src/HOL/GCD.thy
author nipkow
Thu Jun 25 07:34:12 2009 +0200 (2009-06-25)
changeset 31798 fe9a3043d36c
parent 31766 f767c5b1702e
child 31813 4df828bbc411
permissions -rw-r--r--
Cleaned up GCD
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(*  Title:      GCD.thy
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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, and properties of
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primes. Definitions and lemmas are proved uniformly for the natural
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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 Chiaeb.
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Tobias Nipkow cleaned up a lot.
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*)
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header {* GCD *}
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theory GCD
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imports NatTransfer
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begin
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declare One_nat_def [simp del]
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subsection {* gcd *}
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class gcd = one +
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fixes
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  gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and
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  lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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begin
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abbreviation
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  coprime :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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where
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  "coprime x y == (gcd x y = 1)"
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end
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class prime = one +
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fixes
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  prime :: "'a \<Rightarrow> bool"
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(* definitions for the natural numbers *)
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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 nat :: prime
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begin
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definition
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  prime_nat :: "nat \<Rightarrow> bool"
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where
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  [code del]: "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
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instance proof qed
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end
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(* definitions for the integers *)
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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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instantiation int :: prime
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begin
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definition
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  prime_int :: "int \<Rightarrow> bool"
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where
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  [code del]: "prime_int p = prime (nat p)"
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instance proof qed
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end
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subsection {* Set up Transfer *}
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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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  "(x::int) >= 0 \<Longrightarrow> prime (nat x) = prime x"
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  unfolding gcd_int_def lcm_int_def prime_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 TransferMorphism_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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  "prime (int x) = prime x"
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  by (unfold gcd_int_def lcm_int_def prime_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 TransferMorphism_int_nat[transfer add return:
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    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
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subsection {* GCD *}
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(* was gcd_induct *)
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lemma nat_gcd_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 int_gcd_neg1 [simp]: "gcd (-x::int) y = gcd x y"
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  by (simp add: gcd_int_def)
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lemma int_gcd_neg2 [simp]: "gcd (x::int) (-y) = gcd x y"
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  by (simp add: gcd_int_def)
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lemma int_gcd_abs: "gcd (x::int) y = gcd (abs x) (abs y)"
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  by (simp add: gcd_int_def)
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lemma int_gcd_cases:
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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 prems, auto simp add: int_gcd_neg1 int_gcd_neg2, arith)
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lemma int_gcd_ge_0 [simp]: "gcd (x::int) y >= 0"
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  by (simp add: gcd_int_def)
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lemma int_lcm_neg1: "lcm (-x::int) y = lcm x y"
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  by (simp add: lcm_int_def)
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lemma int_lcm_neg2: "lcm (x::int) (-y) = lcm x y"
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  by (simp add: lcm_int_def)
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lemma int_lcm_abs: "lcm (x::int) y = lcm (abs x) (abs y)"
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  by (simp add: lcm_int_def)
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lemma int_lcm_cases:
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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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by (insert prems, auto simp add: int_lcm_neg1 int_lcm_neg2, arith)
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lemma int_lcm_ge_0 [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 nat_gcd_0 [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 int_gcd_0 [simp]: "gcd (x::int) 0 = abs x"
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  by (unfold gcd_int_def, auto)
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lemma nat_gcd_0_left [simp]: "gcd 0 (x::nat) = x"
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  by simp
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lemma int_gcd_0_left [simp]: "gcd 0 (x::int) = abs x"
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  by (unfold gcd_int_def, auto)
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lemma nat_gcd_red: "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 nat_gcd_non_0: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
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  by simp
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lemma nat_gcd_1 [simp]: "gcd (m::nat) 1 = 1"
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  by simp
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lemma nat_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 int_gcd_1 [simp]: "gcd (m::int) 1 = 1"
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  by (simp add: gcd_int_def)
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lemma nat_gcd_idem: "gcd (x::nat) x = x"
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by simp
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lemma int_gcd_idem: "gcd (x::int) x = abs x"
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by (subst int_gcd_abs, 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 nat_gcd_dvd1 [iff]: "(gcd (m::nat)) n dvd m"
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  and nat_gcd_dvd2 [iff]: "(gcd m n) dvd n"
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  apply (induct m n rule: nat_gcd_induct)
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  apply (simp_all add: nat_gcd_non_0)
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  apply (blast dest: dvd_mod_imp_dvd)
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done
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lemma int_gcd_dvd1 [iff]: "gcd (x::int) y dvd x"
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  apply (subst int_gcd_abs)
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  apply (rule dvd_trans)
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  apply (rule nat_gcd_dvd1 [transferred])
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  apply auto
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done
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lemma int_gcd_dvd2 [iff]: "gcd (x::int) y dvd y"
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  apply (subst int_gcd_abs)
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  apply (rule dvd_trans)
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  apply (rule nat_gcd_dvd2 [transferred])
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  apply auto
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done
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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 nat_gcd_dvd1 dvd_trans)
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lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
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by(metis nat_gcd_dvd2 dvd_trans)
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lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
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by(metis int_gcd_dvd1 dvd_trans)
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lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
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by(metis int_gcd_dvd2 dvd_trans)
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lemma nat_gcd_le1 [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 nat_gcd_le2 [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 int_gcd_le1 [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 int_gcd_le2 [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 nat_gcd_greatest: "(k::nat) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
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by (induct m n rule: nat_gcd_induct) (simp_all add: nat_gcd_non_0 dvd_mod)
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lemma int_gcd_greatest:
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  assumes "(k::int) dvd m" and "k dvd n"
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  shows "k dvd gcd m n"
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  apply (subst int_gcd_abs)
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  apply (subst abs_dvd_iff [symmetric])
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  apply (rule nat_gcd_greatest [transferred])
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  using prems apply auto
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done
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lemma nat_gcd_greatest_iff [iff]: "(k dvd gcd (m::nat) n) =
