src/HOL/GCD.thy
author wenzelm
Fri Jan 14 15:44:47 2011 +0100 (2011-01-14)
changeset 41550 efa734d9b221
parent 37770 cddb3106adb8
child 41792 ff3cb0c418b7
permissions -rw-r--r--
eliminated global prems;
tuned proofs;
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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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header {* Greatest common divisor and least common multiple *}
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theory GCD
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imports Fact Parity
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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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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 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 [simp]: "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 [simp]: "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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lemma gcd_dvd1_nat [iff]: "(gcd (m::nat)) n dvd m"
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  and gcd_dvd2_nat [iff]: "(gcd m n) dvd n"
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  apply (induct m n rule: gcd_nat_induct)
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  apply (simp_all add: gcd_non_0_nat)
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  apply (blast dest: dvd_mod_imp_dvd)
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done
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lemma gcd_dvd1_int [iff]: "gcd (x::int) y dvd x"
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by (metis gcd_int_def int_dvd_iff gcd_dvd1_nat)
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lemma gcd_dvd2_int [iff]: "gcd (x::int) y dvd y"
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by (metis gcd_int_def int_dvd_iff gcd_dvd2_nat)
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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_nat 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_nat 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_int 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_int 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_nat: "(k::nat) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> 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)
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lemma gcd_greatest_int:
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  "(k::int) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
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  apply (subst gcd_abs_int)
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  apply (subst abs_dvd_iff [symmetric])
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  apply (rule gcd_greatest_nat [transferred])
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  apply auto
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done
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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_nat 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_int 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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interpretation gcd_nat: abel_semigroup "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat"
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proof
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qed (auto intro: dvd_antisym dvd_trans)
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interpretation gcd_int: abel_semigroup "gcd :: int \<Rightarrow> int \<Rightarrow> int"
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proof
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qed (simp_all add: gcd_int_def gcd_nat.assoc gcd_nat.commute gcd_nat.left_commute)
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lemmas gcd_assoc_nat = gcd_nat.assoc
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lemmas gcd_commute_nat = gcd_nat.commute
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lemmas gcd_left_commute_nat = gcd_nat.left_commute
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lemmas gcd_assoc_int = gcd_int.assoc
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lemmas gcd_commute_int = gcd_int.commute
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lemmas gcd_left_commute_int = gcd_int.left_commute
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lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
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lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
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nipkow@31952
   321
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   322
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   323
  apply auto
nipkow@33657
   324
  apply (rule dvd_antisym)
nipkow@31952
   325
  apply (erule (1) gcd_greatest_nat)
huffman@31706
   326
  apply auto
huffman@31706
   327
done
wenzelm@21256
   328
nipkow@31952
   329
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   330
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
nipkow@33657
   331
apply (case_tac "d = 0")
nipkow@33657
   332
 apply simp
nipkow@33657
   333
apply (rule iffI)
nipkow@33657
   334
 apply (rule zdvd_antisym_nonneg)
nipkow@33657
   335
 apply (auto intro: gcd_greatest_int)
huffman@31706
   336
done
huffman@30082
   337
nipkow@31798
   338
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
nipkow@31952
   339
by (metis dvd.eq_iff gcd_unique_nat)
nipkow@31798
   340
nipkow@31798
   341
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
nipkow@31952
   342
by (metis dvd.eq_iff gcd_unique_nat)
nipkow@31798
   343
nipkow@31798
   344
lemma gcd_proj1_if_dvd_int[simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
nipkow@31952
   345
by (metis abs_dvd_iff abs_eq_0 gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   346
nipkow@31798
   347
lemma gcd_proj2_if_dvd_int[simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
nipkow@31952
   348
by (metis gcd_proj1_if_dvd_int gcd_commute_int)
nipkow@31798
   349
nipkow@31798
   350
wenzelm@21256
   351
text {*
wenzelm@21256
   352
  \medskip Multiplication laws
wenzelm@21256
   353
*}
wenzelm@21256
   354
nipkow@31952
   355
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256
   356
    -- {* \cite[page 27]{davenport92} *}
nipkow@31952
   357
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   358
  apply simp
wenzelm@21256
   359
  apply (case_tac "k = 0")
nipkow@31952
   360
  apply (simp_all add: mod_geq gcd_non_0_nat mod_mult_distrib2)
huffman@31706
   361
done
wenzelm@21256
   362
nipkow@31952
   363
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   364
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   365
  apply (subst (1 2) abs_mult)
nipkow@31952
   366
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   367
  apply auto
huffman@31706
   368
done
wenzelm@21256
   369
nipkow@31952
   370
lemma coprime_dvd_mult_nat: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31952
   371
  apply (insert gcd_mult_distrib_nat [of m k n])
wenzelm@21256
   372
  apply simp
wenzelm@21256
   373
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   374
  apply simp
wenzelm@21256
   375
  done
wenzelm@21256
   376
nipkow@31952
   377
lemma coprime_dvd_mult_int:
nipkow@31813
   378
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31813
   379
apply (subst abs_dvd_iff [symmetric])
nipkow@31813
   380
apply (subst dvd_abs_iff [symmetric])
nipkow@31952
   381
apply (subst (asm) gcd_abs_int)
nipkow@31952
   382
apply (rule coprime_dvd_mult_nat [transferred])
nipkow@31813
   383
    prefer 4 apply assumption
nipkow@31813
   384
   apply auto
nipkow@31813
   385
apply (subst abs_mult [symmetric], auto)
huffman@31706
   386
done
huffman@31706
   387
nipkow@31952
   388
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   389
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   390
  by (auto intro: coprime_dvd_mult_nat)
huffman@31706
   391
nipkow@31952
   392
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   393
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   394
  by (auto intro: coprime_dvd_mult_int)
huffman@31706
   395
nipkow@31952
   396
lemma gcd_mult_cancel_nat: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
nipkow@33657
   397
  apply (rule dvd_antisym)
nipkow@31952
   398
  apply (rule gcd_greatest_nat)
nipkow@31952
   399
  apply (rule_tac n = k in coprime_dvd_mult_nat)
nipkow@31952
   400
  apply (simp add: gcd_assoc_nat)
nipkow@31952
   401
  apply (simp add: gcd_commute_nat)
huffman@31706
   402
  apply (simp_all add: mult_commute)
huffman@31706
   403
done
wenzelm@21256
   404
nipkow@31952
   405
lemma gcd_mult_cancel_int:
nipkow@31813
   406
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
nipkow@31952
   407
apply (subst (1 2) gcd_abs_int)
nipkow@31813
   408
apply (subst abs_mult)
nipkow@31952
   409
apply (rule gcd_mult_cancel_nat [transferred], auto)
huffman@31706
   410
done
wenzelm@21256
   411
haftmann@35368
   412
lemma coprime_crossproduct_nat:
haftmann@35368
   413
  fixes a b c d :: nat
haftmann@35368
   414
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   415
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@35368
   416
proof
haftmann@35368
   417
  assume ?rhs then show ?lhs by simp
haftmann@35368
   418
next
haftmann@35368
   419
  assume ?lhs
haftmann@35368
   420
  from `?lhs` have "a dvd b * d" by (auto intro: dvdI dest: sym)
haftmann@35368
   421
  with `coprime a d` have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
haftmann@35368
   422
  from `?lhs` have "b dvd a * c" by (auto intro: dvdI dest: sym)
haftmann@35368
   423
  with `coprime b c` have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
haftmann@35368
   424
  from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult_commute)
haftmann@35368
   425
  with `coprime b c` have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
haftmann@35368
   426
  from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult_commute)
haftmann@35368
   427
  with `coprime a d` have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
haftmann@35368
   428
  from `a dvd b` `b dvd a` have "a = b" by (rule Nat.dvd.antisym)
haftmann@35368
   429
  moreover from `c dvd d` `d dvd c` have "c = d" by (rule Nat.dvd.antisym)
haftmann@35368
   430
  ultimately show ?rhs ..
