src/HOL/GCD.thy
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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
```
(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow

This file deals with the functions gcd and lcm.  Definitions and
lemmas are proved uniformly for the natural numbers and integers.

This file combines and revises a number of prior developments.

The original theories "GCD" and "Primes" were by Christophe Tabacznyj
and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
gcd, lcm, and prime for the natural numbers.

The original theory "IntPrimes" was by Thomas M. Rasmussen, and
extended gcd, lcm, primes to the integers. Amine Chaieb provided
another extension of the notions to the integers, and added a number
of results to "Primes" and "GCD". IntPrimes also defined and developed
the congruence relations on the integers. The notion was extended to
the natural numbers by Chaieb.

natural numbers and the integers, and added a number of new theorems.

Tobias Nipkow cleaned up a lot.
*)

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 \<Rightarrow> 'a \<Rightarrow> 'a"
and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
begin

abbreviation
coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
where
"coprime x y == (gcd x y = 1)"

end

instantiation nat :: gcd
begin

fun
gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where
"gcd_nat x y =
(if y = 0 then x else gcd y (x mod y))"

definition
lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where
"lcm_nat x y = x * y div (gcd x y)"

instance proof qed

end

instantiation int :: gcd
begin

definition
gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
where
"gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"

definition
lcm_int :: "int \<Rightarrow> int \<Rightarrow> 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 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> 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) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
"x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
by (auto simp add: gcd_int_def lcm_int_def)

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 \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
by (auto simp add: gcd_int_def lcm_int_def)

transfer_int_nat_gcd transfer_int_nat_gcd_closures]

subsection {* GCD properties *}

(* was gcd_induct *)
lemma gcd_nat_induct:
fixes m n :: nat
assumes "\<And>m. P m 0"
and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
shows "P m n"
apply (rule gcd_nat.induct)
apply (case_tac "y = 0")
using assms apply simp_all
done

(* specific to int *)

lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"

lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"

lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"

lemma gcd_abs_int: "gcd (x::int) y = gcd (abs x) (abs y)"

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 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> 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"

lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"

lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"

lemma lcm_abs_int: "lcm (x::int) y = lcm (abs x) (abs y)"

lemma abs_lcm_int [simp]: "abs (lcm i j::int) = lcm i j"

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 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
shows "P (lcm x y)"
using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith

lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"

(* was gcd_0, etc. *)
lemma gcd_0_nat [simp]: "gcd (x::nat) 0 = x"
by simp

(* was igcd_0, etc. *)
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)

(* weaker, but useful for the simplifier *)

lemma gcd_non_0_nat: "y ~= (0::nat) \<Longrightarrow> 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"

lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"

lemma gcd_idem_nat: "gcd (x::nat) x = x"
by simp

lemma gcd_idem_int: "gcd (x::int) x = abs x"

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 (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 \<Longrightarrow> (k::nat) dvd m"
by(metis gcd_dvd1_nat dvd_trans)

lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
by(metis gcd_dvd2_nat dvd_trans)

lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
by(metis gcd_dvd1_int dvd_trans)

lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
by(metis gcd_dvd2_int dvd_trans)

lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
by (rule dvd_imp_le, auto)

lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
by (rule dvd_imp_le, auto)

lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
by (rule zdvd_imp_le, auto)

lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
by (rule zdvd_imp_le, auto)

lemma gcd_greatest_nat: "(k::nat) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> 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 \<Longrightarrow> k dvd n \<Longrightarrow> 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)"

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 \<Rightarrow> nat \<Rightarrow> nat"
proof
qed (auto intro: dvd_antisym dvd_trans)

interpretation gcd_int: abel_semigroup "gcd :: int \<Rightarrow> int \<Rightarrow> 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 \<and> d dvd b \<and>
(\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> 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 \<and> d dvd b \<and>
(\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> 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 \<Longrightarrow> gcd x y = x"
by (metis dvd.eq_iff gcd_unique_nat)

lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
by (metis dvd.eq_iff gcd_unique_nat)

lemma gcd_proj1_if_dvd_int[simp]: "x dvd y \<Longrightarrow> 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 \<Longrightarrow> 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")
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 \<Longrightarrow> k dvd m * n \<Longrightarrow> 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 \<Longrightarrow> k dvd m * n \<Longrightarrow> 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 \<Longrightarrow>
