src/HOL/GCD.thy
author wenzelm
Wed Jun 17 23:01:19 2015 +0200 (2015-06-17)
changeset 60512 e0169291b31c
parent 60357 bc0827281dc1
child 60580 7e741e22d7fc
permissions -rw-r--r--
tuned proofs -- slightly faster;
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with the functions gcd and lcm.  Definitions and
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lemmas are proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on @{cite davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chaieb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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section {* Greatest common divisor and least common multiple *}
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theory GCD
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imports Main
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begin
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declare One_nat_def [simp del]
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subsection {* GCD and LCM definitions *}
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class gcd = zero + one + dvd +
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation
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  coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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where
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  "coprime x y == (gcd x y = 1)"
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end
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class semiring_gcd = comm_semiring_1 + gcd +
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  assumes gcd_dvd1 [iff]: "gcd a b dvd a"
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    and gcd_dvd2 [iff]: "gcd a b dvd b"
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    and gcd_greatest: "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> c dvd gcd a b"
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class ring_gcd = comm_ring_1 + semiring_gcd
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instantiation nat :: gcd
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begin
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fun
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  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "gcd_nat x y =
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   (if y = 0 then x else gcd y (x mod y))"
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definition
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  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "lcm_nat x y = x * y div (gcd x y)"
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instance proof qed
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end
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instantiation int :: gcd
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begin
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definition
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  gcd_int  :: "int \<Rightarrow> int \<Rightarrow> int"
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where
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  "gcd_int x y = int (gcd (nat (abs x)) (nat (abs y)))"
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definition
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  lcm_int :: "int \<Rightarrow> int \<Rightarrow> int"
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where
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  "lcm_int x y = int (lcm (nat (abs x)) (nat (abs y)))"
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instance proof qed
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end
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subsection {* Transfer setup *}
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lemma transfer_nat_int_gcd:
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  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> gcd (nat x) (nat y) = nat (gcd x y)"
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  "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> lcm (nat x) (nat y) = nat (lcm x y)"
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  unfolding gcd_int_def lcm_int_def
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  by auto
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lemma transfer_nat_int_gcd_closures:
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  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
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  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
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  by (auto simp add: gcd_int_def lcm_int_def)
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declare transfer_morphism_nat_int[transfer add return:
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    transfer_nat_int_gcd transfer_nat_int_gcd_closures]
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lemma transfer_int_nat_gcd:
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  "gcd (int x) (int y) = int (gcd x y)"
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  "lcm (int x) (int y) = int (lcm x y)"
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  by (unfold gcd_int_def lcm_int_def, auto)
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lemma transfer_int_nat_gcd_closures:
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  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
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  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
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  by (auto simp add: gcd_int_def lcm_int_def)
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declare transfer_morphism_int_nat[transfer add return:
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    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
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subsection {* GCD properties *}
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(* was gcd_induct *)
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lemma gcd_nat_induct:
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  fixes m n :: nat
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  assumes "\<And>m. P m 0"
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    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
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  shows "P m n"
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  apply (rule gcd_nat.induct)
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  apply (case_tac "y = 0")
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  using assms apply simp_all
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done
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(* specific to int *)
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lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
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  by (simp add: gcd_int_def)
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lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
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  by (simp add: gcd_int_def)
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lemma gcd_neg_numeral_1_int [simp]:
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  "gcd (- numeral n :: int) x = gcd (numeral n) x"
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  by (fact gcd_neg1_int)
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lemma gcd_neg_numeral_2_int [simp]:
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  "gcd x (- numeral n :: int) = gcd x (numeral n)"
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  by (fact gcd_neg2_int)
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lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"
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by(simp add: gcd_int_def)
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lemma gcd_abs_int: "gcd (x::int) y = gcd (abs x) (abs y)"
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by (simp add: gcd_int_def)
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lemma gcd_abs1_int[simp]: "gcd (abs x) (y::int) = gcd x y"
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by (metis abs_idempotent gcd_abs_int)
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lemma gcd_abs2_int[simp]: "gcd x (abs y::int) = gcd x y"
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by (metis abs_idempotent gcd_abs_int)
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lemma gcd_cases_int:
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  fixes x :: int and y
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  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
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      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
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      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
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      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
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  shows "P (gcd x y)"
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by (insert assms, auto, arith)
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lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
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  by (simp add: gcd_int_def)
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lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
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  by (simp add: lcm_int_def)
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lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
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  by (simp add: lcm_int_def)
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lemma lcm_abs_int: "lcm (x::int) y = lcm (abs x) (abs y)"
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  by (simp add: lcm_int_def)
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lemma abs_lcm_int [simp]: "abs (lcm i j::int) = lcm i j"
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by(simp add:lcm_int_def)
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lemma lcm_abs1_int[simp]: "lcm (abs x) (y::int) = lcm x y"
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by (metis abs_idempotent lcm_int_def)
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lemma lcm_abs2_int[simp]: "lcm x (abs y::int) = lcm x y"
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by (metis abs_idempotent lcm_int_def)
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lemma lcm_cases_int:
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  fixes x :: int and y
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  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
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      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
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      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
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      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
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  shows "P (lcm x y)"
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  using assms by (auto simp add: lcm_neg1_int lcm_neg2_int) arith
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lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
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  by (simp add: lcm_int_def)
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(* was gcd_0, etc. *)
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lemma gcd_0_nat: "gcd (x::nat) 0 = x"
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  by simp
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(* was igcd_0, etc. *)
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lemma gcd_0_int [simp]: "gcd (x::int) 0 = abs x"
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  by (unfold gcd_int_def, auto)
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lemma gcd_0_left_nat: "gcd 0 (x::nat) = x"
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  by simp
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lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = abs x"
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  by (unfold gcd_int_def, auto)
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lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
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  by (case_tac "y = 0", auto)
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(* weaker, but useful for the simplifier *)
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lemma gcd_non_0_nat: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
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  by simp
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lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
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  by simp
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lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
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  by (simp add: One_nat_def)
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lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
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  by (simp add: gcd_int_def)
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lemma gcd_idem_nat: "gcd (x::nat) x = x"
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by simp
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lemma gcd_idem_int: "gcd (x::int) x = abs x"
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by (auto simp add: gcd_int_def)
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declare gcd_nat.simps [simp del]
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text {*
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  \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
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  conjunctions don't seem provable separately.
