by (simp add: lcm_int_def)
lemma lcm_abs_int: "lcm (x::int) y = lcm (abs x) (abs y)"
by (simp add: lcm_int_def)
lemma abs_lcm_int [simp]: "abs (lcm i j::int) = lcm i j"
by(simp add:lcm_int_def)
lemma lcm_abs1_int[simp]: "lcm (abs x) (y::int) = lcm x y"
by (metis abs_idempotent lcm_int_def)
lemma lcm_abs2_int[simp]: "lcm x (abs y::int) = lcm x y"
by (metis abs_idempotent lcm_int_def)
lemma lcm_cases_int:
fixes x :: int and y
assumes "x >= 0 ==> y >= 0 ==> P (lcm x y)"
and "x >= 0 ==> y <= 0 ==> P (lcm x (-y))"
and "x <= 0 ==> y >= 0 ==> P (lcm (-x) y)"
and "x <= 0 ==> y <= 0 ==> P (lcm (-x) (-y))"
shows "P (lcm x y)"
by (insert prems, auto simp add: lcm_neg1_int lcm_neg2_int, arith)
lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
by (simp add: lcm_int_def)
lemma gcd_0_nat [simp]: "gcd (x::nat) 0 = x"
by simp
lemma gcd_0_int [simp]: "gcd (x::int) 0 = abs x"
by (unfold gcd_int_def, auto)
lemma gcd_0_left_nat [simp]: "gcd 0 (x::nat) = x"
by simp
lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = abs x"
by (unfold gcd_int_def, auto)
lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
by (case_tac "y = 0", auto)
lemma gcd_non_0_nat: "y ~= (0::nat) ==> gcd (x::nat) y = gcd y (x mod y)"
by simp
lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
by simp
lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
by (simp add: One_nat_def)
lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
by (simp add: gcd_int_def)
lemma gcd_idem_nat: "gcd (x::nat) x = x"
by simp
lemma gcd_idem_int: "gcd (x::int) x = abs x"
by (auto simp add: gcd_int_def)
declare gcd_nat.simps [simp del]
text {*
\medskip @{term "gcd m n"} divides @{text m} and @{text n}. The
conjunctions don't seem provable separately.
*}
lemma gcd_dvd1_nat [iff]: "(gcd (m::nat)) n dvd m"
and gcd_dvd2_nat [iff]: "(gcd m n) dvd n"
apply (induct m n rule: gcd_nat_induct)
apply (simp_all add: gcd_non_0_nat)
apply (blast dest: dvd_mod_imp_dvd)
done
lemma gcd_dvd1_int [iff]: "gcd (x::int) y dvd x"
by (metis gcd_int_def int_dvd_iff gcd_dvd1_nat)
lemma gcd_dvd2_int [iff]: "gcd (x::int) y dvd y"
by (metis gcd_int_def int_dvd_iff gcd_dvd2_nat)
lemma dvd_gcd_D1_nat: "k dvd gcd m n ==> (k::nat) dvd m"
by(metis gcd_dvd1_nat dvd_trans)
lemma dvd_gcd_D2_nat: "k dvd gcd m n ==> (k::nat) dvd n"
by(metis gcd_dvd2_nat dvd_trans)
lemma dvd_gcd_D1_int: "i dvd gcd m n ==> (i::int) dvd m"
by(metis gcd_dvd1_int dvd_trans)
lemma dvd_gcd_D2_int: "i dvd gcd m n ==> (i::int) dvd n"
by(metis gcd_dvd2_int dvd_trans)
lemma gcd_le1_nat [simp]: "a ≠ 0 ==> gcd (a::nat) b ≤ a"
by (rule dvd_imp_le, auto)
lemma gcd_le2_nat [simp]: "b ≠ 0 ==> gcd (a::nat) b ≤ b"
by (rule dvd_imp_le, auto)
lemma gcd_le1_int [simp]: "a > 0 ==> gcd (a::int) b ≤ a"
by (rule zdvd_imp_le, auto)
lemma gcd_le2_int [simp]: "b > 0 ==> gcd (a::int) b ≤ b"
by (rule zdvd_imp_le, auto)
lemma gcd_greatest_nat: "(k::nat) dvd m ==> k dvd n ==> k dvd gcd m n"
by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod)
lemma gcd_greatest_int:
"(k::int) dvd m ==> k dvd n ==> k dvd gcd m n"
apply (subst gcd_abs_int)
apply (subst abs_dvd_iff [symmetric])
apply (rule gcd_greatest_nat [transferred])
apply auto
done
lemma gcd_greatest_iff_nat [iff]: "(k dvd gcd (m::nat) n) =
(k dvd m & k dvd n)"
by (blast intro!: gcd_greatest_nat intro: dvd_trans)
lemma gcd_greatest_iff_int: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
by (blast intro!: gcd_greatest_int intro: dvd_trans)
lemma gcd_zero_nat [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff_nat)
lemma gcd_zero_int [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
by (auto simp add: gcd_int_def)
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
by (insert gcd_zero_nat [of m n], arith)
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
by (insert gcd_zero_int [of m n], insert gcd_ge_0_int [of m n], arith)
interpretation gcd_nat!: abel_semigroup "gcd :: nat => nat => nat"
proof
qed (auto intro: dvd_antisym dvd_trans)
interpretation gcd_int!: abel_semigroup "gcd :: int => int => int"
proof
qed (simp_all add: gcd_int_def gcd_nat.assoc gcd_nat.commute gcd_nat.left_commute)
lemmas gcd_assoc_nat = gcd_nat.assoc
lemmas gcd_commute_nat = gcd_nat.commute
lemmas gcd_left_commute_nat = gcd_nat.left_commute
lemmas gcd_assoc_int = gcd_int.assoc
lemmas gcd_commute_int = gcd_int.commute
lemmas gcd_left_commute_int = gcd_int.left_commute
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
lemma gcd_unique_nat: "(d::nat) dvd a ∧ d dvd b ∧
(∀e. e dvd a ∧ e dvd b --> e dvd d) <-> d = gcd a b"
apply auto
apply (rule dvd_antisym)
apply (erule (1) gcd_greatest_nat)
apply auto
done
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a ∧ d dvd b ∧
(∀e. e dvd a ∧ e dvd b --> e dvd d) <-> d = gcd a b"
apply (case_tac "d = 0")
apply simp
apply (rule iffI)
apply (rule zdvd_antisym_nonneg)
apply (auto intro: gcd_greatest_int)
done
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y ==> gcd x y = x"
by (metis dvd.eq_iff gcd_unique_nat)
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x ==> gcd x y = y"
by (metis dvd.eq_iff gcd_unique_nat)
lemma gcd_proj1_if_dvd_int[simp]: "x dvd y ==> gcd (x::int) y = abs x"
by (metis abs_dvd_iff abs_eq_0 gcd_0_left_int gcd_abs_int gcd_unique_int)
lemma gcd_proj2_if_dvd_int[simp]: "y dvd x ==> gcd (x::int) y = abs y"
by (metis gcd_proj1_if_dvd_int gcd_commute_int)
text {*
\medskip Multiplication laws
*}
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
-- {* \cite[page 27]{davenport92} *}
apply (induct m n rule: gcd_nat_induct)
apply simp
apply (case_tac "k = 0")
apply (simp_all add: mod_geq gcd_non_0_nat mod_mult_distrib2)
done
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
apply (subst (1 2) gcd_abs_int)
apply (subst (1 2) abs_mult)
apply (rule gcd_mult_distrib_nat [transferred])
apply auto
done
lemma coprime_dvd_mult_nat: "coprime (k::nat) n ==> k dvd m * n ==> k dvd m"
apply (insert gcd_mult_distrib_nat [of m k n])
apply simp
apply (erule_tac t = m in ssubst)
apply simp
done
lemma coprime_dvd_mult_int:
