src/HOL/GCD.thy
author haftmann
Fri Nov 15 22:02:01 2013 +0100 (2013-11-15)
changeset 54437 0060957404c7
parent 54257 5c7a3b6b05a9
child 54489 03ff4d1e6784
permissions -rw-r--r--
proper code equations for Gcd and Lcm on nat and int
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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")
huffman@45270
   360
  apply (simp_all add: gcd_non_0_nat)
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"
huffman@44821
   488
by (metis gcd_red_int mod_add_self1 add_commute)
huffman@31706
   489
nipkow@31952
   490
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
huffman@44821
   491
by (metis gcd_add1_int gcd_commute_int add_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"
huffman@44821
   497
by (metis gcd_commute_int gcd_red_int mod_mult_self1 add_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@44890
   528
 apply (fastforce 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)
haftmann@44278
   534
 apply(rule Max_le_iff [THEN iffD2])
haftmann@44278
   535
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
nipkow@31995
   536
done
nipkow@31995
   537
nipkow@31734
   538
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   539
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   540
apply(rule Max_eqI[THEN sym])
nipkow@31995
   541
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   542
 apply simp
nipkow@31952
   543
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
nipkow@31734
   544
apply simp
nipkow@31734
   545
done
nipkow@31734
   546
nipkow@31734
   547
lemma gcd_is_Max_divisors_int:
nipkow@31734
   548
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   549
apply(rule Max_eqI[THEN sym])
nipkow@31995
   550
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
   551
 apply simp
nipkow@31952
   552
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
nipkow@31734
   553
apply simp
nipkow@31734
   554
done
nipkow@31734
   555
haftmann@34030
   556
lemma gcd_code_int [code]:
haftmann@34030
   557
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
haftmann@34030
   558
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
haftmann@34030
   559
chaieb@22027
   560
huffman@31706
   561
subsection {* Coprimality *}
huffman@31706
   562
nipkow@31952
   563
lemma div_gcd_coprime_nat:
huffman@31706
   564
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   565
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   566
proof -
haftmann@27556
   567
  let ?g = "gcd a b"
chaieb@22027
   568
  let ?a' = "a div ?g"
chaieb@22027
   569
  let ?b' = "b div ?g"
haftmann@27556
   570
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   571
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   572
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   573
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   574
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   575
    unfolding dvd_def by blast
huffman@31706
   576
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   577
    by simp_all
wenzelm@22367
   578
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   579
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   580
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@35216
   581
  have "?g \<noteq> 0" using nz by simp
huffman@31706
   582
  then have gp: "?g > 0" by arith
nipkow@31952
   583
  from gcd_greatest_nat [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   584
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   585
qed
chaieb@22027
   586
nipkow@31952
   587
lemma div_gcd_coprime_int:
huffman@31706
   588
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   589
  shows "coprime (a div gcd a b) (b div gcd a b)"
nipkow@31952
   590
apply (subst (1 2 3) gcd_abs_int)
nipkow@31813
   591
apply (subst (1 2) abs_div)
nipkow@31813
   592
  apply simp
nipkow@31813
   593
 apply simp
nipkow@31813
   594
apply(subst (1 2) abs_gcd_int)
nipkow@31952
   595
apply (rule div_gcd_coprime_nat [transferred])
nipkow@31952
   596
using nz apply (auto simp add: gcd_abs_int [symmetric])
huffman@31706
   597
done
huffman@31706
   598
nipkow@31952
   599
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   600
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
   601
nipkow@31952
   602
lemma coprime_Suc_0_nat:
huffman@31706
   603
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
nipkow@31952
   604
  using coprime_nat by (simp add: One_nat_def)
huffman@31706
   605
nipkow@31952
   606
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   607
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   608
  using gcd_unique_int [of 1 a b]
huffman@31706
   609
  apply clarsimp
huffman@31706
   610
  apply (erule subst)
huffman@31706
   611
  apply (rule iffI)
huffman@31706
   612
  apply force
wenzelm@48562
   613
  apply (drule_tac x = "abs ?e" in exI)
wenzelm@48562
   614
  apply (case_tac "(?e::int) >= 0")
huffman@31706
   615
  apply force
huffman@31706
   616
  apply force
huffman@31706
   617
done
huffman@31706
   618
nipkow@31952
   619
lemma gcd_coprime_nat:
huffman@31706
   620
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   621
    b: "b = b' * gcd a b"
huffman@31706
   622
  shows    "coprime a' b'"
huffman@31706
   623
huffman@31706
   624
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   625
  apply (erule ssubst)
huffman@31706
   626
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   627
  apply (erule ssubst)
nipkow@31952
   628
  apply (rule div_gcd_coprime_nat)
wenzelm@41550
   629
  using z apply force
huffman@31706
   630
  apply (subst (1) b)
huffman@31706
   631
  using z apply force
huffman@31706
   632
  apply (subst (1) a)
huffman@31706
   633
  using z apply force
wenzelm@41550
   634
  done
huffman@31706
   635
nipkow@31952
   636
lemma gcd_coprime_int:
huffman@31706
   637
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   638
    b: "b = b' * gcd a b"
huffman@31706
   639
  shows    "coprime a' b'"
huffman@31706
   640
huffman@31706
   641
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   642
  apply (erule ssubst)
huffman@31706
   643
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   644
  apply (erule ssubst)
nipkow@31952
   645
  apply (rule div_gcd_coprime_int)
wenzelm@41550
   646
  using z apply force
huffman@31706
   647
  apply (subst (1) b)
huffman@31706
   648
  using z apply force
huffman@31706
   649
  apply (subst (1) a)
huffman@31706
   650
  using z apply force
wenzelm@41550
   651
  done
huffman@31706
   652
nipkow@31952
   653
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   654
    shows "coprime d (a * b)"
nipkow@31952
   655
  apply (subst gcd_commute_nat)
nipkow@31952
   656
  using da apply (subst gcd_mult_cancel_nat)
nipkow@31952
   657
  apply (subst gcd_commute_nat, assumption)
nipkow@31952
   658
  apply (subst gcd_commute_nat, rule db)
huffman@31706
   659
done
huffman@31706
   660
nipkow@31952
   661
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   662
    shows "coprime d (a * b)"
nipkow@31952
   663
  apply (subst gcd_commute_int)
nipkow@31952
   664
  using da apply (subst gcd_mult_cancel_int)
nipkow@31952
   665
  apply (subst gcd_commute_int, assumption)
nipkow@31952
   666
  apply (subst gcd_commute_int, rule db)
huffman@31706
   667
done
huffman@31706
   668
nipkow@31952
   669
lemma coprime_lmult_nat:
huffman@31706
   670
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   671
proof -
huffman@31706
   672
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   673
    by (rule gcd_greatest_nat, auto)
huffman@31706
   674
  with dab show ?thesis
huffman@31706
   675
    by auto
huffman@31706
   676
qed
huffman@31706
   677
nipkow@31952
   678
lemma coprime_lmult_int:
nipkow@31798
   679
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   680
proof -
huffman@31706
   681
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   682
    by (rule gcd_greatest_int, auto)
nipkow@31798
   683
  with assms show ?thesis
huffman@31706
   684
    by auto
huffman@31706
   685
qed
huffman@31706
   686
nipkow@31952
   687
lemma coprime_rmult_nat:
nipkow@31798
   688
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   689
proof -
huffman@31706
   690
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   691
    by (rule gcd_greatest_nat, auto intro: dvd_mult)
nipkow@31798
   692
  with assms show ?thesis
huffman@31706
   693
    by auto
huffman@31706
   694
qed
huffman@31706