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*}
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instance nat :: semiring_gcd
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proof
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  fix m n :: nat
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  show "gcd m n dvd m" and "gcd m n dvd n"
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  proof (induct m n rule: gcd_nat_induct)
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    fix m n :: nat
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    assume "gcd n (m mod n) dvd m mod n" and "gcd n (m mod n) dvd n"
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    then have "gcd n (m mod n) dvd m"
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      by (rule dvd_mod_imp_dvd)
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    moreover assume "0 < n"
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    ultimately show "gcd m n dvd m"
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      by (simp add: gcd_non_0_nat)
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  qed (simp_all add: gcd_0_nat gcd_non_0_nat)
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next
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  fix m n k :: nat
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  assume "k dvd m" and "k dvd n"
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  then show "k dvd gcd m n"
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    by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod gcd_0_nat)
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qed
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instance int :: ring_gcd
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  by intro_classes (simp_all add: dvd_int_unfold_dvd_nat gcd_int_def gcd_greatest)
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lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
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  by (metis gcd_dvd1 dvd_trans)
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lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
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  by (metis gcd_dvd2 dvd_trans)
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lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
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  by (metis gcd_dvd1 dvd_trans)
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lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
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  by (metis gcd_dvd2 dvd_trans)
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lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
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  by (rule dvd_imp_le, auto)
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lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
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  by (rule dvd_imp_le, auto)
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lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
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  by (rule zdvd_imp_le, auto)
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lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
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  by (rule zdvd_imp_le, auto)
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lemma gcd_greatest_iff_nat [iff]: "(k dvd gcd (m::nat) n) =
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    (k dvd m & k dvd n)"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemma gcd_greatest_iff_int: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
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  by (blast intro!: gcd_greatest intro: dvd_trans)
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lemma gcd_zero_nat [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
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  by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff_nat)
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lemma gcd_zero_int [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
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  by (auto simp add: gcd_int_def)
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lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
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  by (insert gcd_zero_nat [of m n], arith)
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lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
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  by (insert gcd_zero_int [of m n], insert gcd_ge_0_int [of m n], arith)
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lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
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    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
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  apply auto
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  apply (rule dvd_antisym)
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  apply (erule (1) gcd_greatest)
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  apply auto
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   322
done
wenzelm@21256
   323
nipkow@31952
   324
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   325
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
nipkow@33657
   326
apply (case_tac "d = 0")
nipkow@33657
   327
 apply simp
nipkow@33657
   328
apply (rule iffI)
nipkow@33657
   329
 apply (rule zdvd_antisym_nonneg)
haftmann@59008
   330
 apply (auto intro: gcd_greatest)
huffman@31706
   331
done
huffman@30082
   332
haftmann@54867
   333
interpretation gcd_nat: abel_semigroup "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@54867
   334
  + gcd_nat: semilattice_neutr_order "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat" 0 "op dvd" "(\<lambda>m n. m dvd n \<and> \<not> n dvd m)"
haftmann@54867
   335
apply default
haftmann@54867
   336
apply (auto intro: dvd_antisym dvd_trans)[4]
haftmann@59545
   337
apply (metis dvd.dual_order.refl gcd_unique_nat)+
haftmann@54867
   338
done
haftmann@54867
   339
haftmann@54867
   340
interpretation gcd_int: abel_semigroup "gcd :: int \<Rightarrow> int \<Rightarrow> int"
haftmann@54867
   341
proof
haftmann@54867
   342
qed (simp_all add: gcd_int_def gcd_nat.assoc gcd_nat.commute gcd_nat.left_commute)
haftmann@54867
   343
haftmann@54867
   344
lemmas gcd_assoc_nat = gcd_nat.assoc
haftmann@54867
   345
lemmas gcd_commute_nat = gcd_nat.commute
haftmann@54867
   346
lemmas gcd_left_commute_nat = gcd_nat.left_commute
haftmann@54867
   347
lemmas gcd_assoc_int = gcd_int.assoc
haftmann@54867
   348
lemmas gcd_commute_int = gcd_int.commute
haftmann@54867
   349
lemmas gcd_left_commute_int = gcd_int.left_commute
haftmann@54867
   350
haftmann@54867
   351
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
haftmann@54867
   352
haftmann@54867
   353
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
haftmann@54867
   354
nipkow@31798
   355
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
haftmann@54867
   356
  by (fact gcd_nat.absorb1)
nipkow@31798
   357
nipkow@31798
   358
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
haftmann@54867
   359
  by (fact gcd_nat.absorb2)
nipkow@31798
   360
haftmann@54867
   361
lemma gcd_proj1_if_dvd_int [simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
haftmann@54867
   362
  by (metis abs_dvd_iff gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   363
haftmann@54867
   364
lemma gcd_proj2_if_dvd_int [simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
haftmann@54867
   365
  by (metis gcd_proj1_if_dvd_int gcd_commute_int)
nipkow@31798
   366
wenzelm@21256
   367
text {*
wenzelm@21256
   368
  \medskip Multiplication laws
wenzelm@21256
   369
*}
wenzelm@21256
   370
nipkow@31952
   371
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@58623
   372
    -- {* @{cite \<open>page 27\<close> davenport92} *}
nipkow@31952
   373
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   374
  apply simp
wenzelm@21256
   375
  apply (case_tac "k = 0")
huffman@45270
   376
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   377
done
wenzelm@21256
   378
nipkow@31952
   379
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   380
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   381
  apply (subst (1 2) abs_mult)
nipkow@31952
   382
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   383
  apply auto
huffman@31706
   384
done
wenzelm@21256
   385
nipkow@31952
   386
lemma coprime_dvd_mult_nat: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31952
   387
  apply (insert gcd_mult_distrib_nat [of m k n])
wenzelm@21256
   388
  apply simp
wenzelm@21256
   389
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   390
  apply simp
wenzelm@21256
   391
  done
wenzelm@21256
   392
nipkow@31952
   393
lemma coprime_dvd_mult_int:
nipkow@31813
   394
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31813
   395
apply (subst abs_dvd_iff [symmetric])
nipkow@31813
   396
apply (subst dvd_abs_iff [symmetric])
nipkow@31952
   397
apply (subst (asm) gcd_abs_int)
nipkow@31952
   398
apply (rule coprime_dvd_mult_nat [transferred])
nipkow@31813
   399
    prefer 4 apply assumption
nipkow@31813
   400
   apply auto
nipkow@31813
   401
apply (subst abs_mult [symmetric], auto)
huffman@31706
   402
done
huffman@31706
   403
nipkow@31952
   404
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   405
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   406
  by (auto intro: coprime_dvd_mult_nat)
huffman@31706
   407
nipkow@31952
   408
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   409
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   410
  by (auto intro: coprime_dvd_mult_int)
huffman@31706
   411
nipkow@31952
   412
lemma gcd_mult_cancel_nat: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
nipkow@33657
   413
  apply (rule dvd_antisym)
haftmann@59008
   414
  apply (rule gcd_greatest)
nipkow@31952
   415
  apply (rule_tac n = k in coprime_dvd_mult_nat)
nipkow@31952
   416
  apply (simp add: gcd_assoc_nat)
nipkow@31952
   417
  apply (simp add: gcd_commute_nat)
haftmann@57512
   418
  apply (simp_all add: mult.commute)
huffman@31706
   419
done
wenzelm@21256
   420
nipkow@31952
   421
lemma gcd_mult_cancel_int:
nipkow@31813
   422
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
nipkow@31952
   423
apply (subst (1 2) gcd_abs_int)
nipkow@31813
   424
apply (subst abs_mult)
nipkow@31952
   425
apply (rule gcd_mult_cancel_nat [transferred], auto)
huffman@31706
   426
done
wenzelm@21256
   427
haftmann@35368
   428
lemma coprime_crossproduct_nat:
haftmann@35368
   429
  fixes a b c d :: nat
haftmann@35368
   430
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   431
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35368
   432
proof
haftmann@35368
   433
  assume ?rhs then show ?lhs by simp
haftmann@35368
   434
next
haftmann@35368
   435
  assume ?lhs
haftmann@35368
   436
  from `?lhs` have "a dvd b * d" by (auto intro: dvdI dest: sym)
haftmann@35368
   437
  with `coprime a d` have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
haftmann@35368
   438
  from `?lhs` have "b dvd a * c" by (auto intro: dvdI dest: sym)
haftmann@35368
   439
  with `coprime b c` have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
haftmann@57512
   440
  from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@35368
   441
  with `coprime b c` have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
haftmann@57512
   442
  from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult.commute)
haftmann@35368
   443
  with `coprime a d` have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
haftmann@35368
   444
  from `a dvd b` `b dvd a` have "a = b" by (rule Nat.dvd.antisym)
haftmann@35368
   445
  moreover from `c dvd d` `d dvd c` have "c = d" by (rule Nat.dvd.antisym)
haftmann@35368
   446
  ultimately show ?rhs ..
haftmann@35368
   447
qed
haftmann@35368
   448
haftmann@35368
   449
lemma coprime_crossproduct_int:
haftmann@35368
   450
  fixes a b c d :: int
haftmann@35368
   451
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   452
  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>"
haftmann@35368
   453
  using assms by (intro coprime_crossproduct_nat [transferred]) auto
haftmann@35368
   454
wenzelm@21256
   455
text {* \medskip Addition laws *}
wenzelm@21256
   456
nipkow@31952
   457
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   458
  apply (case_tac "n = 0")
nipkow@31952
   459
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   460
done
huffman@31706
   461
nipkow@31952
   462
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
nipkow@31952
   463
  apply (subst (1 2) gcd_commute_nat)
haftmann@57512
   464
  apply (subst add.commute)
huffman@31706
   465
  apply simp
huffman@31706
   466
done
huffman@31706
   467
huffman@31706
   468
(* to do: add the other variations? *)
huffman@31706
   469
nipkow@31952
   470
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
nipkow@31952
   471
  by (subst gcd_add1_nat [symmetric], auto)
huffman@31706
   472
nipkow@31952
   473
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
nipkow@31952
   474
  apply (subst gcd_commute_nat)
nipkow@31952
   475
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
   476
  apply auto
nipkow@31952
   477
  apply (subst gcd_commute_nat)
nipkow@31952
   478
  apply (subst gcd_diff1_nat)
huffman@31706
   479
  apply assumption
nipkow@31952
   480
  apply (rule gcd_commute_nat)
huffman@31706
   481
done
huffman@31706
   482
nipkow@31952
   483
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   484
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   485
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
   486
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
   487
    zmod_zminus1_eq_if)
huffman@31706
   488
  apply (frule_tac a = x in pos_mod_bound)
nipkow@31952
   489
  apply (subst (1 2) gcd_commute_nat)
nipkow@31952
   490
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