"coprime (k::int) n ==> k dvd m * n ==> k dvd m"
apply (subst abs_dvd_iff [symmetric])
apply (subst dvd_abs_iff [symmetric])
apply (subst (asm) gcd_abs_int)
apply (rule coprime_dvd_mult_nat [transferred])
prefer 4 apply assumption
apply auto
apply (subst abs_mult [symmetric], auto)
done
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n ==>
(k dvd m * n) = (k dvd m)"
by (auto intro: coprime_dvd_mult_nat)
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n ==>
(k dvd m * n) = (k dvd m)"
by (auto intro: coprime_dvd_mult_int)
lemma gcd_mult_cancel_nat: "coprime k n ==> gcd ((k::nat) * m) n = gcd m n"
apply (rule dvd_antisym)
apply (rule gcd_greatest_nat)
apply (rule_tac n = k in coprime_dvd_mult_nat)
apply (simp add: gcd_assoc_nat)
apply (simp add: gcd_commute_nat)
apply (simp_all add: mult_commute)
done
lemma gcd_mult_cancel_int:
"coprime (k::int) n ==> gcd (k * m) n = gcd m n"
apply (subst (1 2) gcd_abs_int)
apply (subst abs_mult)
apply (rule gcd_mult_cancel_nat [transferred], auto)
done
lemma coprime_crossproduct_nat:
fixes a b c d :: nat
assumes "coprime a d" and "coprime b c"
shows "a * c = b * d <-> a = b ∧ c = d" (is "?lhs <-> ?rhs")
proof
assume ?rhs then show ?lhs by simp
next
assume ?lhs
from `?lhs` have "a dvd b * d" by (auto intro: dvdI dest: sym)
with `coprime a d` have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
from `?lhs` have "b dvd a * c" by (auto intro: dvdI dest: sym)
with `coprime b c` have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult_commute)
with `coprime b c` have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult_commute)
with `coprime a d` have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
from `a dvd b` `b dvd a` have "a = b" by (rule Nat.dvd.antisym)
moreover from `c dvd d` `d dvd c` have "c = d" by (rule Nat.dvd.antisym)
ultimately show ?rhs ..
qed
lemma coprime_crossproduct_int:
fixes a b c d :: int
assumes "coprime a d" and "coprime b c"
shows "¦a¦ * ¦c¦ = ¦b¦ * ¦d¦ <-> ¦a¦ = ¦b¦ ∧ ¦c¦ = ¦d¦"
using assms by (intro coprime_crossproduct_nat [transferred]) auto
text {* \medskip Addition laws *}
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
apply (case_tac "n = 0")
apply (simp_all add: gcd_non_0_nat)
done
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
apply (subst (1 2) gcd_commute_nat)
apply (subst add_commute)
apply simp
done
lemma gcd_diff1_nat: "(m::nat) >= n ==> gcd (m - n) n = gcd m n"
by (subst gcd_add1_nat [symmetric], auto)
lemma gcd_diff2_nat: "(n::nat) >= m ==> gcd (n - m) n = gcd m n"
apply (subst gcd_commute_nat)
apply (subst gcd_diff1_nat [symmetric])
apply auto
apply (subst gcd_commute_nat)
apply (subst gcd_diff1_nat)
apply assumption
apply (rule gcd_commute_nat)
done
lemma gcd_non_0_int: "(y::int) > 0 ==> gcd x y = gcd y (x mod y)"
apply (frule_tac b = y and a = x in pos_mod_sign)
apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
zmod_zminus1_eq_if)
apply (frule_tac a = x in pos_mod_bound)
apply (subst (1 2) gcd_commute_nat)
apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
nat_le_eq_zle)
done
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
apply (case_tac "y = 0")
apply force
apply (case_tac "y > 0")
apply (subst gcd_non_0_int, auto)
apply (insert gcd_non_0_int [of "-y" "-x"])
apply auto
done
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
by (metis gcd_red_int mod_add_self1 zadd_commute)
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
by (metis gcd_add1_int gcd_commute_int zadd_commute)
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
by (metis gcd_commute_int gcd_red_int mod_mult_self1 zadd_commute)
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
using mult_dvd_mono [of 1] by auto
lemma finite_divisors_nat[simp]:
assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
proof-
have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
from finite_subset[OF _ this] show ?thesis using assms
by(bestsimp intro!:dvd_imp_le)
qed
lemma finite_divisors_int[simp]:
assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
proof-
have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
hence "finite{d. abs d <= abs i}" by simp
from finite_subset[OF _ this] show ?thesis using assms
by(bestsimp intro!:dvd_imp_le_int)
qed
lemma Max_divisors_self_nat[simp]: "n≠0 ==> Max{d::nat. d dvd n} = n"
apply(rule antisym)
apply (fastsimp intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
apply simp
done
lemma Max_divisors_self_int[simp]: "n≠0 ==> Max{d::int. d dvd n} = abs n"
apply(rule antisym)
apply(rule Max_le_iff[THEN iffD2])
apply simp
apply fastsimp
apply (metis Collect_def abs_ge_self dvd_imp_le_int mem_def zle_trans)
apply simp
done
lemma gcd_is_Max_divisors_nat:
"m ~= 0 ==> n ~= 0 ==> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
apply(rule Max_eqI[THEN sym])
apply (metis finite_Collect_conjI finite_divisors_nat)
apply simp
apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
apply simp
done
lemma gcd_is_Max_divisors_int:
"m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
apply(rule Max_eqI[THEN sym])
apply (metis finite_Collect_conjI finite_divisors_int)
apply simp
apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
apply simp
done
lemma gcd_code_int [code]:
"gcd k l = ¦if l = (0::int) then k else gcd l (¦k¦ mod ¦l¦)¦"
by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
subsection {* Coprimality *}
lemma div_gcd_coprime_nat:
assumes nz: "(a::nat) ≠ 0 ∨ b ≠ 0"
shows "coprime (a div gcd a b) (b div gcd a b)"
proof -
let ?g = "gcd a b"
let ?a' = "a div ?g"
let ?b' = "b div ?g"
let ?g' = "gcd ?a' ?b'"
have dvdg: "?g dvd a" "?g dvd b" by simp_all
have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
from dvdg dvdg' obtain ka kb ka' kb' where
kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
unfolding dvd_def by blast
then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
by simp_all
then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
have "?g ≠ 0" using nz by simp
then have gp: "?g > 0" by arith
from gcd_greatest_nat [OF dvdgg'] have "?g * ?g' dvd ?g" .