haftmann@35368
   431
qed
haftmann@35368
   432
haftmann@35368
   433
lemma coprime_crossproduct_int:
haftmann@35368
   434
  fixes a b c d :: int
haftmann@35368
   435
  assumes "coprime a d" and "coprime b c"
haftmann@35368
   436
  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
   437
  using assms by (intro coprime_crossproduct_nat [transferred]) auto
haftmann@35368
   438
wenzelm@21256
   439
text {* \medskip Addition laws *}
wenzelm@21256
   440
nipkow@31952
   441
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   442
  apply (case_tac "n = 0")
nipkow@31952
   443
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   444
done
huffman@31706
   445
nipkow@31952
   446
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
nipkow@31952
   447
  apply (subst (1 2) gcd_commute_nat)
huffman@31706
   448
  apply (subst add_commute)
huffman@31706
   449
  apply simp
huffman@31706
   450
done
huffman@31706
   451
huffman@31706
   452
(* to do: add the other variations? *)
huffman@31706
   453
nipkow@31952
   454
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
nipkow@31952
   455
  by (subst gcd_add1_nat [symmetric], auto)
huffman@31706
   456
nipkow@31952
   457
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
nipkow@31952
   458
  apply (subst gcd_commute_nat)
nipkow@31952
   459
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
   460
  apply auto
nipkow@31952
   461
  apply (subst gcd_commute_nat)
nipkow@31952
   462
  apply (subst gcd_diff1_nat)
huffman@31706
   463
  apply assumption
nipkow@31952
   464
  apply (rule gcd_commute_nat)
huffman@31706
   465
done
huffman@31706
   466
nipkow@31952
   467
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   468
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   469
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
   470
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
   471
    zmod_zminus1_eq_if)
huffman@31706
   472
  apply (frule_tac a = x in pos_mod_bound)
nipkow@31952
   473
  apply (subst (1 2) gcd_commute_nat)
nipkow@31952
   474
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
   475
    nat_le_eq_zle)
huffman@31706
   476
done
wenzelm@21256
   477
nipkow@31952
   478
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   479
  apply (case_tac "y = 0")
huffman@31706
   480
  apply force
huffman@31706
   481
  apply (case_tac "y > 0")
nipkow@31952
   482
  apply (subst gcd_non_0_int, auto)
nipkow@31952
   483
  apply (insert gcd_non_0_int [of "-y" "-x"])
huffman@35216
   484
  apply auto
huffman@31706
   485
done
huffman@31706
   486
nipkow@31952
   487
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
nipkow@31952
   488
by (metis gcd_red_int mod_add_self1 zadd_commute)
huffman@31706
   489
nipkow@31952
   490
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
nipkow@31952
   491
by (metis gcd_add1_int gcd_commute_int zadd_commute)
wenzelm@21256
   492
nipkow@31952
   493
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31952
   494
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
wenzelm@21256
   495
nipkow@31952
   496
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
nipkow@31952
   497
by (metis gcd_commute_int gcd_red_int mod_mult_self1 zadd_commute)
nipkow@31798
   498
wenzelm@21256
   499
huffman@31706
   500
(* to do: differences, and all variations of addition rules
huffman@31706
   501
    as simplification rules for nat and int *)
huffman@31706
   502
nipkow@31798
   503
(* FIXME remove iff *)
nipkow@31952
   504
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   505
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   506
huffman@31706
   507
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   508
nipkow@31992
   509
lemma finite_divisors_nat[simp]:
nipkow@31992
   510
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
nipkow@31734
   511
proof-
nipkow@31734
   512
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
nipkow@31734
   513
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   514
    by(bestsimp intro!:dvd_imp_le)
nipkow@31734
   515
qed
nipkow@31734
   516
nipkow@31995
   517
lemma finite_divisors_int[simp]:
nipkow@31734
   518
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
   519
proof-
nipkow@31734
   520
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
   521
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
   522
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   523
    by(bestsimp intro!:dvd_imp_le_int)
nipkow@31734
   524
qed
nipkow@31734
   525
nipkow@31995
   526
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
   527
apply(rule antisym)
nipkow@31995
   528
 apply (fastsimp intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
   529
apply simp
nipkow@31995
   530
done
nipkow@31995
   531
nipkow@31995
   532
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
nipkow@31995
   533
apply(rule antisym)
nipkow@31995
   534
 apply(rule Max_le_iff[THEN iffD2])
nipkow@31995
   535
   apply simp
nipkow@31995
   536
  apply fastsimp
nipkow@31995
   537
 apply (metis Collect_def abs_ge_self dvd_imp_le_int mem_def zle_trans)
nipkow@31995
   538
apply simp
nipkow@31995
   539
done
nipkow@31995
   540
nipkow@31734
   541
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   542
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   543
apply(rule Max_eqI[THEN sym])
nipkow@31995
   544
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   545
 apply simp
nipkow@31952
   546
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
nipkow@31734
   547
apply simp
nipkow@31734
   548
done
nipkow@31734
   549
nipkow@31734
   550
lemma gcd_is_Max_divisors_int:
nipkow@31734
   551
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   552
apply(rule Max_eqI[THEN sym])
nipkow@31995
   553
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
   554
 apply simp
nipkow@31952
   555
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
nipkow@31734
   556
apply simp
nipkow@31734
   557
done
nipkow@31734
   558
haftmann@34030
   559
lemma gcd_code_int [code]:
haftmann@34030
   560
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@34030
   561
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
   562
chaieb@22027
   563
huffman@31706
   564
subsection {* Coprimality *}
huffman@31706
   565
nipkow@31952
   566
lemma div_gcd_coprime_nat:
huffman@31706
   567
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   568
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   569
proof -
haftmann@27556
   570
  let ?g = "gcd a b"
chaieb@22027
   571
  let ?a' = "a div ?g"
chaieb@22027
   572
  let ?b' = "b div ?g"
haftmann@27556
   573
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   574
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   575
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   576
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   577
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   578
    unfolding dvd_def by blast
huffman@31706
   579
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   580
    by simp_all
wenzelm@22367
   581
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   582
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   583
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@35216
   584
  have "?g \<noteq> 0" using nz by simp
huffman@31706
   585
  then have gp: "?g > 0" by arith
nipkow@31952
   586
  from gcd_greatest_nat [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   587
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   588
qed
chaieb@22027
   589
nipkow@31952
   590
lemma div_gcd_coprime_int:
huffman@31706
   591
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   592
  shows "coprime (a div gcd a b) (b div gcd a b)"
nipkow@31952
   593
apply (subst (1 2 3) gcd_abs_int)
nipkow@31813
   594
apply (subst (1 2) abs_div)
nipkow@31813
   595
  apply simp
nipkow@31813
   596
 apply simp
nipkow@31813
   597
apply(subst (1 2) abs_gcd_int)
nipkow@31952
   598
apply (rule div_gcd_coprime_nat [transferred])
nipkow@31952
   599
using nz apply (auto simp add: gcd_abs_int [symmetric])
huffman@31706
   600
done
huffman@31706
   601
nipkow@31952
   602
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   603
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
   604
nipkow@31952
   605