   695
nipkow@31952
   696
lemma coprime_rmult_int:
huffman@31706
   697
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   698
proof -
huffman@31706
   699
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   700
    by (rule gcd_greatest_int, auto intro: dvd_mult)
huffman@31706
   701
  with dab show ?thesis
huffman@31706
   702
    by auto
huffman@31706
   703
qed
huffman@31706
   704
nipkow@31952
   705
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   706
    coprime d a \<and>  coprime d b"
nipkow@31952
   707
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
nipkow@31952
   708
    coprime_mult_nat[of d a b]
huffman@31706
   709
  by blast
huffman@31706
   710
nipkow@31952
   711
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   712
    coprime d a \<and>  coprime d b"
nipkow@31952
   713
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
nipkow@31952
   714
    coprime_mult_int[of d a b]
huffman@31706
   715
  by blast
huffman@31706
   716
noschinl@52397
   717
lemma coprime_power_int:
noschinl@52397
   718
  assumes "0 < n" shows "coprime (a :: int) (b ^ n) \<longleftrightarrow> coprime a b"
noschinl@52397
   719
  using assms
noschinl@52397
   720
proof (induct n)
noschinl@52397
   721
  case (Suc n) then show ?case
noschinl@52397
   722
    by (cases n) (simp_all add: coprime_mul_eq_int)
noschinl@52397
   723
qed simp
noschinl@52397
   724
nipkow@31952
   725
lemma gcd_coprime_exists_nat:
huffman@31706
   726
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   727
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   728
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   729
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   730
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
huffman@31706
   731
done
huffman@31706
   732
nipkow@31952
   733
lemma gcd_coprime_exists_int:
huffman@31706
   734
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   735
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   736
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   737
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   738
  using nz apply (auto simp add: div_gcd_coprime_int dvd_div_mult_self)
huffman@31706
   739
done
huffman@31706
   740
nipkow@31952
   741
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   742
  by (induct n, simp_all add: coprime_mult_nat)
huffman@31706
   743
nipkow@31952
   744
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   745
  by (induct n, simp_all add: coprime_mult_int)
huffman@31706
   746
nipkow@31952
   747
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   748
  apply (rule coprime_exp_nat)
nipkow@31952
   749
  apply (subst gcd_commute_nat)
nipkow@31952
   750
  apply (rule coprime_exp_nat)
nipkow@31952
   751
  apply (subst gcd_commute_nat, assumption)
huffman@31706
   752
done
huffman@31706
   753
nipkow@31952
   754
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   755
  apply (rule coprime_exp_int)
nipkow@31952
   756
  apply (subst gcd_commute_int)
nipkow@31952
   757
  apply (rule coprime_exp_int)
nipkow@31952
   758
  apply (subst gcd_commute_int, assumption)
huffman@31706
   759
done
huffman@31706
   760
nipkow@31952
   761
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   762
proof (cases)
huffman@31706
   763
  assume "a = 0 & b = 0"
huffman@31706
   764
  thus ?thesis by simp
huffman@31706
   765
  next assume "~(a = 0 & b = 0)"
huffman@31706
   766
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
nipkow@31952
   767
    by (auto simp:div_gcd_coprime_nat)
huffman@31706
   768
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   769
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   770
    apply (subst (1 2) mult_commute)
nipkow@31952
   771
    apply (subst gcd_mult_distrib_nat [symmetric])
huffman@31706
   772
    apply simp
huffman@31706
   773
    done
huffman@31706
   774
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
huffman@31706
   775
    apply (subst div_power)
huffman@31706
   776
    apply auto
huffman@31706
   777
    apply (rule dvd_div_mult_self)
huffman@31706
   778
    apply (rule dvd_power_same)
huffman@31706
   779
    apply auto
huffman@31706
   780
    done
huffman@31706
   781
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   782
    apply (subst div_power)
huffman@31706
   783
    apply auto
huffman@31706
   784
    apply (rule dvd_div_mult_self)
huffman@31706
   785
    apply (rule dvd_power_same)
huffman@31706
   786
    apply auto
huffman@31706
   787
    done
huffman@31706
   788
  finally show ?thesis .
huffman@31706
   789
qed
huffman@31706
   790
nipkow@31952
   791
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
   792
  apply (subst (1 2) gcd_abs_int)
huffman@31706
   793
  apply (subst (1 2) power_abs)
nipkow@31952
   794
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
   795
  apply auto
huffman@31706
   796
done
huffman@31706
   797
nipkow@31952
   798
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   799
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   800
proof-
huffman@31706
   801
  let ?g = "gcd a b"
huffman@31706
   802
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   803
  moreover
huffman@31706
   804
  {assume z: "?g \<noteq> 0"
nipkow@31952
   805
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   806
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   807
      by blast
huffman@31706
   808
    have thb: "?g dvd b" by auto
huffman@31706
   809
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   810
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   811
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   812
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   813
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   814
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
huffman@31706
   815
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   816
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   817
    with thb thc have ?thesis by blast }
huffman@31706
   818
  ultimately show ?thesis by blast
huffman@31706
   819
qed
huffman@31706
   820
nipkow@31952
   821
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
   822
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   823
proof-
huffman@31706
   824
  let ?g = "gcd a b"
huffman@31706
   825
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   826
  moreover
huffman@31706
   827
  {assume z: "?g \<noteq> 0"
nipkow@31952
   828
    from gcd_coprime_exists_int[OF z]
huffman@31706
   829
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   830
      by blast
huffman@31706
   831
    have thb: "?g dvd b" by auto
huffman@31706
   832
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   833
    with dc have th0: "a' dvd b*c"
huffman@31706
   834
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   835
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   836
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   837
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   838
    from coprime_dvd_mult_int[OF ab'(3)] th_1
huffman@31706
   839
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   840
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   841
    with thb thc have ?thesis by blast }
huffman@31706
   842
  ultimately show ?thesis by blast
chaieb@27669
   843
qed
chaieb@27669
   844
nipkow@31952
   845
lemma pow_divides_pow_nat:
huffman@31706
   846
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   847
  shows "a dvd b"
huffman@31706
   848
proof-
huffman@31706
   849
  let ?g = "gcd a b"
huffman@31706
   850
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   851
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   852
  moreover
huffman@31706
   853
  {assume z: "?g \<noteq> 0"
huffman@35216
   854
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
   855
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   856
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   857
      by blast
huffman@31706
   858
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   859
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   860
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   861
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   862
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   863
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   864
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   865