with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
qed
lemma div_gcd_coprime_int:
assumes nz: "(a::int) ≠ 0 ∨ b ≠ 0"
shows "coprime (a div gcd a b) (b div gcd a b)"
apply (subst (1 2 3) gcd_abs_int)
apply (subst (1 2) abs_div)
apply simp
apply simp
apply(subst (1 2) abs_gcd_int)
apply (rule div_gcd_coprime_nat [transferred])
using nz apply (auto simp add: gcd_abs_int [symmetric])
done
lemma coprime_nat: "coprime (a::nat) b <-> (∀d. d dvd a ∧ d dvd b <-> d = 1)"
using gcd_unique_nat[of 1 a b, simplified] by auto
lemma coprime_Suc_0_nat:
"coprime (a::nat) b <-> (∀d. d dvd a ∧ d dvd b <-> d = Suc 0)"
using coprime_nat by (simp add: One_nat_def)
lemma coprime_int: "coprime (a::int) b <->
(∀d. d >= 0 ∧ d dvd a ∧ d dvd b <-> d = 1)"
using gcd_unique_int [of 1 a b]
apply clarsimp
apply (erule subst)
apply (rule iffI)
apply force
apply (drule_tac x = "abs e" in exI)
apply (case_tac "e >= 0")
apply force
apply force
done
lemma gcd_coprime_nat:
assumes z: "gcd (a::nat) b ≠ 0" and a: "a = a' * gcd a b" and
b: "b = b' * gcd a b"
shows "coprime a' b'"
apply (subgoal_tac "a' = a div gcd a b")
apply (erule ssubst)
apply (subgoal_tac "b' = b div gcd a b")
apply (erule ssubst)
apply (rule div_gcd_coprime_nat)
using prems
apply force
apply (subst (1) b)
using z apply force
apply (subst (1) a)
using z apply force
done
lemma gcd_coprime_int:
assumes z: "gcd (a::int) b ≠ 0" and a: "a = a' * gcd a b" and
b: "b = b' * gcd a b"
shows "coprime a' b'"
apply (subgoal_tac "a' = a div gcd a b")
apply (erule ssubst)
apply (subgoal_tac "b' = b div gcd a b")
apply (erule ssubst)
apply (rule div_gcd_coprime_int)
using prems
apply force
apply (subst (1) b)
using z apply force
apply (subst (1) a)
using z apply force
done
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
shows "coprime d (a * b)"
apply (subst gcd_commute_nat)
using da apply (subst gcd_mult_cancel_nat)
apply (subst gcd_commute_nat, assumption)
apply (subst gcd_commute_nat, rule db)
done
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
shows "coprime d (a * b)"
apply (subst gcd_commute_int)
using da apply (subst gcd_mult_cancel_int)
apply (subst gcd_commute_int, assumption)
apply (subst gcd_commute_int, rule db)
done
lemma coprime_lmult_nat:
assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
proof -
have "gcd d a dvd gcd d (a * b)"
by (rule gcd_greatest_nat, auto)
with dab show ?thesis
by auto
qed
lemma coprime_lmult_int:
assumes "coprime (d::int) (a * b)" shows "coprime d a"
proof -
have "gcd d a dvd gcd d (a * b)"
by (rule gcd_greatest_int, auto)
with assms show ?thesis
by auto
qed
lemma coprime_rmult_nat:
assumes "coprime (d::nat) (a * b)" shows "coprime d b"
proof -
have "gcd d b dvd gcd d (a * b)"
by (rule gcd_greatest_nat, auto intro: dvd_mult)
with assms show ?thesis
by auto
qed
lemma coprime_rmult_int:
assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
proof -
have "gcd d b dvd gcd d (a * b)"
by (rule gcd_greatest_int, auto intro: dvd_mult)
with dab show ?thesis
by auto
qed
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) <->
coprime d a ∧ coprime d b"
using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
coprime_mult_nat[of d a b]
by blast
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) <->
coprime d a ∧ coprime d b"
using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
coprime_mult_int[of d a b]
by blast
lemma gcd_coprime_exists_nat:
assumes nz: "gcd (a::nat) b ≠ 0"
shows "∃a' b'. a = a' * gcd a b ∧ b = b' * gcd a b ∧ coprime a' b'"
apply (rule_tac x = "a div gcd a b" in exI)
apply (rule_tac x = "b div gcd a b" in exI)
using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
done
lemma gcd_coprime_exists_int:
assumes nz: "gcd (a::int) b ≠ 0"
shows "∃a' b'. a = a' * gcd a b ∧ b = b' * gcd a b ∧ coprime a' b'"
apply (rule_tac x = "a div gcd a b" in exI)
apply (rule_tac x = "b div gcd a b" in exI)
using nz apply (auto simp add: div_gcd_coprime_int dvd_div_mult_self)
done
lemma coprime_exp_nat: "coprime (d::nat) a ==> coprime d (a^n)"
by (induct n, simp_all add: coprime_mult_nat)
lemma coprime_exp_int: "coprime (d::int) a ==> coprime d (a^n)"
by (induct n, simp_all add: coprime_mult_int)
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b ==> coprime (a^n) (b^m)"
apply (rule coprime_exp_nat)
apply (subst gcd_commute_nat)
apply (rule coprime_exp_nat)
apply (subst gcd_commute_nat, assumption)
done
lemma coprime_exp2_int [intro]: "coprime (a::int) b ==> coprime (a^n) (b^m)"
apply (rule coprime_exp_int)
apply (subst gcd_commute_int)
apply (rule coprime_exp_int)
apply (subst gcd_commute_int, assumption)
done
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
proof (cases)
assume "a = 0 & b = 0"
thus ?thesis by simp
next assume "~(a = 0 & b = 0)"
hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
by (auto simp:div_gcd_coprime_nat)
hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
apply (subst (1 2) mult_commute)
apply (subst gcd_mult_distrib_nat [symmetric])
apply simp
done
also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
apply (subst div_power)
apply auto
apply (rule dvd_div_mult_self)
apply (rule dvd_power_same)
apply auto
done
also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
apply (subst div_power)
apply auto
apply (rule dvd_div_mult_self)
apply (rule dvd_power_same)
apply auto
done
finally show ?thesis .