   491
    nat_le_eq_zle)
huffman@31706
   492
done
wenzelm@21256
   493
nipkow@31952
   494
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   495
  apply (case_tac "y = 0")
huffman@31706
   496
  apply force
huffman@31706
   497
  apply (case_tac "y > 0")
nipkow@31952
   498
  apply (subst gcd_non_0_int, auto)
nipkow@31952
   499
  apply (insert gcd_non_0_int [of "-y" "-x"])
huffman@35216
   500
  apply auto
huffman@31706
   501
done
huffman@31706
   502
nipkow@31952
   503
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
haftmann@57512
   504
by (metis gcd_red_int mod_add_self1 add.commute)
huffman@31706
   505
nipkow@31952
   506
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
haftmann@57512
   507
by (metis gcd_add1_int gcd_commute_int add.commute)
wenzelm@21256
   508
nipkow@31952
   509
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31952
   510
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
wenzelm@21256
   511
nipkow@31952
   512
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
haftmann@57512
   513
by (metis gcd_commute_int gcd_red_int mod_mult_self1 add.commute)
nipkow@31798
   514
wenzelm@21256
   515
huffman@31706
   516
(* to do: differences, and all variations of addition rules
huffman@31706
   517
    as simplification rules for nat and int *)
huffman@31706
   518
nipkow@31798
   519
(* FIXME remove iff *)
nipkow@31952
   520
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   521
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   522
huffman@31706
   523
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   524
nipkow@31992
   525
lemma finite_divisors_nat[simp]:
nipkow@31992
   526
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
nipkow@31734
   527
proof-
wenzelm@60512
   528
  have "finite{d. d <= m}"
wenzelm@60512
   529
    by (blast intro: bounded_nat_set_is_finite)
nipkow@31734
   530
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
   531
    by (metis Collect_mono dvd_imp_le neq0_conv)
nipkow@31734
   532
qed
nipkow@31734
   533
nipkow@31995
   534
lemma finite_divisors_int[simp]:
nipkow@31734
   535
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
   536
proof-
nipkow@31734
   537
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
   538
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
   539
  from finite_subset[OF _ this] show ?thesis using assms
wenzelm@60512
   540
    by (simp add: dvd_imp_le_int subset_iff)
nipkow@31734
   541
qed
nipkow@31734
   542
nipkow@31995
   543
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
   544
apply(rule antisym)
nipkow@44890
   545
 apply (fastforce intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
   546
apply simp
nipkow@31995
   547
done
nipkow@31995
   548
nipkow@31995
   549
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
nipkow@31995
   550
apply(rule antisym)
haftmann@44278
   551
 apply(rule Max_le_iff [THEN iffD2])
haftmann@44278
   552
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
nipkow@31995
   553
done
nipkow@31995
   554
nipkow@31734
   555
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   556
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   557
apply(rule Max_eqI[THEN sym])
nipkow@31995
   558
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   559
 apply simp
nipkow@31952
   560
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
nipkow@31734
   561
apply simp
nipkow@31734
   562
done
nipkow@31734
   563
nipkow@31734
   564
lemma gcd_is_Max_divisors_int:
nipkow@31734
   565
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   566
apply(rule Max_eqI[THEN sym])
nipkow@31995
   567
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
   568
 apply simp
nipkow@31952
   569
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
nipkow@31734
   570
apply simp
nipkow@31734
   571
done
nipkow@31734
   572
haftmann@34030
   573
lemma gcd_code_int [code]:
haftmann@34030
   574
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@34030
   575
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
   576
chaieb@22027
   577
huffman@31706
   578
subsection {* Coprimality *}
huffman@31706
   579
nipkow@31952
   580
lemma div_gcd_coprime_nat:
huffman@31706
   581
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   582
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   583
proof -
haftmann@27556
   584
  let ?g = "gcd a b"
chaieb@22027
   585
  let ?a' = "a div ?g"
chaieb@22027
   586
  let ?b' = "b div ?g"
haftmann@27556
   587
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   588
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   589
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   590
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   591
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   592
    unfolding dvd_def by blast
haftmann@58834
   593
  from this [symmetric] have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
haftmann@58834
   594
    by (simp_all add: mult.assoc mult.left_commute [of "gcd a b"])
wenzelm@22367
   595
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   596
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   597
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@35216
   598
  have "?g \<noteq> 0" using nz by simp
huffman@31706
   599
  then have gp: "?g > 0" by arith
haftmann@59008
   600
  from gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   601
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   602
qed
chaieb@22027
   603
nipkow@31952
   604
lemma div_gcd_coprime_int:
huffman@31706
   605
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   606
  shows "coprime (a div gcd a b) (b div gcd a b)"
nipkow@31952
   607
apply (subst (1 2 3) gcd_abs_int)
nipkow@31813
   608
apply (subst (1 2) abs_div)
nipkow@31813
   609
  apply simp
nipkow@31813
   610
 apply simp
nipkow@31813
   611
apply(subst (1 2) abs_gcd_int)
nipkow@31952
   612
apply (rule div_gcd_coprime_nat [transferred])
nipkow@31952
   613
using nz apply (auto simp add: gcd_abs_int [symmetric])
huffman@31706
   614
done
huffman@31706
   615
nipkow@31952
   616
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   617
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
   618
nipkow@31952
   619
lemma coprime_Suc_0_nat:
huffman@31706
   620
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
nipkow@31952
   621
  using coprime_nat by (simp add: One_nat_def)
huffman@31706
   622
nipkow@31952
   623
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   624
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   625
  using gcd_unique_int [of 1 a b]
huffman@31706
   626
  apply clarsimp
huffman@31706
   627
  apply (erule subst)
huffman@31706
   628
  apply (rule iffI)
huffman@31706
   629
  apply force
wenzelm@59807
   630
  apply (drule_tac x = "abs e" for e in exI)
wenzelm@59807
   631
  apply (case_tac "e >= 0" for e :: int)
huffman@31706
   632
  apply force
huffman@31706
   633
  apply force
wenzelm@59807
   634
  done
huffman@31706
   635
nipkow@31952
   636
lemma gcd_coprime_nat:
huffman@31706
   637
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   638
    b: "b = b' * gcd a b"
huffman@31706
   639
  shows    "coprime a' b'"
huffman@31706
   640
huffman@31706
   641
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   642
  apply (erule ssubst)
huffman@31706
   643
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   644
  apply (erule ssubst)
nipkow@31952
   645
  apply (rule div_gcd_coprime_nat)
wenzelm@41550
   646
  using z apply force
huffman@31706
   647
  apply (subst (1) b)
huffman@31706
   648
  using z apply force
huffman@31706
   649
  apply (subst (1) a)
huffman@31706
   650
  using z apply force
wenzelm@41550
   651
  done
huffman@31706
   652
nipkow@31952
   653
lemma gcd_coprime_int:
huffman@31706
   654
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   655
    b: "b = b' * gcd a b"
huffman@31706
   656
  shows    "coprime a' b'"
huffman@31706
   657
huffman@31706
   658
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   659
  apply (erule ssubst)
huffman@31706
   660
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   661
  apply (erule ssubst)
nipkow@31952
   662
  apply (rule div_gcd_coprime_int)
wenzelm@41550
   663
  using z apply force
huffman@31706
   664
  apply (subst (1) b)
huffman@31706
   665
  using z apply force
huffman@31706
   666
  apply (subst (1) a)
huffman@31706
   667
  using z apply force
wenzelm@41550
   668
  done
huffman@31706
   669
nipkow@31952
   670
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   671
    shows "coprime d (a * b)"
nipkow@31952
   672
  apply (subst gcd_commute_nat)
nipkow@31952
   673
  using da apply (subst gcd_mult_cancel_nat)
nipkow@31952
   674
  apply (subst gcd_commute_nat, assumption)
nipkow@31952
   675
  apply (subst gcd_commute_nat, rule db)
huffman@31706
   676
done
huffman@31706
   677
nipkow@31952
   678
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   679
    shows "coprime d (a * b)"
nipkow@31952
   680
  apply (subst gcd_commute_int)
nipkow@31952
   681
  using da apply (subst gcd_mult_cancel_int)
nipkow@31952
   682
  apply (subst gcd_commute_int, assumption)
nipkow@31952
   683
  apply (subst gcd_commute_int, rule db)
huffman@31706
   684
done
huffman@31706
   685
nipkow@31952
   686
lemma coprime_lmult_nat:
huffman@31706
   687
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   688
proof -
huffman@31706
   689
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
   690
    by (rule gcd_greatest, auto)
huffman@31706
   691
  with dab show ?thesis
huffman@31706
   692
    by auto
huffman@31706
   693
qed
huffman@31706
   694
nipkow@31952
   695
lemma coprime_lmult_int:
nipkow@31798
   696
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   697
proof -
huffman@31706
   698
  have "gcd d a dvd gcd d (a * b)"
haftmann@59008
   699
    by (rule gcd_greatest, auto)
nipkow@31798
   700
  with assms show ?thesis
huffman@31706
   701
    by auto
huffman@31706
   702
qed
huffman@31706
   703
nipkow@31952
   704
lemma coprime_rmult_nat:
nipkow@31798
   705
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   706
proof -
huffman@31706
   707
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
   708
    by (rule gcd_greatest, auto intro: dvd_mult)
nipkow@31798
   709
  with assms show ?thesis
huffman@31706
   710
    by auto
huffman@31706
   711
qed
huffman@31706
   712
nipkow@31952
   713
lemma coprime_rmult_int:
huffman@31706
   714
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   715
proof -
huffman@31706
   716
  have "gcd d b dvd gcd d (a * b)"
haftmann@59008
   717
    by (rule gcd_greatest, auto intro: dvd_mult)
huffman@31706
   718
  with dab show ?thesis
huffman@31706
   719
    by auto
huffman@31706
   720
qed
huffman@31706
   721
nipkow@31952
   722
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   723
    coprime d a \<and>  coprime d b"
nipkow@31952
   724
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
nipkow@31952
   725
    coprime_mult_nat[of d a b]
huffman@31706
   726
  by blast
huffman@31706
   727
nipkow@31952
   728
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   729
    coprime d a \<and>  coprime d b"
nipkow@31952
   730
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
nipkow@31952
   731
    coprime_mult_int[of d a b]
huffman@31706
   732
  by blast
huffman@31706
   733
noschinl@52397
   734
lemma coprime_power_int:
noschinl@52397
   735
  assumes "0 < n" shows "coprime (a :: int) (b ^ n) \<longleftrightarrow> coprime a b"
noschinl@52397
   736
  using assms
noschinl@52397
   737
proof (induct n)
noschinl@52397
   738
  case (Suc n) then show ?case
noschinl@52397
   739
    by (cases n) (simp_all add: coprime_mul_eq_int)
noschinl@52397
   740
qed simp
noschinl@52397
   741
nipkow@31952
   742
lemma gcd_coprime_exists_nat:
huffman@31706
   743
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   744
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   745
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   746
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   747
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
huffman@31706
   748
done
huffman@31706
   749
nipkow@31952
   750
lemma gcd_coprime_exists_int:
huffman@31706
   751
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   752
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   753
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   754
  apply (rule_tac x = "b div gcd a b" in exI)
haftmann@59008
   755
  using nz apply (auto simp add: div_gcd_coprime_int)
huffman@31706
   756
done
huffman@31706
   757
nipkow@31952
   758
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
lp15@60162
   759
  by (induct n, simp_all add: power_Suc coprime_mult_nat)
huffman@31706
   760
nipkow@31952
   761
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
lp15@60162
   762
  by (induct n, simp_all add: power_Suc coprime_mult_int)
huffman@31706
   763
nipkow@31952
   764
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
lp15@60162
   765
  by (simp add: coprime_exp_nat gcd_nat.commute)
huffman@31706
   766
nipkow@31952
   767
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
lp15@60162
   768
  by (simp add: coprime_exp_int gcd_int.commute)
huffman@31706
   769
nipkow@31952
   770
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   771
proof (cases)
huffman@31706
   772
  assume "a = 0 & b = 0"
huffman@31706
   773
  thus ?thesis by simp
huffman@31706
   774
  next assume "~(a = 0 & b = 0)"
huffman@31706
   775
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
nipkow@31952
   776
    by (auto simp:div_gcd_coprime_nat)
huffman@31706
   777
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   778
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
lp15@60162
   779
    by (metis gcd_mult_distrib_nat mult.commute mult.right_neutral)
huffman@31706
   780
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
lp15@60162
   781
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
   782
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
lp15@60162
   783
    by (metis dvd_div_mult_self gcd_unique_nat power_mult_distrib)
huffman@31706
   784
  finally show ?thesis .