lemma coprime_Suc_0_nat:
huffman@31706
   606
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
nipkow@31952
   607
  using coprime_nat by (simp add: One_nat_def)
huffman@31706
   608
nipkow@31952
   609
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   610
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   611
  using gcd_unique_int [of 1 a b]
huffman@31706
   612
  apply clarsimp
huffman@31706
   613
  apply (erule subst)
huffman@31706
   614
  apply (rule iffI)
huffman@31706
   615
  apply force
huffman@31706
   616
  apply (drule_tac x = "abs e" in exI)
huffman@31706
   617
  apply (case_tac "e >= 0")
huffman@31706
   618
  apply force
huffman@31706
   619
  apply force
huffman@31706
   620
done
huffman@31706
   621
nipkow@31952
   622
lemma gcd_coprime_nat:
huffman@31706
   623
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   624
    b: "b = b' * gcd a b"
huffman@31706
   625
  shows    "coprime a' b'"
huffman@31706
   626
huffman@31706
   627
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   628
  apply (erule ssubst)
huffman@31706
   629
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   630
  apply (erule ssubst)
nipkow@31952
   631
  apply (rule div_gcd_coprime_nat)
wenzelm@41550
   632
  using z apply force
huffman@31706
   633
  apply (subst (1) b)
huffman@31706
   634
  using z apply force
huffman@31706
   635
  apply (subst (1) a)
huffman@31706
   636
  using z apply force
wenzelm@41550
   637
  done
huffman@31706
   638
nipkow@31952
   639
lemma gcd_coprime_int:
huffman@31706
   640
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   641
    b: "b = b' * gcd a b"
huffman@31706
   642
  shows    "coprime a' b'"
huffman@31706
   643
huffman@31706
   644
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   645
  apply (erule ssubst)
huffman@31706
   646
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   647
  apply (erule ssubst)
nipkow@31952
   648
  apply (rule div_gcd_coprime_int)
wenzelm@41550
   649
  using z apply force
huffman@31706
   650
  apply (subst (1) b)
huffman@31706
   651
  using z apply force
huffman@31706
   652
  apply (subst (1) a)
huffman@31706
   653
  using z apply force
wenzelm@41550
   654
  done
huffman@31706
   655
nipkow@31952
   656
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   657
    shows "coprime d (a * b)"
nipkow@31952
   658
  apply (subst gcd_commute_nat)
nipkow@31952
   659
  using da apply (subst gcd_mult_cancel_nat)
nipkow@31952
   660
  apply (subst gcd_commute_nat, assumption)
nipkow@31952
   661
  apply (subst gcd_commute_nat, rule db)
huffman@31706
   662
done
huffman@31706
   663
nipkow@31952
   664
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   665
    shows "coprime d (a * b)"
nipkow@31952
   666
  apply (subst gcd_commute_int)
nipkow@31952
   667
  using da apply (subst gcd_mult_cancel_int)
nipkow@31952
   668
  apply (subst gcd_commute_int, assumption)
nipkow@31952
   669
  apply (subst gcd_commute_int, rule db)
huffman@31706
   670
done
huffman@31706
   671
nipkow@31952
   672
lemma coprime_lmult_nat:
huffman@31706
   673
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   674
proof -
huffman@31706
   675
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   676
    by (rule gcd_greatest_nat, auto)
huffman@31706
   677
  with dab show ?thesis
huffman@31706
   678
    by auto
huffman@31706
   679
qed
huffman@31706
   680
nipkow@31952
   681
lemma coprime_lmult_int:
nipkow@31798
   682
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   683
proof -
huffman@31706
   684
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   685
    by (rule gcd_greatest_int, auto)
nipkow@31798
   686
  with assms show ?thesis
huffman@31706
   687
    by auto
huffman@31706
   688
qed
huffman@31706
   689
nipkow@31952
   690
lemma coprime_rmult_nat:
nipkow@31798
   691
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   692
proof -
huffman@31706
   693
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   694
    by (rule gcd_greatest_nat, auto intro: dvd_mult)
nipkow@31798
   695
  with assms show ?thesis
huffman@31706
   696
    by auto
huffman@31706
   697
qed
huffman@31706
   698
nipkow@31952
   699
lemma coprime_rmult_int:
huffman@31706
   700
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   701
proof -
huffman@31706
   702
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   703
    by (rule gcd_greatest_int, auto intro: dvd_mult)
huffman@31706
   704
  with dab show ?thesis
huffman@31706
   705
    by auto
huffman@31706
   706
qed
huffman@31706
   707
nipkow@31952
   708
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   709
    coprime d a \<and>  coprime d b"
nipkow@31952
   710
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
nipkow@31952
   711
    coprime_mult_nat[of d a b]
huffman@31706
   712
  by blast
huffman@31706
   713
nipkow@31952
   714
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   715
    coprime d a \<and>  coprime d b"
nipkow@31952
   716
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
nipkow@31952
   717
    coprime_mult_int[of d a b]
huffman@31706
   718
  by blast
huffman@31706
   719
nipkow@31952
   720
lemma gcd_coprime_exists_nat:
huffman@31706
   721
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   722
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   723
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   724
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   725
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
huffman@31706
   726
done
huffman@31706
   727
nipkow@31952
   728
lemma gcd_coprime_exists_int:
huffman@31706
   729
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   730
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   731
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   732
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   733
  using nz apply (auto simp add: div_gcd_coprime_int dvd_div_mult_self)
huffman@31706
   734
done
huffman@31706
   735
nipkow@31952
   736
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   737
  by (induct n, simp_all add: coprime_mult_nat)
huffman@31706
   738
nipkow@31952
   739
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   740
  by (induct n, simp_all add: coprime_mult_int)
huffman@31706
   741
nipkow@31952
   742
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   743
  apply (rule coprime_exp_nat)
nipkow@31952
   744
  apply (subst gcd_commute_nat)
nipkow@31952
   745
  apply (rule coprime_exp_nat)
nipkow@31952
   746
  apply (subst gcd_commute_nat, assumption)
huffman@31706
   747
done
huffman@31706
   748
nipkow@31952
   749
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   750
  apply (rule coprime_exp_int)
nipkow@31952
   751
  apply (subst gcd_commute_int)
nipkow@31952
   752
  apply (rule coprime_exp_int)
nipkow@31952
   753
  apply (subst gcd_commute_int, assumption)
huffman@31706
   754
done
huffman@31706
   755
nipkow@31952
   756
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   757
proof (cases)
huffman@31706
   758
  assume "a = 0 & b = 0"
huffman@31706
   759
  thus ?thesis by simp
huffman@31706
   760
  next assume "~(a = 0 & b = 0)"
huffman@31706
   761
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
nipkow@31952
   762
    by (auto simp:div_gcd_coprime_nat)
huffman@31706
   763
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   764
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   765
    apply (subst (1 2) mult_commute)
nipkow@31952
   766
    apply (subst gcd_mult_distrib_nat [symmetric])
huffman@31706
   767
    apply simp
huffman@31706
   768
    done
huffman@31706
   769
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
huffman@31706
   770
    apply (subst div_power)
huffman@31706
   771
    apply auto
huffman@31706
   772
    apply (rule dvd_div_mult_self)
huffman@31706
   773
    apply (rule dvd_power_same)
huffman@31706
   774
    apply auto
huffman@31706
   775
    done
huffman@31706
   776
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   777
    apply (subst div_power)
huffman@31706
   778
    apply auto
huffman@31706
   779
    apply (rule dvd_div_mult_self)
huffman@31706
   780
    apply (rule dvd_power_same)
huffman@31706
   781
    apply auto
huffman@31706
   782
    done
huffman@31706
   783
  finally show ?thesis .