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   866
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
huffman@31706
   867
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   868
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   869
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   870
  ultimately show ?thesis by blast
huffman@31706
   871
qed
huffman@31706
   872
nipkow@31952
   873
lemma pow_divides_pow_int:
huffman@31706
   874
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   875
  shows "a dvd b"
chaieb@27669
   876
proof-
huffman@31706
   877
  let ?g = "gcd a b"
huffman@31706
   878
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   879
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   880
  moreover
huffman@31706
   881
  {assume z: "?g \<noteq> 0"
huffman@35216
   882
    hence zn: "?g ^ n \<noteq> 0" using n by simp
nipkow@31952
   883
    from gcd_coprime_exists_int[OF z]
huffman@31706
   884
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   885
      by blast
huffman@31706
   886
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   887
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   888
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   889
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   890
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   891
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   892
    with th0 have "a' dvd b'^n"
huffman@31706
   893
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   894
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   895
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
huffman@31706
   896
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   897
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   898
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   899
  ultimately show ?thesis by blast
huffman@31706
   900
qed
huffman@31706
   901
nipkow@31952
   902
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
   903
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
   904
nipkow@31952
   905
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
   906
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
   907
nipkow@31952
   908
lemma divides_mult_nat:
huffman@31706
   909
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   910
  shows "m * n dvd r"
huffman@31706
   911
proof-
huffman@31706
   912
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   913
    unfolding dvd_def by blast
huffman@31706
   914
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   915
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
huffman@31706
   916
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   917
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   918
qed
huffman@31706
   919
nipkow@31952
   920
lemma divides_mult_int:
huffman@31706
   921
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   922
  shows "m * n dvd r"
huffman@31706
   923
proof-
huffman@31706
   924
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   925
    unfolding dvd_def by blast
huffman@31706
   926
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   927
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
huffman@31706
   928
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   929
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   930
qed
chaieb@27669
   931
nipkow@31952
   932
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   933
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   934
  apply force
nipkow@31952
   935
  apply (rule dvd_diff_nat)
huffman@31706
   936
  apply auto
huffman@31706
   937
done
huffman@31706
   938
nipkow@31952
   939
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
nipkow@31952
   940
  using coprime_plus_one_nat by (simp add: One_nat_def)
huffman@31706
   941
nipkow@31952
   942
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   943
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   944
  apply force
huffman@31706
   945
  apply (rule dvd_diff)
huffman@31706
   946
  apply auto
huffman@31706
   947
done
huffman@31706
   948
nipkow@31952
   949
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
   950
  using coprime_plus_one_nat [of "n - 1"]
nipkow@31952
   951
    gcd_commute_nat [of "n - 1" n] by auto
huffman@31706
   952
nipkow@31952
   953
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
   954
  using coprime_plus_one_int [of "n - 1"]
nipkow@31952
   955
    gcd_commute_int [of "n - 1" n] by auto
huffman@31706
   956
nipkow@31952
   957
lemma setprod_coprime_nat [rule_format]:
huffman@31706
   958
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   959
  apply (case_tac "finite A")
huffman@31706
   960
  apply (induct set: finite)
nipkow@31952
   961
  apply (auto simp add: gcd_mult_cancel_nat)
huffman@31706
   962
done
huffman@31706
   963
nipkow@31952
   964
lemma setprod_coprime_int [rule_format]:
huffman@31706
   965
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
   966
  apply (case_tac "finite A")
huffman@31706
   967
  apply (induct set: finite)
nipkow@31952
   968
  apply (auto simp add: gcd_mult_cancel_int)
huffman@31706
   969
done
huffman@31706
   970
nipkow@31952
   971
lemma coprime_common_divisor_nat: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
   972
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
   973
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
   974
  apply simp
nipkow@31952
   975
  apply (erule (1) gcd_greatest_nat)
huffman@31706
   976
done
huffman@31706
   977
nipkow@31952
   978
lemma coprime_common_divisor_int: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
   979
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
   980
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
   981
  apply simp
nipkow@31952
   982
  apply (erule (1) gcd_greatest_int)
huffman@31706
   983
done
huffman@31706
   984
nipkow@31952
   985
lemma coprime_divisors_nat: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
   986
    coprime d e"
huffman@31706
   987
  apply (auto simp add: dvd_def)
nipkow@31952
   988
  apply (frule coprime_lmult_int)
nipkow@31952
   989
  apply (subst gcd_commute_int)
nipkow@31952
   990
  apply (subst (asm) (2) gcd_commute_int)
nipkow@31952
   991
  apply (erule coprime_lmult_int)
huffman@31706
   992
done
huffman@31706
   993
nipkow@31952
   994
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
   995
apply (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
huffman@31706
   996
done
huffman@31706
   997
nipkow@31952
   998
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
   999
apply (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
huffman@31706
  1000
done
huffman@31706
  1001
huffman@31706
  1002
huffman@31706
  1003
subsection {* Bezout's theorem *}
huffman@31706
  1004
huffman@31706
  1005
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1006
   see the theorems that follow the definition. *)
huffman@31706
  1007
fun
huffman@31706
  1008
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1009
where
huffman@31706
  1010
  "bezw x y =
huffman@31706
  1011
  (if y = 0 then (1, 0) else
huffman@31706
  1012
      (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
huffman@31706
  1015
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1016
huffman@31706
  1017
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1018
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1019
  by simp
huffman@31706
  1020
huffman@31706
  1021
declare bezw.simps [simp del]
huffman@31706
  1022
huffman@31706
  1023
lemma bezw_aux [rule_format]:
huffman@31706
  1024
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1025
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1026
  fix m :: nat
huffman@31706
  1027
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1028
    by auto
huffman@31706
  1029
  next fix m :: nat and n
huffman@31706
  1030
    assume ngt0: "n > 0" and
huffman@31706
  1031
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1032
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1033
        int (gcd n (m mod n))"
huffman@31706
  1034
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1035
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1036
      apply (erule subst)
haftmann@36350
  1037
      apply (simp add: field_simps)
huffman@31706
  1038
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1039
      (* applying simp here undoes the last substitution!