huffman@31706
   785
qed
huffman@31706
   786
nipkow@31952
   787
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
   788
  apply (subst (1 2) gcd_abs_int)
huffman@31706
   789
  apply (subst (1 2) power_abs)
nipkow@31952
   790
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
   791
  apply auto
huffman@31706
   792
done
huffman@31706
   793
nipkow@31952
   794
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   795
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   796
proof-
huffman@31706
   797
  let ?g = "gcd a b"
huffman@31706
   798
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   799
  moreover
huffman@31706
   800
  {assume z: "?g \<noteq> 0"
nipkow@31952
   801
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   802
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   803
      by blast
huffman@31706
   804
    have thb: "?g dvd b" by auto
huffman@31706
   805
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   806
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   807
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@57512
   808
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
huffman@31706
   809
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   810
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
haftmann@57512
   811
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
   812
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   813
    with thb thc have ?thesis by blast }
huffman@31706
   814
  ultimately show ?thesis by blast
huffman@31706
   815
qed
huffman@31706
   816
nipkow@31952
   817
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
   818
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   819
proof-
huffman@31706
   820
  let ?g = "gcd a b"
huffman@31706
   821
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   822
  moreover
huffman@31706
   823
  {assume z: "?g \<noteq> 0"
nipkow@31952
   824
    from gcd_coprime_exists_int[OF z]
huffman@31706
   825
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   826
      by blast
huffman@31706
   827
    have thb: "?g dvd b" by auto
huffman@31706
   828
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   829
    with dc have th0: "a' dvd b*c"
huffman@31706
   830
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   831
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
haftmann@57512
   832
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult.assoc)
huffman@31706
   833
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   834
    from coprime_dvd_mult_int[OF ab'(3)] th_1
haftmann@57512
   835
    have thc: "a' dvd c" by (subst (asm) mult.commute, blast)
huffman@31706
   836
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   837
    with thb thc have ?thesis by blast }
huffman@31706
   838
  ultimately show ?thesis by blast
chaieb@27669
   839
qed
chaieb@27669
   840
nipkow@31952
   841
lemma pow_divides_pow_nat:
huffman@31706
   842
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   843
  shows "a dvd b"
huffman@31706
   844
proof-
huffman@31706
   845
  let ?g = "gcd a b"
huffman@31706
   846
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   847
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   848
  moreover
huffman@31706
   849
  {assume z: "?g \<noteq> 0"
huffman@35216
   850
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
   851
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   852
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   853
      by blast
huffman@31706
   854
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   855
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   856
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
   857
      by (simp only: power_mult_distrib mult.commute)
haftmann@58787
   858
    then have th0: "a'^n dvd b'^n"
haftmann@58787
   859
      using zn by auto
huffman@31706
   860
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   861
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
haftmann@57512
   862
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute)
nipkow@31952
   863
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
haftmann@57512
   864
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
   865
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   866
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   867
  ultimately show ?thesis by blast
huffman@31706
   868
qed
huffman@31706
   869
nipkow@31952
   870
lemma pow_divides_pow_int:
huffman@31706
   871
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   872
  shows "a dvd b"
chaieb@27669
   873
proof-
huffman@31706
   874
  let ?g = "gcd a b"
huffman@31706
   875
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   876
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   877
  moreover
huffman@31706
   878
  {assume z: "?g \<noteq> 0"
huffman@35216
   879
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
   880
    from gcd_coprime_exists_int[OF z]
huffman@31706
   881
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   882
      by blast
huffman@31706
   883
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   884
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   885
    hence "?g^n*a'^n dvd ?g^n *b'^n"
haftmann@57512
   886
      by (simp only: power_mult_distrib mult.commute)
huffman@31706
   887
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   888
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   889
    with th0 have "a' dvd b'^n"
huffman@31706
   890
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
lp15@60162
   891
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult.commute power_Suc)
nipkow@31952
   892
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
haftmann@57512
   893
    have "a' dvd b'" by (subst (asm) mult.commute, blast)
huffman@31706
   894
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   895
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   896
  ultimately show ?thesis by blast
huffman@31706
   897
qed
huffman@31706
   898
nipkow@31952
   899
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
   900
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
   901
nipkow@31952
   902
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
   903
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
   904
nipkow@31952
   905
lemma divides_mult_nat:
huffman@31706
   906
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   907
  shows "m * n dvd r"
huffman@31706
   908
proof-
huffman@31706
   909
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   910
    unfolding dvd_def by blast
haftmann@57512
   911
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
nipkow@31952
   912
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
huffman@31706
   913
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   914
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   915
qed
huffman@31706
   916
nipkow@31952
   917
lemma divides_mult_int:
huffman@31706
   918
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   919
  shows "m * n dvd r"
huffman@31706
   920
proof-
huffman@31706
   921
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   922
    unfolding dvd_def by blast
haftmann@57512
   923
  from mr n' have "m dvd n'*n" by (simp add: mult.commute)
nipkow@31952
   924
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
huffman@31706
   925
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   926
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   927
qed
chaieb@27669
   928
nipkow@31952
   929
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
lp15@60162
   930
  by (simp add: gcd_nat.commute)
huffman@31706
   931
nipkow@31952
   932
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
nipkow@31952
   933
  using coprime_plus_one_nat by (simp add: One_nat_def)
huffman@31706
   934
nipkow@31952
   935
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
lp15@60162
   936
  by (simp add: gcd_int.commute)
huffman@31706
   937
nipkow@31952
   938
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
   939
  using coprime_plus_one_nat [of "n - 1"]
nipkow@31952
   940
    gcd_commute_nat [of "n - 1" n] by auto
huffman@31706
   941
nipkow@31952
   942
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
   943
  using coprime_plus_one_int [of "n - 1"]
nipkow@31952
   944
    gcd_commute_int [of "n - 1" n] by auto
huffman@31706
   945
nipkow@31952
   946
lemma setprod_coprime_nat [rule_format]:
huffman@31706
   947
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   948
  apply (case_tac "finite A")
huffman@31706
   949
  apply (induct set: finite)
nipkow@31952
   950
  apply (auto simp add: gcd_mult_cancel_nat)
huffman@31706
   951
done
huffman@31706
   952
nipkow@31952
   953
lemma setprod_coprime_int [rule_format]:
huffman@31706
   954
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
   955
  apply (case_tac "finite A")
huffman@31706
   956
  apply (induct set: finite)
nipkow@31952
   957
  apply (auto simp add: gcd_mult_cancel_int)
huffman@31706
   958
done
huffman@31706
   959
lp15@60162
   960
lemma coprime_common_divisor_nat: 
lp15@60162
   961
    "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> x = 1"
lp15@60162
   962
  by (metis gcd_greatest_iff_nat nat_dvd_1_iff_1)
huffman@31706
   963
lp15@60162
   964
lemma coprime_common_divisor_int:
lp15@60162
   965
    "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow> x dvd b \<Longrightarrow> abs x = 1"
lp15@60162
   966
  using gcd_greatest by fastforce
huffman@31706
   967
lp15@60162
   968
lemma coprime_divisors_nat:
lp15@60162
   969
    "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow> coprime d e"
lp15@60162
   970
  by (meson coprime_int dvd_trans gcd_dvd1 gcd_dvd2 gcd_ge_0_int)
huffman@31706
   971
nipkow@31952
   972
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
lp15@60162
   973
by (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
huffman@31706
   974
nipkow@31952
   975
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
lp15@60162
   976
by (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
huffman@31706
   977
huffman@31706
   978
huffman@31706
   979
subsection {* Bezout's theorem *}
huffman@31706
   980
huffman@31706
   981
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
   982
   see the theorems that follow the definition. *)
huffman@31706
   983
fun
huffman@31706
   984
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
   985
where
huffman@31706
   986
  "bezw x y =
huffman@31706
   987
  (if y = 0 then (1, 0) else
huffman@31706
   988
      (snd (bezw y (x mod y)),
huffman@31706
   989
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
   990
huffman@31706
   991
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
   992
huffman@31706
   993
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
   994
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
   995
  by simp
huffman@31706
   996
huffman@31706
   997
declare bezw.simps [simp del]
huffman@31706
   998
huffman@31706
   999
lemma bezw_aux [rule_format]:
huffman@31706
  1000
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1001
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1002
  fix m :: nat
huffman@31706
  1003
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1004
    by auto
huffman@31706
  1005
  next fix m :: nat and n
huffman@31706
  1006
    assume ngt0: "n > 0" and
huffman@31706
  1007
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1008
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1009
        int (gcd n (m mod n))"
huffman@31706
  1010
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1011
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1012
      apply (erule subst)
haftmann@36350
  1013
      apply (simp add: field_simps)
huffman@31706
  1014
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1015
      (* applying simp here undoes the last substitution!