qed
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
apply (subst (1 2) gcd_abs_int)
apply (subst (1 2) power_abs)
apply (rule gcd_exp_nat [where n = n, transferred])
apply auto
done
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
shows "∃b' c'. a = b' * c' ∧ b' dvd b ∧ c' dvd c"
proof-
let ?g = "gcd a b"
{assume "?g = 0" with dc have ?thesis by auto}
moreover
{assume z: "?g ≠ 0"
from gcd_coprime_exists_nat[OF z]
obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
by blast
have thb: "?g dvd b" by auto
from ab'(1) have "a' dvd a" unfolding dvd_def by blast
with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
with z have th_1: "a' dvd b' * c" by auto
from coprime_dvd_mult_nat[OF ab'(3)] th_1
have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
from ab' have "a = ?g*a'" by algebra
with thb thc have ?thesis by blast }
ultimately show ?thesis by blast
qed
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
shows "∃b' c'. a = b' * c' ∧ b' dvd b ∧ c' dvd c"
proof-
let ?g = "gcd a b"
{assume "?g = 0" with dc have ?thesis by auto}
moreover
{assume z: "?g ≠ 0"
from gcd_coprime_exists_int[OF z]
obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
by blast
have thb: "?g dvd b" by auto
from ab'(1) have "a' dvd a" unfolding dvd_def by blast
with dc have th0: "a' dvd b*c"
using dvd_trans[of a' a "b*c"] by simp
from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
with z have th_1: "a' dvd b' * c" by auto
from coprime_dvd_mult_int[OF ab'(3)] th_1
have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
from ab' have "a = ?g*a'" by algebra
with thb thc have ?thesis by blast }
ultimately show ?thesis by blast
qed
lemma pow_divides_pow_nat:
assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n ≠ 0"
shows "a dvd b"
proof-
let ?g = "gcd a b"
from n obtain m where m: "n = Suc m" by (cases n, simp_all)
{assume "?g = 0" with ab n have ?thesis by auto }
moreover
{assume z: "?g ≠ 0"
hence zn: "?g ^ n ≠ 0" using n by simp
from gcd_coprime_exists_nat[OF z]
obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
by blast
from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
by (simp add: ab'(1,2)[symmetric])
hence "?g^n*a'^n dvd ?g^n *b'^n"
by (simp only: power_mult_distrib mult_commute)
with zn z n have th0:"a'^n dvd b'^n" by auto
have "a' dvd a'^n" by (simp add: m)
with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
have "a' dvd b'" by (subst (asm) mult_commute, blast)
hence "a'*?g dvd b'*?g" by simp
with ab'(1,2) have ?thesis by simp }
ultimately show ?thesis by blast
qed
lemma pow_divides_pow_int:
assumes ab: "(a::int) ^ n dvd b ^n" and n:"n ≠ 0"
shows "a dvd b"
proof-
let ?g = "gcd a b"
from n obtain m where m: "n = Suc m" by (cases n, simp_all)
{assume "?g = 0" with ab n have ?thesis by auto }
moreover
{assume z: "?g ≠ 0"
hence zn: "?g ^ n ≠ 0" using n by simp
from gcd_coprime_exists_int[OF z]
obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
by blast
from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
by (simp add: ab'(1,2)[symmetric])
hence "?g^n*a'^n dvd ?g^n *b'^n"
by (simp only: power_mult_distrib mult_commute)
with zn z n have th0:"a'^n dvd b'^n" by auto
have "a' dvd a'^n" by (simp add: m)
with th0 have "a' dvd b'^n"
using dvd_trans[of a' "a'^n" "b'^n"] by simp
hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
have "a' dvd b'" by (subst (asm) mult_commute, blast)
hence "a'*?g dvd b'*?g" by simp
with ab'(1,2) have ?thesis by simp }
ultimately show ?thesis by blast
qed
lemma pow_divides_eq_nat [simp]: "n ~= 0 ==> ((a::nat)^n dvd b^n) = (a dvd b)"
by (auto intro: pow_divides_pow_nat dvd_power_same)
lemma pow_divides_eq_int [simp]: "n ~= 0 ==> ((a::int)^n dvd b^n) = (a dvd b)"
by (auto intro: pow_divides_pow_int dvd_power_same)
lemma divides_mult_nat:
assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
shows "m * n dvd r"
proof-
from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
unfolding dvd_def by blast
from mr n' have "m dvd n'*n" by (simp add: mult_commute)
hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
then obtain k where k: "n' = m*k" unfolding dvd_def by blast
from n' k show ?thesis unfolding dvd_def by auto
qed
lemma divides_mult_int:
assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
shows "m * n dvd r"
proof-
from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
unfolding dvd_def by blast
from mr n' have "m dvd n'*n" by (simp add: mult_commute)
hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
then obtain k where k: "n' = m*k" unfolding dvd_def by blast
from n' k show ?thesis unfolding dvd_def by auto
qed
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
apply force
apply (rule dvd_diff_nat)
apply auto
done
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
using coprime_plus_one_nat by (simp add: One_nat_def)
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