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    (k dvd m & k dvd n)"
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  by (blast intro!: nat_gcd_greatest intro: dvd_trans)
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lemma int_gcd_greatest_iff: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
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  by (blast intro!: int_gcd_greatest intro: dvd_trans)
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lemma nat_gcd_zero [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
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  by (simp only: dvd_0_left_iff [symmetric] nat_gcd_greatest_iff)
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lemma int_gcd_zero [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 nat_gcd_pos [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
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  by (insert nat_gcd_zero [of m n], arith)
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lemma int_gcd_pos [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
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  by (insert int_gcd_zero [of m n], insert int_gcd_ge_0 [of m n], arith)
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lemma nat_gcd_commute: "gcd (m::nat) n = gcd n m"
huffman@31706
   342
  by (rule dvd_anti_sym, auto)
haftmann@23687
   343
huffman@31706
   344
lemma int_gcd_commute: "gcd (m::int) n = gcd n m"
huffman@31706
   345
  by (auto simp add: gcd_int_def nat_gcd_commute)
huffman@31706
   346
huffman@31706
   347
lemma nat_gcd_assoc: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
huffman@31706
   348
  apply (rule dvd_anti_sym)
huffman@31706
   349
  apply (blast intro: dvd_trans)+
huffman@31706
   350
done
wenzelm@21256
   351
huffman@31706
   352
lemma int_gcd_assoc: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
huffman@31706
   353
  by (auto simp add: gcd_int_def nat_gcd_assoc)
huffman@31706
   354
nipkow@31766
   355
lemmas nat_gcd_left_commute =
nipkow@31766
   356
  mk_left_commute[of gcd, OF nat_gcd_assoc nat_gcd_commute]
huffman@31706
   357
nipkow@31766
   358
lemmas int_gcd_left_commute =
nipkow@31766
   359
  mk_left_commute[of gcd, OF int_gcd_assoc int_gcd_commute]
huffman@31706
   360
huffman@31706
   361
lemmas nat_gcd_ac = nat_gcd_assoc nat_gcd_commute nat_gcd_left_commute
huffman@31706
   362
  -- {* gcd is an AC-operator *}
wenzelm@21256
   363
huffman@31706
   364
lemmas int_gcd_ac = int_gcd_assoc int_gcd_commute int_gcd_left_commute
huffman@31706
   365
huffman@31706
   366
lemma nat_gcd_unique: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   367
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   368
  apply auto
huffman@31706
   369
  apply (rule dvd_anti_sym)
huffman@31706
   370
  apply (erule (1) nat_gcd_greatest)
huffman@31706
   371
  apply auto
huffman@31706
   372
done
wenzelm@21256
   373
huffman@31706
   374
lemma int_gcd_unique: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   375
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   376
  apply (case_tac "d = 0")
huffman@31706
   377
  apply force
huffman@31706
   378
  apply (rule iffI)
huffman@31706
   379
  apply (rule zdvd_anti_sym)
huffman@31706
   380
  apply arith
huffman@31706
   381
  apply (subst int_gcd_pos)
huffman@31706
   382
  apply clarsimp
huffman@31706
   383
  apply (drule_tac x = "d + 1" in spec)
huffman@31706
   384
  apply (frule zdvd_imp_le)
huffman@31706
   385
  apply (auto intro: int_gcd_greatest)
huffman@31706
   386
done
huffman@30082
   387
nipkow@31798
   388
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
nipkow@31798
   389
by (metis dvd.eq_iff nat_gcd_unique)
nipkow@31798
   390
nipkow@31798
   391
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
nipkow@31798
   392
by (metis dvd.eq_iff nat_gcd_unique)
nipkow@31798
   393
nipkow@31798
   394
lemma gcd_proj1_if_dvd_int[simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
nipkow@31798
   395
by (metis abs_dvd_iff abs_eq_0 int_gcd_0_left int_gcd_abs int_gcd_unique)
nipkow@31798
   396
nipkow@31798
   397
lemma gcd_proj2_if_dvd_int[simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
nipkow@31798
   398
by (metis gcd_proj1_if_dvd_int int_gcd_commute)
nipkow@31798
   399
nipkow@31798
   400
wenzelm@21256
   401
text {*
wenzelm@21256
   402
  \medskip Multiplication laws
wenzelm@21256
   403
*}
wenzelm@21256
   404
huffman@31706
   405
lemma nat_gcd_mult_distrib: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256
   406
    -- {* \cite[page 27]{davenport92} *}
huffman@31706
   407
  apply (induct m n rule: nat_gcd_induct)
huffman@31706
   408
  apply simp
wenzelm@21256
   409
  apply (case_tac "k = 0")
huffman@31706
   410
  apply (simp_all add: mod_geq nat_gcd_non_0 mod_mult_distrib2)
huffman@31706
   411
done
wenzelm@21256
   412
huffman@31706
   413
lemma int_gcd_mult_distrib: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
huffman@31706
   414
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   415
  apply (simp add: abs_mult)
huffman@31706
   416
  apply (rule nat_gcd_mult_distrib [transferred])
huffman@31706
   417
  apply auto
huffman@31706
   418
done
wenzelm@21256
   419
huffman@31706
   420
lemma nat_coprime_dvd_mult: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
huffman@31706
   421
  apply (insert nat_gcd_mult_distrib [of m k n])
wenzelm@21256
   422
  apply simp
wenzelm@21256
   423
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   424
  apply simp
wenzelm@21256
   425
  done
wenzelm@21256
   426
huffman@31706
   427
lemma int_coprime_dvd_mult:
huffman@31706
   428
  assumes "coprime (k::int) n" and "k dvd m * n"
huffman@31706
   429
  shows "k dvd m"
wenzelm@21256
   430
huffman@31706
   431
  using prems
huffman@31706
   432
  apply (subst abs_dvd_iff [symmetric])
huffman@31706
   433
  apply (subst dvd_abs_iff [symmetric])
huffman@31706
   434
  apply (subst (asm) int_gcd_abs)
huffman@31706
   435
  apply (rule nat_coprime_dvd_mult [transferred])
huffman@31706
   436
  apply auto
huffman@31706
   437
  apply (subst abs_mult [symmetric], auto)
huffman@31706
   438
done
huffman@31706
   439
huffman@31706
   440
lemma nat_coprime_dvd_mult_iff: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   441
    (k dvd m * n) = (k dvd m)"
huffman@31706
   442
  by (auto intro: nat_coprime_dvd_mult)
huffman@31706
   443
huffman@31706
   444
lemma int_coprime_dvd_mult_iff: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   445
    (k dvd m * n) = (k dvd m)"
huffman@31706
   446
  by (auto intro: int_coprime_dvd_mult)
huffman@31706
   447
huffman@31706
   448
lemma nat_gcd_mult_cancel: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
wenzelm@21256
   449
  apply (rule dvd_anti_sym)
huffman@31706
   450
  apply (rule nat_gcd_greatest)
huffman@31706
   451
  apply (rule_tac n = k in nat_coprime_dvd_mult)
huffman@31706
   452
  apply (simp add: nat_gcd_assoc)
huffman@31706
   453
  apply (simp add: nat_gcd_commute)
huffman@31706
   454
  apply (simp_all add: mult_commute)
huffman@31706
   455
done
wenzelm@21256
   456
huffman@31706
   457
lemma int_gcd_mult_cancel:
huffman@31706
   458
  assumes "coprime (k::int) n"
huffman@31706
   459
  shows "gcd (k * m) n = gcd m n"
huffman@31706
   460
huffman@31706
   461
  using prems
huffman@31706
   462
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   463
  apply (subst abs_mult)
huffman@31706
   464
  apply (rule nat_gcd_mult_cancel [transferred])
huffman@31706
   465
  apply (auto simp add: int_gcd_abs [symmetric])
huffman@31706
   466
done
wenzelm@21256
   467
wenzelm@21256
   468
text {* \medskip Addition laws *}
wenzelm@21256
   469
huffman@31706
   470
lemma nat_gcd_add1 [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   471
  apply (case_tac "n = 0")
huffman@31706
   472
  apply (simp_all add: nat_gcd_non_0)
huffman@31706
   473
done
huffman@31706
   474
huffman@31706
   475
lemma nat_gcd_add2 [simp]: "gcd (m::nat) (m + n) = gcd m n"
huffman@31706
   476
  apply (subst (1 2) nat_gcd_commute)
huffman@31706
   477
  apply (subst add_commute)
huffman@31706
   478
  apply simp
huffman@31706
   479
done
huffman@31706
   480
huffman@31706
   481
(* to do: add the other variations? *)
huffman@31706
   482
huffman@31706
   483
lemma nat_gcd_diff1: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
huffman@31706
   484
  by (subst nat_gcd_add1 [symmetric], auto)
huffman@31706
   485
huffman@31706
   486
lemma nat_gcd_diff2: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
huffman@31706
   487
  apply (subst nat_gcd_commute)
huffman@31706
   488
  apply (subst nat_gcd_diff1 [symmetric])
huffman@31706
   489
  apply auto
huffman@31706
   490
  apply (subst nat_gcd_commute)
huffman@31706
   491
  apply (subst nat_gcd_diff1)
huffman@31706
   492
  apply assumption
huffman@31706
   493
  apply (rule nat_gcd_commute)
huffman@31706
   494
done
huffman@31706
   495
huffman@31706
   496
lemma int_gcd_non_0: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   497
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   498
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
huffman@31706
   499
  apply (auto simp add: nat_gcd_non_0 nat_mod_distrib [symmetric]
huffman@31706
   500
    zmod_zminus1_eq_if)
huffman@31706
   501
  apply (frule_tac a = x in pos_mod_bound)
huffman@31706
   502
  apply (subst (1 2) nat_gcd_commute)
huffman@31706
   503
  apply (simp del: pos_mod_bound add: nat_diff_distrib nat_gcd_diff2
huffman@31706
   504
    nat_le_eq_zle)
huffman@31706
   505
done
wenzelm@21256
   506
huffman@31706
   507
lemma int_gcd_red: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   508
  apply (case_tac "y = 0")
huffman@31706
   509
  apply force
huffman@31706
   510
  apply (case_tac "y > 0")
huffman@31706
   511
  apply (subst int_gcd_non_0, auto)
huffman@31706
   512
  apply (insert int_gcd_non_0 [of "-y" "-x"])
huffman@31706
   513
  apply (auto simp add: int_gcd_neg1 int_gcd_neg2)
huffman@31706
   514
done
huffman@31706
   515
huffman@31706
   516
lemma int_gcd_add1 [simp]: "gcd ((m::int) + n) n = gcd m n"
nipkow@31798
   517
by (metis int_gcd_red mod_add_self1 zadd_commute)
huffman@31706
   518
huffman@31706
   519
lemma int_gcd_add2 [simp]: "gcd m ((m::int) + n) = gcd m n"
nipkow@31798
   520
by (metis int_gcd_add1 int_gcd_commute zadd_commute)
wenzelm@21256
   521
huffman@31706
   522
lemma nat_gcd_add_mult: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31798
   523
by (metis mod_mult_self3 nat_gcd_commute nat_gcd_red)
wenzelm@21256
   524
huffman@31706
   525
lemma int_gcd_add_mult: "gcd (m::int) (k * m + n) = gcd m n"
nipkow@31798
   526
by (metis int_gcd_commute int_gcd_red mod_mult_self1 zadd_commute)
nipkow@31798
   527
wenzelm@21256
   528
huffman@31706
   529
(* to do: differences, and all variations of addition rules
huffman@31706
   530
    as simplification rules for nat and int *)
huffman@31706
   531
nipkow@31798
   532
(* FIXME remove iff *)
huffman@31706
   533
lemma nat_gcd_dvd_prod [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   534
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   535
huffman@31706
   536
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   537
nipkow@31734
   538
lemma finite_divisors_nat:
nipkow@31734
   539
  assumes "(m::nat)~= 0" shows "finite{d. d dvd m}"
nipkow@31734
   540
proof-
nipkow@31734
   541
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
nipkow@31734
   542
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   543
    by(bestsimp intro!:dvd_imp_le)
nipkow@31734
   544
qed
nipkow@31734
   545
nipkow@31734
   546
lemma finite_divisors_int:
nipkow@31734
   547
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
   548