(k dvd m * n) = (k dvd m)"
by (auto intro: coprime_dvd_mult_nat)

lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
(k dvd m * n) = (k dvd m)"
by (auto intro: coprime_dvd_mult_int)

lemma gcd_mult_cancel_nat: "coprime k n \<Longrightarrow> 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)
done

lemma gcd_mult_cancel_int:
"coprime (k::int) n \<Longrightarrow> 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 \<longleftrightarrow> a = b \<and> c = d" (is "?lhs \<longleftrightarrow> ?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 "\<bar>a\<bar> * \<bar>c\<bar> = \<bar>b\<bar> * \<bar>d\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>b\<bar> \<and> \<bar>c\<bar> = \<bar>d\<bar>"
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")
done

lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
apply (subst (1 2) gcd_commute_nat)
apply simp
done

(* to do: add the other variations? *)

lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"

lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> 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 \<Longrightarrow> 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"

lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"

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 add_commute)

(* to do: differences, and all variations of addition rules
as simplification rules for nat and int *)

(* FIXME remove iff *)
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
using mult_dvd_mono [of 1] by auto

(* to do: add the three variations of these, and for ints? *)

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\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
apply(rule antisym)
apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
apply simp
done

lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
apply(rule antisym)
apply(rule Max_le_iff [THEN iffD2])
apply (auto intro: abs_le_D1 dvd_imp_le_int)
done

lemma gcd_is_Max_divisors_nat:
"m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> 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 = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)

subsection {* Coprimality *}

lemma div_gcd_coprime_nat:
assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 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 \<noteq> 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) \<noteq> 0 \<or> b \<noteq> 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 \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
using gcd_unique_nat[of 1 a b, simplified] by auto

lemma coprime_Suc_0_nat:
"coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
using coprime_nat by (simp add: One_nat_def)

lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
(\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> 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 \<noteq> 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 z 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 \<noteq> 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 z 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) \<longleftrightarrow>
coprime d a \<and>  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) \<longleftrightarrow>
coprime d a \<and>  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 \<noteq> 0"
shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> 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 \<noteq> 0"
shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> 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 \<Longrightarrow> coprime d (a^n)"
by (induct n, simp_all add: coprime_mult_nat)

lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
by (induct n, simp_all add: coprime_mult_int)

lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> 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 \<Longrightarrow> 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 "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
proof-
let ?g = "gcd a b"
{assume "?g = 0" with dc have ?thesis by auto}
moreover
{assume z: "?g \<noteq> 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 "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
proof-
let ?g = "gcd a b"
{assume "?g = 0" with dc have ?thesis by auto}
moreover
{assume z: "?g \<noteq> 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 \<noteq> 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 \<noteq> 0"
hence zn: "?g ^ n \<noteq> 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"
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 \<noteq> 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 \<noteq> 0"
hence zn: "?g ^ n \<noteq> 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"
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 \<Longrightarrow> ((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 \<Longrightarrow> ((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) \<noteq> 0 \<Longrightarrow> 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)
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)
done

lemma coprime_common_divisor_nat: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
x dvd b \<Longrightarrow> 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 \<Longrightarrow> x dvd a \<Longrightarrow>
x dvd b \<Longrightarrow> 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 \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
coprime d e"
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 \<Longrightarrow> 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 \<Longrightarrow> coprime x m"
apply (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
done

subsection {* Bezout's theorem *}

(* Function bezw returns a pair of witnesses to Bezout's theorem --
see the theorems that follow the definition. *)
fun
bezw  :: "nat \<Rightarrow> nat \<Rightarrow> 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 \<Longrightarrow> 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 (erule subst)
apply (subst mod_div_equality [of m n, symmetric])
(* applying simp here undoes the last substitution!