apply force
apply (rule dvd_diff)
apply auto
done
lemma coprime_minus_one_nat: "(n::nat) ≠ 0 ==> coprime (n - 1) n"
using coprime_plus_one_nat [of "n - 1"]
gcd_commute_nat [of "n - 1" n] by auto
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
using coprime_plus_one_int [of "n - 1"]
gcd_commute_int [of "n - 1" n] by auto
lemma setprod_coprime_nat [rule_format]:
"(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
apply (case_tac "finite A")
apply (induct set: finite)
apply (auto simp add: gcd_mult_cancel_nat)
done
lemma setprod_coprime_int [rule_format]:
"(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
apply (case_tac "finite A")
apply (induct set: finite)
apply (auto simp add: gcd_mult_cancel_int)
done
lemma coprime_common_divisor_nat: "coprime (a::nat) b ==> x dvd a ==>
x dvd b ==> x = 1"
apply (subgoal_tac "x dvd gcd a b")
apply simp
apply (erule (1) gcd_greatest_nat)
done
lemma coprime_common_divisor_int: "coprime (a::int) b ==> x dvd a ==>
x dvd b ==> abs x = 1"
apply (subgoal_tac "x dvd gcd a b")
apply simp
apply (erule (1) gcd_greatest_int)
done
lemma coprime_divisors_nat: "(d::int) dvd a ==> e dvd b ==> coprime a b ==>
coprime d e"
apply (auto simp add: dvd_def)
apply (frule coprime_lmult_int)
apply (subst gcd_commute_int)
apply (subst (asm) (2) gcd_commute_int)
apply (erule coprime_lmult_int)
done
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 ==> coprime x m"
apply (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
done
lemma invertible_coprime_int: "(x::int) * y mod m = 1 ==> coprime x m"
apply (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
done
subsection {* Bezout's theorem *}
fun
bezw :: "nat => nat => int * int"
where
"bezw x y =
(if y = 0 then (1, 0) else
(snd (bezw y (x mod y)),
fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
lemma bezw_non_0: "y > 0 ==> bezw x y = (snd (bezw y (x mod y)),
fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
by simp
declare bezw.simps [simp del]
lemma bezw_aux [rule_format]:
"fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
proof (induct x y rule: gcd_nat_induct)
fix m :: nat
show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
by auto
next fix m :: nat and n
assume ngt0: "n > 0" and
ih: "fst (bezw n (m mod n)) * int n +
snd (bezw n (m mod n)) * int (m mod n) =
int (gcd n (m mod n))"
thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
apply (simp add: bezw_non_0 gcd_non_0_nat)
apply (erule subst)
apply (simp add: field_simps)
apply (subst mod_div_equality [of m n, symmetric])
apply (simp only: field_simps zadd_int [symmetric]
zmult_int [symmetric])
done
qed
lemma bezout_int:
fixes x y
shows "EX u v. u * (x::int) + v * y = gcd x y"
proof -
have bezout_aux: "!!x y. x ≥ (0::int) ==> y ≥ 0 ==>
EX u v. u * x + v * y = gcd x y"
apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
apply (unfold gcd_int_def)
apply simp
apply (subst bezw_aux [symmetric])
apply auto
done
have "(x ≥ 0 ∧ y ≥ 0) | (x ≥ 0 ∧ y ≤ 0) | (x ≤ 0 ∧ y ≥ 0) |
(x ≤ 0 ∧ y ≤ 0)"
by auto
moreover have "x ≥ 0 ==> y ≥ 0 ==> ?thesis"
by (erule (1) bezout_aux)
moreover have "x >= 0 ==> y <= 0 ==> ?thesis"
apply (insert bezout_aux [of x "-y"])
apply auto
apply (rule_tac x = u in exI)
apply (rule_tac x = "-v" in exI)
apply (subst gcd_neg2_int [symmetric])
apply auto
done
moreover have "x <= 0 ==> y >= 0 ==> ?thesis"
apply (insert bezout_aux [of "-x" y])
apply auto
apply (rule_tac x = "-u" in exI)
apply (rule_tac x = v in exI)
apply (subst gcd_neg1_int [symmetric])
apply auto
done
moreover have "x <= 0 ==> y <= 0 ==> ?thesis"
apply (insert bezout_aux [of "-x" "-y"])
apply auto
apply (rule_tac x = "-u" in exI)
apply (rule_tac x = "-v" in exI)
apply (subst gcd_neg1_int [symmetric])
apply (subst gcd_neg2_int [symmetric])
apply auto
done
ultimately show ?thesis by blast
qed
text {* versions of Bezout for nat, by Amine Chaieb *}
lemma ind_euclid:
assumes c: " ∀a b. P (a::nat) b <-> P b a" and z: "∀a. P a 0"
and add: "∀a b. P a b --> P a (a + b)"
shows "P a b"
proof(induct "a + b" arbitrary: a b rule: less_induct)
case less
have "a = b ∨ a < b ∨ b < a" by arith
moreover {assume eq: "a= b"
from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
by simp}
moreover
{assume lt: "a < b"
hence "a + b - a < a + b ∨ a = 0" by arith
moreover
{assume "a =0" with z c have "P a b" by blast }
moreover
{assume "a + b - a < a + b"
also have th0: "a + b - a = a + (b - a)" using lt by arith
finally have "a + (b - a) < a + b" .
then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
then have "P a b" by (simp add: th0[symmetric])}
ultimately have "P a b" by blast}
moreover
{assume lt: "a > b"
hence "b + a - b < a + b ∨ b = 0" by arith
moreover
{assume "b =0" with z c have "P a b" by blast }
moreover
{assume "b + a - b < a + b"
also have th0: "b + a - b = b + (a - b)" using lt by arith
finally have "b + (a - b) < a + b" .