proof-
nipkow@31734
   549
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
   550
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
   551
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   552
    by(bestsimp intro!:dvd_imp_le_int)
nipkow@31734
   553
qed
nipkow@31734
   554
nipkow@31734
   555
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   556
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   557
apply(rule Max_eqI[THEN sym])
nipkow@31734
   558
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   559
 apply simp
nipkow@31734
   560
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le nat_gcd_greatest_iff nat_gcd_pos)
nipkow@31734
   561
apply simp
nipkow@31734
   562
done
nipkow@31734
   563
nipkow@31734
   564
lemma gcd_is_Max_divisors_int:
nipkow@31734
   565
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   566
apply(rule Max_eqI[THEN sym])
nipkow@31734
   567
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_int)
nipkow@31734
   568
 apply simp
nipkow@31734
   569
 apply (metis int_gcd_greatest_iff int_gcd_pos zdvd_imp_le)
nipkow@31734
   570
apply simp
nipkow@31734
   571
done
nipkow@31734
   572
chaieb@22027
   573
huffman@31706
   574
subsection {* Coprimality *}
huffman@31706
   575
huffman@31706
   576
lemma nat_div_gcd_coprime [intro]:
huffman@31706
   577
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   578
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   579
proof -
haftmann@27556
   580
  let ?g = "gcd a b"
chaieb@22027
   581
  let ?a' = "a div ?g"
chaieb@22027
   582
  let ?b' = "b div ?g"
haftmann@27556
   583
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   584
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   585
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   586
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   587
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   588
    unfolding dvd_def by blast
huffman@31706
   589
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   590
    by simp_all
wenzelm@22367
   591
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   592
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   593
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@31706
   594
  have "?g \<noteq> 0" using nz by (simp add: nat_gcd_zero)
huffman@31706
   595
  then have gp: "?g > 0" by arith
huffman@31706
   596
  from nat_gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   597
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   598
qed
chaieb@22027
   599
huffman@31706
   600
lemma int_div_gcd_coprime [intro]:
huffman@31706
   601
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   602
  shows "coprime (a div gcd a b) (b div gcd a b)"
chaieb@27669
   603
huffman@31706
   604
  apply (subst (1 2 3) int_gcd_abs)
huffman@31706
   605
  apply (subst (1 2) abs_div)
huffman@31706
   606
  apply auto
huffman@31706
   607
  prefer 3
huffman@31706
   608
  apply (rule nat_div_gcd_coprime [transferred])
huffman@31706
   609
  using nz apply (auto simp add: int_gcd_abs [symmetric])
huffman@31706
   610
done
huffman@31706
   611
huffman@31706
   612
lemma nat_coprime: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
huffman@31706
   613
  using nat_gcd_unique[of 1 a b, simplified] by auto
huffman@31706
   614
huffman@31706
   615
lemma nat_coprime_Suc_0:
huffman@31706
   616
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
huffman@31706
   617
  using nat_coprime by (simp add: One_nat_def)
huffman@31706
   618
huffman@31706
   619
lemma int_coprime: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   620
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
huffman@31706
   621
  using int_gcd_unique [of 1 a b]
huffman@31706
   622
  apply clarsimp
huffman@31706
   623
  apply (erule subst)
huffman@31706
   624
  apply (rule iffI)
huffman@31706
   625
  apply force
huffman@31706
   626
  apply (drule_tac x = "abs e" in exI)
huffman@31706
   627
  apply (case_tac "e >= 0")
huffman@31706
   628
  apply force
huffman@31706
   629
  apply force
huffman@31706
   630
done
huffman@31706
   631
huffman@31706
   632
lemma nat_gcd_coprime:
huffman@31706
   633
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   634
    b: "b = b' * gcd a b"
huffman@31706
   635
  shows    "coprime a' b'"
huffman@31706
   636
huffman@31706
   637
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   638
  apply (erule ssubst)
huffman@31706
   639
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   640
  apply (erule ssubst)
huffman@31706
   641
  apply (rule nat_div_gcd_coprime)
huffman@31706
   642
  using prems
huffman@31706
   643
  apply force
huffman@31706
   644
  apply (subst (1) b)
huffman@31706
   645
  using z apply force
huffman@31706
   646
  apply (subst (1) a)
huffman@31706
   647
  using z apply force
huffman@31706
   648
done
huffman@31706
   649
huffman@31706
   650
lemma int_gcd_coprime:
huffman@31706
   651
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   652
    b: "b = b' * gcd a b"
huffman@31706
   653
  shows    "coprime a' b'"
huffman@31706
   654
huffman@31706
   655
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   656
  apply (erule ssubst)
huffman@31706
   657
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   658
  apply (erule ssubst)
huffman@31706
   659
  apply (rule int_div_gcd_coprime)
huffman@31706
   660
  using prems
huffman@31706
   661
  apply force
huffman@31706
   662
  apply (subst (1) b)
huffman@31706
   663
  using z apply force
huffman@31706
   664
  apply (subst (1) a)
huffman@31706
   665
  using z apply force
huffman@31706
   666
done
huffman@31706
   667
huffman@31706
   668
lemma nat_coprime_mult: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   669
    shows "coprime d (a * b)"
huffman@31706
   670
  apply (subst nat_gcd_commute)
huffman@31706
   671
  using da apply (subst nat_gcd_mult_cancel)
huffman@31706
   672
  apply (subst nat_gcd_commute, assumption)
huffman@31706
   673
  apply (subst nat_gcd_commute, rule db)
huffman@31706
   674
done
huffman@31706
   675
huffman@31706
   676
lemma int_coprime_mult: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   677
    shows "coprime d (a * b)"
huffman@31706
   678
  apply (subst int_gcd_commute)
huffman@31706
   679
  using da apply (subst int_gcd_mult_cancel)
huffman@31706
   680
  apply (subst int_gcd_commute, assumption)
huffman@31706
   681
  apply (subst int_gcd_commute, rule db)
huffman@31706
   682
done
huffman@31706
   683
huffman@31706
   684
lemma nat_coprime_lmult:
huffman@31706
   685
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   686
proof -
huffman@31706
   687
  have "gcd d a dvd gcd d (a * b)"
huffman@31706
   688
    by (rule nat_gcd_greatest, auto)
huffman@31706
   689
  with dab show ?thesis
huffman@31706
   690
    by auto
huffman@31706
   691
qed
huffman@31706
   692
huffman@31706
   693
lemma int_coprime_lmult:
nipkow@31798
   694
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   695
proof -
huffman@31706
   696
  have "gcd d a dvd gcd d (a * b)"
huffman@31706
   697
    by (rule int_gcd_greatest, auto)
nipkow@31798
   698
  with assms show ?thesis
huffman@31706
   699
    by auto
huffman@31706
   700
qed
huffman@31706
   701
huffman@31706
   702
lemma nat_coprime_rmult:
nipkow@31798
   703
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   704
proof -
huffman@31706
   705
  have "gcd d b dvd gcd d (a * b)"
huffman@31706
   706
    by (rule nat_gcd_greatest, auto intro: dvd_mult)
nipkow@31798
   707
  with assms show ?thesis
huffman@31706
   708
    by auto
huffman@31706
   709
qed
huffman@31706
   710
huffman@31706
   711
lemma int_coprime_rmult:
huffman@31706
   712
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   713
proof -
huffman@31706
   714
  have "gcd d b dvd gcd d (a * b)"
huffman@31706
   715
    by (rule int_gcd_greatest, auto intro: dvd_mult)
huffman@31706
   716
  with dab show ?thesis
huffman@31706
   717
    by auto
huffman@31706
   718
qed
huffman@31706
   719
huffman@31706
   720
lemma nat_coprime_mul_eq: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   721
    coprime d a \<and>  coprime d b"
huffman@31706
   722
  using nat_coprime_rmult[of d a b] nat_coprime_lmult[of d a b]
huffman@31706
   723
    nat_coprime_mult[of d a b]
huffman@31706
   724
  by blast
huffman@31706
   725
huffman@31706
   726
lemma int_coprime_mul_eq: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   727
    coprime d a \<and>  coprime d b"
huffman@31706
   728
  using int_coprime_rmult[of d a b] int_coprime_lmult[of d a b]
huffman@31706
   729
    int_coprime_mult[of d a b]
huffman@31706
   730
  by blast
huffman@31706
   731
huffman@31706
   732
lemma nat_gcd_coprime_exists:
huffman@31706
   733
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   734
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   735
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   736
  apply (rule_tac x = "b div gcd a b" in exI)
huffman@31706
   737
  using nz apply (auto simp add: nat_div_gcd_coprime dvd_div_mult)
huffman@31706
   738
done
huffman@31706
   739
huffman@31706
   740
lemma int_gcd_coprime_exists:
huffman@31706
   741
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   742
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   743
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   744
  apply (rule_tac x = "b div gcd a b" in exI)
huffman@31706
   745
  using nz apply (auto simp add: int_div_gcd_coprime dvd_div_mult_self)
huffman@31706
   746
done
huffman@31706
   747
huffman@31706
   748
lemma nat_coprime_exp: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
huffman@31706
   749
  by (induct n, simp_all add: nat_coprime_mult)
huffman@31706
   750
huffman@31706
   751
lemma int_coprime_exp: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
huffman@31706
   752
  by (induct n, simp_all add: int_coprime_mult)
huffman@31706
   753
huffman@31706
   754
lemma nat_coprime_exp2 [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
huffman@31706
   755
  apply (rule nat_coprime_exp)
huffman@31706
   756
  apply (subst nat_gcd_commute)
huffman@31706
   757
  apply (rule nat_coprime_exp)
huffman@31706
   758
  apply (subst nat_gcd_commute, assumption)
huffman@31706
   759
done
huffman@31706
   760
huffman@31706
   761
lemma int_coprime_exp2 [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
huffman@31706
   762
  apply (rule int_coprime_exp)
huffman@31706
   763
  apply (subst int_gcd_commute)
huffman@31706
   764
  apply (rule int_coprime_exp)
huffman@31706
   765
  apply (subst int_gcd_commute, assumption)
huffman@31706
   766
done
huffman@31706
   767
huffman@31706
   768
lemma nat_gcd_exp: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   769
proof (cases)
huffman@31706
   770
  assume "a = 0 & b = 0"
huffman@31706
   771
  thus ?thesis by simp
huffman@31706
   772
  next assume "~(a = 0 & b = 0)"
huffman@31706
   773
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
huffman@31706
   774
    by auto
huffman@31706
   775
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   776
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   777
    apply (subst (1 2) mult_commute)
huffman@31706
   778
    apply (subst nat_gcd_mult_distrib [symmetric])
huffman@31706
   779
    apply simp
huffman@31706
   780
    done
huffman@31706
   781
  also have "(a div gcd a b)^n * (gcd a b)^n = a^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
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   789
    apply (subst div_power)
huffman@31706
   790
    apply auto
huffman@31706
   791
    apply (rule dvd_div_mult_self)
huffman@31706
   792
    apply (rule dvd_power_same)
huffman@31706
   793
    apply auto
huffman@31706
   794
    done
huffman@31706
   795
  finally show ?thesis .