what is procedure cancel_div_mod? *)
apply (simp only: field_simps of_nat_add of_nat_mult)
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 \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
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 \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
(x \<le> 0 \<and> y \<le> 0)"
by auto
moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
by (erule (1) bezout_aux)
moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?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 \<Longrightarrow> y >= 0 \<Longrightarrow> ?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 \<Longrightarrow> y <= 0 \<Longrightarrow> ?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: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
shows "P a b"
proof(induct "a + b" arbitrary: a b rule: less_induct)
case less
have "a = b \<or> a < b \<or> 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 \<or> 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 \<or> 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: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
(a * x = b * y + d \<or> b * x = a * y + d)"
shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
(a * x = (a + b) * y + d \<or> (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: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
(a * x = b * y + d \<or> 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: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
(a * x - b * y = d \<or> b * x - a * y = d)"
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 \<noteq> (0::nat)"
shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
proof-
from nz have ap: "a > 0" by simp
have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
(\<exists>d x y. d dvd a \<and> d dvd b \<and> 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 \<noteq> 0" hence bp: "b > 0" by simp
from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
by auto
moreover
{assume db: "d=b"
with nz H 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 nz H by simp }
moreover
{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
from db have "d \<le> b - 1" by simp
hence "d*b \<le> b*(b - 1)" by simp
with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
have dble: "d*b \<le> 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) \<noteq> 0"
shows "\<exists>x y. a * x = b * y + gcd a b"
proof-
let ?g = "gcd a 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
of_nat_mult 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])
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 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)

lemma lcm_pos_int:
"(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> 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)"
also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
by (simp add: k_m [symmetric] k_n [symmetric])
also have "\<dots> = k * p * q * gcd m n"
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"
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"
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 \<Longrightarrow> n dvd k \<Longrightarrow> 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"
also have "\<dots> = 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 \<Longrightarrow> k dvd lcm m n"
by(metis lcm_dvd1_nat dvd_trans)

lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
by(metis lcm_dvd2_nat dvd_trans)

lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
by(metis lcm_dvd1_int dvd_trans)

lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
by(metis lcm_dvd2_int dvd_trans)

lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
(\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
by (auto intro: dvd_antisym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)

lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
(\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
by (auto intro: dvd_antisym [transferred] lcm_least_int)

interpretation lcm_nat: abel_semigroup "lcm :: nat \<Rightarrow> nat \<Rightarrow> 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 \<Rightarrow> int \<Rightarrow> 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"
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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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) \<longleftrightarrow> n dvd m"
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)

lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> 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) \<longleftrightarrow> 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) \<longleftrightarrow> m dvd n"
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)

lemma comp_fun_idem_gcd_nat: "comp_fun_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
proof qed (auto simp add: gcd_ac_nat)

lemma comp_fun_idem_gcd_int: "comp_fun_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
proof qed (auto simp add: gcd_ac_int)

lemma comp_fun_idem_lcm_nat: "comp_fun_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
proof qed (auto simp add: lcm_ac_nat)

lemma comp_fun_idem_lcm_int: "comp_fun_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
proof qed (auto simp add: lcm_ac_int)

(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)

lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> 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 \<longleftrightarrow> m=0 \<or> n=0"
by (metis lcm_0_int lcm_0_left_int lcm_pos_int less_le)

lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> 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 \<longleftrightarrow> (m=1 \<or> m = -1) \<and> (n=1 \<or> n = -1)"
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)

subsection {* The complete divisibility lattice *}

interpretation gcd_semilattice_nat: semilattice_inf gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
proof
case goal3 thus ?case by(metis gcd_unique_nat)
qed auto

interpretation lcm_semilattice_nat: semilattice_sup lcm "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
proof
case goal3 thus ?case by(metis lcm_unique_nat)
qed auto

interpretation gcd_lcm_lattice_nat: lattice gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm ..

text{* Lifting gcd and lcm to sets (Gcd/Lcm).
Gcd is defined via Lcm to facilitate the proof that we have a complete lattice.
*}

class Gcd = gcd +
fixes Gcd :: "'a set \<Rightarrow> 'a"
fixes Lcm :: "'a set \<Rightarrow> 'a"

instantiation nat :: Gcd
begin

definition
"Lcm (M::nat set) = (if finite M then Finite_Set.fold lcm 1 M else 0)"

definition
"Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"

instance ..
end

lemma dvd_Lcm_nat [simp]:
fixes M :: "nat set" assumes "m \<in> M" shows "m dvd Lcm M"
using lcm_semilattice_nat.sup_le_fold_sup[OF _ assms, of 1]

lemma Lcm_dvd_nat [simp]:
fixes M :: "nat set" assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
proof (cases "n = 0")
assume "n \<noteq> 0"
hence "finite {d. d dvd n}" by (rule finite_divisors_nat)
moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
ultimately have "finite M" by (rule rev_finite_subset)
thus ?thesis
using lcm_semilattice_nat.fold_sup_le_sup [OF _ assms, of 1]
qed simp

interpretation gcd_lcm_complete_lattice_nat:
complete_lattice Gcd Lcm gcd "op dvd" "%m n::nat. m dvd n & ~ n dvd m" lcm 1 0
proof
case goal1 show ?case by simp
next
case goal2 show ?case by simp
next
case goal5 thus ?case by (rule dvd_Lcm_nat)
next
case goal6 thus ?case by simp
next
case goal3 thus ?case by (simp add: Gcd_nat_def)
next
case goal4 thus ?case by (simp add: Gcd_nat_def)
qed
(* bh: This interpretation generates many lemmas about
"complete_lattice.SUPR Lcm" and "complete_lattice.INFI Gcd".