then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
then have "P b a" by (simp add: th0[symmetric])
hence "P a b" using c by blast }
ultimately have "P a b" by blast}
ultimately show "P a b" by blast
qed
lemma bezout_lemma_nat:
assumes ex: "∃(d::nat) x y. d dvd a ∧ d dvd b ∧
(a * x = b * y + d ∨ b * x = a * y + d)"
shows "∃d x y. d dvd a ∧ d dvd a + b ∧
(a * x = (a + b) * y + d ∨ (a + b) * x = a * y + d)"
using ex
apply clarsimp
apply (rule_tac x="d" in exI, simp)
apply (case_tac "a * x = b * y + d" , simp_all)
apply (rule_tac x="x + y" in exI)
apply (rule_tac x="y" in exI)
apply algebra
apply (rule_tac x="x" in exI)
apply (rule_tac x="x + y" in exI)
apply algebra
done
lemma bezout_add_nat: "∃(d::nat) x y. d dvd a ∧ d dvd b ∧
(a * x = b * y + d ∨ b * x = a * y + d)"
apply(induct a b rule: ind_euclid)
apply blast
apply clarify
apply (rule_tac x="a" in exI, simp)
apply clarsimp
apply (rule_tac x="d" in exI)
apply (case_tac "a * x = b * y + d", simp_all)
apply (rule_tac x="x+y" in exI)
apply (rule_tac x="y" in exI)
apply algebra
apply (rule_tac x="x" in exI)
apply (rule_tac x="x+y" in exI)
apply algebra
done
lemma bezout1_nat: "∃(d::nat) x y. d dvd a ∧ d dvd b ∧
(a * x - b * y = d ∨ b * x - a * y = d)"
using bezout_add_nat[of a b]
apply clarsimp
apply (rule_tac x="d" in exI, simp)
apply (rule_tac x="x" in exI)
apply (rule_tac x="y" in exI)
apply auto
done
lemma bezout_add_strong_nat: assumes nz: "a ≠ (0::nat)"
shows "∃d x y. d dvd a ∧ d dvd b ∧ a * x = b * y + d"
proof-
from nz have ap: "a > 0" by simp
from bezout_add_nat[of a b]
have "(∃d x y. d dvd a ∧ d dvd b ∧ a * x = b * y + d) ∨
(∃d x y. d dvd a ∧ d dvd b ∧ b * x = a * y + d)" by blast
moreover
{fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
from H have ?thesis by blast }
moreover
{fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
{assume b0: "b = 0" with H have ?thesis by simp}
moreover
{assume b: "b ≠ 0" hence bp: "b > 0" by simp
from b dvd_imp_le [OF H(2)] have "d < b ∨ d = b"
by auto
moreover
{assume db: "d=b"
from prems have ?thesis apply simp
apply (rule exI[where x = b], simp)
apply (rule exI[where x = b])
by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
moreover
{assume db: "d < b"
{assume "x=0" hence ?thesis using prems by simp }
moreover
{assume x0: "x ≠ 0" hence xp: "x > 0" by simp
from db have "d ≤ b - 1" by simp
hence "d*b ≤ b*(b - 1)" by simp
with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
have dble: "d*b ≤ x*b*(b - 1)" using bp by simp
from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
by simp
hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
by (simp only: mult_assoc right_distrib)
hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
by algebra
hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
by (simp only: diff_add_assoc[OF dble, of d, symmetric])
hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
by (simp only: diff_mult_distrib2 add_commute mult_ac)
hence ?thesis using H(1,2)
apply -
apply (rule exI[where x=d], simp)
apply (rule exI[where x="(b - 1) * y"])
by (rule exI[where x="x*(b - 1) - d"], simp)}
ultimately have ?thesis by blast}
ultimately have ?thesis by blast}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed
lemma bezout_nat: assumes a: "(a::nat) ≠ 0"
shows "∃x y. a * x = b * y + gcd a b"
proof-
let ?g = "gcd a b"
from bezout_add_strong_nat[OF a, of b]
obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
from d(1,2) have "d dvd ?g" by simp
then obtain k where k: "?g = d*k" unfolding dvd_def by blast
from d(3) have "a * x * k = (b * y + d) *k " by auto
hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
thus ?thesis by blast
qed
subsection {* LCM properties *}
lemma lcm_altdef_int [code]: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
by (simp add: lcm_int_def lcm_nat_def zdiv_int
zmult_int [symmetric] gcd_int_def)
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
unfolding lcm_nat_def
by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
unfolding lcm_int_def gcd_int_def
apply (subst int_mult [symmetric])
apply (subst prod_gcd_lcm_nat [symmetric])
apply (subst nat_abs_mult_distrib [symmetric])
apply (simp, simp add: abs_mult)
done
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
unfolding lcm_nat_def by simp
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
unfolding lcm_int_def by simp
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
unfolding lcm_nat_def by simp
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
unfolding lcm_int_def by simp
lemma lcm_pos_nat:
"(m::nat) > 0 ==> n>0 ==> lcm m n > 0"
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
lemma lcm_pos_int:
"(m::int) ~= 0 ==> n ~= 0 ==> lcm m n > 0"
apply (subst lcm_abs_int)
apply (rule lcm_pos_nat [transferred])
apply auto
done
lemma dvd_pos_nat:
fixes n m :: nat
assumes "n > 0" and "m dvd n"
shows "m > 0"
using assms by (cases m) auto
lemma lcm_least_nat:
assumes "(m::nat) dvd k" and "n dvd k"
shows "lcm m n dvd k"
proof (cases k)
case 0 then show ?thesis by auto
next
case (Suc _) then have pos_k: "k > 0" by auto
from assms dvd_pos_nat [OF this] have pos_mn: "m > 0" "n > 0" by auto
with gcd_zero_nat [of m n] have pos_gcd: "gcd m n > 0" by simp
from assms obtain p where k_m: "k = m * p" using dvd_def by blast
from assms obtain q where k_n: "k = n * q" using dvd_def by blast
from pos_k k_m have pos_p: "p > 0" by auto
from pos_k k_n have pos_q: "q > 0" by auto
have "k * k * gcd q p = k * gcd (k * q) (k * p)"
by (simp add: mult_ac gcd_mult_distrib_nat)
also have "… = k * gcd (m * p * q) (n * q * p)"
by (simp add: k_m [symmetric] k_n [symmetric])
also have "… = k * p * q * gcd m n"
by (simp add: mult_ac gcd_mult_distrib_nat)
finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
by (simp only: k_m [symmetric] k_n [symmetric])
then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
by (simp add: mult_ac)
with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
by simp
with prod_gcd_lcm_nat [of m n]
have "lcm m n * gcd q p * gcd m n = k * gcd m n"
by (simp add: mult_ac)
with pos_gcd have "lcm m n * gcd q p = k" by auto