huffman@31706
   796
qed
huffman@31706
   797
huffman@31706
   798
lemma int_gcd_exp: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
huffman@31706
   799
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   800
  apply (subst (1 2) power_abs)
huffman@31706
   801
  apply (rule nat_gcd_exp [where n = n, transferred])
huffman@31706
   802
  apply auto
huffman@31706
   803
done
huffman@31706
   804
huffman@31706
   805
lemma nat_coprime_divprod: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
huffman@31706
   806
  using nat_coprime_dvd_mult_iff[of d a b]
huffman@31706
   807
  by (auto simp add: mult_commute)
huffman@31706
   808
huffman@31706
   809
lemma int_coprime_divprod: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
huffman@31706
   810
  using int_coprime_dvd_mult_iff[of d a b]
huffman@31706
   811
  by (auto simp add: mult_commute)
huffman@31706
   812
huffman@31706
   813
lemma nat_division_decomp: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   814
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   815
proof-
huffman@31706
   816
  let ?g = "gcd a b"
huffman@31706
   817
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   818
  moreover
huffman@31706
   819
  {assume z: "?g \<noteq> 0"
huffman@31706
   820
    from nat_gcd_coprime_exists[OF z]
huffman@31706
   821
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   822
      by blast
huffman@31706
   823
    have thb: "?g dvd b" by auto
huffman@31706
   824
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   825
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   826
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   827
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   828
    with z have th_1: "a' dvd b' * c" by auto
huffman@31706
   829
    from nat_coprime_dvd_mult[OF ab'(3)] th_1
huffman@31706
   830
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   831
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   832
    with thb thc have ?thesis by blast }
huffman@31706
   833
  ultimately show ?thesis by blast
huffman@31706
   834
qed
huffman@31706
   835
huffman@31706
   836
lemma int_division_decomp: assumes dc: "(a::int) dvd b * c"
huffman@31706
   837
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   838
proof-
huffman@31706
   839
  let ?g = "gcd a b"
huffman@31706
   840
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   841
  moreover
huffman@31706
   842
  {assume z: "?g \<noteq> 0"
huffman@31706
   843
    from int_gcd_coprime_exists[OF z]
huffman@31706
   844
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   845
      by blast
huffman@31706
   846
    have thb: "?g dvd b" by auto
huffman@31706
   847
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   848
    with dc have th0: "a' dvd b*c"
huffman@31706
   849
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   850
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   851
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   852
    with z have th_1: "a' dvd b' * c" by auto
huffman@31706
   853
    from int_coprime_dvd_mult[OF ab'(3)] th_1
huffman@31706
   854
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   855
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   856
    with thb thc have ?thesis by blast }
huffman@31706
   857
  ultimately show ?thesis by blast
chaieb@27669
   858
qed
chaieb@27669
   859
huffman@31706
   860
lemma nat_pow_divides_pow:
huffman@31706
   861
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   862
  shows "a dvd b"
huffman@31706
   863
proof-
huffman@31706
   864
  let ?g = "gcd a b"
huffman@31706
   865
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   866
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   867
  moreover
huffman@31706
   868
  {assume z: "?g \<noteq> 0"
huffman@31706
   869
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
huffman@31706
   870
    from nat_gcd_coprime_exists[OF z]
huffman@31706
   871
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   872
      by blast
huffman@31706
   873
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   874
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   875
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   876
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   877
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   878
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   879
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   880
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
huffman@31706
   881
    from nat_coprime_dvd_mult[OF nat_coprime_exp [OF ab'(3), of m]] th1
huffman@31706
   882
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   883
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   884
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   885
  ultimately show ?thesis by blast
huffman@31706
   886
qed
huffman@31706
   887
huffman@31706
   888
lemma int_pow_divides_pow:
huffman@31706
   889
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   890
  shows "a dvd b"
chaieb@27669
   891
proof-
huffman@31706
   892
  let ?g = "gcd a b"
huffman@31706
   893
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   894
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   895
  moreover
huffman@31706
   896
  {assume z: "?g \<noteq> 0"
huffman@31706
   897
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
huffman@31706
   898
    from int_gcd_coprime_exists[OF z]
huffman@31706
   899
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   900
      by blast
huffman@31706
   901
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   902
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   903
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   904
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   905
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   906
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   907
    with th0 have "a' dvd b'^n"
huffman@31706
   908
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   909
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
huffman@31706
   910
    from int_coprime_dvd_mult[OF int_coprime_exp [OF ab'(3), of m]] th1
huffman@31706
   911
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   912
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   913
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   914
  ultimately show ?thesis by blast
huffman@31706
   915
qed
huffman@31706
   916
nipkow@31798
   917
(* FIXME move to Divides(?) *)
huffman@31706
   918
lemma nat_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
huffman@31706
   919
  by (auto intro: nat_pow_divides_pow dvd_power_same)
huffman@31706
   920
huffman@31706
   921
lemma int_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
huffman@31706
   922
  by (auto intro: int_pow_divides_pow dvd_power_same)
huffman@31706
   923
huffman@31706
   924
lemma nat_divides_mult:
huffman@31706
   925
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   926
  shows "m * n dvd r"
huffman@31706
   927
proof-
huffman@31706
   928
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   929
    unfolding dvd_def by blast
huffman@31706
   930
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
huffman@31706
   931
  hence "m dvd n'" using nat_coprime_dvd_mult_iff[OF mn] by simp
huffman@31706
   932
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   933
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   934
qed
huffman@31706
   935
huffman@31706
   936
lemma int_divides_mult:
huffman@31706
   937
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   938
  shows "m * n dvd r"
huffman@31706
   939
proof-
huffman@31706
   940
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   941
    unfolding dvd_def by blast
huffman@31706
   942
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
huffman@31706
   943
  hence "m dvd n'" using int_coprime_dvd_mult_iff[OF mn] by simp
huffman@31706
   944
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   945
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   946
qed
chaieb@27669
   947
huffman@31706
   948
lemma nat_coprime_plus_one [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   949
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   950
  apply force
huffman@31706
   951
  apply (rule nat_dvd_diff)
huffman@31706
   952
  apply auto
huffman@31706
   953
done
huffman@31706
   954
huffman@31706
   955
lemma nat_coprime_Suc [simp]: "coprime (Suc n) n"
huffman@31706
   956
  using nat_coprime_plus_one by (simp add: One_nat_def)
huffman@31706
   957
huffman@31706
   958
lemma int_coprime_plus_one [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   959
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   960
  apply force
huffman@31706
   961
  apply (rule dvd_diff)
huffman@31706
   962
  apply auto
huffman@31706
   963
done
huffman@31706
   964
huffman@31706
   965
lemma nat_coprime_minus_one: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
huffman@31706
   966
  using nat_coprime_plus_one [of "n - 1"]
huffman@31706
   967
    nat_gcd_commute [of "n - 1" n] by auto
huffman@31706
   968
huffman@31706
   969
lemma int_coprime_minus_one: "coprime ((n::int) - 1) n"
huffman@31706
   970
  using int_coprime_plus_one [of "n - 1"]
huffman@31706
   971
    int_gcd_commute [of "n - 1" n] by auto
huffman@31706
   972
huffman@31706
   973
lemma nat_setprod_coprime [rule_format]:
huffman@31706
   974
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   975
  apply (case_tac "finite A")
huffman@31706
   976