huffman@31706
   784
qed
huffman@31706
   785
nipkow@31952
   786
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
   787
  apply (subst (1 2) gcd_abs_int)
huffman@31706
   788
  apply (subst (1 2) power_abs)
nipkow@31952
   789
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
   790
  apply auto
huffman@31706
   791
done
huffman@31706
   792
nipkow@31952
   793
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   794
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   795
proof-
huffman@31706
   796
  let ?g = "gcd a b"
huffman@31706
   797
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   798
  moreover
huffman@31706
   799
  {assume z: "?g \<noteq> 0"
nipkow@31952
   800
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   801
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   802
      by blast
huffman@31706
   803
    have thb: "?g dvd b" by auto
huffman@31706
   804
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   805
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   806
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   807
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   808
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   809
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
huffman@31706
   810
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   811
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   812
    with thb thc have ?thesis by blast }
huffman@31706
   813
  ultimately show ?thesis by blast
huffman@31706
   814
qed
huffman@31706
   815
nipkow@31952
   816
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
   817
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   818
proof-
huffman@31706
   819
  let ?g = "gcd a b"
huffman@31706
   820
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   821
  moreover
huffman@31706
   822
  {assume z: "?g \<noteq> 0"
nipkow@31952
   823
    from gcd_coprime_exists_int[OF z]
huffman@31706
   824
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   825
      by blast
huffman@31706
   826
    have thb: "?g dvd b" by auto
huffman@31706
   827
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   828
    with dc have th0: "a' dvd b*c"
huffman@31706
   829
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   830
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   831
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   832
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   833
    from coprime_dvd_mult_int[OF ab'(3)] th_1
huffman@31706
   834
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   835
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   836
    with thb thc have ?thesis by blast }
huffman@31706
   837
  ultimately show ?thesis by blast
chaieb@27669
   838
qed
chaieb@27669
   839
nipkow@31952
   840
lemma pow_divides_pow_nat:
huffman@31706
   841
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   842
  shows "a dvd b"
huffman@31706
   843
proof-
huffman@31706
   844
  let ?g = "gcd a b"
huffman@31706
   845
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   846
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   847
  moreover
huffman@31706
   848
  {assume z: "?g \<noteq> 0"
huffman@35216
   849
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
   850
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   851
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   852
      by blast
huffman@31706
   853
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   854
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   855
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   856
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   857
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   858
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   859
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   860
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   861
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
huffman@31706
   862
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   863
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   864
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   865
  ultimately show ?thesis by blast
huffman@31706
   866
qed
huffman@31706
   867
nipkow@31952
   868
lemma pow_divides_pow_int:
huffman@31706
   869
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   870
  shows "a dvd b"
chaieb@27669
   871
proof-
huffman@31706
   872
  let ?g = "gcd a b"
huffman@31706
   873
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   874
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   875
  moreover
huffman@31706
   876
  {assume z: "?g \<noteq> 0"
huffman@35216
   877
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
   878
    from gcd_coprime_exists_int[OF z]
huffman@31706
   879
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   880
      by blast
huffman@31706
   881
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   882
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   883
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   884
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   885
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   886
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   887
    with th0 have "a' dvd b'^n"
huffman@31706
   888
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   889
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   890
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
huffman@31706
   891
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   892
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   893
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   894
  ultimately show ?thesis by blast
huffman@31706
   895
qed
huffman@31706
   896
nipkow@31952
   897
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
   898
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
   899
nipkow@31952
   900
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
   901
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
   902
nipkow@31952
   903
lemma divides_mult_nat:
huffman@31706
   904
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   905
  shows "m * n dvd r"
huffman@31706
   906
proof-
huffman@31706
   907
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   908
    unfolding dvd_def by blast
huffman@31706
   909
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   910
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
huffman@31706
   911
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   912
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   913
qed
huffman@31706
   914
nipkow@31952
   915
lemma divides_mult_int:
huffman@31706
   916
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   917
  shows "m * n dvd r"
huffman@31706
   918
proof-
huffman@31706
   919
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   920
    unfolding dvd_def by blast
huffman@31706
   921
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   922
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
huffman@31706
   923
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   924
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   925
qed
chaieb@27669
   926
nipkow@31952
   927
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   928
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   929
  apply force
nipkow@31952
   930
  apply (rule dvd_diff_nat)
huffman@31706
   931
  apply auto
huffman@31706
   932
done
huffman@31706
   933
nipkow@31952
   934
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
nipkow@31952
   935
  using coprime_plus_one_nat by (simp add: One_nat_def)
huffman@31706
   936
nipkow@31952
   937
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   938
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   939
  apply force
huffman@31706
   940
  apply (rule dvd_diff)
huffman@31706
   941
  apply auto
huffman@31706
   942
done
huffman@31706
   943
nipkow@31952
   944
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
   945
  using coprime_plus_one_nat [of "n - 1"]
nipkow@31952
   946
    gcd_commute_nat [of "n - 1" n] by auto
huffman@31706
   947
nipkow@31952
   948
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
   949
  using coprime_plus_one_int [of "n - 1"]
nipkow@31952
   950
    gcd_commute_int [of "n - 1" n] by auto
huffman@31706
   951
nipkow@31952
   952
lemma setprod_coprime_nat [rule_format]:
huffman@31706
   953
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   954
  apply (case_tac "finite A")
huffman@31706
   955
  apply (induct set: finite)
nipkow@31952
   956
  apply (auto simp add: gcd_mult_cancel_nat)
huffman@31706
   957
done
huffman@31706
   958
nipkow@31952
   959
lemma setprod_coprime_int [rule_format]:
huffman@31706
   960
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
   961
  apply (case_tac "finite A")
huffman@31706
   962
  apply (induct set: finite)
nipkow@31952
   963
  apply (auto simp add: gcd_mult_cancel_int)
huffman@31706
   964
done
huffman@31706
   965
nipkow@31952
   966
lemma coprime_common_divisor_nat: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
   967
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
   968
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
   969
  apply simp
nipkow@31952
   970
  apply (erule (1) gcd_greatest_nat)
huffman@31706
   971
done
huffman@31706
   972
nipkow@31952
   973
lemma coprime_common_divisor_int: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
   974
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
   975
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
   976
  apply simp
nipkow@31952
   977
  apply (erule (1) gcd_greatest_int)
huffman@31706
   978
done
huffman@31706
   979
nipkow@31952
   980
lemma coprime_divisors_nat: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
   981
    coprime d e"
huffman@31706
   982
  apply (auto simp add: dvd_def)
nipkow@31952
   983
  apply (frule coprime_lmult_int)
nipkow@31952
   984
  apply (subst gcd_commute_int)
nipkow@31952
   985
  apply (subst (asm) (2) gcd_commute_int)
nipkow@31952
   986
  apply (erule coprime_lmult_int)
huffman@31706
   987
done
huffman@31706
   988
nipkow@31952
   989
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
   990
apply (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
huffman@31706
   991
done
huffman@31706
   992
nipkow@31952
   993
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
   994
apply (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
huffman@31706
   995
done
huffman@31706
   996
huffman@31706
   997
huffman@31706
   998
subsection {* Bezout's theorem *}
huffman@31706
   999
huffman@31706
  1000
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1001
   see the theorems that follow the definition. *)
huffman@31706
  1002
fun
huffman@31706
  1003
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1004
where
huffman@31706
  1005
  "bezw x y =
huffman@31706
  1006
  (if y = 0 then (1, 0) else
huffman@31706
  1007
      (snd (bezw y (x mod y)),
huffman@31706
  1008
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1009
huffman@31706
  1010
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1011
huffman@31706
  1012
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1013
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1014
  by simp
huffman@31706
  1015
huffman@31706
  1016
declare bezw.simps [simp del]
huffman@31706
  1017
huffman@31706
  1018
lemma bezw_aux [rule_format]:
huffman@31706
  1019
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1020
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1021
  fix m :: nat
huffman@31706
  1022
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1023
    by auto
huffman@31706
  1024
  next fix m :: nat and n
huffman@31706
  1025
    assume ngt0: "n > 0" and
huffman@31706
  1026
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1027
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1028
        int (gcd n (m mod n))"
huffman@31706
  1029
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1030
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1031
      apply (erule subst)
haftmann@36350
  1032
      apply (simp add: field_simps)
huffman@31706
  1033
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1034
      (* applying simp here undoes the last substitution!
huffman@31706
  1035
         what is procedure cancel_div_mod? *)
haftmann@36350
  1036
      apply (simp only: field_simps zadd_int [symmetric]
huffman@31706
  1037
        zmult_int [symmetric])
huffman@31706
  1038
      done
huffman@31706
  1039
qed
huffman@31706
  1040
nipkow@31952
  1041
lemma bezout_int:
huffman@31706
  1042
  fixes x y
huffman@31706
  1043
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1044
proof -
huffman@31706
  1045
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1046
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1047
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1048
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1049
    apply (unfold gcd_int_def)
huffman@31706
  1050
    apply simp
huffman@31706
  1051
    apply (subst bezw_aux [symmetric])
huffman@31706
  1052
    apply auto
huffman@31706
  1053
    done
huffman@31706
  1054
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1055
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1056
    by auto
huffman@31706
  1057
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1058
    by (erule (1) bezout_aux)
huffman@31706
  1059
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1060
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1061
    apply auto
huffman@31706
  1062
    apply (rule_tac x = u in exI)
huffman@31706
  1063
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1064
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1065
    apply auto
huffman@31706
  1066
    done
huffman@31706
  1067
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1068
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1069
    apply auto
huffman@31706
  1070
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1071
    apply (rule_tac x = v in exI)
nipkow@31952
  1072
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1073
    apply auto
huffman@31706
  1074
    done
huffman@31706
  1075
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1076
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1077
    apply auto
huffman@31706
  1078
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1079
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1080
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1081
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1082
    apply auto
huffman@31706
  1083
    done
huffman@31706
  1084
  ultimately show ?thesis by blast
huffman@31706
  1085
qed
huffman@31706
  1086
huffman@31706
  1087
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1088
huffman@31706
  1089
lemma ind_euclid:
huffman@31706
  1090
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1091
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1092
  shows "P a b"
berghofe@34915
  1093
proof(induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1094
  case less
chaieb@27669
  1095
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1096
  moreover {assume eq: "a= b"
huffman@31706
  1097
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1098
    by simp}
chaieb@27669
  1099
  moreover
chaieb@27669
  1100
  {assume lt: "a < b"
berghofe@34915
  1101
    hence "a + b - a < a + b \<or> a = 0" by arith
chaieb@27669
  1102
    moreover
chaieb@27669
  1103
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1104
    moreover
berghofe@34915
  1105
    {assume "a + b - a < a + b"
berghofe@34915
  1106
      also have th0: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1107
      finally have "a + (b - a) < a + b" .