huffman@31706
  1016
         what is procedure cancel_div_mod? *)
hoelzl@58776
  1017
      apply (simp only: NO_MATCH_def field_simps of_nat_add of_nat_mult)
huffman@31706
  1018
      done
huffman@31706
  1019
qed
huffman@31706
  1020
nipkow@31952
  1021
lemma bezout_int:
huffman@31706
  1022
  fixes x y
huffman@31706
  1023
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1024
proof -
huffman@31706
  1025
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1026
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1027
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1028
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1029
    apply (unfold gcd_int_def)
huffman@31706
  1030
    apply simp
huffman@31706
  1031
    apply (subst bezw_aux [symmetric])
huffman@31706
  1032
    apply auto
huffman@31706
  1033
    done
huffman@31706
  1034
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1035
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1036
    by auto
huffman@31706
  1037
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1038
    by (erule (1) bezout_aux)
huffman@31706
  1039
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1040
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1041
    apply auto
huffman@31706
  1042
    apply (rule_tac x = u in exI)
huffman@31706
  1043
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1044
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1045
    apply auto
huffman@31706
  1046
    done
huffman@31706
  1047
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1048
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1049
    apply auto
huffman@31706
  1050
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1051
    apply (rule_tac x = v in exI)
nipkow@31952
  1052
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1053
    apply auto
huffman@31706
  1054
    done
huffman@31706
  1055
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1056
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1057
    apply auto
huffman@31706
  1058
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1059
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1060
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1061
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1062
    apply auto
huffman@31706
  1063
    done
huffman@31706
  1064
  ultimately show ?thesis by blast
huffman@31706
  1065
qed
huffman@31706
  1066
huffman@31706
  1067
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1068
huffman@31706
  1069
lemma ind_euclid:
huffman@31706
  1070
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1071
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1072
  shows "P a b"
berghofe@34915
  1073
proof(induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1074
  case less
chaieb@27669
  1075
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1076
  moreover {assume eq: "a= b"
huffman@31706
  1077
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1078
    by simp}
chaieb@27669
  1079
  moreover
chaieb@27669
  1080
  {assume lt: "a < b"
berghofe@34915
  1081
    hence "a + b - a < a + b \<or> a = 0" by arith
chaieb@27669
  1082
    moreover
chaieb@27669
  1083
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1084
    moreover
berghofe@34915
  1085
    {assume "a + b - a < a + b"
berghofe@34915
  1086
      also have th0: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1087
      finally have "a + (b - a) < a + b" .
berghofe@34915
  1088
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
berghofe@34915
  1089
      then have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1090
    ultimately have "P a b" by blast}
chaieb@27669
  1091
  moreover
chaieb@27669
  1092
  {assume lt: "a > b"
berghofe@34915
  1093
    hence "b + a - b < a + b \<or> b = 0" by arith
chaieb@27669
  1094
    moreover
chaieb@27669
  1095
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1096
    moreover
berghofe@34915
  1097
    {assume "b + a - b < a + b"
berghofe@34915
  1098
      also have th0: "b + a - b = b + (a - b)" using lt by arith
berghofe@34915
  1099
      finally have "b + (a - b) < a + b" .
berghofe@34915
  1100
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
berghofe@34915
  1101
      then have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1102
      hence "P a b" using c by blast }
chaieb@27669
  1103
    ultimately have "P a b" by blast}
chaieb@27669
  1104
ultimately  show "P a b" by blast
chaieb@27669
  1105
qed
chaieb@27669
  1106
nipkow@31952
  1107
lemma bezout_lemma_nat:
huffman@31706
  1108
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1109
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1110
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1111
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1112
  using ex
huffman@31706
  1113
  apply clarsimp
huffman@35216
  1114
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1115
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1116
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1117
  apply (rule_tac x="y" in exI)
huffman@31706
  1118
  apply algebra
huffman@31706
  1119
  apply (rule_tac x="x" in exI)
huffman@31706
  1120
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1121
  apply algebra
chaieb@27669
  1122
done
chaieb@27669
  1123
nipkow@31952
  1124
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1125
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1126
  apply(induct a b rule: ind_euclid)
huffman@31706
  1127
  apply blast
huffman@31706
  1128
  apply clarify
huffman@35216
  1129
  apply (rule_tac x="a" in exI, simp)
huffman@31706
  1130
  apply clarsimp
huffman@31706
  1131
  apply (rule_tac x="d" in exI)
huffman@35216
  1132
  apply (case_tac "a * x = b * y + d", simp_all)
huffman@31706
  1133
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1134
  apply (rule_tac x="y" in exI)
huffman@31706
  1135
  apply algebra
huffman@31706
  1136
  apply (rule_tac x="x" in exI)
huffman@31706
  1137
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1138
  apply algebra
chaieb@27669
  1139
done
chaieb@27669
  1140
nipkow@31952
  1141
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1142
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1143
  using bezout_add_nat[of a b]
huffman@31706
  1144
  apply clarsimp
huffman@31706
  1145
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1146
  apply (rule_tac x="x" in exI)
huffman@31706
  1147
  apply (rule_tac x="y" in exI)
huffman@31706
  1148
  apply auto
chaieb@27669
  1149
done
chaieb@27669
  1150
nipkow@31952
  1151
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1152
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1153
proof-
huffman@31706
  1154
 from nz have ap: "a > 0" by simp
nipkow@31952
  1155
 from bezout_add_nat[of a b]
huffman@31706
  1156
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1157
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1158
 moreover
huffman@31706
  1159
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1160
     from H have ?thesis by blast }
chaieb@27669
  1161
 moreover
chaieb@27669
  1162
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1163
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1164
   moreover
chaieb@27669
  1165
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1166
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1167
       by auto
chaieb@27669
  1168
     moreover
chaieb@27669
  1169
     {assume db: "d=b"
wenzelm@41550
  1170
       with nz H have ?thesis apply simp
wenzelm@32960
  1171
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1172
         apply (rule exI[where x = b])
wenzelm@32960
  1173
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1174
    moreover
huffman@31706
  1175
    {assume db: "d < b"
wenzelm@41550
  1176
        {assume "x=0" hence ?thesis using nz H by simp }
wenzelm@32960
  1177
        moreover
wenzelm@32960
  1178
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1179
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1180
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1181
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1182
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1183
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1184
            by simp
wenzelm@32960
  1185
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
haftmann@57512
  1186
            by (simp only: mult.assoc distrib_left)
wenzelm@32960
  1187
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1188
            by algebra
wenzelm@32960
  1189
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1190
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1191
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1192
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
haftmann@59008
  1193
            by (simp only: diff_mult_distrib2 ac_simps)
wenzelm@32960
  1194
          hence ?thesis using H(1,2)
wenzelm@32960
  1195
            apply -
wenzelm@32960
  1196
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1197
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1198
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1199
        ultimately have ?thesis by blast}
chaieb@27669
  1200
    ultimately have ?thesis by blast}
chaieb@27669
  1201
  ultimately have ?thesis by blast}
chaieb@27669
  1202
 ultimately show ?thesis by blast
chaieb@27669
  1203
qed
chaieb@27669
  1204
nipkow@31952
  1205
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1206
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1207
proof-
chaieb@27669
  1208
  let ?g = "gcd a b"
nipkow@31952
  1209
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1210
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1211
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1212
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1213
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1214
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1215
  thus ?thesis by blast
chaieb@27669
  1216
qed
chaieb@27669
  1217
huffman@31706
  1218
haftmann@34030
  1219
subsection {* LCM properties *}
huffman@31706
  1220
haftmann@34030
  1221
lemma lcm_altdef_int [code]: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1222
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@44821
  1223
    of_nat_mult gcd_int_def)
huffman@31706
  1224
nipkow@31952
  1225
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1226
  unfolding lcm_nat_def
nipkow@31952
  1227
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1228
nipkow@31952
  1229
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1230
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1231
  apply (subst int_mult [symmetric])
nipkow@31952
  1232
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1233
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1234
  apply (simp, simp add: abs_mult)
huffman@31706
  1235
done
huffman@31706
  1236
nipkow@31952
  1237
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1238
  unfolding lcm_nat_def by simp
huffman@31706
  1239
nipkow@31952
  1240
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1241
  unfolding lcm_int_def by simp
huffman@31706
  1242
nipkow@31952
  1243
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1244
  unfolding lcm_nat_def by simp
chaieb@27669
  1245
nipkow@31952
  1246
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1247