huffman@31706
  1040
         what is procedure cancel_div_mod? *)
huffman@44821
  1041
      apply (simp only: field_simps of_nat_add of_nat_mult)
huffman@31706
  1042
      done
huffman@31706
  1043
qed
huffman@31706
  1044
nipkow@31952
  1045
lemma bezout_int:
huffman@31706
  1046
  fixes x y
huffman@31706
  1047
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1048
proof -
huffman@31706
  1049
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1050
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1051
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1052
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1053
    apply (unfold gcd_int_def)
huffman@31706
  1054
    apply simp
huffman@31706
  1055
    apply (subst bezw_aux [symmetric])
huffman@31706
  1056
    apply auto
huffman@31706
  1057
    done
huffman@31706
  1058
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1059
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1060
    by auto
huffman@31706
  1061
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1062
    by (erule (1) bezout_aux)
huffman@31706
  1063
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1064
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1065
    apply auto
huffman@31706
  1066
    apply (rule_tac x = u in exI)
huffman@31706
  1067
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1068
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1069
    apply auto
huffman@31706
  1070
    done
huffman@31706
  1071
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1072
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1073
    apply auto
huffman@31706
  1074
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1075
    apply (rule_tac x = v in exI)
nipkow@31952
  1076
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1077
    apply auto
huffman@31706
  1078
    done
huffman@31706
  1079
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1080
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1081
    apply auto
huffman@31706
  1082
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1083
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1084
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1085
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1086
    apply auto
huffman@31706
  1087
    done
huffman@31706
  1088
  ultimately show ?thesis by blast
huffman@31706
  1089
qed
huffman@31706
  1090
huffman@31706
  1091
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1092
huffman@31706
  1093
lemma ind_euclid:
huffman@31706
  1094
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1095
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1096
  shows "P a b"
berghofe@34915
  1097
proof(induct "a + b" arbitrary: a b rule: less_induct)
berghofe@34915
  1098
  case less
chaieb@27669
  1099
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1100
  moreover {assume eq: "a= b"
huffman@31706
  1101
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1102
    by simp}
chaieb@27669
  1103
  moreover
chaieb@27669
  1104
  {assume lt: "a < b"
berghofe@34915
  1105
    hence "a + b - a < a + b \<or> a = 0" by arith
chaieb@27669
  1106
    moreover
chaieb@27669
  1107
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1108
    moreover
berghofe@34915
  1109
    {assume "a + b - a < a + b"
berghofe@34915
  1110
      also have th0: "a + b - a = a + (b - a)" using lt by arith
berghofe@34915
  1111
      finally have "a + (b - a) < a + b" .
berghofe@34915
  1112
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
berghofe@34915
  1113
      then have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1114
    ultimately have "P a b" by blast}
chaieb@27669
  1115
  moreover
chaieb@27669
  1116
  {assume lt: "a > b"
berghofe@34915
  1117
    hence "b + a - b < a + b \<or> b = 0" by arith
chaieb@27669
  1118
    moreover
chaieb@27669
  1119
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1120
    moreover
berghofe@34915
  1121
    {assume "b + a - b < a + b"
berghofe@34915
  1122
      also have th0: "b + a - b = b + (a - b)" using lt by arith
berghofe@34915
  1123
      finally have "b + (a - b) < a + b" .
berghofe@34915
  1124
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
berghofe@34915
  1125
      then have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1126
      hence "P a b" using c by blast }
chaieb@27669
  1127
    ultimately have "P a b" by blast}
chaieb@27669
  1128
ultimately  show "P a b" by blast
chaieb@27669
  1129
qed
chaieb@27669
  1130
nipkow@31952
  1131
lemma bezout_lemma_nat:
huffman@31706
  1132
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1133
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1134
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1135
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1136
  using ex
huffman@31706
  1137
  apply clarsimp
huffman@35216
  1138
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1139
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1140
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1141
  apply (rule_tac x="y" in exI)
huffman@31706
  1142
  apply algebra
huffman@31706
  1143
  apply (rule_tac x="x" in exI)
huffman@31706
  1144
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1145
  apply algebra
chaieb@27669
  1146
done
chaieb@27669
  1147
nipkow@31952
  1148
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1149
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1150
  apply(induct a b rule: ind_euclid)
huffman@31706
  1151
  apply blast
huffman@31706
  1152
  apply clarify
huffman@35216
  1153
  apply (rule_tac x="a" in exI, simp)
huffman@31706
  1154
  apply clarsimp
huffman@31706
  1155
  apply (rule_tac x="d" in exI)
huffman@35216
  1156
  apply (case_tac "a * x = b * y + d", simp_all)
huffman@31706
  1157
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1158
  apply (rule_tac x="y" in exI)
huffman@31706
  1159
  apply algebra
huffman@31706
  1160
  apply (rule_tac x="x" in exI)
huffman@31706
  1161
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1162
  apply algebra
chaieb@27669
  1163
done
chaieb@27669
  1164
nipkow@31952
  1165
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1166
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1167
  using bezout_add_nat[of a b]
huffman@31706
  1168
  apply clarsimp
huffman@31706
  1169
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1170
  apply (rule_tac x="x" in exI)
huffman@31706
  1171
  apply (rule_tac x="y" in exI)
huffman@31706
  1172
  apply auto
chaieb@27669
  1173
done
chaieb@27669
  1174
nipkow@31952
  1175
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1176
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1177
proof-
huffman@31706
  1178
 from nz have ap: "a > 0" by simp
nipkow@31952
  1179
 from bezout_add_nat[of a b]
huffman@31706
  1180
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1181
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1182
 moreover
huffman@31706
  1183
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1184
     from H have ?thesis by blast }
chaieb@27669
  1185
 moreover
chaieb@27669
  1186
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1187
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1188
   moreover
chaieb@27669
  1189
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1190
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1191
       by auto
chaieb@27669
  1192
     moreover
chaieb@27669
  1193
     {assume db: "d=b"
wenzelm@41550
  1194
       with nz H have ?thesis apply simp
wenzelm@32960
  1195
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1196
         apply (rule exI[where x = b])
wenzelm@32960
  1197
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1198
    moreover
huffman@31706
  1199
    {assume db: "d < b"
wenzelm@41550
  1200
        {assume "x=0" hence ?thesis using nz H by simp }
wenzelm@32960
  1201
        moreover
wenzelm@32960
  1202
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1203
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1204
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1205