berghofe@34915
  1108
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
berghofe@34915
  1109
      then have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1110
    ultimately have "P a b" by blast}
chaieb@27669
  1111
  moreover
chaieb@27669
  1112
  {assume lt: "a > b"
berghofe@34915
  1113
    hence "b + a - b < a + b \<or> b = 0" by arith
chaieb@27669
  1114
    moreover
chaieb@27669
  1115
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1116
    moreover
berghofe@34915
  1117
    {assume "b + a - b < a + b"
berghofe@34915
  1118
      also have th0: "b + a - b = b + (a - b)" using lt by arith
berghofe@34915
  1119
      finally have "b + (a - b) < a + b" .
berghofe@34915
  1120
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
berghofe@34915
  1121
      then have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1122
      hence "P a b" using c by blast }
chaieb@27669
  1123
    ultimately have "P a b" by blast}
chaieb@27669
  1124
ultimately  show "P a b" by blast
chaieb@27669
  1125
qed
chaieb@27669
  1126
nipkow@31952
  1127
lemma bezout_lemma_nat:
huffman@31706
  1128
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1129
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1130
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1131
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1132
  using ex
huffman@31706
  1133
  apply clarsimp
huffman@35216
  1134
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1135
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1136
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1137
  apply (rule_tac x="y" in exI)
huffman@31706
  1138
  apply algebra
huffman@31706
  1139
  apply (rule_tac x="x" in exI)
huffman@31706
  1140
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1141
  apply algebra
chaieb@27669
  1142
done
chaieb@27669
  1143
nipkow@31952
  1144
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1145
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1146
  apply(induct a b rule: ind_euclid)
huffman@31706
  1147
  apply blast
huffman@31706
  1148
  apply clarify
huffman@35216
  1149
  apply (rule_tac x="a" in exI, simp)
huffman@31706
  1150
  apply clarsimp
huffman@31706
  1151
  apply (rule_tac x="d" in exI)
huffman@35216
  1152
  apply (case_tac "a * x = b * y + d", simp_all)
huffman@31706
  1153
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1154
  apply (rule_tac x="y" in exI)
huffman@31706
  1155
  apply algebra
huffman@31706
  1156
  apply (rule_tac x="x" in exI)
huffman@31706
  1157
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1158
  apply algebra
chaieb@27669
  1159
done
chaieb@27669
  1160
nipkow@31952
  1161
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1162
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1163
  using bezout_add_nat[of a b]
huffman@31706
  1164
  apply clarsimp
huffman@31706
  1165
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1166
  apply (rule_tac x="x" in exI)
huffman@31706
  1167
  apply (rule_tac x="y" in exI)
huffman@31706
  1168
  apply auto
chaieb@27669
  1169
done
chaieb@27669
  1170
nipkow@31952
  1171
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1172
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1173
proof-
huffman@31706
  1174
 from nz have ap: "a > 0" by simp
nipkow@31952
  1175
 from bezout_add_nat[of a b]
huffman@31706
  1176
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1177
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1178
 moreover
huffman@31706
  1179
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1180
     from H have ?thesis by blast }
chaieb@27669
  1181
 moreover
chaieb@27669
  1182
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1183
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1184
   moreover
chaieb@27669
  1185
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1186
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1187
       by auto
chaieb@27669
  1188
     moreover
chaieb@27669
  1189
     {assume db: "d=b"
wenzelm@41550
  1190
       with nz H have ?thesis apply simp
wenzelm@32960
  1191
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1192
         apply (rule exI[where x = b])
wenzelm@32960
  1193
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1194
    moreover
huffman@31706
  1195
    {assume db: "d < b"
wenzelm@41550
  1196
        {assume "x=0" hence ?thesis using nz H by simp }
wenzelm@32960
  1197
        moreover
wenzelm@32960
  1198
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1199
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1200
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1201
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1202
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1203
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1204
            by simp
wenzelm@32960
  1205
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
wenzelm@32960
  1206
            by (simp only: mult_assoc right_distrib)
wenzelm@32960
  1207
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1208
            by algebra
wenzelm@32960
  1209
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1210
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1211
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1212
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
wenzelm@32960
  1213
            by (simp only: diff_mult_distrib2 add_commute mult_ac)
wenzelm@32960
  1214
          hence ?thesis using H(1,2)
wenzelm@32960
  1215
            apply -
wenzelm@32960
  1216
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1217
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1218
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1219
        ultimately have ?thesis by blast}
chaieb@27669
  1220
    ultimately have ?thesis by blast}
chaieb@27669
  1221
  ultimately have ?thesis by blast}
chaieb@27669
  1222
 ultimately show ?thesis by blast
chaieb@27669
  1223
qed
chaieb@27669
  1224
nipkow@31952
  1225
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1226
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1227
proof-
chaieb@27669
  1228
  let ?g = "gcd a b"
nipkow@31952
  1229
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1230
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1231
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1232
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1233
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1234
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1235
  thus ?thesis by blast
chaieb@27669
  1236
qed
chaieb@27669
  1237
huffman@31706
  1238
haftmann@34030
  1239
subsection {* LCM properties *}
huffman@31706
  1240
haftmann@34030
  1241
lemma lcm_altdef_int [code]: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1242
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@31706
  1243
    zmult_int [symmetric] gcd_int_def)
huffman@31706
  1244
nipkow@31952
  1245
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1246
  unfolding lcm_nat_def
nipkow@31952
  1247
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1248
nipkow@31952
  1249
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1250
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1251
  apply (subst int_mult [symmetric])
nipkow@31952
  1252
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1253
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1254
  apply (simp, simp add: abs_mult)
huffman@31706
  1255
done
huffman@31706
  1256
nipkow@31952
  1257
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1258
  unfolding lcm_nat_def by simp
huffman@31706
  1259
nipkow@31952
  1260
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1261
  unfolding lcm_int_def by simp
huffman@31706
  1262
nipkow@31952
  1263
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1264
  unfolding lcm_nat_def by simp
chaieb@27669
  1265
nipkow@31952
  1266
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1267
  unfolding lcm_int_def by simp
huffman@31706
  1268
nipkow@31952
  1269
lemma lcm_pos_nat:
nipkow@31798
  1270
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1271
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1272
nipkow@31952
  1273
lemma lcm_pos_int:
nipkow@31798
  1274
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1275
  apply (subst lcm_abs_int)
nipkow@31952
  1276
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1277
  apply auto
huffman@31706
  1278
done
haftmann@23687
  1279
nipkow@31952
  1280
lemma dvd_pos_nat:
haftmann@23687
  1281
  fixes n m :: nat
haftmann@23687
  1282
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1283
  shows "m > 0"
haftmann@23687
  1284
using assms by (cases m) auto
haftmann@23687
  1285
nipkow@31952
  1286
lemma lcm_least_nat:
huffman@31706
  1287
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1288
  shows "lcm m n dvd k"
haftmann@23687
  1289
proof (cases k)
haftmann@23687
  1290
  case 0 then show ?thesis by auto
haftmann@23687
  1291
next
haftmann@23687
  1292
  case (Suc _) then have pos_k: "k > 0" by auto
nipkow@31952
  1293
  from assms dvd_pos_nat [OF this] have pos_mn: "m > 0" "n > 0" by auto
nipkow@31952
  1294
  with gcd_zero_nat [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1295
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1296
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1297
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1298
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1299