then show ?thesis using dvd_def by auto
qed
lemma lcm_least_int:
"(m::int) dvd k ==> n dvd k ==> lcm m n dvd k"
apply (subst lcm_abs_int)
apply (rule dvd_trans)
apply (rule lcm_least_nat [transferred, of _ "abs k" _])
apply auto
done
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
proof (cases m)
case 0 then show ?thesis by simp
next
case (Suc _)
then have mpos: "m > 0" by simp
show ?thesis
proof (cases n)
case 0 then show ?thesis by simp
next
case (Suc _)
then have npos: "n > 0" by simp
have "gcd m n dvd n" by simp
then obtain k where "n = gcd m n * k" using dvd_def by auto
then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
by (simp add: mult_ac)
also have "… = m * k" using mpos npos gcd_zero_nat by simp
finally show ?thesis by (simp add: lcm_nat_def)
qed
qed
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
apply (subst lcm_abs_int)
apply (rule dvd_trans)
prefer 2
apply (rule lcm_dvd1_nat [transferred])
apply auto
done
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
using lcm_dvd1_nat [of n m] by (simp only: lcm_nat_def mult.commute gcd_nat.commute)
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
using lcm_dvd1_int [of n m] by (simp only: lcm_int_def lcm_nat_def mult.commute gcd_nat.commute)
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m ==> k dvd lcm m n"
by(metis lcm_dvd1_nat dvd_trans)
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n ==> k dvd lcm m n"
by(metis lcm_dvd2_nat dvd_trans)
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m ==> i dvd lcm m n"
by(metis lcm_dvd1_int dvd_trans)
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n ==> i dvd lcm m n"
by(metis lcm_dvd2_int dvd_trans)
lemma lcm_unique_nat: "(a::nat) dvd d ∧ b dvd d ∧
(∀e. a dvd e ∧ b dvd e --> d dvd e) <-> d = lcm a b"
by (auto intro: dvd_antisym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
lemma lcm_unique_int: "d >= 0 ∧ (a::int) dvd d ∧ b dvd d ∧
(∀e. a dvd e ∧ b dvd e --> d dvd e) <-> d = lcm a b"
by (auto intro: dvd_antisym [transferred] lcm_least_int)
interpretation lcm_nat!: abel_semigroup "lcm :: nat => nat => nat"
proof
fix n m p :: nat
show "lcm (lcm n m) p = lcm n (lcm m p)"
by (rule lcm_unique_nat [THEN iffD1]) (metis dvd.order_trans lcm_unique_nat)
show "lcm m n = lcm n m"
by (simp add: lcm_nat_def gcd_commute_nat field_simps)
qed
interpretation lcm_int!: abel_semigroup "lcm :: int => int => int"
proof
fix n m p :: int
show "lcm (lcm n m) p = lcm n (lcm m p)"
by (rule lcm_unique_int [THEN iffD1]) (metis dvd_trans lcm_unique_int)
show "lcm m n = lcm n m"
by (simp add: lcm_int_def lcm_nat.commute)
qed
lemmas lcm_assoc_nat = lcm_nat.assoc
lemmas lcm_commute_nat = lcm_nat.commute
lemmas lcm_left_commute_nat = lcm_nat.left_commute
lemmas lcm_assoc_int = lcm_int.assoc
lemmas lcm_commute_int = lcm_int.commute
lemmas lcm_left_commute_int = lcm_int.left_commute
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y ==> lcm x y = y"
apply (rule sym)
apply (subst lcm_unique_nat [symmetric])
apply auto
done
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y ==> lcm x y = abs y"
apply (rule sym)
apply (subst lcm_unique_int [symmetric])
apply auto
done
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y ==> lcm y x = y"
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y ==> lcm y x = abs y"
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) <-> n dvd m"
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) <-> m dvd n"
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) <-> n dvd m"
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) <-> m dvd n"
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
lemma fun_left_comm_idem_gcd_nat: "fun_left_comm_idem (gcd :: nat=>nat=>nat)"
proof qed (auto simp add: gcd_ac_nat)
lemma fun_left_comm_idem_gcd_int: "fun_left_comm_idem (gcd :: int=>int=>int)"
proof qed (auto simp add: gcd_ac_int)
lemma fun_left_comm_idem_lcm_nat: "fun_left_comm_idem (lcm :: nat=>nat=>nat)"
proof qed (auto simp add: lcm_ac_nat)
lemma fun_left_comm_idem_lcm_int: "fun_left_comm_idem (lcm :: int=>int=>int)"
proof qed (auto simp add: lcm_ac_int)
lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 <-> m=0 ∨ n=0"
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 <-> m=0 ∨ n=0"
by (metis lcm_0_int lcm_0_left_int lcm_pos_int zless_le)
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 <-> m=1 ∧ n=1"
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
lemma lcm_1_iff_int[simp]: "lcm (m::int) n = 1 <-> (m=1 ∨ m = -1) ∧ (n=1 ∨ n = -1)"
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
subsubsection {* The complete divisibility lattice *}
interpretation gcd_semilattice_nat: semilattice_inf "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd
proof
case goal3 thus ?case by(metis gcd_unique_nat)
qed auto
interpretation lcm_semilattice_nat: semilattice_sup "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm
proof
case goal3 thus ?case by(metis lcm_unique_nat)
qed auto
interpretation gcd_lcm_lattice_nat: lattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd lcm ..
text{* Lifting gcd and lcm to finite (Gcd/Lcm) and infinite sets (GCD/LCM).
GCD is defined via LCM to facilitate the proof that we have a complete lattice.
Later on we show that GCD and Gcd coincide on finite sets.
*}
context gcd
begin
definition Gcd :: "'a set => 'a"
where "Gcd = fold gcd 0"
definition Lcm :: "'a set => 'a"
where "Lcm = fold lcm 1"
definition LCM :: "'a set => 'a" where
"LCM M = (if finite M then Lcm M else 0)"
definition GCD :: "'a set => 'a" where
"GCD M = LCM(INT m:M. {d. d dvd m})"
end
lemma Gcd_empty[simp]: "Gcd {} = 0"
by(simp add:Gcd_def)
lemma Lcm_empty[simp]: "Lcm {} = 1"
by(simp add:Lcm_def)
lemma GCD_empty_nat[simp]: "GCD {} = (0::nat)"
by(simp add:GCD_def LCM_def)
lemma LCM_eq_Lcm[simp]: "finite M ==> LCM M = Lcm M"
by(simp add:LCM_def)
lemma Lcm_insert_nat [simp]:
assumes "finite N"
shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
proof -
interpret fun_left_comm_idem "lcm::nat=>nat=>nat"
by (rule fun_left_comm_idem_lcm_nat)
from assms show ?thesis by(simp add: Lcm_def)
