src/HOL/GCD.thy

Made "prime" executable

(*  Title:      GCD.thy
    Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow


This file deals with the functions gcd and lcm, and properties of
primes. 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 Chiaeb.

Tobias Nipkow cleaned up a lot.
*)


header {* GCD *}

theory GCD
imports NatTransfer
begin

declare One_nat_def [simp del]

subsection {* gcd *}

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

class prime = one +

fixes
  prime :: "'a \<Rightarrow> bool"


(* definitions for the natural numbers *)

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 nat :: prime

begin

definition
  prime_nat :: "nat \<Rightarrow> bool"
where
  [code del]: "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"

instance proof qed

end


(* definitions for the integers *)

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


instantiation int :: prime

begin

definition
  prime_int :: "int \<Rightarrow> bool"
where
  [code del]: "prime_int p = prime (nat p)"

instance proof qed

end


subsection {* Set up Transfer *}


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)"
  "(x::int) >= 0 \<Longrightarrow> prime (nat x) = prime x"
  unfolding gcd_int_def lcm_int_def prime_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)

declare TransferMorphism_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)"
  "prime (int x) = prime x"
  by (unfold gcd_int_def lcm_int_def prime_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)

declare TransferMorphism_int_nat[transfer add return:
    transfer_int_nat_gcd transfer_int_nat_gcd_closures]


subsection {* GCD *}

(* 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"
  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 \<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 prems, auto simp add: gcd_neg1_int gcd_neg2_int, 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 \<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)"
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)

(* 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"
  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 \<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)"
  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)

lemma gcd_commute_nat: "gcd (m::nat) n = gcd n m"
  by (rule dvd_anti_sym, auto)

lemma gcd_commute_int: "gcd (m::int) n = gcd n m"
  by (auto simp add: gcd_int_def gcd_commute_nat)

lemma gcd_assoc_nat: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
  apply (rule dvd_anti_sym)
  apply (blast intro: dvd_trans)+
done

lemma gcd_assoc_int: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
  by (auto simp add: gcd_int_def gcd_assoc_nat)

lemmas gcd_left_commute_nat =
  mk_left_commute[of gcd, OF gcd_assoc_nat gcd_commute_nat]

lemmas gcd_left_commute_int =
  mk_left_commute[of gcd, OF gcd_assoc_int gcd_commute_int]

lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
  -- {* gcd is an AC-operator *}

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_anti_sym)
  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 force
  apply (rule iffI)
  apply (rule zdvd_anti_sym)
  apply arith
  apply (subst gcd_pos_int)
  apply clarsimp
  apply (drule_tac x = "d + 1" in spec)
  apply (frule zdvd_imp_le)
  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")
  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 \<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_anti_sym)
  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 \<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

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

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

lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
  by (subst gcd_add1_nat [symmetric], auto)

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 simp add: gcd_neg1_int gcd_neg2_int)
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)


(* 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 (fastsimp 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 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 \<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


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 add: gcd_zero_nat)
  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 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 \<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 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) \<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 coprime_divprod_nat: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
  using coprime_dvd_mult_iff_nat[of d a b]
  by (auto simp add: mult_commute)

lemma coprime_divprod_int: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
  using coprime_dvd_mult_iff_int[of d a b]
  by (auto simp add: mult_commute)

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 add: neq0_conv)
    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 \<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 add: neq0_conv)
    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

(* FIXME move to Divides(?) *)
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)
  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 prime_odd_nat: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
  unfolding prime_nat_def
  apply (subst even_mult_two_ex)
  apply clarify
  apply (drule_tac x = 2 in spec)
  apply auto
done

lemma prime_odd_int: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
  unfolding prime_int_def
  apply (frule prime_odd_nat)
  apply (auto simp add: even_nat_def)
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 (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 \<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 (simp add: bezw_non_0 gcd_non_0_nat)
      apply (erule subst)
      apply (simp add: ring_simps)
      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: ring_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 \<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 n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
  fix n a b
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
  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 < n \<or> a = 0"  using H(2) by arith
    moreover
    {assume "a =0" with z c have "P a b" by blast }
    moreover
    {assume ab: "a + b - a < n"
      have th0: "a + b - a = a + (b - a)" using lt by arith
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
      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 < n \<or> b = 0"  using H(2) by arith
    moreover
    {assume "b =0" with z c have "P a b" by blast }
    moreover
    {assume ab: "b + a - b < n"
      have th0: "b + a - b = b + (a - b)" using lt by arith
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
      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 add: dvd_add)
  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 add: dvd_add)
  apply clarsimp
  apply (rule_tac x="d" in exI)
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
  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)"
  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 \<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
 from bezout_add_nat[of a b]
 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"
       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 \<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"
  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 *}

lemma lcm_altdef_int: "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_commute_nat: "lcm (m::nat) n = lcm n m"
  unfolding lcm_nat_def by (simp add: gcd_commute_nat ring_simps)

lemma lcm_commute_int: "lcm (m::int) n = lcm n m"
  unfolding lcm_int_def by (subst lcm_commute_nat, rule refl)