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1206
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1207
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1208
            by simp
wenzelm@32960
  1209
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
webertj@49962
  1210
            by (simp only: mult_assoc distrib_left)
wenzelm@32960
  1211
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1212
            by algebra
wenzelm@32960
  1213
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1214
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1215
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1216
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
wenzelm@32960
  1217
            by (simp only: diff_mult_distrib2 add_commute mult_ac)
wenzelm@32960
  1218
          hence ?thesis using H(1,2)
wenzelm@32960
  1219
            apply -
wenzelm@32960
  1220
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1221
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1222
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1223
        ultimately have ?thesis by blast}
chaieb@27669
  1224
    ultimately have ?thesis by blast}
chaieb@27669
  1225
  ultimately have ?thesis by blast}
chaieb@27669
  1226
 ultimately show ?thesis by blast
chaieb@27669
  1227
qed
chaieb@27669
  1228
nipkow@31952
  1229
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1230
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1231
proof-
chaieb@27669
  1232
  let ?g = "gcd a b"
nipkow@31952
  1233
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1234
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1235
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1236
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1237
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1238
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1239
  thus ?thesis by blast
chaieb@27669
  1240
qed
chaieb@27669
  1241
huffman@31706
  1242
haftmann@34030
  1243
subsection {* LCM properties *}
huffman@31706
  1244
haftmann@34030
  1245
lemma lcm_altdef_int [code]: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1246
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@44821
  1247
    of_nat_mult gcd_int_def)
huffman@31706
  1248
nipkow@31952
  1249
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1250
  unfolding lcm_nat_def
nipkow@31952
  1251
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1252
nipkow@31952
  1253
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1254
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1255
  apply (subst int_mult [symmetric])
nipkow@31952
  1256
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1257
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1258
  apply (simp, simp add: abs_mult)
huffman@31706
  1259
done
huffman@31706
  1260
nipkow@31952
  1261
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1262
  unfolding lcm_nat_def by simp
huffman@31706
  1263
nipkow@31952
  1264
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1265
  unfolding lcm_int_def by simp
huffman@31706
  1266
nipkow@31952
  1267
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1268
  unfolding lcm_nat_def by simp
chaieb@27669
  1269
nipkow@31952
  1270
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1271
  unfolding lcm_int_def by simp
huffman@31706
  1272
nipkow@31952
  1273
lemma lcm_pos_nat:
nipkow@31798
  1274
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1275
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1276
nipkow@31952
  1277
lemma lcm_pos_int:
nipkow@31798
  1278
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1279
  apply (subst lcm_abs_int)
nipkow@31952
  1280
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1281
  apply auto
huffman@31706
  1282
done
haftmann@23687
  1283
nipkow@31952
  1284
lemma dvd_pos_nat:
haftmann@23687
  1285
  fixes n m :: nat
haftmann@23687
  1286
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1287
  shows "m > 0"
haftmann@23687
  1288
using assms by (cases m) auto
haftmann@23687
  1289
nipkow@31952
  1290
lemma lcm_least_nat:
huffman@31706
  1291
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1292
  shows "lcm m n dvd k"
haftmann@23687
  1293
proof (cases k)
haftmann@23687
  1294
  case 0 then show ?thesis by auto
haftmann@23687
  1295
next
haftmann@23687
  1296
  case (Suc _) then have pos_k: "k > 0" by auto
nipkow@31952
  1297
  from assms dvd_pos_nat [OF this] have pos_mn: "m > 0" "n > 0" by auto
nipkow@31952
  1298
  with gcd_zero_nat [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1299
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1300
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1301
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1302
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1303
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
nipkow@31952
  1304
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1305
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1306
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1307
  also have "\<dots> = k * p * q * gcd m n"
nipkow@31952
  1308
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1309
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1310
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1311
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1312
    by (simp add: mult_ac)
haftmann@27556
  1313
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1314
    by simp
nipkow@31952
  1315
  with prod_gcd_lcm_nat [of m n]
haftmann@27556
  1316
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1317
    by (simp add: mult_ac)
huffman@31706
  1318
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1319
  then show ?thesis using dvd_def by auto
haftmann@23687
  1320
qed
haftmann@23687
  1321
nipkow@31952
  1322
lemma lcm_least_int:
nipkow@31798
  1323
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
nipkow@31952
  1324
apply (subst lcm_abs_int)
nipkow@31798
  1325
apply (rule dvd_trans)
nipkow@31952
  1326
apply (rule lcm_least_nat [transferred, of _ "abs k" _])
nipkow@31798
  1327
apply auto
huffman@31706
  1328
done
huffman@31706
  1329
nipkow@31952
  1330
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
haftmann@23687
  1331
proof (cases m)
haftmann@23687
  1332
  case 0 then show ?thesis by simp
haftmann@23687
  1333
next
haftmann@23687
  1334
  case (Suc _)
haftmann@23687
  1335
  then have mpos: "m > 0" by simp
haftmann@23687
  1336
  show ?thesis
haftmann@23687
  1337
  proof (cases n)
haftmann@23687
  1338
    case 0 then show ?thesis by simp
haftmann@23687
  1339
  next
haftmann@23687
  1340
    case (Suc _)
haftmann@23687
  1341
    then have npos: "n > 0" by simp
haftmann@27556
  1342
    have "gcd m n dvd n" by simp
haftmann@27556
  1343
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1344
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1345
      by (simp add: mult_ac)
nipkow@31952
  1346
    also have "\<dots> = m * k" using mpos npos gcd_zero_nat by simp
huffman@31706
  1347
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1348
  qed
haftmann@23687
  1349
qed
haftmann@23687
  1350
nipkow@31952
  1351
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
nipkow@31952
  1352
  apply (subst lcm_abs_int)
huffman@31706
  1353
  apply (rule dvd_trans)
huffman@31706
  1354
  prefer 2
nipkow@31952
  1355
  apply (rule lcm_dvd1_nat [transferred])
huffman@31706
  1356
  apply auto
huffman@31706
  1357
done
huffman@31706
  1358
nipkow@31952
  1359
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
haftmann@35726
  1360
  using lcm_dvd1_nat [of n m] by (simp only: lcm_nat_def mult.commute gcd_nat.commute)
huffman@31706
  1361
nipkow@31952
  1362
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
haftmann@35726
  1363
  using lcm_dvd1_int [of n m] by (simp only: lcm_int_def lcm_nat_def mult.commute gcd_nat.commute)
huffman@31706
  1364
nipkow@31730
  1365
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1366
by(metis lcm_dvd1_nat dvd_trans)
nipkow@31729
  1367
nipkow@31730
  1368
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1369
by(metis lcm_dvd2_nat dvd_trans)