qed
lemma Lcm_insert_int [simp]:
assumes "finite N"
shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"
proof -
interpret fun_left_comm_idem "lcm::int=>int=>int"
by (rule fun_left_comm_idem_lcm_int)
from assms show ?thesis by(simp add: Lcm_def)
qed
lemma Gcd_insert_nat [simp]:
assumes "finite N"
shows "Gcd (insert (n::nat) N) = gcd n (Gcd N)"
proof -
interpret fun_left_comm_idem "gcd::nat=>nat=>nat"
by (rule fun_left_comm_idem_gcd_nat)
from assms show ?thesis by(simp add: Gcd_def)
qed
lemma Gcd_insert_int [simp]:
assumes "finite N"
shows "Gcd (insert (n::int) N) = gcd n (Gcd N)"
proof -
interpret fun_left_comm_idem "gcd::int=>int=>int"
by (rule fun_left_comm_idem_gcd_int)
from assms show ?thesis by(simp add: Gcd_def)
qed
lemma Lcm0_iff[simp]: "finite (M::nat set) ==> M ≠ {} ==> Lcm M = 0 <-> 0 : M"
by(induct rule:finite_ne_induct) auto
lemma Lcm_eq_0[simp]: "finite (M::nat set) ==> 0 : M ==> Lcm M = 0"
by (metis Lcm0_iff empty_iff)
lemma Gcd_dvd_nat [simp]:
assumes "finite M" and "(m::nat) ∈ M"
shows "Gcd M dvd m"
proof -
show ?thesis using gcd_semilattice_nat.fold_inf_le_inf[OF assms, of 0] by (simp add: Gcd_def)
qed
lemma dvd_Gcd_nat[simp]:
assumes "finite M" and "ALL (m::nat) : M. n dvd m"
shows "n dvd Gcd M"
proof -
show ?thesis using gcd_semilattice_nat.inf_le_fold_inf[OF assms, of 0] by (simp add: Gcd_def)
qed
lemma dvd_Lcm_nat [simp]:
assumes "finite M" and "(m::nat) ∈ M"
shows "m dvd Lcm M"
proof -
show ?thesis using lcm_semilattice_nat.sup_le_fold_sup[OF assms, of 1] by (simp add: Lcm_def)
qed
lemma Lcm_dvd_nat[simp]:
assumes "finite M" and "ALL (m::nat) : M. m dvd n"
shows "Lcm M dvd n"
proof -
show ?thesis using lcm_semilattice_nat.fold_sup_le_sup[OF assms, of 1] by (simp add: Lcm_def)
qed
interpretation gcd_lcm_complete_lattice_nat:
complete_lattice GCD LCM "op dvd" "%m n::nat. m dvd n & ~ n dvd m" gcd lcm 1 0
proof
case goal1 show ?case by simp
next
case goal2 show ?case by simp
next
case goal5 thus ?case by (auto simp: LCM_def)
next
case goal6 thus ?case
by(auto simp: LCM_def)(metis finite_nat_set_iff_bounded_le gcd_proj2_if_dvd_nat gcd_le1_nat)
next
case goal3 thus ?case by (auto simp: GCD_def LCM_def)(metis finite_INT finite_divisors_nat)
next
case goal4 thus ?case by(auto simp: LCM_def GCD_def)
qed
text{* Alternative characterizations of Gcd and GCD: *}
lemma Gcd_eq_Max: "finite(M::nat set) ==> M ≠ {} ==> 0 ∉ M ==> Gcd M = Max(\<Inter>m∈M. {d. d dvd m})"
apply(rule antisym)
apply(rule Max_ge)
apply (metis all_not_in_conv finite_divisors_nat finite_INT)
apply simp
apply (rule Max_le_iff[THEN iffD2])
apply (metis all_not_in_conv finite_divisors_nat finite_INT)
apply fastsimp
apply clarsimp
apply (metis Gcd_dvd_nat Max_in dvd_0_left dvd_Gcd_nat dvd_imp_le linorder_antisym_conv3 not_less0)
done
lemma Gcd_remove0_nat: "finite M ==> Gcd M = Gcd (M - {0::nat})"
apply(induct pred:finite)
apply simp
apply(case_tac "x=0")
apply simp
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
apply simp
apply blast
done
lemma Lcm_in_lcm_closed_set_nat:
"finite M ==> M ≠ {} ==> ALL m n :: nat. m:M --> n:M --> lcm m n : M ==> Lcm M : M"
apply(induct rule:finite_linorder_min_induct)
apply simp
apply simp
apply(subgoal_tac "ALL m n :: nat. m:A --> n:A --> lcm m n : A")
apply simp
apply(case_tac "A={}")
apply simp
apply simp
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
done
lemma Lcm_eq_Max_nat:
"finite M ==> M ≠ {} ==> 0 ∉ M ==> ALL m n :: nat. m:M --> n:M --> lcm m n : M ==> Lcm M = Max M"
apply(rule antisym)
apply(rule Max_ge, assumption)
apply(erule (2) Lcm_in_lcm_closed_set_nat)
apply clarsimp
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
done
text{* Finally GCD is Gcd: *}
lemma GCD_eq_Gcd[simp]: assumes "finite(M::nat set)" shows "GCD M = Gcd M"
proof-
have divisors_remove0_nat: "(\<Inter>m∈M. {d::nat. d dvd m}) = (\<Inter>m∈M-{0}. {d::nat. d dvd m})" by auto
show ?thesis
proof cases
assume "M={}" thus ?thesis by simp
next
assume "M≠{}"
show ?thesis
proof cases
assume "M={0}" thus ?thesis by(simp add:GCD_def LCM_def)
next
assume "M≠{0}"
with `M≠{}` assms show ?thesis
apply(subst Gcd_remove0_nat[OF assms])
apply(simp add:GCD_def)
apply(subst divisors_remove0_nat)
apply(simp add:LCM_def)
apply rule
apply rule
apply(subst Gcd_eq_Max)
apply simp
apply blast
apply blast
apply(rule Lcm_eq_Max_nat)
apply simp
apply blast
apply fastsimp
apply clarsimp
apply(fastsimp intro: finite_divisors_nat intro!: finite_INT)
done
qed
qed
qed
lemma Lcm_set_nat [code_unfold]:
"Lcm (set ns) = foldl lcm (1::nat) ns"
proof -
interpret fun_left_comm_idem "lcm::nat=>nat=>nat" by (rule fun_left_comm_idem_lcm_nat)
show ?thesis by(simp add: Lcm_def fold_set lcm_commute_nat)
qed
lemma Lcm_set_int [code_unfold]:
"Lcm (set is) = foldl lcm (1::int) is"
proof -
interpret fun_left_comm_idem "lcm::int=>int=>int" by (rule fun_left_comm_idem_lcm_int)
show ?thesis by(simp add: Lcm_def fold_set lcm_commute_int)
qed
lemma Gcd_set_nat [code_unfold]:
"Gcd (set ns) = foldl gcd (0::nat) ns"
proof -
interpret fun_left_comm_idem "gcd::nat=>nat=>nat" by (rule fun_left_comm_idem_gcd_nat)
show ?thesis by(simp add: Gcd_def fold_set gcd_commute_nat)
qed
lemma Gcd_set_int [code_unfold]:
"Gcd (set ns) = foldl gcd (0::int) ns"
proof -
interpret fun_left_comm_idem "gcd::int=>int=>int" by (rule fun_left_comm_idem_gcd_int)
show ?thesis by(simp add: Gcd_def fold_set gcd_commute_int)
qed
lemma mult_inj_if_coprime_nat:
"inj_on f A ==> inj_on g B ==> ALL a:A. ALL b:B. coprime (f a) (g b)
==> inj_on (%(a,b). f a * g b::nat) (A × B)"
apply(auto simp add:inj_on_def)
apply (metis coprime_dvd_mult_iff_nat dvd.neq_le_trans dvd_triv_left)
apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
dvd.neq_le_trans dvd_triv_right mult_commute)
done
text{* Nitpick: *}
lemma gcd_eq_nitpick_gcd [nitpick_def]: "gcd x y ≡ Nitpick.nat_gcd x y"
apply (rule eq_reflection)
apply (induct x y rule: nat_gcd.induct)
by (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
lemma lcm_eq_nitpick_lcm [nitpick_def]: "lcm x y ≡ Nitpick.nat_lcm x y"
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
end