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
nipkow@31952
  1300
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1301
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1302
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1303
  also have "\<dots> = k * p * q * gcd m n"
nipkow@31952
  1304
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1305
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1306
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1307
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1308
    by (simp add: mult_ac)
haftmann@27556
  1309
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1310
    by simp
nipkow@31952
  1311
  with prod_gcd_lcm_nat [of m n]
haftmann@27556
  1312
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1313
    by (simp add: mult_ac)
huffman@31706
  1314
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1315
  then show ?thesis using dvd_def by auto
haftmann@23687
  1316
qed
haftmann@23687
  1317
nipkow@31952
  1318
lemma lcm_least_int:
nipkow@31798
  1319
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
nipkow@31952
  1320
apply (subst lcm_abs_int)
nipkow@31798
  1321
apply (rule dvd_trans)
nipkow@31952
  1322
apply (rule lcm_least_nat [transferred, of _ "abs k" _])
nipkow@31798
  1323
apply auto
huffman@31706
  1324
done
huffman@31706
  1325
nipkow@31952
  1326
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
haftmann@23687
  1327
proof (cases m)
haftmann@23687
  1328
  case 0 then show ?thesis by simp
haftmann@23687
  1329
next
haftmann@23687
  1330
  case (Suc _)
haftmann@23687
  1331
  then have mpos: "m > 0" by simp
haftmann@23687
  1332
  show ?thesis
haftmann@23687
  1333
  proof (cases n)
haftmann@23687
  1334
    case 0 then show ?thesis by simp
haftmann@23687
  1335
  next
haftmann@23687
  1336
    case (Suc _)
haftmann@23687
  1337
    then have npos: "n > 0" by simp
haftmann@27556
  1338
    have "gcd m n dvd n" by simp
haftmann@27556
  1339
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1340
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1341
      by (simp add: mult_ac)
nipkow@31952
  1342
    also have "\<dots> = m * k" using mpos npos gcd_zero_nat by simp
huffman@31706
  1343
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1344
  qed
haftmann@23687
  1345
qed
haftmann@23687
  1346
nipkow@31952
  1347
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
nipkow@31952
  1348
  apply (subst lcm_abs_int)
huffman@31706
  1349
  apply (rule dvd_trans)
huffman@31706
  1350
  prefer 2
nipkow@31952
  1351
  apply (rule lcm_dvd1_nat [transferred])
huffman@31706
  1352
  apply auto
huffman@31706
  1353
done
huffman@31706
  1354
nipkow@31952
  1355
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
haftmann@35726
  1356
  using lcm_dvd1_nat [of n m] by (simp only: lcm_nat_def mult.commute gcd_nat.commute)
huffman@31706
  1357
nipkow@31952
  1358
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
haftmann@35726
  1359
  using lcm_dvd1_int [of n m] by (simp only: lcm_int_def lcm_nat_def mult.commute gcd_nat.commute)
huffman@31706
  1360
nipkow@31730
  1361
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1362
by(metis lcm_dvd1_nat dvd_trans)
nipkow@31729
  1363
nipkow@31730
  1364
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1365
by(metis lcm_dvd2_nat dvd_trans)
nipkow@31729
  1366
nipkow@31730
  1367
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1368
by(metis lcm_dvd1_int dvd_trans)
nipkow@31729
  1369
nipkow@31730
  1370
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1371
by(metis lcm_dvd2_int dvd_trans)
nipkow@31729
  1372
nipkow@31952
  1373
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1374
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@33657
  1375
  by (auto intro: dvd_antisym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
chaieb@27568
  1376
nipkow@31952
  1377
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1378
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@33657
  1379
  by (auto intro: dvd_antisym [transferred] lcm_least_int)
huffman@31706
  1380
haftmann@37770
  1381
interpretation lcm_nat: abel_semigroup "lcm :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@34973
  1382
proof
haftmann@34973
  1383
  fix n m p :: nat
haftmann@34973
  1384
  show "lcm (lcm n m) p = lcm n (lcm m p)"
haftmann@34973
  1385
    by (rule lcm_unique_nat [THEN iffD1]) (metis dvd.order_trans lcm_unique_nat)
haftmann@34973
  1386
  show "lcm m n = lcm n m"
haftmann@36350
  1387
    by (simp add: lcm_nat_def gcd_commute_nat field_simps)
haftmann@34973
  1388
qed
haftmann@34973
  1389
haftmann@37770
  1390
interpretation lcm_int: abel_semigroup "lcm :: int \<Rightarrow> int \<Rightarrow> int"
haftmann@34973
  1391
proof
haftmann@34973
  1392
  fix n m p :: int
haftmann@34973
  1393
  show "lcm (lcm n m) p = lcm n (lcm m p)"
haftmann@34973
  1394
    by (rule lcm_unique_int [THEN iffD1]) (metis dvd_trans lcm_unique_int)
haftmann@34973
  1395
  show "lcm m n = lcm n m"
haftmann@34973
  1396
    by (simp add: lcm_int_def lcm_nat.commute)
haftmann@34973
  1397
qed
haftmann@34973
  1398
haftmann@34973
  1399
lemmas lcm_assoc_nat = lcm_nat.assoc
haftmann@34973
  1400
lemmas lcm_commute_nat = lcm_nat.commute
haftmann@34973
  1401
lemmas lcm_left_commute_nat = lcm_nat.left_commute
haftmann@34973
  1402
lemmas lcm_assoc_int = lcm_int.assoc
haftmann@34973
  1403
lemmas lcm_commute_int = lcm_int.commute
haftmann@34973
  1404
lemmas lcm_left_commute_int = lcm_int.left_commute
haftmann@34973
  1405
haftmann@34973
  1406
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
haftmann@34973
  1407
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
haftmann@34973
  1408
nipkow@31798
  1409
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1410
  apply (rule sym)
nipkow@31952
  1411
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1412
  apply auto
huffman@31706
  1413
done
huffman@31706
  1414
nipkow@31798
  1415
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1416
  apply (rule sym)
nipkow@31952
  1417
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1418
  apply auto
huffman@31706
  1419
done
huffman@31706
  1420
nipkow@31798
  1421
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31952
  1422
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1423
nipkow@31798
  1424
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31952
  1425
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
huffman@31706
  1426
nipkow@31992
  1427
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1428
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1429
nipkow@31992
  1430
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1431
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1432
nipkow@31992
  1433
lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) \<longleftrightarrow> n dvd m"
nipkow@31992
  1434
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1435
nipkow@31992
  1436
lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) \<longleftrightarrow> m dvd n"
nipkow@31992
  1437
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1438
nipkow@31992
  1439
lemma fun_left_comm_idem_gcd_nat: "fun_left_comm_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1440
proof qed (auto simp add: gcd_ac_nat)
nipkow@31992
  1441
nipkow@31992
  1442
lemma fun_left_comm_idem_gcd_int: "fun_left_comm_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1443
proof qed (auto simp add: gcd_ac_int)
nipkow@31992
  1444
nipkow@31992
  1445
lemma fun_left_comm_idem_lcm_nat: "fun_left_comm_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1446
proof qed (auto simp add: lcm_ac_nat)
nipkow@31992
  1447
nipkow@31992
  1448
lemma fun_left_comm_idem_lcm_int: "fun_left_comm_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1449
proof qed (auto simp add: lcm_ac_int)
nipkow@31992
  1450
haftmann@23687
  1451
nipkow@31995
  1452
(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)
nipkow@31995
  1453
nipkow@31995
  1454
lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1455
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
nipkow@31995
  1456
nipkow@31995
  1457
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1458
by (metis lcm_0_int lcm_0_left_int lcm_pos_int zless_le)
nipkow@31995
  1459
nipkow@31995
  1460
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
nipkow@31995
  1461
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
nipkow@31995
  1462
nipkow@31995
  1463
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
  1464
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
nipkow@31995
  1465
haftmann@34030
  1466
nipkow@32112
  1467
subsubsection {* The complete divisibility lattice *}
nipkow@32112
  1468
nipkow@32112
  1469
haftmann@35028
  1470
interpretation gcd_semilattice_nat: semilattice_inf "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd
nipkow@32112
  1471
proof
nipkow@32112
  1472
  case goal3 thus ?case by(metis gcd_unique_nat)
nipkow@32112
  1473
qed auto
nipkow@32112
  1474
haftmann@35028
  1475
interpretation lcm_semilattice_nat: semilattice_sup "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm
nipkow@32112
  1476
proof
nipkow@32112
  1477
  case goal3 thus ?case by(metis lcm_unique_nat)
nipkow@32112
  1478
qed auto
nipkow@32112
  1479
nipkow@32112
  1480
interpretation gcd_lcm_lattice_nat: lattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd lcm ..