nipkow@31729
  1370
nipkow@31730
  1371
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1372
by(metis lcm_dvd1_int dvd_trans)
nipkow@31729
  1373
nipkow@31730
  1374
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1375
by(metis lcm_dvd2_int dvd_trans)
nipkow@31729
  1376
nipkow@31952
  1377
lemma lcm_unique_nat: "(a::nat) 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 lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
chaieb@27568
  1380
nipkow@31952
  1381
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1382
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@33657
  1383
  by (auto intro: dvd_antisym [transferred] lcm_least_int)
huffman@31706
  1384
haftmann@37770
  1385
interpretation lcm_nat: abel_semigroup "lcm :: nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@34973
  1386
proof
haftmann@34973
  1387
  fix n m p :: nat
haftmann@34973
  1388
  show "lcm (lcm n m) p = lcm n (lcm m p)"
haftmann@34973
  1389
    by (rule lcm_unique_nat [THEN iffD1]) (metis dvd.order_trans lcm_unique_nat)
haftmann@34973
  1390
  show "lcm m n = lcm n m"
haftmann@36350
  1391
    by (simp add: lcm_nat_def gcd_commute_nat field_simps)
haftmann@34973
  1392
qed
haftmann@34973
  1393
haftmann@37770
  1394
interpretation lcm_int: abel_semigroup "lcm :: int \<Rightarrow> int \<Rightarrow> int"
haftmann@34973
  1395
proof
haftmann@34973
  1396
  fix n m p :: int
haftmann@34973
  1397
  show "lcm (lcm n m) p = lcm n (lcm m p)"
haftmann@34973
  1398
    by (rule lcm_unique_int [THEN iffD1]) (metis dvd_trans lcm_unique_int)
haftmann@34973
  1399
  show "lcm m n = lcm n m"
haftmann@34973
  1400
    by (simp add: lcm_int_def lcm_nat.commute)
haftmann@34973
  1401
qed
haftmann@34973
  1402
haftmann@34973
  1403
lemmas lcm_assoc_nat = lcm_nat.assoc
haftmann@34973
  1404
lemmas lcm_commute_nat = lcm_nat.commute
haftmann@34973
  1405
lemmas lcm_left_commute_nat = lcm_nat.left_commute
haftmann@34973
  1406
lemmas lcm_assoc_int = lcm_int.assoc
haftmann@34973
  1407
lemmas lcm_commute_int = lcm_int.commute
haftmann@34973
  1408
lemmas lcm_left_commute_int = lcm_int.left_commute
haftmann@34973
  1409
haftmann@34973
  1410
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
haftmann@34973
  1411
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
haftmann@34973
  1412
nipkow@31798
  1413
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1414
  apply (rule sym)
nipkow@31952
  1415
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1416
  apply auto
huffman@31706
  1417
done
huffman@31706
  1418
nipkow@31798
  1419
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1420
  apply (rule sym)
nipkow@31952
  1421
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1422
  apply auto
huffman@31706
  1423
done
huffman@31706
  1424
nipkow@31798
  1425
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31952
  1426
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1427
nipkow@31798
  1428
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31952
  1429
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
huffman@31706
  1430
nipkow@31992
  1431
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1432
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1433
nipkow@31992
  1434
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1435
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1436
nipkow@31992
  1437
lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) \<longleftrightarrow> n dvd m"
nipkow@31992
  1438
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1439
nipkow@31992
  1440
lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) \<longleftrightarrow> m dvd n"
nipkow@31992
  1441
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1442
haftmann@42871
  1443
lemma comp_fun_idem_gcd_nat: "comp_fun_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1444
proof qed (auto simp add: gcd_ac_nat)
nipkow@31992
  1445
haftmann@42871
  1446
lemma comp_fun_idem_gcd_int: "comp_fun_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1447
proof qed (auto simp add: gcd_ac_int)
nipkow@31992
  1448
haftmann@42871
  1449
lemma comp_fun_idem_lcm_nat: "comp_fun_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1450
proof qed (auto simp add: lcm_ac_nat)
nipkow@31992
  1451
haftmann@42871
  1452
lemma comp_fun_idem_lcm_int: "comp_fun_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1453
proof qed (auto simp add: lcm_ac_int)
nipkow@31992
  1454
haftmann@23687
  1455
nipkow@31995
  1456
(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)
nipkow@31995
  1457
nipkow@31995
  1458
lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1459
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
nipkow@31995
  1460
nipkow@31995
  1461
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
huffman@44766
  1462
by (metis lcm_0_int lcm_0_left_int lcm_pos_int less_le)
nipkow@31995
  1463
nipkow@31995
  1464
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
nipkow@31995
  1465
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
nipkow@31995
  1466
nipkow@31995
  1467
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
  1468
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
nipkow@31995
  1469
haftmann@34030
  1470
huffman@45264
  1471
subsection {* The complete divisibility lattice *}
nipkow@32112
  1472
haftmann@51489
  1473
lemma semilattice_neutr_set_lcm_one_nat:
haftmann@51489
  1474
  "semilattice_neutr_set lcm (1::nat)"
haftmann@51489
  1475
  by default simp_all
haftmann@51489
  1476
krauss@44845
  1477
interpretation gcd_semilattice_nat: semilattice_inf gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
nipkow@32112
  1478
proof
nipkow@32112
  1479
  case goal3 thus ?case by(metis gcd_unique_nat)
nipkow@32112
  1480
qed auto
nipkow@32112
  1481
krauss@44845
  1482
interpretation lcm_semilattice_nat: semilattice_sup lcm "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)"
nipkow@32112
  1483
proof
nipkow@32112
  1484
  case goal3 thus ?case by(metis lcm_unique_nat)
nipkow@32112
  1485
qed auto
nipkow@32112
  1486
krauss@44845
  1487
interpretation gcd_lcm_lattice_nat: lattice gcd "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm ..
nipkow@32112
  1488
huffman@45264
  1489
text{* Lifting gcd and lcm to sets (Gcd/Lcm).
huffman@45264
  1490
Gcd is defined via Lcm to facilitate the proof that we have a complete lattice.
nipkow@32112
  1491
*}
huffman@45264
  1492
huffman@45264
  1493
class Gcd = gcd +
huffman@45264
  1494
  fixes Gcd :: "'a set \<Rightarrow> 'a"
huffman@45264
  1495
  fixes Lcm :: "'a set \<Rightarrow> 'a"
huffman@45264
  1496
huffman@45264
  1497
instantiation nat :: Gcd
nipkow@32112
  1498
begin
nipkow@32112
  1499
huffman@45264
  1500
definition
haftmann@51489
  1501
  "Lcm (M::nat set) = (if finite M then semilattice_neutr_set.F lcm 1 M else 0)"
haftmann@51489
  1502
haftmann@51489
  1503
lemma Lcm_nat_infinite:
haftmann@51489
  1504
  "\<not> finite M \<Longrightarrow> Lcm M = (0::nat)"
haftmann@51489
  1505
  by (simp add: Lcm_nat_def)
haftmann@51489
  1506
haftmann@51489
  1507
lemma Lcm_nat_empty:
haftmann@51489
  1508
  "Lcm {} = (1::nat)"
haftmann@51489
  1509
proof -
haftmann@51489
  1510
  interpret semilattice_neutr_set lcm "1::nat"
haftmann@51489
  1511
    by (fact semilattice_neutr_set_lcm_one_nat)
haftmann@51489
  1512
  show ?thesis by (simp add: Lcm_nat_def)
haftmann@51489
  1513
qed
haftmann@51489
  1514
haftmann@51489
  1515
lemma Lcm_nat_insert:
haftmann@51489
  1516
  "Lcm (insert n M) = lcm (n::nat) (Lcm M)"
haftmann@51489
  1517
proof (cases "finite M")
haftmann@51489
  1518
  interpret semilattice_neutr_set lcm "1::nat"
haftmann@51489
  1519
    by (fact semilattice_neutr_set_lcm_one_nat)
haftmann@51489
  1520
  case True
haftmann@51489
  1521
  then show ?thesis by (simp add: Lcm_nat_def)
haftmann@51489
  1522
next
haftmann@51489
  1523
  case False then show ?thesis by (simp add: Lcm_nat_infinite)
haftmann@51489
  1524
qed
nipkow@32112
  1525
huffman@45264
  1526
definition
huffman@45264
  1527
  "Gcd (M::nat set) = Lcm {d. \<forall>m\<in>M. d dvd m}"
nipkow@32112
  1528
huffman@45264
  1529
instance ..