Should we define binder versions of LCM and GCD to correspond
with these? *)

lemma Lcm_empty_nat: "Lcm {} = (1::nat)"
by (fact gcd_lcm_complete_lattice_nat.Sup_empty) (* already simp *)

lemma Gcd_empty_nat: "Gcd {} = (0::nat)"
by (fact gcd_lcm_complete_lattice_nat.Inf_empty) (* already simp *)

lemma Lcm_insert_nat [simp]:
shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
by (fact gcd_lcm_complete_lattice_nat.Sup_insert)

lemma Gcd_insert_nat [simp]:
shows "Gcd (insert (n::nat) N) = gcd n (Gcd N)"
by (fact gcd_lcm_complete_lattice_nat.Inf_insert)

lemma Lcm0_iff[simp]: "finite (M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> Lcm M = 0 \<longleftrightarrow> 0 : M"
by(induct rule:finite_ne_induct) auto

lemma Lcm_eq_0[simp]: "finite (M::nat set) \<Longrightarrow> 0 : M \<Longrightarrow> Lcm M = 0"
by (metis Lcm0_iff empty_iff)

lemma Gcd_dvd_nat [simp]:
fixes M :: "nat set"
assumes "m \<in> M" shows "Gcd M dvd m"
using assms by (fact gcd_lcm_complete_lattice_nat.Inf_lower)

lemma dvd_Gcd_nat[simp]:
fixes M :: "nat set"
assumes "\<forall>m\<in>M. n dvd m" shows "n dvd Gcd M"
using assms by (simp only: gcd_lcm_complete_lattice_nat.Inf_greatest)

text{* Alternative characterizations of Gcd: *}

lemma Gcd_eq_Max: "finite(M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> Gcd M = Max(\<Inter>m\<in>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 fastforce
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 \<Longrightarrow> 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 \<Longrightarrow> M \<noteq> {} \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> Lcm M : M"
apply(induct rule:finite_linorder_min_induct)
apply simp
apply simp
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> 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 \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> 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

lemma Lcm_set_nat [code_unfold]:
"Lcm (set ns) = fold lcm ns (1::nat)"
by (fact gcd_lcm_complete_lattice_nat.Sup_set_fold)

lemma Gcd_set_nat [code_unfold]:
"Gcd (set ns) = fold gcd ns (0::nat)"
by (fact gcd_lcm_complete_lattice_nat.Inf_set_fold)

lemma mult_inj_if_coprime_nat:
"inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
\<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
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_unfold]: "gcd x y = Nitpick.nat_gcd x y"
by (induct x y rule: nat_gcd.induct)

lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)

subsubsection {* Setwise gcd and lcm for integers *}

instantiation int :: Gcd
begin

definition
"Lcm M = int (Lcm (nat ` abs ` M))"

definition
"Gcd M = int (Gcd (nat ` abs ` M))"

instance ..
end

lemma Lcm_empty_int [simp]: "Lcm {} = (1::int)"

lemma Gcd_empty_int [simp]: "Gcd {} = (0::int)"

lemma Lcm_insert_int [simp]:
shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"

lemma Gcd_insert_int [simp]:
shows "Gcd (insert (n::int) N) = gcd n (Gcd N)"

lemma dvd_int_iff: "x dvd y \<longleftrightarrow> nat (abs x) dvd nat (abs y)"

lemma dvd_Lcm_int [simp]:
fixes M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
using assms by (simp add: Lcm_int_def dvd_int_iff)

lemma Lcm_dvd_int [simp]:
fixes M :: "int set"
assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
using assms by (simp add: Lcm_int_def dvd_int_iff)

lemma Gcd_dvd_int [simp]:
fixes M :: "int set"
assumes "m \<in> M" shows "Gcd M dvd m"
using assms by (simp add: Gcd_int_def dvd_int_iff)

lemma dvd_Gcd_int[simp]:
fixes M :: "int set"
assumes "\<forall>m\<in>M. n dvd m" shows "n dvd Gcd M"
using assms by (simp add: Gcd_int_def dvd_int_iff)

lemma Lcm_set_int [code_unfold]:
"Lcm (set xs) = foldl lcm (1::int) xs"
by (induct xs rule: rev_induct, simp_all add: lcm_commute_int)

lemma Gcd_set_int [code_unfold]:
"Gcd (set xs) = foldl gcd (0::int) xs"
by (induct xs rule: rev_induct, simp_all add: gcd_commute_int)

end
```