  unfolding lcm_int_def by simp
huffman@31706
  1248
nipkow@31952
  1249
lemma lcm_pos_nat:
nipkow@31798
  1250
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1251
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1252
nipkow@31952
  1253
lemma lcm_pos_int:
nipkow@31798
  1254
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1255
  apply (subst lcm_abs_int)
nipkow@31952
  1256
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1257
  apply auto
huffman@31706
  1258
done
haftmann@23687
  1259
nipkow@31952
  1260
lemma dvd_pos_nat:
haftmann@23687
  1261
  fixes n m :: nat
haftmann@23687
  1262
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1263
  shows "m > 0"
haftmann@23687
  1264
using assms by (cases m) auto
haftmann@23687
  1265
nipkow@31952
  1266
lemma lcm_least_nat:
huffman@31706
  1267
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1268
  shows "lcm m n dvd k"
haftmann@23687
  1269
proof (cases k)
haftmann@23687
  1270
  case 0 then show ?thesis by auto
haftmann@23687
  1271
next
haftmann@23687
  1272
  case (Suc _) then have pos_k: "k > 0" by auto
nipkow@31952
  1273
  from assms dvd_pos_nat [OF this] have pos_mn: "m > 0" "n > 0" by auto
nipkow@31952
  1274
  with gcd_zero_nat [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1275
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1276
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1277
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1278
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1279
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
haftmann@57514
  1280
    by (simp add: ac_simps gcd_mult_distrib_nat)
haftmann@27556
  1281
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1282
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1283
  also have "\<dots> = k * p * q * gcd m n"
haftmann@57514
  1284
    by (simp add: ac_simps gcd_mult_distrib_nat)
haftmann@27556
  1285
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1286
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1287
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@57514
  1288
    by (simp add: ac_simps)
haftmann@27556
  1289
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1290
    by simp
nipkow@31952
  1291
  with prod_gcd_lcm_nat [of m n]
haftmann@27556
  1292
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@57514
  1293
    by (simp add: ac_simps)
huffman@31706
  1294
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1295
  then show ?thesis using dvd_def by auto
haftmann@23687
  1296
qed
haftmann@23687
  1297
nipkow@31952
  1298
lemma lcm_least_int:
nipkow@31798
  1299
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
nipkow@31952
  1300
apply (subst lcm_abs_int)
nipkow@31798
  1301
apply (rule dvd_trans)
nipkow@31952
  1302
apply (rule lcm_least_nat [transferred, of _ "abs k" _])
nipkow@31798
  1303
apply auto
huffman@31706
  1304
done
huffman@31706
  1305
nipkow@31952
  1306
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
haftmann@23687
  1307
proof (cases m)
haftmann@23687
  1308
  case 0 then show ?thesis by simp
haftmann@23687
  1309
next
haftmann@23687
  1310
  case (Suc _)
haftmann@23687
  1311
  then have mpos: "m > 0" by simp
haftmann@23687
  1312
  show ?thesis
haftmann@23687
  1313
  proof (cases n)
haftmann@23687
  1314
    case 0 then show ?thesis by simp
haftmann@23687
  1315
  next
haftmann@23687
  1316
    case (Suc _)
haftmann@23687
  1317
    then have npos: "n > 0" by simp
haftmann@27556
  1318
    have "gcd m n dvd n" by simp
haftmann@27556
  1319
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1320
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
haftmann@57514
  1321
      by (simp add: ac_simps)
nipkow@31952
  1322
    also have "\<dots> = m * k" using mpos npos gcd_zero_nat by simp
huffman@31706
  1323
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1324
  qed
haftmann@23687
  1325
qed
haftmann@23687
  1326
nipkow@31952
  1327
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
nipkow@31952
  1328
  apply (subst lcm_abs_int)
huffman@31706
  1329
  apply (rule dvd_trans)
huffman@31706
  1330
  prefer 2
nipkow@31952
  1331
  apply (rule lcm_dvd1_nat [transferred])
huffman@31706
  1332
  apply auto
huffman@31706
  1333
done
huffman@31706
  1334
nipkow@31952
  1335
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
haftmann@35726
  1336
  using lcm_dvd1_nat [of n m] by (simp only: lcm_nat_def mult.commute gcd_nat.commute)
huffman@31706
  1337
nipkow@31952
  1338
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
haftmann@35726
  1339
  using lcm_dvd1_int [of n m] by (simp only: lcm_int_def lcm_nat_def mult.commute gcd_nat.commute)
huffman@31706
  1340
nipkow@31730
  1341
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1342
by(metis lcm_dvd1_nat dvd_trans)
nipkow@31729
  1343
nipkow@31730
  1344
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1345
by(metis lcm_dvd2_nat dvd_trans)
nipkow@31729
  1346
nipkow@31730
  1347
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1348
by(metis lcm_dvd1_int dvd_trans)
nipkow@31729
  1349
nipkow@31730
  1350
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1351
by(metis lcm_dvd2_int dvd_trans)
nipkow@31729
  1352
nipkow@31952
  1353
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1354
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@33657
  1355
  by (auto intro: dvd_antisym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
chaieb@27568
  1356
nipkow@31952
  1357
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1358
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
wenzelm@60357
  1359
  using lcm_least_int zdvd_antisym_nonneg by auto
huffman@31706
  1360
haftmann@37770
  1361
interpretation lcm_nat: abel_semigroup "lcm :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@54867
  1362
  + lcm_nat: semilattice_neutr "lcm :: nat \<Rightarrow> nat \<Rightarrow> nat" 1
haftmann@34973
  1363
proof
haftmann@34973
  1364
  fix n m p :: nat
haftmann@34973
  1365
  show "lcm (lcm n m) p = lcm n (lcm m p)"
haftmann@34973
  1366
    by (rule lcm_unique_nat [THEN iffD1]) (metis dvd.order_trans lcm_unique_nat)
haftmann@34973
  1367
  show "lcm m n = lcm n m"
haftmann@36350
  1368
    by (simp add: lcm_nat_def gcd_commute_nat field_simps)
haftmann@54867
  1369
  show "lcm m m = m"
haftmann@54867
  1370
    by (metis dvd.order_refl lcm_unique_nat)
haftmann@54867
  1371
  show "lcm m 1 = m"
haftmann@54867
  1372
    by (metis dvd.dual_order.refl lcm_unique_nat one_dvd)
haftmann@34973
  1373
qed
haftmann@34973
  1374
haftmann@37770
  1375
interpretation lcm_int: abel_semigroup "lcm :: int \<Rightarrow> int \<Rightarrow> int"
haftmann@34973
  1376
proof
haftmann@34973
  1377
  fix n m p :: int
haftmann@34973
  1378
  show "lcm (lcm n m) p = lcm n (lcm m p)"
haftmann@34973
  1379
    by (rule lcm_unique_int [THEN iffD1]) (metis dvd_trans lcm_unique_int)
haftmann@34973
  1380
  show "lcm m n = lcm n m"
haftmann@34973
  1381
    by (simp add: lcm_int_def lcm_nat.commute)
haftmann@34973
  1382
qed
haftmann@34973
  1383
haftmann@34973
  1384
lemmas lcm_assoc_nat = lcm_nat.assoc
haftmann@34973
  1385
lemmas lcm_commute_nat = lcm_nat.commute
haftmann@34973
  1386
lemmas lcm_left_commute_nat = lcm_nat.left_commute
haftmann@34973
  1387
lemmas lcm_assoc_int = lcm_int.assoc
haftmann@34973
  1388
lemmas lcm_commute_int = lcm_int.commute
haftmann@34973
  1389
lemmas lcm_left_commute_int = lcm_int.left_commute
haftmann@34973
  1390
haftmann@34973
  1391
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
haftmann@34973
  1392
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
haftmann@34973
  1393
nipkow@31798
  1394
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1395
  apply (rule sym)
nipkow@31952
  1396
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1397
  apply auto
huffman@31706
  1398
done
huffman@31706
  1399
nipkow@31798
  1400
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1401
  apply (rule sym)
nipkow@31952
  1402
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1403
  apply auto
huffman@31706
  1404
done
huffman@31706
  1405
nipkow@31798
  1406
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31952
  1407
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1408
nipkow@31798
  1409
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31952
  1410
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
huffman@31706
  1411
nipkow@31992
  1412
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1413
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1414
nipkow@31992
  1415
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1416
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1417
nipkow@31992
  1418
lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) \<longleftrightarrow> n dvd m"
nipkow@31992
  1419
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1420
nipkow@31992
  1421
lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) \<longleftrightarrow> m dvd n"
nipkow@31992
  1422
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1423
haftmann@42871
  1424
lemma comp_fun_idem_gcd_nat: "comp_fun_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1425
proof qed (auto simp add: gcd_ac_nat)
nipkow@31992
  1426
haftmann@42871
  1427
lemma comp_fun_idem_gcd_int: "comp_fun_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1428
proof qed (auto simp add: gcd_ac_int)
nipkow@31992
  1429
haftmann@42871
  1430
lemma comp_fun_idem_lcm_nat: "comp_fun_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1431
proof qed (auto simp add: lcm_ac_nat)
nipkow@31992
  1432
haftmann@42871
  1433
lemma comp_fun_idem_lcm_int: "comp_fun_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1434
proof qed (auto simp add: lcm_ac_int)
nipkow@31992
  1435
haftmann@23687
  1436
nipkow@31995
  1437
(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)
nipkow@31995
  1438
nipkow@31995
  1439
lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1440
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
nipkow@31995
  1441
nipkow@31995
  1442
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
huffman@44766
  1443
by (metis lcm_0_int lcm_0_left_int lcm_pos_int less_le)
nipkow@31995
  1444
nipkow@31995
  1445
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
nipkow@31995
  1446
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
nipkow@31995
  1447
nipkow@31995
  1448
lemma lcm_1_iff_int[simp]: "lcm (m::int) n = 1 \<longleftrightarrow> (m=1 \<or> m = -1) \<and> (n=1 \<or> n = -1)"
berghofe@31996
  1449
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
nipkow@31995
  1450
haftmann@34030
  1451
huffman@45264
  1452
subsection {* The complete divisibility lattice *}
nipkow@32112
  1453
krauss@44845
  1454
interpretation gcd_semilattice_nat: semilattice_inf gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
nipkow@32112
  1455
proof
nipkow@32112
  1456
  case goal3 thus ?case by(metis gcd_unique_nat)
nipkow@32112
  1457
qed auto
nipkow@32112
  1458
krauss@44845
  1459
interpretation lcm_semilattice_nat: semilattice_sup lcm "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
nipkow@32112
  1460
proof
nipkow@32112
  1461
  case goal3 thus ?case by(metis lcm_unique_nat)
nipkow@32112
  1462
qed auto
nipkow@32112
  1463
krauss@44845
  1464
interpretation gcd_lcm_lattice_nat: lattice gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm ..