  apply (induct set: finite)
huffman@31706
   977
  apply (auto simp add: nat_gcd_mult_cancel)
huffman@31706
   978
done
huffman@31706
   979
huffman@31706
   980
lemma int_setprod_coprime [rule_format]:
huffman@31706
   981
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
   982
  apply (case_tac "finite A")
huffman@31706
   983
  apply (induct set: finite)
huffman@31706
   984
  apply (auto simp add: int_gcd_mult_cancel)
huffman@31706
   985
done
huffman@31706
   986
huffman@31706
   987
lemma nat_prime_odd: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
   988
  unfolding prime_nat_def
huffman@31706
   989
  apply (subst even_mult_two_ex)
huffman@31706
   990
  apply clarify
huffman@31706
   991
  apply (drule_tac x = 2 in spec)
huffman@31706
   992
  apply auto
huffman@31706
   993
done
huffman@31706
   994
huffman@31706
   995
lemma int_prime_odd: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
   996
  unfolding prime_int_def
huffman@31706
   997
  apply (frule nat_prime_odd)
huffman@31706
   998
  apply (auto simp add: even_nat_def)
huffman@31706
   999
done
huffman@31706
  1000
huffman@31706
  1001
lemma nat_coprime_common_divisor: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1002
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
  1003
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1004
  apply simp
huffman@31706
  1005
  apply (erule (1) nat_gcd_greatest)
huffman@31706
  1006
done
huffman@31706
  1007
huffman@31706
  1008
lemma int_coprime_common_divisor: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1009
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
  1010
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1011
  apply simp
huffman@31706
  1012
  apply (erule (1) int_gcd_greatest)
huffman@31706
  1013
done
huffman@31706
  1014
huffman@31706
  1015
lemma nat_coprime_divisors: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
  1016
    coprime d e"
huffman@31706
  1017
  apply (auto simp add: dvd_def)
huffman@31706
  1018
  apply (frule int_coprime_lmult)
huffman@31706
  1019
  apply (subst int_gcd_commute)
huffman@31706
  1020
  apply (subst (asm) (2) int_gcd_commute)
huffman@31706
  1021
  apply (erule int_coprime_lmult)
huffman@31706
  1022
done
huffman@31706
  1023
huffman@31706
  1024
lemma nat_invertible_coprime: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
huffman@31706
  1025
apply (metis nat_coprime_lmult nat_gcd_1 nat_gcd_commute nat_gcd_red)
huffman@31706
  1026
done
huffman@31706
  1027
huffman@31706
  1028
lemma int_invertible_coprime: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
huffman@31706
  1029
apply (metis int_coprime_lmult int_gcd_1 int_gcd_commute int_gcd_red)
huffman@31706
  1030
done
huffman@31706
  1031
huffman@31706
  1032
huffman@31706
  1033
subsection {* Bezout's theorem *}
huffman@31706
  1034
huffman@31706
  1035
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1036
   see the theorems that follow the definition. *)
huffman@31706
  1037
fun
huffman@31706
  1038
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1039
where
huffman@31706
  1040
  "bezw x y =
huffman@31706
  1041
  (if y = 0 then (1, 0) else
huffman@31706
  1042
      (snd (bezw y (x mod y)),
huffman@31706
  1043
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1044
huffman@31706
  1045
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1046
huffman@31706
  1047
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1048
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1049
  by simp
huffman@31706
  1050
huffman@31706
  1051
declare bezw.simps [simp del]
huffman@31706
  1052
huffman@31706
  1053
lemma bezw_aux [rule_format]:
huffman@31706
  1054
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
huffman@31706
  1055
proof (induct x y rule: nat_gcd_induct)
huffman@31706
  1056
  fix m :: nat
huffman@31706
  1057
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1058
    by auto
huffman@31706
  1059
  next fix m :: nat and n
huffman@31706
  1060
    assume ngt0: "n > 0" and
huffman@31706
  1061
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1062
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1063
        int (gcd n (m mod n))"
huffman@31706
  1064
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
huffman@31706
  1065
      apply (simp add: bezw_non_0 nat_gcd_non_0)
huffman@31706
  1066
      apply (erule subst)
huffman@31706
  1067
      apply (simp add: ring_simps)
huffman@31706
  1068
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1069
      (* applying simp here undoes the last substitution!
huffman@31706
  1070
         what is procedure cancel_div_mod? *)
huffman@31706
  1071
      apply (simp only: ring_simps zadd_int [symmetric]
huffman@31706
  1072
        zmult_int [symmetric])
huffman@31706
  1073
      done
huffman@31706
  1074
qed
huffman@31706
  1075
huffman@31706
  1076
lemma int_bezout:
huffman@31706
  1077
  fixes x y
huffman@31706
  1078
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1079
proof -
huffman@31706
  1080
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1081
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1082
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1083
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1084
    apply (unfold gcd_int_def)
huffman@31706
  1085
    apply simp
huffman@31706
  1086
    apply (subst bezw_aux [symmetric])
huffman@31706
  1087
    apply auto
huffman@31706
  1088
    done
huffman@31706
  1089
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1090
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1091
    by auto
huffman@31706
  1092
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1093
    by (erule (1) bezout_aux)
huffman@31706
  1094
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1095
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1096
    apply auto
huffman@31706
  1097
    apply (rule_tac x = u in exI)
huffman@31706
  1098
    apply (rule_tac x = "-v" in exI)
huffman@31706
  1099
    apply (subst int_gcd_neg2 [symmetric])
huffman@31706
  1100
    apply auto
huffman@31706
  1101
    done
huffman@31706
  1102
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1103
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1104
    apply auto
huffman@31706
  1105
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1106
    apply (rule_tac x = v in exI)
huffman@31706
  1107
    apply (subst int_gcd_neg1 [symmetric])
huffman@31706
  1108
    apply auto
huffman@31706
  1109
    done
huffman@31706
  1110
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1111
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1112
    apply auto
huffman@31706
  1113
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1114
    apply (rule_tac x = "-v" in exI)
huffman@31706
  1115
    apply (subst int_gcd_neg1 [symmetric])
huffman@31706
  1116
    apply (subst int_gcd_neg2 [symmetric])
huffman@31706
  1117
    apply auto
huffman@31706
  1118
    done
huffman@31706
  1119
  ultimately show ?thesis by blast
huffman@31706
  1120
qed
huffman@31706
  1121
huffman@31706
  1122
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1123
huffman@31706
  1124
lemma ind_euclid:
huffman@31706
  1125
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1126
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1127
  shows "P a b"
chaieb@27669
  1128
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
chaieb@27669
  1129
  fix n a b
chaieb@27669
  1130
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
chaieb@27669
  1131
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1132
  moreover {assume eq: "a= b"
huffman@31706
  1133
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1134
    by simp}
chaieb@27669
  1135
  moreover
chaieb@27669
  1136
  {assume lt: "a < b"
chaieb@27669
  1137
    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
chaieb@27669
  1138
    moreover
chaieb@27669
  1139
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1140
    moreover
chaieb@27669
  1141
    {assume ab: "a + b - a < n"
chaieb@27669
  1142
      have th0: "a + b - a = a + (b - a)" using lt by arith
chaieb@27669
  1143
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1144
      have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1145
    ultimately have "P a b" by blast}
chaieb@27669
  1146
  moreover
chaieb@27669
  1147
  {assume lt: "a > b"
chaieb@27669
  1148
    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
chaieb@27669
  1149
    moreover
chaieb@27669
  1150
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1151
    moreover
chaieb@27669
  1152
    {assume ab: "b + a - b < n"
chaieb@27669
  1153
      have th0: "b + a - b = b + (a - b)" using lt by arith
chaieb@27669
  1154
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1155
      have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1156
      hence "P a b" using c by blast }
chaieb@27669
  1157
    ultimately have "P a b" by blast}
chaieb@27669
  1158
ultimately  show "P a b" by blast
chaieb@27669
  1159
qed
chaieb@27669
  1160
huffman@31706
  1161
lemma nat_bezout_lemma:
huffman@31706
  1162
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1163
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1164
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1165
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1166
  using ex
huffman@31706
  1167
  apply clarsimp
huffman@31706
  1168
  apply (rule_tac x="d" in exI, simp add: dvd_add)
huffman@31706
  1169
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1170