nipkow@32112
  1481
nipkow@32112
  1482
text{* Lifting gcd and lcm to finite (Gcd/Lcm) and infinite sets (GCD/LCM).
nipkow@32112
  1483
GCD is defined via LCM to facilitate the proof that we have a complete lattice.
nipkow@32112
  1484
Later on we show that GCD and Gcd coincide on finite sets.
nipkow@32112
  1485
*}
nipkow@32112
  1486
context gcd
nipkow@32112
  1487
begin
nipkow@32112
  1488
nipkow@32112
  1489
definition Gcd :: "'a set \<Rightarrow> 'a"
nipkow@32112
  1490
where "Gcd = fold gcd 0"
nipkow@32112
  1491
nipkow@32112
  1492
definition Lcm :: "'a set \<Rightarrow> 'a"
nipkow@32112
  1493
where "Lcm = fold lcm 1"
nipkow@32112
  1494
nipkow@32112
  1495
definition LCM :: "'a set \<Rightarrow> 'a" where
nipkow@32112
  1496
"LCM M = (if finite M then Lcm M else 0)"
nipkow@32112
  1497
nipkow@32112
  1498
definition GCD :: "'a set \<Rightarrow> 'a" where
nipkow@32112
  1499
"GCD M = LCM(INT m:M. {d. d dvd m})"
nipkow@32112
  1500
nipkow@32112
  1501
end
nipkow@32112
  1502
nipkow@32112
  1503
lemma Gcd_empty[simp]: "Gcd {} = 0"
nipkow@32112
  1504
by(simp add:Gcd_def)
nipkow@32112
  1505
nipkow@32112
  1506
lemma Lcm_empty[simp]: "Lcm {} = 1"
nipkow@32112
  1507
by(simp add:Lcm_def)
nipkow@32112
  1508
nipkow@32112
  1509
lemma GCD_empty_nat[simp]: "GCD {} = (0::nat)"
nipkow@32112
  1510
by(simp add:GCD_def LCM_def)
nipkow@32112
  1511
nipkow@32112
  1512
lemma LCM_eq_Lcm[simp]: "finite M \<Longrightarrow> LCM M = Lcm M"
nipkow@32112
  1513
by(simp add:LCM_def)
nipkow@32112
  1514
nipkow@32112
  1515
lemma Lcm_insert_nat [simp]:
nipkow@32112
  1516
  assumes "finite N"
nipkow@32112
  1517
  shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
nipkow@32112
  1518
proof -
nipkow@32112
  1519
  interpret fun_left_comm_idem "lcm::nat\<Rightarrow>nat\<Rightarrow>nat"
nipkow@32112
  1520
    by (rule fun_left_comm_idem_lcm_nat)
nipkow@32112
  1521
  from assms show ?thesis by(simp add: Lcm_def)
nipkow@32112
  1522
qed
nipkow@32112
  1523
nipkow@32112
  1524
lemma Lcm_insert_int [simp]:
nipkow@32112
  1525
  assumes "finite N"
nipkow@32112
  1526
  shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"
nipkow@32112
  1527
proof -
nipkow@32112
  1528
  interpret fun_left_comm_idem "lcm::int\<Rightarrow>int\<Rightarrow>int"
nipkow@32112
  1529
    by (rule fun_left_comm_idem_lcm_int)
nipkow@32112
  1530
  from assms show ?thesis by(simp add: Lcm_def)
nipkow@32112
  1531
qed
nipkow@32112
  1532
nipkow@32112
  1533
lemma Gcd_insert_nat [simp]:
nipkow@32112
  1534
  assumes "finite N"
nipkow@32112
  1535
  shows "Gcd (insert (n::nat) N) = gcd n (Gcd N)"
nipkow@32112
  1536
proof -
nipkow@32112
  1537
  interpret fun_left_comm_idem "gcd::nat\<Rightarrow>nat\<Rightarrow>nat"
nipkow@32112
  1538
    by (rule fun_left_comm_idem_gcd_nat)
nipkow@32112
  1539
  from assms show ?thesis by(simp add: Gcd_def)
nipkow@32112
  1540
qed
nipkow@32112
  1541
nipkow@32112
  1542
lemma Gcd_insert_int [simp]:
nipkow@32112
  1543
  assumes "finite N"
nipkow@32112
  1544
  shows "Gcd (insert (n::int) N) = gcd n (Gcd N)"
nipkow@32112
  1545
proof -
nipkow@32112
  1546
  interpret fun_left_comm_idem "gcd::int\<Rightarrow>int\<Rightarrow>int"
nipkow@32112
  1547
    by (rule fun_left_comm_idem_gcd_int)
nipkow@32112
  1548
  from assms show ?thesis by(simp add: Gcd_def)
nipkow@32112
  1549
qed
nipkow@32112
  1550
nipkow@32112
  1551
lemma Lcm0_iff[simp]: "finite (M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> Lcm M = 0 \<longleftrightarrow> 0 : M"
nipkow@32112
  1552
by(induct rule:finite_ne_induct) auto
nipkow@32112
  1553
nipkow@32112
  1554
lemma Lcm_eq_0[simp]: "finite (M::nat set) \<Longrightarrow> 0 : M \<Longrightarrow> Lcm M = 0"
nipkow@32112
  1555
by (metis Lcm0_iff empty_iff)
nipkow@32112
  1556
nipkow@32112
  1557
lemma Gcd_dvd_nat [simp]:
nipkow@32112
  1558
  assumes "finite M" and "(m::nat) \<in> M"
nipkow@32112
  1559
  shows "Gcd M dvd m"
nipkow@32112
  1560
proof -
nipkow@32112
  1561
  show ?thesis using gcd_semilattice_nat.fold_inf_le_inf[OF assms, of 0] by (simp add: Gcd_def)
nipkow@32112
  1562
qed
nipkow@32112
  1563
nipkow@32112
  1564
lemma dvd_Gcd_nat[simp]:
nipkow@32112
  1565
  assumes "finite M" and "ALL (m::nat) : M. n dvd m"
nipkow@32112
  1566
  shows "n dvd Gcd M"
nipkow@32112
  1567
proof -
nipkow@32112
  1568
  show ?thesis using gcd_semilattice_nat.inf_le_fold_inf[OF assms, of 0] by (simp add: Gcd_def)
nipkow@32112
  1569
qed
nipkow@32112
  1570
nipkow@32112
  1571
lemma dvd_Lcm_nat [simp]:
nipkow@32112
  1572
  assumes "finite M" and "(m::nat) \<in> M"
nipkow@32112
  1573
  shows "m dvd Lcm M"
nipkow@32112
  1574
proof -
nipkow@32112
  1575
  show ?thesis using lcm_semilattice_nat.sup_le_fold_sup[OF assms, of 1] by (simp add: Lcm_def)
nipkow@32112
  1576
qed
nipkow@32112
  1577
nipkow@32112
  1578
lemma Lcm_dvd_nat[simp]:
nipkow@32112
  1579
  assumes "finite M" and "ALL (m::nat) : M. m dvd n"
nipkow@32112
  1580
  shows "Lcm M dvd n"
nipkow@32112
  1581
proof -
nipkow@32112
  1582
  show ?thesis using lcm_semilattice_nat.fold_sup_le_sup[OF assms, of 1] by (simp add: Lcm_def)
nipkow@32112
  1583
qed
nipkow@32112
  1584
nipkow@32112
  1585
interpretation gcd_lcm_complete_lattice_nat:
haftmann@32879
  1586
  complete_lattice GCD LCM "op dvd" "%m n::nat. m dvd n & ~ n dvd m" gcd lcm 1 0
nipkow@32112
  1587
proof
nipkow@32112
  1588
  case goal1 show ?case by simp
nipkow@32112
  1589
next
nipkow@32112
  1590
  case goal2 show ?case by simp
nipkow@32112
  1591
next
nipkow@32112
  1592
  case goal5 thus ?case by (auto simp: LCM_def)
nipkow@32112
  1593
next
nipkow@32112
  1594
  case goal6 thus ?case
nipkow@32112
  1595
    by(auto simp: LCM_def)(metis finite_nat_set_iff_bounded_le gcd_proj2_if_dvd_nat gcd_le1_nat)
nipkow@32112
  1596
next
nipkow@32112
  1597
  case goal3 thus ?case by (auto simp: GCD_def LCM_def)(metis finite_INT finite_divisors_nat)
nipkow@32112
  1598
next
nipkow@32112
  1599
  case goal4 thus ?case by(auto simp: LCM_def GCD_def)
nipkow@32112
  1600
qed
nipkow@32112
  1601
nipkow@32112
  1602
text{* Alternative characterizations of Gcd and GCD: *}
nipkow@32112
  1603
nipkow@32112
  1604
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
  1605
apply(rule antisym)
nipkow@32112
  1606
 apply(rule Max_ge)
nipkow@32112
  1607