haftmann@51489
  1530
nipkow@32112
  1531
end
nipkow@32112
  1532
huffman@45264
  1533
lemma dvd_Lcm_nat [simp]:
haftmann@51489
  1534
  fixes M :: "nat set"
haftmann@51489
  1535
  assumes "m \<in> M"
haftmann@51489
  1536
  shows "m dvd Lcm M"
haftmann@51489
  1537
proof (cases "finite M")
haftmann@51489
  1538
  case False then show ?thesis by (simp add: Lcm_nat_infinite)
haftmann@51489
  1539
next
haftmann@51489
  1540
  case True then show ?thesis using assms by (induct M) (auto simp add: Lcm_nat_insert)
haftmann@51489
  1541
qed
nipkow@32112
  1542
huffman@45264
  1543
lemma Lcm_dvd_nat [simp]:
haftmann@51489
  1544
  fixes M :: "nat set"
haftmann@51489
  1545
  assumes "\<forall>m\<in>M. m dvd n"
haftmann@51489
  1546
  shows "Lcm M dvd n"
huffman@45264
  1547
proof (cases "n = 0")
huffman@45264
  1548
  assume "n \<noteq> 0"
huffman@45264
  1549
  hence "finite {d. d dvd n}" by (rule finite_divisors_nat)
huffman@45264
  1550
  moreover have "M \<subseteq> {d. d dvd n}" using assms by fast
huffman@45264
  1551
  ultimately have "finite M" by (rule rev_finite_subset)
haftmann@51489
  1552
  then show ?thesis using assms by (induct M) (simp_all add: Lcm_nat_empty Lcm_nat_insert)
huffman@45264
  1553
qed simp
nipkow@32112
  1554
huffman@45264
  1555
interpretation gcd_lcm_complete_lattice_nat:
haftmann@51547
  1556
  complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat"
haftmann@51547
  1557
where
hoelzl@54257
  1558
  "Inf.INFI Gcd A f = Gcd (f ` A :: nat set)"
hoelzl@54257
  1559
  and "Sup.SUPR Lcm A f = Lcm (f ` A)"
haftmann@51547
  1560
proof -
haftmann@51547
  1561
  show "class.complete_lattice Gcd Lcm gcd Rings.dvd (\<lambda>m n. m dvd n \<and> \<not> n dvd m) lcm 1 (0::nat)"
haftmann@51547
  1562
  proof
haftmann@52729
  1563
    case goal1 thus ?case by (simp add: Gcd_nat_def)
haftmann@51547
  1564
  next
haftmann@52729
  1565
    case goal2 thus ?case by (simp add: Gcd_nat_def)
haftmann@51547
  1566
  next
haftmann@52729
  1567
    case goal5 show ?case by (simp add: Gcd_nat_def Lcm_nat_infinite)
haftmann@51547
  1568
  next
haftmann@52729
  1569
    case goal6 show ?case by (simp add: Lcm_nat_empty)
haftmann@51547
  1570
  next
haftmann@52729
  1571
    case goal3 thus ?case by simp
haftmann@51547
  1572
  next
haftmann@52729
  1573
    case goal4 thus ?case by simp
haftmann@51547
  1574
  qed
haftmann@51547
  1575
  then interpret gcd_lcm_complete_lattice_nat:
haftmann@51547
  1576
    complete_lattice Gcd Lcm gcd Rings.dvd "\<lambda>m n. m dvd n \<and> \<not> n dvd m" lcm 1 "0::nat" .
hoelzl@54257
  1577
  from gcd_lcm_complete_lattice_nat.INF_def show "Inf.INFI Gcd A f = Gcd (f ` A)" .
hoelzl@54257
  1578
  from gcd_lcm_complete_lattice_nat.SUP_def show "Sup.SUPR Lcm A f = Lcm (f ` A)" .
huffman@45264
  1579
qed
nipkow@32112
  1580
huffman@45264
  1581
lemma Lcm_empty_nat: "Lcm {} = (1::nat)"
huffman@45264
  1582
  by (fact gcd_lcm_complete_lattice_nat.Sup_empty) (* already simp *)
huffman@45264
  1583
huffman@45264
  1584
lemma Gcd_empty_nat: "Gcd {} = (0::nat)"
huffman@45264
  1585
  by (fact gcd_lcm_complete_lattice_nat.Inf_empty) (* already simp *)
nipkow@32112
  1586
nipkow@32112
  1587
lemma Lcm_insert_nat [simp]:
nipkow@32112
  1588
  shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
huffman@45264
  1589
  by (fact gcd_lcm_complete_lattice_nat.Sup_insert)
nipkow@32112
  1590
nipkow@32112
  1591
lemma Gcd_insert_nat [simp]:
nipkow@32112
  1592
  shows "Gcd (insert (n::nat) N) = gcd n (Gcd N)"
huffman@45264
  1593
  by (fact gcd_lcm_complete_lattice_nat.Inf_insert)
nipkow@32112
  1594
nipkow@32112
  1595
lemma Lcm0_iff[simp]: "finite (M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> Lcm M = 0 \<longleftrightarrow> 0 : M"
nipkow@32112
  1596
by(induct rule:finite_ne_induct) auto
nipkow@32112
  1597
nipkow@32112
  1598
lemma Lcm_eq_0[simp]: "finite (M::nat set) \<Longrightarrow> 0 : M \<Longrightarrow> Lcm M = 0"
nipkow@32112
  1599
by (metis Lcm0_iff empty_iff)
nipkow@32112
  1600
nipkow@32112
  1601
lemma Gcd_dvd_nat [simp]:
huffman@45264
  1602
  fixes M :: "nat set"
huffman@45264
  1603
  assumes "m \<in> M" shows "Gcd M dvd m"
huffman@45264
  1604
  using assms by (fact gcd_lcm_complete_lattice_nat.Inf_lower)
nipkow@32112
  1605
nipkow@32112
  1606
lemma dvd_Gcd_nat[simp]:
huffman@45264
  1607
  fixes M :: "nat set"
huffman@45264
  1608
  assumes "\<forall>m\<in>M. n dvd m" shows "n dvd Gcd M"
huffman@45264
  1609
  using assms by (simp only: gcd_lcm_complete_lattice_nat.Inf_greatest)
nipkow@32112
  1610
huffman@45264
  1611
text{* Alternative characterizations of Gcd: *}
nipkow@32112
  1612
nipkow@32112
  1613
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
  1614
apply(rule antisym)
nipkow@32112
  1615
 apply(rule Max_ge)
nipkow@32112
  1616
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@32112
  1617
 apply simp
nipkow@32112
  1618
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  1619
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@44890
  1620
 apply fastforce
nipkow@32112
  1621
apply clarsimp
nipkow@32112
  1622
apply (metis Gcd_dvd_nat Max_in dvd_0_left dvd_Gcd_nat dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  1623
done
nipkow@32112
  1624
nipkow@32112
  1625
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  1626
apply(induct pred:finite)
nipkow@32112
  1627
 apply simp
nipkow@32112
  1628
apply(case_tac "x=0")
nipkow@32112
  1629
 apply simp
nipkow@32112
  1630