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1171
  apply (rule_tac x="y" in exI)
huffman@31706
  1172
  apply algebra
huffman@31706
  1173
  apply (rule_tac x="x" in exI)
huffman@31706
  1174
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1175
  apply algebra
chaieb@27669
  1176
done
chaieb@27669
  1177
huffman@31706
  1178
lemma nat_bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1179
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1180
  apply(induct a b rule: ind_euclid)
huffman@31706
  1181
  apply blast
huffman@31706
  1182
  apply clarify
huffman@31706
  1183
  apply (rule_tac x="a" in exI, simp add: dvd_add)
huffman@31706
  1184
  apply clarsimp
huffman@31706
  1185
  apply (rule_tac x="d" in exI)
huffman@31706
  1186
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
huffman@31706
  1187
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1188
  apply (rule_tac x="y" in exI)
huffman@31706
  1189
  apply algebra
huffman@31706
  1190
  apply (rule_tac x="x" in exI)
huffman@31706
  1191
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1192
  apply algebra
chaieb@27669
  1193
done
chaieb@27669
  1194
huffman@31706
  1195
lemma nat_bezout1: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1196
    (a * x - b * y = d \<or> b * x - a * y = d)"
huffman@31706
  1197
  using nat_bezout_add[of a b]
huffman@31706
  1198
  apply clarsimp
huffman@31706
  1199
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1200
  apply (rule_tac x="x" in exI)
huffman@31706
  1201
  apply (rule_tac x="y" in exI)
huffman@31706
  1202
  apply auto
chaieb@27669
  1203
done
chaieb@27669
  1204
huffman@31706
  1205
lemma nat_bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1206
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1207
proof-
huffman@31706
  1208
 from nz have ap: "a > 0" by simp
huffman@31706
  1209
 from nat_bezout_add[of a b]
huffman@31706
  1210
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1211
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1212
 moreover
huffman@31706
  1213
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1214
     from H have ?thesis by blast }
chaieb@27669
  1215
 moreover
chaieb@27669
  1216
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1217
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1218
   moreover
chaieb@27669
  1219
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1220
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1221
       by auto
chaieb@27669
  1222
     moreover
chaieb@27669
  1223
     {assume db: "d=b"
chaieb@27669
  1224
       from prems have ?thesis apply simp
chaieb@27669
  1225
	 apply (rule exI[where x = b], simp)
chaieb@27669
  1226
	 apply (rule exI[where x = b])
chaieb@27669
  1227
	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1228
    moreover
huffman@31706
  1229
    {assume db: "d < b"
chaieb@27669
  1230
	{assume "x=0" hence ?thesis  using prems by simp }
chaieb@27669
  1231
	moreover
chaieb@27669
  1232
	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
chaieb@27669
  1233
	  from db have "d \<le> b - 1" by simp
chaieb@27669
  1234
	  hence "d*b \<le> b*(b - 1)" by simp
chaieb@27669
  1235
	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
chaieb@27669
  1236
	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
huffman@31706
  1237
	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1238
            by simp
huffman@31706
  1239
	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
huffman@31706
  1240
	    by (simp only: mult_assoc right_distrib)
huffman@31706
  1241
	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1242
            by algebra
chaieb@27669
  1243
	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
huffman@31706
  1244
	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
chaieb@27669
  1245
	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
chaieb@27669
  1246
	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
chaieb@27669
  1247
	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
chaieb@27669
  1248
	  hence ?thesis using H(1,2)
chaieb@27669
  1249
	    apply -
chaieb@27669
  1250
	    apply (rule exI[where x=d], simp)
chaieb@27669
  1251
	    apply (rule exI[where x="(b - 1) * y"])
chaieb@27669
  1252
	    by (rule exI[where x="x*(b - 1) - d"], simp)}
chaieb@27669
  1253
	ultimately have ?thesis by blast}
chaieb@27669
  1254
    ultimately have ?thesis by blast}
chaieb@27669
  1255
  ultimately have ?thesis by blast}
chaieb@27669
  1256
 ultimately show ?thesis by blast
chaieb@27669
  1257
qed
chaieb@27669
  1258
huffman@31706
  1259
lemma nat_bezout: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1260
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1261
proof-
chaieb@27669
  1262
  let ?g = "gcd a b"
huffman@31706
  1263
  from nat_bezout_add_strong[OF a, of b]
chaieb@27669
  1264
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1265
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1266
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1267
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1268
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1269
  thus ?thesis by blast
chaieb@27669
  1270
qed
chaieb@27669
  1271
huffman@31706
  1272
huffman@31706
  1273
subsection {* LCM *}
huffman@31706
  1274
huffman@31706
  1275
lemma int_lcm_altdef: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1276
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@31706
  1277
    zmult_int [symmetric] gcd_int_def)
huffman@31706
  1278
huffman@31706
  1279
lemma nat_prod_gcd_lcm: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1280
  unfolding lcm_nat_def
huffman@31706
  1281
  by (simp add: dvd_mult_div_cancel [OF nat_gcd_dvd_prod])
huffman@31706
  1282
huffman@31706
  1283
lemma int_prod_gcd_lcm: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1284
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1285
  apply (subst int_mult [symmetric])
huffman@31706
  1286
  apply (subst nat_prod_gcd_lcm [symmetric])
huffman@31706
  1287
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1288
  apply (simp, simp add: abs_mult)
huffman@31706
  1289
done
huffman@31706
  1290
huffman@31706
  1291
lemma nat_lcm_0 [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1292
  unfolding lcm_nat_def by simp
huffman@31706
  1293
huffman@31706
  1294
lemma int_lcm_0 [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1295
  unfolding lcm_int_def by simp
huffman@31706
  1296
huffman@31706
  1297
lemma nat_lcm_0_left [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1298
  unfolding lcm_nat_def by simp
chaieb@27669
  1299
huffman@31706
  1300
lemma int_lcm_0_left [simp]: "lcm (0::int) n = 0"
huffman@31706
  1301
  unfolding lcm_int_def by simp
huffman@31706
  1302
huffman@31706
  1303
lemma nat_lcm_commute: "lcm (m::nat) n = lcm n m"
huffman@31706
  1304
  unfolding lcm_nat_def by (simp add: nat_gcd_commute ring_simps)
huffman@31706
  1305
huffman@31706
  1306
lemma int_lcm_commute: "lcm (m::int) n = lcm n m"
huffman@31706
  1307
  unfolding lcm_int_def by (subst nat_lcm_commute, rule refl)
huffman@31706
  1308
huffman@31706
  1309
huffman@31706
  1310
lemma nat_lcm_pos:
nipkow@31798
  1311
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31798
  1312
by (metis gr0I mult_is_0 nat_prod_gcd_lcm)
chaieb@27669
  1313
huffman@31706
  1314
lemma int_lcm_pos:
nipkow@31798
  1315
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
huffman@31706
  1316
  apply (subst int_lcm_abs)
huffman@31706
  1317
  apply (rule nat_lcm_pos [transferred])
nipkow@31798
  1318
  apply auto
huffman@31706
  1319
done
haftmann@23687
  1320
huffman@31706
  1321
lemma nat_dvd_pos:
haftmann@23687
  1322
  fixes n m :: nat
haftmann@23687
  1323
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1324
  shows "m > 0"
haftmann@23687
  1325
using assms by (cases m) auto
haftmann@23687
  1326
huffman@31706
  1327
lemma nat_lcm_least:
huffman@31706
  1328
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1329
  shows "lcm m n dvd k"
haftmann@23687
  1330
proof (cases k)
haftmann@23687
  1331
  case 0 then show ?thesis by auto
haftmann@23687
  1332
next
haftmann@23687
  1333
  case (Suc _) then have pos_k: "k > 0" by auto
huffman@31706
  1334
  from assms nat_dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
huffman@31706
  1335
  with nat_gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1336
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1337
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1338
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1339
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1340
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
huffman@31706
  1341
    by (simp add: mult_ac nat_gcd_mult_distrib)
haftmann@27556
  1342
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1343
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1344
  also have "\<dots> = k * p * q * gcd m n"
huffman@31706
  1345
    by (simp add: mult_ac nat_gcd_mult_distrib)
haftmann@27556
  1346
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1347
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1348
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1349
    by (simp add: mult_ac)
haftmann@27556
  1350
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1351
    by simp
huffman@31706
  1352
  with nat_prod_gcd_lcm [of m n]
haftmann@27556
  1353
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1354
    by (simp add: mult_ac)
huffman@31706
  1355
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1356
  then show ?thesis using dvd_def by auto