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@32112
  1608
 apply simp
nipkow@32112
  1609
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  1610
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@32112
  1611
 apply fastsimp
nipkow@32112
  1612
apply clarsimp
nipkow@32112
  1613
apply (metis Gcd_dvd_nat Max_in dvd_0_left dvd_Gcd_nat dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  1614
done
nipkow@32112
  1615
nipkow@32112
  1616
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  1617
apply(induct pred:finite)
nipkow@32112
  1618
 apply simp
nipkow@32112
  1619
apply(case_tac "x=0")
nipkow@32112
  1620
 apply simp
nipkow@32112
  1621
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  1622
 apply simp
nipkow@32112
  1623
apply blast
nipkow@32112
  1624
done
nipkow@32112
  1625
nipkow@32112
  1626
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  1627
  "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
  1628
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  1629
 apply simp
nipkow@32112
  1630
apply simp
nipkow@32112
  1631
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  1632
 apply simp
nipkow@32112
  1633
 apply(case_tac "A={}")
nipkow@32112
  1634
  apply simp
nipkow@32112
  1635
 apply simp
nipkow@32112
  1636
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  1637
done
nipkow@32112
  1638
nipkow@32112
  1639
lemma Lcm_eq_Max_nat:
nipkow@32112
  1640
  "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
  1641
apply(rule antisym)
nipkow@32112
  1642
 apply(rule Max_ge, assumption)
nipkow@32112
  1643
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
nipkow@32112
  1644
apply clarsimp
nipkow@32112
  1645
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
nipkow@32112
  1646
done
nipkow@32112
  1647
nipkow@32112
  1648
text{* Finally GCD is Gcd: *}
nipkow@32112
  1649
nipkow@32112
  1650
lemma GCD_eq_Gcd[simp]: assumes "finite(M::nat set)" shows "GCD M = Gcd M"
nipkow@32112
  1651
proof-
nipkow@32112
  1652
  have divisors_remove0_nat: "(\<Inter>m\<in>M. {d::nat. d dvd m}) = (\<Inter>m\<in>M-{0}. {d::nat. d dvd m})" by auto
nipkow@32112
  1653
  show ?thesis
nipkow@32112
  1654
  proof cases
nipkow@32112
  1655
    assume "M={}" thus ?thesis by simp
nipkow@32112
  1656
  next
nipkow@32112
  1657
    assume "M\<noteq>{}"
nipkow@32112
  1658
    show ?thesis
nipkow@32112
  1659
    proof cases
nipkow@32112
  1660
      assume "M={0}" thus ?thesis by(simp add:GCD_def LCM_def)
nipkow@32112
  1661
    next
nipkow@32112
  1662
      assume "M\<noteq>{0}"
nipkow@32112
  1663
      with `M\<noteq>{}` assms show ?thesis
wenzelm@32960
  1664
        apply(subst Gcd_remove0_nat[OF assms])
wenzelm@32960
  1665
        apply(simp add:GCD_def)
wenzelm@32960
  1666
        apply(subst divisors_remove0_nat)
wenzelm@32960
  1667
        apply(simp add:LCM_def)
wenzelm@32960
  1668
        apply rule
wenzelm@32960
  1669
         apply rule
wenzelm@32960
  1670
         apply(subst Gcd_eq_Max)
wenzelm@32960
  1671
            apply simp
wenzelm@32960
  1672
           apply blast
wenzelm@32960
  1673
          apply blast
wenzelm@32960
  1674
         apply(rule Lcm_eq_Max_nat)
wenzelm@32960
  1675
            apply simp
wenzelm@32960
  1676
           apply blast
wenzelm@32960
  1677
          apply fastsimp
wenzelm@32960
  1678
         apply clarsimp
wenzelm@32960
  1679
        apply(fastsimp intro: finite_divisors_nat intro!: finite_INT)
wenzelm@32960
  1680
        done
nipkow@32112
  1681
    qed
nipkow@32112
  1682
  qed
nipkow@32112
  1683
qed
nipkow@32112
  1684
nipkow@32112
  1685
lemma Lcm_set_nat [code_unfold]:
nipkow@32112
  1686
  "Lcm (set ns) = foldl lcm (1::nat) ns"
nipkow@32112
  1687
proof -
nipkow@32112
  1688
  interpret fun_left_comm_idem "lcm::nat\<Rightarrow>nat\<Rightarrow>nat" by (rule fun_left_comm_idem_lcm_nat)
nipkow@32112
  1689
  show ?thesis by(simp add: Lcm_def fold_set lcm_commute_nat)
nipkow@32112
  1690
qed
nipkow@32112
  1691
nipkow@32112
  1692
lemma Lcm_set_int [code_unfold]:
nipkow@32112
  1693
  "Lcm (set is) = foldl lcm (1::int) is"
nipkow@32112
  1694
proof -
nipkow@32112
  1695
  interpret fun_left_comm_idem "lcm::int\<Rightarrow>int\<Rightarrow>int" by (rule fun_left_comm_idem_lcm_int)
nipkow@32112
  1696
  show ?thesis by(simp add: Lcm_def fold_set lcm_commute_int)
nipkow@32112
  1697
qed
nipkow@32112
  1698
nipkow@32112
  1699
lemma Gcd_set_nat [code_unfold]:
nipkow@32112
  1700
  "Gcd (set ns) = foldl gcd (0::nat) ns"
nipkow@32112
  1701
proof -
nipkow@32112
  1702
  interpret fun_left_comm_idem "gcd::nat\<Rightarrow>nat\<Rightarrow>nat" by (rule fun_left_comm_idem_gcd_nat)
nipkow@32112
  1703
  show ?thesis by(simp add: Gcd_def fold_set gcd_commute_nat)
nipkow@32112
  1704
qed
nipkow@32112
  1705
nipkow@32112
  1706
lemma Gcd_set_int [code_unfold]:
nipkow@32112
  1707
  "Gcd (set ns) = foldl gcd (0::int) ns"
nipkow@32112
  1708
proof -
nipkow@32112
  1709
  interpret fun_left_comm_idem "gcd::int\<Rightarrow>int\<Rightarrow>int" by (rule fun_left_comm_idem_gcd_int)
nipkow@32112
  1710
  show ?thesis by(simp add: Gcd_def fold_set gcd_commute_int)
nipkow@32112
  1711
qed
nipkow@32112
  1712
nipkow@34222
  1713
nipkow@34222
  1714
lemma mult_inj_if_coprime_nat:
nipkow@34222
  1715
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34222
  1716
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
nipkow@34222
  1717
apply(auto simp add:inj_on_def)
huffman@35216
  1718
apply (metis coprime_dvd_mult_iff_nat dvd.neq_le_trans dvd_triv_left)
nipkow@34223
  1719
apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
nipkow@34223
  1720
             dvd.neq_le_trans dvd_triv_right mult_commute)
nipkow@34222
  1721
done
nipkow@34222
  1722
nipkow@34222
  1723
text{* Nitpick: *}
nipkow@34222
  1724
blanchet@33197
  1725
lemma gcd_eq_nitpick_gcd [nitpick_def]: "gcd x y \<equiv> Nitpick.nat_gcd x y"
blanchet@33197
  1726
apply (rule eq_reflection)
blanchet@33197
  1727
apply (induct x y rule: nat_gcd.induct)
blanchet@33197
  1728
by (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  1729
blanchet@33197
  1730
lemma lcm_eq_nitpick_lcm [nitpick_def]: "lcm x y \<equiv> Nitpick.nat_lcm x y"
blanchet@33197
  1731
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  1732
wenzelm@21256
  1733
end