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  1631
 apply simp
nipkow@32112
  1632
apply blast
nipkow@32112
  1633
done
nipkow@32112
  1634
nipkow@32112
  1635
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  1636
  "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
  1637
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  1638
 apply simp
nipkow@32112
  1639
apply simp
nipkow@32112
  1640
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  1641
 apply simp
nipkow@32112
  1642
 apply(case_tac "A={}")
nipkow@32112
  1643
  apply simp
nipkow@32112
  1644
 apply simp
nipkow@32112
  1645
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  1646
done
nipkow@32112
  1647
nipkow@32112
  1648
lemma Lcm_eq_Max_nat:
nipkow@32112
  1649
  "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
  1650
apply(rule antisym)
nipkow@32112
  1651
 apply(rule Max_ge, assumption)
nipkow@32112
  1652
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
nipkow@32112
  1653
apply clarsimp
nipkow@32112
  1654
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
nipkow@32112
  1655
done
nipkow@32112
  1656
haftmann@54437
  1657
lemma Lcm_set_nat [code, code_unfold]:
haftmann@45992
  1658
  "Lcm (set ns) = fold lcm ns (1::nat)"
huffman@45264
  1659
  by (fact gcd_lcm_complete_lattice_nat.Sup_set_fold)
nipkow@32112
  1660
haftmann@54437
  1661
lemma Gcd_set_nat [code, code_unfold]:
haftmann@45992
  1662
  "Gcd (set ns) = fold gcd ns (0::nat)"
huffman@45264
  1663
  by (fact gcd_lcm_complete_lattice_nat.Inf_set_fold)
nipkow@34222
  1664
nipkow@34222
  1665
lemma mult_inj_if_coprime_nat:
nipkow@34222
  1666
  "inj_on f A \<Longrightarrow> inj_on g B \<Longrightarrow> ALL a:A. ALL b:B. coprime (f a) (g b)
nipkow@34222
  1667
   \<Longrightarrow> inj_on (%(a,b). f a * g b::nat) (A \<times> B)"
nipkow@34222
  1668
apply(auto simp add:inj_on_def)
huffman@35216
  1669
apply (metis coprime_dvd_mult_iff_nat dvd.neq_le_trans dvd_triv_left)
nipkow@34223
  1670
apply (metis gcd_semilattice_nat.inf_commute coprime_dvd_mult_iff_nat
nipkow@34223
  1671
             dvd.neq_le_trans dvd_triv_right mult_commute)
nipkow@34222
  1672
done
nipkow@34222
  1673
nipkow@34222
  1674
text{* Nitpick: *}
nipkow@34222
  1675
blanchet@41792
  1676
lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
blanchet@41792
  1677
by (induct x y rule: nat_gcd.induct)
blanchet@41792
  1678
   (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  1679
blanchet@41792
  1680
lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
blanchet@33197
  1681
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  1682
huffman@45264
  1683
subsubsection {* Setwise gcd and lcm for integers *}
huffman@45264
  1684
huffman@45264
  1685
instantiation int :: Gcd
huffman@45264
  1686
begin
huffman@45264
  1687
huffman@45264
  1688
definition
huffman@45264
  1689
  "Lcm M = int (Lcm (nat ` abs ` M))"
huffman@45264
  1690
huffman@45264
  1691
definition
huffman@45264
  1692
  "Gcd M = int (Gcd (nat ` abs ` M))"
huffman@45264
  1693
huffman@45264
  1694
instance ..
wenzelm@21256
  1695
end
huffman@45264
  1696
huffman@45264
  1697
lemma Lcm_empty_int [simp]: "Lcm {} = (1::int)"
huffman@45264
  1698
  by (simp add: Lcm_int_def)
huffman@45264
  1699
huffman@45264
  1700
lemma Gcd_empty_int [simp]: "Gcd {} = (0::int)"
huffman@45264
  1701
  by (simp add: Gcd_int_def)
huffman@45264
  1702
huffman@45264
  1703
lemma Lcm_insert_int [simp]:
huffman@45264
  1704
  shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"
huffman@45264
  1705
  by (simp add: Lcm_int_def lcm_int_def)
huffman@45264
  1706
huffman@45264
  1707
lemma Gcd_insert_int [simp]:
huffman@45264
  1708
  shows "Gcd (insert (n::int) N) = gcd n (Gcd N)"
huffman@45264
  1709
  by (simp add: Gcd_int_def gcd_int_def)
huffman@45264
  1710
huffman@45264
  1711
lemma dvd_int_iff: "x dvd y \<longleftrightarrow> nat (abs x) dvd nat (abs y)"
huffman@45264
  1712
  by (simp add: zdvd_int)
huffman@45264
  1713
huffman@45264
  1714
lemma dvd_Lcm_int [simp]:
huffman@45264
  1715
  fixes M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
huffman@45264
  1716
  using assms by (simp add: Lcm_int_def dvd_int_iff)
huffman@45264
  1717
huffman@45264
  1718
lemma Lcm_dvd_int [simp]:
huffman@45264
  1719
  fixes M :: "int set"
huffman@45264
  1720
  assumes "\<forall>m\<in>M. m dvd n" shows "Lcm M dvd n"
huffman@45264
  1721
  using assms by (simp add: Lcm_int_def dvd_int_iff)
huffman@45264
  1722
huffman@45264
  1723
lemma Gcd_dvd_int [simp]:
huffman@45264
  1724
  fixes M :: "int set"
huffman@45264
  1725
  assumes "m \<in> M" shows "Gcd M dvd m"
huffman@45264
  1726
  using assms by (simp add: Gcd_int_def dvd_int_iff)
huffman@45264
  1727
huffman@45264
  1728
lemma dvd_Gcd_int[simp]:
huffman@45264
  1729
  fixes M :: "int set"
huffman@45264
  1730
  assumes "\<forall>m\<in>M. n dvd m" shows "n dvd Gcd M"
huffman@45264
  1731
  using assms by (simp add: Gcd_int_def dvd_int_iff)
huffman@45264
  1732
haftmann@54437
  1733
lemma Lcm_set_int [code, code_unfold]:
haftmann@51547
  1734
  "Lcm (set xs) = fold lcm xs (1::int)"
huffman@45264
  1735
  by (induct xs rule: rev_induct, simp_all add: lcm_commute_int)
huffman@45264
  1736
haftmann@54437
  1737
lemma Gcd_set_int [code, code_unfold]:
haftmann@51547
  1738
  "Gcd (set xs) = fold gcd xs (0::int)"
huffman@45264
  1739
  by (induct xs rule: rev_induct, simp_all add: gcd_commute_int)
huffman@45264
  1740
huffman@45264
  1741
end
haftmann@51547
  1742