haftmann@23687
  1357
qed
haftmann@23687
  1358
huffman@31706
  1359
lemma int_lcm_least:
nipkow@31798
  1360
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
nipkow@31798
  1361
apply (subst int_lcm_abs)
nipkow@31798
  1362
apply (rule dvd_trans)
nipkow@31798
  1363
apply (rule nat_lcm_least [transferred, of _ "abs k" _])
nipkow@31798
  1364
apply auto
huffman@31706
  1365
done
huffman@31706
  1366
nipkow@31730
  1367
lemma nat_lcm_dvd1: "(m::nat) dvd lcm m n"
haftmann@23687
  1368
proof (cases m)
haftmann@23687
  1369
  case 0 then show ?thesis by simp
haftmann@23687
  1370
next
haftmann@23687
  1371
  case (Suc _)
haftmann@23687
  1372
  then have mpos: "m > 0" by simp
haftmann@23687
  1373
  show ?thesis
haftmann@23687
  1374
  proof (cases n)
haftmann@23687
  1375
    case 0 then show ?thesis by simp
haftmann@23687
  1376
  next
haftmann@23687
  1377
    case (Suc _)
haftmann@23687
  1378
    then have npos: "n > 0" by simp
haftmann@27556
  1379
    have "gcd m n dvd n" by simp
haftmann@27556
  1380
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1381
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1382
      by (simp add: mult_ac)
huffman@31706
  1383
    also have "\<dots> = m * k" using mpos npos nat_gcd_zero by simp
huffman@31706
  1384
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1385
  qed
haftmann@23687
  1386
qed
haftmann@23687
  1387
nipkow@31730
  1388
lemma int_lcm_dvd1: "(m::int) dvd lcm m n"
huffman@31706
  1389
  apply (subst int_lcm_abs)
huffman@31706
  1390
  apply (rule dvd_trans)
huffman@31706
  1391
  prefer 2
huffman@31706
  1392
  apply (rule nat_lcm_dvd1 [transferred])
huffman@31706
  1393
  apply auto
huffman@31706
  1394
done
huffman@31706
  1395
nipkow@31730
  1396
lemma nat_lcm_dvd2: "(n::nat) dvd lcm m n"
huffman@31706
  1397
  by (subst nat_lcm_commute, rule nat_lcm_dvd1)
huffman@31706
  1398
nipkow@31730
  1399
lemma int_lcm_dvd2: "(n::int) dvd lcm m n"
huffman@31706
  1400
  by (subst int_lcm_commute, rule int_lcm_dvd1)
huffman@31706
  1401
nipkow@31730
  1402
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31729
  1403
by(metis nat_lcm_dvd1 dvd_trans)
nipkow@31729
  1404
nipkow@31730
  1405
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31729
  1406
by(metis nat_lcm_dvd2 dvd_trans)
nipkow@31729
  1407
nipkow@31730
  1408
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31729
  1409
by(metis int_lcm_dvd1 dvd_trans)
nipkow@31729
  1410
nipkow@31730
  1411
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31729
  1412
by(metis int_lcm_dvd2 dvd_trans)
nipkow@31729
  1413
huffman@31706
  1414
lemma nat_lcm_unique: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1415
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31730
  1416
  by (auto intro: dvd_anti_sym nat_lcm_least nat_lcm_dvd1 nat_lcm_dvd2)
chaieb@27568
  1417
huffman@31706
  1418
lemma int_lcm_unique: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1419
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
huffman@31706
  1420
  by (auto intro: dvd_anti_sym [transferred] int_lcm_least)
huffman@31706
  1421
nipkow@31798
  1422
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1423
  apply (rule sym)
huffman@31706
  1424
  apply (subst nat_lcm_unique [symmetric])
huffman@31706
  1425
  apply auto
huffman@31706
  1426
done
huffman@31706
  1427
nipkow@31798
  1428
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1429
  apply (rule sym)
huffman@31706
  1430
  apply (subst int_lcm_unique [symmetric])
huffman@31706
  1431
  apply auto
huffman@31706
  1432
done
huffman@31706
  1433
nipkow@31798
  1434
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31798
  1435
by (subst nat_lcm_commute, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1436
nipkow@31798
  1437
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31798
  1438
by (subst int_lcm_commute, erule lcm_proj2_if_dvd_int)
huffman@31706
  1439
chaieb@27568
  1440
nipkow@31766
  1441
lemma lcm_assoc_nat: "lcm (lcm n m) (p::nat) = lcm n (lcm m p)"
nipkow@31766
  1442
apply(rule nat_lcm_unique[THEN iffD1])
nipkow@31766
  1443
apply (metis dvd.order_trans nat_lcm_unique)
nipkow@31766
  1444
done
nipkow@31766
  1445
nipkow@31766
  1446
lemma lcm_assoc_int: "lcm (lcm n m) (p::int) = lcm n (lcm m p)"
nipkow@31766
  1447
apply(rule int_lcm_unique[THEN iffD1])
nipkow@31766
  1448
apply (metis dvd_trans int_lcm_unique)
nipkow@31766
  1449
done
nipkow@31766
  1450
nipkow@31766
  1451
lemmas lcm_left_commute_nat =
nipkow@31766
  1452
  mk_left_commute[of lcm, OF lcm_assoc_nat nat_lcm_commute]
nipkow@31766
  1453
nipkow@31766
  1454
lemmas lcm_left_commute_int =
nipkow@31766
  1455
  mk_left_commute[of lcm, OF lcm_assoc_int int_lcm_commute]
nipkow@31766
  1456
nipkow@31766
  1457
lemmas lcm_ac_nat = lcm_assoc_nat nat_lcm_commute lcm_left_commute_nat
nipkow@31766
  1458
lemmas lcm_ac_int = lcm_assoc_int int_lcm_commute lcm_left_commute_int
nipkow@31766
  1459
haftmann@23687
  1460
huffman@31706
  1461
subsection {* Primes *}
wenzelm@22367
  1462
huffman@31706
  1463
(* Is there a better way to handle these, rather than making them
huffman@31706
  1464
   elim rules? *)
chaieb@22027
  1465
huffman@31706
  1466
lemma nat_prime_ge_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
huffman@31706
  1467
  by (unfold prime_nat_def, auto)
chaieb@22027
  1468
huffman@31706
  1469
lemma nat_prime_gt_0 [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
huffman@31706
  1470
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1471
huffman@31706
  1472
lemma nat_prime_ge_1 [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
huffman@31706
  1473
  by (unfold prime_nat_def, auto)
chaieb@22027
  1474
huffman@31706
  1475
lemma nat_prime_gt_1 [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
huffman@31706
  1476
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1477
huffman@31706
  1478
lemma nat_prime_ge_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
huffman@31706
  1479
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1480
huffman@31706
  1481
lemma nat_prime_gt_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
huffman@31706
  1482
  by (unfold prime_nat_def, auto)
huffman@31706
  1483
huffman@31706
  1484
lemma nat_prime_ge_2 [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
huffman@31706
  1485
  by (unfold prime_nat_def, auto)
huffman@31706
  1486
huffman@31706
  1487
lemma int_prime_ge_0 [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
huffman@31706
  1488
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1489
huffman@31706
  1490
lemma int_prime_gt_0 [elim]: "prime (p::int) \<Longrightarrow> p > 0"
huffman@31706
  1491
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1492
huffman@31706
  1493
lemma int_prime_ge_1 [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
huffman@31706
  1494
  by (unfold prime_int_def prime_nat_def, auto)
chaieb@22027
  1495
huffman@31706
  1496
lemma int_prime_gt_1 [elim]: "prime (p::int) \<Longrightarrow> p > 1"
huffman@31706
  1497
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1498
huffman@31706
  1499
lemma int_prime_ge_2 [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
huffman@31706
  1500
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1501
huffman@31706
  1502
thm prime_nat_def;
huffman@31706
  1503
thm prime_nat_def [transferred];
huffman@31706
  1504
huffman@31706
  1505
lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
huffman@31706
  1506
    m = 1 \<or> m = p))"
huffman@31706
  1507
  using prime_nat_def [transferred]
huffman@31706
  1508
    apply (case_tac "p >= 0")
huffman@31706
  1509
    by (blast, auto simp add: int_prime_ge_0)
huffman@31706
  1510
huffman@31706
  1511
(* To do: determine primality of any numeral *)
huffman@31706
  1512
huffman@31706
  1513
lemma nat_zero_not_prime [simp]: "~prime (0::nat)"
huffman@31706
  1514
  by (simp add: prime_nat_def)
huffman@31706
  1515
huffman@31706
  1516
lemma int_zero_not_prime [simp]: "~prime (0::int)"
huffman@31706
  1517
  by (simp add: prime_int_def)
huffman@31706
  1518
huffman@31706
  1519
lemma nat_one_not_prime [simp]: "~prime (1::nat)"
huffman@31706
  1520
  by (simp add: prime_nat_def)
chaieb@22027
  1521
huffman@31706
  1522
lemma nat_Suc_0_not_prime [simp]: "~prime (Suc 0)"
huffman@31706
  1523
  by (simp add: prime_nat_def One_nat_def)
huffman@31706
  1524
huffman@31706
  1525
lemma int_one_not_prime [simp]: "~prime (1::int)"
huffman@31706
  1526
  by (simp add: prime_int_def)
huffman@31706
  1527
huffman@31706
  1528
lemma nat_two_is_prime [simp]: "prime (2::nat)"
huffman@31706
  1529
  apply (auto simp add: prime_nat_def)
huffman@31706
  1530
  apply (case_tac m)
huffman@31706
  1531
  apply (auto dest!: dvd_imp_le)
huffman@31706
  1532
  done
chaieb@22027
  1533
huffman@31706
  1534
lemma int_two_is_prime [simp]: "prime (2::int)"
huffman@31706
  1535
  by (rule nat_two_is_prime [transferred direction: nat "op <= (0::int)"])
chaieb@27568
  1536
huffman@31706
  1537
lemma nat_prime_imp_coprime: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1538
  apply (unfold prime_nat_def)
huffman@31706
  1539
  apply (metis nat_gcd_dvd1 nat_gcd_dvd2)
huffman@31706
  1540
  done
huffman@31706
  1541
huffman@31706
  1542
lemma int_prime_imp_coprime: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1543
  apply (unfold prime_int_altdef)
huffman@31706
  1544
  apply (metis int_gcd_dvd1 int_gcd_dvd2 int_gcd_ge_0)
chaieb@27568
  1545
  done
chaieb@27568
  1546
huffman@31706
  1547
lemma nat_prime_dvd_mult: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