nipkow@32112
  1465
huffman@45264
  1466
text{* Lifting gcd and lcm to sets (Gcd/Lcm).
huffman@45264
  1467
Gcd is defined via Lcm to facilitate the proof that we have a complete lattice.
nipkow@32112
  1468
*}
huffman@45264
  1469
huffman@45264
  1470
class Gcd = gcd +
huffman@45264
  1471
  fixes Gcd :: "'a set \<Rightarrow> 'a"
huffman@45264
  1472
  fixes Lcm :: "'a set \<Rightarrow> 'a"
huffman@45264
  1473
huffman@45264
  1474
instantiation nat :: Gcd
nipkow@32112
  1475
begin
nipkow@32112
  1476
huffman@45264
  1477
definition
haftmann@51489
  1478
  "Lcm (M::nat set) = (if finite M then semilattice_neutr_set.F lcm 1 M else 0)"
haftmann@51489
  1479
haftmann@54867
  1480
interpretation semilattice_neutr_set lcm "1::nat" ..
haftmann@54867
  1481
haftmann@51489
  1482
lemma Lcm_nat_infinite:
haftmann@51489
  1483
  "\<not> finite M \<Longrightarrow> Lcm M = (0::nat)"
haftmann@51489
  1484
  by (simp add: Lcm_nat_def)
haftmann@51489
  1485
haftmann@51489
  1486
lemma Lcm_nat_empty:
haftmann@51489
  1487
  "Lcm {} = (1::nat)"
haftmann@54867
  1488
  by (simp add: Lcm_nat_def)
haftmann@51489
  1489
haftmann@51489
  1490
lemma Lcm_nat_insert:
haftmann@51489
  1491
  "Lcm (insert n M) = lcm (n::nat) (Lcm M)"
haftmann@54867
  1492
  by (cases "finite M") (simp_all add: Lcm_nat_def Lcm_nat_infinite)
nipkow@32112
  1493
huffman@45264
  1494
definition
huffman@45264
  1495
  "Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"
nipkow@32112
  1496
huffman@45264
  1497
instance ..
haftmann@51489
  1498
nipkow@32112
  1499
end
nipkow@32112
  1500
huffman@45264
  1501
lemma dvd_Lcm_nat [simp]:
haftmann@51489
  1502
  fixes M :: "nat set"
haftmann@51489
  1503
  assumes "m \<in> M"
haftmann@51489
  1504
  shows "m dvd Lcm M"
haftmann@51489
  1505
proof (cases "finite M")
haftmann@51489
  1506
  case False then show ?thesis by (simp add: Lcm_nat_infinite)
haftmann@51489
  1507
next
haftmann@51489
  1508
  case True then show ?thesis using assms by (induct M) (auto simp add: Lcm_nat_insert)
haftmann@51489
  1509
qed
nipkow@32112
  1510
huffman@45264
  1511
lemma Lcm_dvd_nat [simp]:
haftmann@51489
  1512
  fixes M :: "nat set"
haftmann@51489
  1513
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@51489
  1514
  shows "Lcm M dvd n"
huffman@45264
  1515
proof (cases "n = 0")
huffman@45264
  1516
  assume "n \<noteq> 0"
huffman@45264
  1517
  hence "finite {d. d dvd n}" by (rule finite_divisors_nat)
huffman@45264
  1518
  moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
huffman@45264
  1519
  ultimately have "finite M" by (rule rev_finite_subset)
haftmann@51489
  1520
  then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
huffman@45264
  1521
qed simp
nipkow@32112
  1522
huffman@45264
  1523
interpretation gcd_lcm_complete_lattice_nat:
haftmann@51547
  1524
  complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat"
haftmann@51547
  1525
where
haftmann@56218
  1526
  "Inf.INFIMUM Gcd A f = Gcd (f ` A :: nat set)"
haftmann@56218
  1527
  and "Sup.SUPREMUM Lcm A f = Lcm (f ` A)"
haftmann@51547
  1528
proof -
haftmann@51547
  1529
  show "class.complete_lattice Gcd Lcm gcd Rings.dvd (\<lambda>m n. m dvd n \<and> \<not> n dvd m) lcm 1 (0::nat)"
haftmann@51547
  1530
  proof
haftmann@52729
  1531
    case goal1 thus ?case by (simp add: Gcd_nat_def)
haftmann@51547
  1532
  next
haftmann@52729
  1533
    case goal2 thus ?case by (simp add: Gcd_nat_def)
haftmann@51547
  1534
  next
haftmann@52729
  1535
    case goal5 show ?case by (simp add: Gcd_nat_def Lcm_nat_infinite)
haftmann@51547
  1536
  next
haftmann@52729
  1537
    case goal6 show ?case by (simp add: Lcm_nat_empty)
haftmann@51547
  1538
  next
haftmann@52729
  1539
    case goal3 thus ?case by simp
haftmann@51547
  1540
  next
haftmann@52729
  1541
    case goal4 thus ?case by simp
haftmann@51547
  1542
  qed
haftmann@51547
  1543
  then interpret gcd_lcm_complete_lattice_nat:
haftmann@51547
  1544
    complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat" .
haftmann@56218
  1545
  from gcd_lcm_complete_lattice_nat.INF_def show "Inf.INFIMUM Gcd A f = Gcd (f ` A)" .
haftmann@56218
  1546
  from gcd_lcm_complete_lattice_nat.SUP_def show "Sup.SUPREMUM Lcm A f = Lcm (f ` A)" .
huffman@45264
  1547
qed
nipkow@32112
  1548
haftmann@56166
  1549
declare gcd_lcm_complete_lattice_nat.Inf_image_eq [simp del]
haftmann@56166
  1550
declare gcd_lcm_complete_lattice_nat.Sup_image_eq [simp del]
haftmann@56166
  1551
huffman@45264
  1552
lemma Lcm_empty_nat: "Lcm {} = (1::nat)"
haftmann@54867
  1553
  by (fact Lcm_nat_empty)
huffman@45264
  1554
huffman@45264
  1555
lemma Gcd_empty_nat: "Gcd {} = (0::nat)"
huffman@45264
  1556
  by (fact gcd_lcm_complete_lattice_nat.Inf_empty) (* already simp *)
nipkow@32112
  1557
nipkow@32112
  1558
lemma Lcm_insert_nat [simp]:
nipkow@32112
  1559
  shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
huffman@45264
  1560
  by (fact gcd_lcm_complete_lattice_nat.Sup_insert)
nipkow@32112
  1561
nipkow@32112
  1562
lemma Gcd_insert_nat [simp]:
nipkow@32112
  1563
  shows "Gcd (insert (n::nat) N) = gcd n (Gcd N)"
huffman@45264
  1564
  by (fact gcd_lcm_complete_lattice_nat.Inf_insert)
nipkow@32112
  1565
nipkow@32112
  1566
lemma Lcm0_iff[simp]: "finite (M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> Lcm M = 0 \<longleftrightarrow> 0 : M"
nipkow@32112
  1567
by(induct rule:finite_ne_induct) auto
nipkow@32112
  1568
nipkow@32112
  1569
lemma Lcm_eq_0[simp]: "finite (M::nat set) \<Longrightarrow> 0 : M \<Longrightarrow> Lcm M = 0"
nipkow@32112
  1570
by (metis Lcm0_iff empty_iff)
nipkow@32112
  1571
nipkow@32112
  1572
lemma Gcd_dvd_nat [simp]:
huffman@45264
  1573
  fixes M :: "nat set"
huffman@45264
  1574
  assumes "m \<in> M" shows "Gcd M dvd m"
huffman@45264
  1575
  using assms by (fact gcd_lcm_complete_lattice_nat.Inf_lower)
nipkow@32112
  1576
nipkow@32112
  1577
lemma dvd_Gcd_nat[simp]:
huffman@45264
  1578
  fixes M :: "nat set"
huffman@45264
  1579
  assumes "\<forall>m\<in>M. n dvd m" shows "n dvd Gcd M"
huffman@45264
  1580
  using assms by (simp only: gcd_lcm_complete_lattice_nat.Inf_greatest)
nipkow@32112
  1581
huffman@45264
  1582
text{* Alternative characterizations of Gcd: *}
nipkow@32112
  1583
nipkow@32112
  1584
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})"
nipkow@32112
  1585
apply(rule antisym)
nipkow@32112
  1586
 apply(rule Max_ge)
nipkow@32112
  1587
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@32112
  1588
 apply simp
nipkow@32112
  1589
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  1590
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@44890
  1591
 apply fastforce
nipkow@32112
  1592
apply clarsimp
nipkow@32112
  1593
apply (metis Gcd_dvd_nat Max_in dvd_0_left dvd_Gcd_nat dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  1594
done
nipkow@32112
  1595
nipkow@32112
  1596
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  1597
apply(induct pred:finite)
nipkow@32112
  1598
 apply simp
nipkow@32112
  1599
apply(case_tac "x=0")
nipkow@32112
  1600
 apply simp
nipkow@32112
  1601
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  1602
 apply simp
nipkow@32112
  1603
apply blast
nipkow@32112
  1604
done
nipkow@32112
  1605
nipkow@32112
  1606
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  1607
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> Lcm M : M"
nipkow@32112
  1608
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  1609
 apply simp
nipkow@32112
  1610
apply simp
nipkow@32112
  1611
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  1612
 apply simp
nipkow@32112
  1613
 apply(case_tac "A={}")
nipkow@32112
  1614
  apply simp
nipkow@32112
  1615
 apply simp
nipkow@32112
  1616
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  1617
done
nipkow@32112
  1618
nipkow@32112
  1619
lemma Lcm_eq_Max_nat:
nipkow@32112
  1620
  "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"
nipkow@32112
  1621
apply(rule antisym)
nipkow@32112
  1622
 apply(rule Max_ge, assumption)
nipkow@32112
  1623
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
nipkow@32112
  1624
apply clarsimp
nipkow@32112
  1625
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
nipkow@32112
  1626
done
nipkow@32112
  1627
haftmann@54437
  1628
lemma Lcm_set_nat [code, code_unfold]:
haftmann@45992
  1629
  "Lcm (set ns) = fold lcm ns (1::nat)"
huffman@45264
  1630
  by (fact gcd_lcm_complete_lattice_nat.Sup_set_fold)
nipkow@32112
  1631
haftmann@54437
  1632
lemma Gcd_set_nat [code, code_unfold]:
haftmann@45992
  1633
  "Gcd (set ns) = fold gcd ns (0::nat)"
huffman@45264
  1634
  by (fact gcd_lcm_complete_lattice_nat.Inf_set_fold)
nipkow@34222
  1635
nipkow@34222
  1636
lemma mult_inj_if_coprime_nat:
nipkow@34222
  1637
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34222
  1638
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
nipkow@34222
  1639
apply(auto simp add:inj_on_def)
huffman@35216
  1640
apply (metis coprime_dvd_mult_iff_nat dvd.neq_le_trans dvd_triv_left)
nipkow@34223
  1641
apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
haftmann@57512
  1642
             dvd.neq_le_trans dvd_triv_right mult.commute)
nipkow@34222
  1643
done
nipkow@34222
  1644
nipkow@34222
  1645
text{* Nitpick: *}
nipkow@34222
  1646
blanchet@41792
  1647
lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
blanchet@41792
  1648
by (induct x y rule: nat_gcd.induct)
blanchet@41792
  1649
   (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  1650
blanchet@41792
  1651
lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
blanchet@33197
  1652
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  1653
haftmann@54867
  1654
huffman@45264
  1655
subsubsection {* Setwise gcd and lcm for integers *}
huffman@45264
  1656
huffman@45264
  1657
instantiation int :: Gcd
huffman@45264
  1658
begin
huffman@45264
  1659
huffman@45264
  1660
definition
huffman@45264
  1661
  "Lcm M = int (Lcm (nat ` abs ` M))"
huffman@45264
  1662
huffman@45264
  1663
definition
huffman@45264
  1664
  "Gcd M = int (Gcd (nat ` abs ` M))"
huffman@45264
  1665
huffman@45264
  1666
instance ..
wenzelm@21256
  1667
end
huffman@45264
  1668
huffman@45264
  1669
lemma Lcm_empty_int [simp]: "Lcm {} = (1::int)"
huffman@45264
  1670
  by (simp add: Lcm_int_def)
huffman@45264
  1671
huffman@45264
  1672
lemma Gcd_empty_int [simp]: "Gcd {} = (0::int)"
huffman@45264
  1673
  by (simp add: Gcd_int_def)
huffman@45264
  1674
huffman@45264
  1675
lemma Lcm_insert_int [simp]:
huffman@45264
  1676
  shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"
huffman@45264
  1677
  by (simp add: Lcm_int_def lcm_int_def)
huffman@45264
  1678
huffman@45264
  1679
lemma Gcd_insert_int [simp]:
huffman@45264
  1680
  shows "Gcd (insert (n::int) N) = gcd n (Gcd N)"
huffman@45264
  1681
  by (simp add: Gcd_int_def gcd_int_def)
huffman@45264
  1682
huffman@45264
  1683
lemma dvd_int_iff: "x dvd y \<longleftrightarrow> nat (abs x) dvd nat (abs y)"
huffman@45264
  1684
  by (simp add: zdvd_int)
huffman@45264
  1685
huffman@45264
  1686
lemma dvd_Lcm_int [simp]:
huffman@45264
  1687
  fixes M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
huffman@45264
  1688
  using assms by (simp add: Lcm_int_def dvd_int_iff)
huffman@45264
  1689
huffman@45264
  1690
lemma Lcm_dvd_int [simp]:
huffman@45264
  1691
  fixes M :: "int set"
huffman@45264
  1692
  assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
huffman@45264
  1693
  using assms by (simp add: Lcm_int_def dvd_int_iff)
huffman@45264
  1694
huffman@45264
  1695
lemma Gcd_dvd_int [simp]:
huffman@45264
  1696
  fixes M :: "int set"
huffman@45264
  1697
  assumes "m \<in> M" shows "Gcd M dvd m"
huffman@45264
  1698
  using assms by (simp add: Gcd_int_def dvd_int_iff)
huffman@45264
  1699
huffman@45264
  1700
lemma dvd_Gcd_int[simp]:
huffman@45264
  1701
  fixes M :: "int set"
huffman@45264
  1702
  assumes "\<forall>m\<in>M. n dvd m" shows "n dvd Gcd M"
huffman@45264
  1703
  using assms by (simp add: Gcd_int_def dvd_int_iff)
huffman@45264
  1704
haftmann@54437
  1705
lemma Lcm_set_int [code, code_unfold]:
haftmann@51547
  1706
  "Lcm (set xs) = fold lcm xs (1::int)"
haftmann@56166
  1707
  by (induct xs rule: rev_induct) (simp_all add: lcm_commute_int)
huffman@45264
  1708
haftmann@54437
  1709
lemma Gcd_set_int [code, code_unfold]:
haftmann@51547
  1710
  "Gcd (set xs) = fold gcd xs (0::int)"
haftmann@56166
  1711
  by (induct xs rule: rev_induct) (simp_all add: gcd_commute_int)
huffman@45264
  1712
haftmann@59008
  1713
haftmann@59008
  1714
text \<open>Fact aliasses\<close>
lp15@59667
  1715
lp15@59667
  1716
lemmas gcd_dvd1_nat = gcd_dvd1 [where ?'a = nat]
haftmann@59008
  1717
  and gcd_dvd2_nat = gcd_dvd2 [where ?'a = nat]
haftmann@59008
  1718
  and gcd_greatest_nat = gcd_greatest [where ?'a = nat]
haftmann@59008
  1719
lp15@59667
  1720
lemmas gcd_dvd1_int = gcd_dvd1 [where ?'a = int]
haftmann@59008
  1721
  and gcd_dvd2_int = gcd_dvd2 [where ?'a = int]
haftmann@59008
  1722
  and gcd_greatest_int = gcd_greatest [where ?'a = int]
haftmann@59008
  1723
huffman@45264
  1724
end
haftmann@51547
  1725