huffman@31706
  1548
  by (blast intro: nat_coprime_dvd_mult nat_prime_imp_coprime)
huffman@31706
  1549
huffman@31706
  1550
lemma int_prime_dvd_mult: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
huffman@31706
  1551
  by (blast intro: int_coprime_dvd_mult int_prime_imp_coprime)
huffman@31706
  1552
huffman@31706
  1553
lemma nat_prime_dvd_mult_eq [simp]: "prime (p::nat) \<Longrightarrow>
huffman@31706
  1554
    p dvd m * n = (p dvd m \<or> p dvd n)"
huffman@31706
  1555
  by (rule iffI, rule nat_prime_dvd_mult, auto)
chaieb@27568
  1556
huffman@31706
  1557
lemma int_prime_dvd_mult_eq [simp]: "prime (p::int) \<Longrightarrow>
huffman@31706
  1558
    p dvd m * n = (p dvd m \<or> p dvd n)"
huffman@31706
  1559
  by (rule iffI, rule int_prime_dvd_mult, auto)
chaieb@27568
  1560
huffman@31706
  1561
lemma nat_not_prime_eq_prod: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1562
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1563
  unfolding prime_nat_def dvd_def apply auto
huffman@31706
  1564
  apply (subgoal_tac "k > 1")
huffman@31706
  1565
  apply force
huffman@31706
  1566
  apply (subgoal_tac "k ~= 0")
huffman@31706
  1567
  apply force
huffman@31706
  1568
  apply (rule notI)
huffman@31706
  1569
  apply force
huffman@31706
  1570
done
chaieb@27568
  1571
huffman@31706
  1572
(* there's a lot of messing around with signs of products here --
huffman@31706
  1573
   could this be made more automatic? *)
huffman@31706
  1574
lemma int_not_prime_eq_prod: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1575
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1576
  unfolding prime_int_altdef dvd_def
huffman@31706
  1577
  apply auto
huffman@31706
  1578
  apply (rule_tac x = m in exI)
huffman@31706
  1579
  apply (rule_tac x = k in exI)
huffman@31706
  1580
  apply (auto simp add: mult_compare_simps)
huffman@31706
  1581
  apply (subgoal_tac "k > 0")
huffman@31706
  1582
  apply arith
huffman@31706
  1583
  apply (case_tac "k <= 0")
huffman@31706
  1584
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1585
  apply force
huffman@31706
  1586
  apply (subst zero_compare_simps(8))
huffman@31706
  1587
  apply auto
huffman@31706
  1588
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1589
  apply force
huffman@31706
  1590
  apply (subst zero_compare_simps(8))
huffman@31706
  1591
  apply auto
huffman@31706
  1592
done
chaieb@27568
  1593
huffman@31706
  1594
lemma nat_prime_dvd_power [rule_format]: "prime (p::nat) -->
huffman@31706
  1595
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1596
  by (induct n rule: nat_induct, auto)
chaieb@27568
  1597
huffman@31706
  1598
lemma int_prime_dvd_power [rule_format]: "prime (p::int) -->
huffman@31706
  1599
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1600
  apply (induct n rule: nat_induct, auto)
huffman@31706
  1601
  apply (frule int_prime_ge_0)
huffman@31706
  1602
  apply auto
huffman@31706
  1603
done
huffman@31706
  1604
huffman@31706
  1605
lemma nat_prime_imp_power_coprime: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1606
    coprime a (p^m)"
huffman@31706
  1607
  apply (rule nat_coprime_exp)
huffman@31706
  1608
  apply (subst nat_gcd_commute)
huffman@31706
  1609
  apply (erule (1) nat_prime_imp_coprime)
huffman@31706
  1610
done
chaieb@27568
  1611
huffman@31706
  1612
lemma int_prime_imp_power_coprime: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1613
    coprime a (p^m)"
huffman@31706
  1614
  apply (rule int_coprime_exp)
huffman@31706
  1615
  apply (subst int_gcd_commute)
huffman@31706
  1616
  apply (erule (1) int_prime_imp_coprime)
huffman@31706
  1617
done
chaieb@27568
  1618
huffman@31706
  1619
lemma nat_primes_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
huffman@31706
  1620
  apply (rule nat_prime_imp_coprime, assumption)
huffman@31706
  1621
  apply (unfold prime_nat_def, auto)
huffman@31706
  1622
done
chaieb@27568
  1623
huffman@31706
  1624
lemma int_primes_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
huffman@31706
  1625
  apply (rule int_prime_imp_coprime, assumption)
huffman@31706
  1626
  apply (unfold prime_int_altdef, clarify)
huffman@31706
  1627
  apply (drule_tac x = q in spec)
huffman@31706
  1628
  apply (drule_tac x = p in spec)
huffman@31706
  1629
  apply auto
huffman@31706
  1630
done
chaieb@27568
  1631
huffman@31706
  1632
lemma nat_primes_imp_powers_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1633
    coprime (p^m) (q^n)"
huffman@31706
  1634
  by (rule nat_coprime_exp2, rule nat_primes_coprime)
chaieb@27568
  1635
huffman@31706
  1636
lemma int_primes_imp_powers_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1637
    coprime (p^m) (q^n)"
huffman@31706
  1638
  by (rule int_coprime_exp2, rule int_primes_coprime)
chaieb@27568
  1639
huffman@31706
  1640
lemma nat_prime_factor: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
huffman@31706
  1641
  apply (induct n rule: nat_less_induct)
huffman@31706
  1642
  apply (case_tac "n = 0")
huffman@31706
  1643
  using nat_two_is_prime apply blast
huffman@31706
  1644
  apply (case_tac "prime n")
huffman@31706
  1645
  apply blast
huffman@31706
  1646
  apply (subgoal_tac "n > 1")
huffman@31706
  1647
  apply (frule (1) nat_not_prime_eq_prod)
huffman@31706
  1648
  apply (auto intro: dvd_mult dvd_mult2)
huffman@31706
  1649
done
chaieb@23244
  1650
huffman@31706
  1651
(* An Isar version:
huffman@31706
  1652
huffman@31706
  1653
lemma nat_prime_factor_b:
huffman@31706
  1654
  fixes n :: nat
huffman@31706
  1655
  assumes "n \<noteq> 1"
huffman@31706
  1656
  shows "\<exists>p. prime p \<and> p dvd n"
nipkow@23983
  1657
huffman@31706
  1658
using `n ~= 1`
huffman@31706
  1659
proof (induct n rule: nat_less_induct)
huffman@31706
  1660
  fix n :: nat
huffman@31706
  1661
  assume "n ~= 1" and
huffman@31706
  1662
    ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
huffman@31706
  1663
  thus "\<exists>p. prime p \<and> p dvd n"
huffman@31706
  1664
  proof -
huffman@31706
  1665
  {
huffman@31706
  1666
    assume "n = 0"
huffman@31706
  1667
    moreover note nat_two_is_prime
huffman@31706
  1668
    ultimately have ?thesis
huffman@31706
  1669
      by (auto simp del: nat_two_is_prime)
huffman@31706
  1670
  }
huffman@31706
  1671
  moreover
huffman@31706
  1672
  {
huffman@31706
  1673
    assume "prime n"
huffman@31706
  1674
    hence ?thesis by auto
huffman@31706
  1675
  }
huffman@31706
  1676
  moreover
huffman@31706
  1677
  {
huffman@31706
  1678
    assume "n ~= 0" and "~ prime n"
huffman@31706
  1679
    with `n ~= 1` have "n > 1" by auto
huffman@31706
  1680
    with `~ prime n` and nat_not_prime_eq_prod obtain m k where
huffman@31706
  1681
      "n = m * k" and "1 < m" and "m < n" by blast
huffman@31706
  1682
    with ih obtain p where "prime p" and "p dvd m" by blast
huffman@31706
  1683
    with `n = m * k` have ?thesis by auto
huffman@31706
  1684
  }
huffman@31706
  1685
  ultimately show ?thesis by blast
huffman@31706
  1686
  qed
nipkow@23983
  1687
qed
nipkow@23983
  1688
huffman@31706
  1689
*)
huffman@31706
  1690
huffman@31706
  1691
text {* One property of coprimality is easier to prove via prime factors. *}
huffman@31706
  1692
huffman@31706
  1693
lemma nat_prime_divprod_pow:
huffman@31706
  1694
  assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
huffman@31706
  1695
  shows "p^n dvd a \<or> p^n dvd b"
huffman@31706
  1696
proof-
huffman@31706
  1697
  {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
huffman@31706
  1698
      apply (cases "n=0", simp_all)
huffman@31706
  1699
      apply (cases "a=1", simp_all) done}
huffman@31706
  1700
  moreover
huffman@31706
  1701
  {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
huffman@31706
  1702
    then obtain m where m: "n = Suc m" by (cases n, auto)
huffman@31706
  1703
    from n have "p dvd p^n" by (intro dvd_power, auto)
huffman@31706
  1704
    also note pab
huffman@31706
  1705
    finally have pab': "p dvd a * b".
huffman@31706
  1706
    from nat_prime_dvd_mult[OF p pab']
huffman@31706
  1707
    have "p dvd a \<or> p dvd b" .
huffman@31706
  1708
    moreover
huffman@31706
  1709
    {assume pa: "p dvd a"
huffman@31706
  1710
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
huffman@31706
  1711
      from nat_coprime_common_divisor [OF ab, OF pa] p have "\<not> p dvd b" by auto
huffman@31706
  1712
      with p have "coprime b p"
huffman@31706
  1713
        by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
huffman@31706
  1714
      hence pnb: "coprime (p^n) b"
huffman@31706
  1715
        by (subst nat_gcd_commute, rule nat_coprime_exp)
huffman@31706
  1716
      from nat_coprime_divprod[OF pnba pnb] have ?thesis by blast }
huffman@31706
  1717
    moreover
huffman@31706
  1718
    {assume pb: "p dvd b"
huffman@31706
  1719
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
huffman@31706
  1720
      from nat_coprime_common_divisor [OF ab, of p] pb p have "\<not> p dvd a"
huffman@31706
  1721
        by auto
huffman@31706
  1722
      with p have "coprime a p"
huffman@31706
  1723
        by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
huffman@31706
  1724
      hence pna: "coprime (p^n) a"
huffman@31706
  1725
        by (subst nat_gcd_commute, rule nat_coprime_exp)
huffman@31706
  1726
      from nat_coprime_divprod[OF pab pna] have ?thesis by blast }
huffman@31706
  1727
    ultimately have ?thesis by blast}
huffman@31706
  1728
  ultimately show ?thesis by blast
nipkow@23983
  1729
qed
nipkow@23983
  1730
wenzelm@21256
  1731
end