src/HOL/GCD.thy
author nipkow
Sat Jun 20 01:53:39 2009 +0200 (2009-06-20)
changeset 31729 b9299916d618
parent 31709 061f01ee9978
child 31730 d74830dc3e4a
permissions -rw-r--r--
new lemmas and tuning
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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
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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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*)
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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_self [simp]: "gcd (x::nat) x = x"
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  by simp
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lemma int_gcd_def [simp]: "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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thm nat_gcd_dvd1 [transferred]
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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 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"
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  by (rule dvd_anti_sym, auto)
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lemma int_gcd_commute: "gcd (m::int) n = gcd n m"
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  by (auto simp add: gcd_int_def nat_gcd_commute)
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lemma nat_gcd_assoc: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
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  apply (rule dvd_anti_sym)
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  apply (blast intro: dvd_trans)+
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done
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lemma int_gcd_assoc: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
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  by (auto simp add: gcd_int_def nat_gcd_assoc)
huffman@31706
   343
huffman@31706
   344
lemma nat_gcd_left_commute: "gcd (k::nat) (gcd m n) = gcd m (gcd k n)"
huffman@31706
   345
  apply (rule nat_gcd_commute [THEN trans])
huffman@31706
   346
  apply (rule nat_gcd_assoc [THEN trans])
huffman@31706
   347
  apply (rule nat_gcd_commute [THEN arg_cong])
huffman@31706
   348
done
huffman@31706
   349
huffman@31706
   350
lemma int_gcd_left_commute: "gcd (k::int) (gcd m n) = gcd m (gcd k n)"
huffman@31706
   351
  apply (rule int_gcd_commute [THEN trans])
huffman@31706
   352
  apply (rule int_gcd_assoc [THEN trans])
huffman@31706
   353
  apply (rule int_gcd_commute [THEN arg_cong])
huffman@31706
   354
done
huffman@31706
   355
huffman@31706
   356
lemmas nat_gcd_ac = nat_gcd_assoc nat_gcd_commute nat_gcd_left_commute
huffman@31706
   357
  -- {* gcd is an AC-operator *}
wenzelm@21256
   358
huffman@31706
   359
lemmas int_gcd_ac = int_gcd_assoc int_gcd_commute int_gcd_left_commute
huffman@31706
   360
huffman@31706
   361
lemma nat_gcd_1_left [simp]: "gcd (1::nat) m = 1"
huffman@31706
   362
  by (subst nat_gcd_commute, simp)
huffman@31706
   363
huffman@31706
   364
lemma nat_gcd_Suc_0_left [simp]: "gcd (Suc 0) m = Suc 0"
huffman@31706
   365
  by (subst nat_gcd_commute, simp add: One_nat_def)
huffman@31706
   366
huffman@31706
   367
lemma int_gcd_1_left [simp]: "gcd (1::int) m = 1"
huffman@31706
   368
  by (subst int_gcd_commute, simp)
wenzelm@21256
   369
huffman@31706
   370
lemma nat_gcd_unique: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   371
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   372
  apply auto
huffman@31706
   373
  apply (rule dvd_anti_sym)
huffman@31706
   374
  apply (erule (1) nat_gcd_greatest)
huffman@31706
   375
  apply auto
huffman@31706
   376
done
wenzelm@21256
   377
huffman@31706
   378
lemma int_gcd_unique: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   379
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   380
  apply (case_tac "d = 0")
huffman@31706
   381
  apply force
huffman@31706
   382
  apply (rule iffI)
huffman@31706
   383
  apply (rule zdvd_anti_sym)
huffman@31706
   384
  apply arith
huffman@31706
   385
  apply (subst int_gcd_pos)
huffman@31706
   386
  apply clarsimp
huffman@31706
   387
  apply (drule_tac x = "d + 1" in spec)
huffman@31706
   388
  apply (frule zdvd_imp_le)
huffman@31706
   389
  apply (auto intro: int_gcd_greatest)
huffman@31706
   390
done
huffman@30082
   391
wenzelm@21256
   392
text {*
wenzelm@21256
   393
  \medskip Multiplication laws
wenzelm@21256
   394
*}
wenzelm@21256
   395
huffman@31706
   396
lemma nat_gcd_mult_distrib: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256
   397
    -- {* \cite[page 27]{davenport92} *}
huffman@31706
   398
  apply (induct m n rule: nat_gcd_induct)
huffman@31706
   399
  apply simp
wenzelm@21256
   400
  apply (case_tac "k = 0")
huffman@31706
   401
  apply (simp_all add: mod_geq nat_gcd_non_0 mod_mult_distrib2)
huffman@31706
   402
done
wenzelm@21256
   403
huffman@31706
   404
lemma int_gcd_mult_distrib: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
huffman@31706
   405
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   406
  apply (simp add: abs_mult)
huffman@31706
   407
  apply (rule nat_gcd_mult_distrib [transferred])
huffman@31706
   408
  apply auto
huffman@31706
   409
done
wenzelm@21256
   410
huffman@31706
   411
lemma nat_gcd_mult [simp]: "gcd (k::nat) (k * n) = k"
huffman@31706
   412
  by (rule nat_gcd_mult_distrib [of k 1 n, simplified, symmetric])
wenzelm@21256
   413
huffman@31706
   414
lemma int_gcd_mult [simp]: "gcd (k::int) (k * n) = abs k"
huffman@31706
   415
  by (rule int_gcd_mult_distrib [of k 1 n, simplified, symmetric])
huffman@31706
   416
huffman@31706
   417
lemma nat_coprime_dvd_mult: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
huffman@31706
   418
  apply (insert nat_gcd_mult_distrib [of m k n])
wenzelm@21256
   419
  apply simp
wenzelm@21256
   420
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   421
  apply simp
wenzelm@21256
   422
  done
wenzelm@21256
   423
huffman@31706
   424
lemma int_coprime_dvd_mult:
huffman@31706
   425
  assumes "coprime (k::int) n" and "k dvd m * n"
huffman@31706
   426
  shows "k dvd m"
wenzelm@21256
   427
huffman@31706
   428
  using prems
huffman@31706
   429
  apply (subst abs_dvd_iff [symmetric])
huffman@31706
   430
  apply (subst dvd_abs_iff [symmetric])
huffman@31706
   431
  apply (subst (asm) int_gcd_abs)
huffman@31706
   432
  apply (rule nat_coprime_dvd_mult [transferred])
huffman@31706
   433
  apply auto
huffman@31706
   434
  apply (subst abs_mult [symmetric], auto)
huffman@31706
   435
done
huffman@31706
   436
huffman@31706
   437
lemma nat_coprime_dvd_mult_iff: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   438
    (k dvd m * n) = (k dvd m)"
huffman@31706
   439
  by (auto intro: nat_coprime_dvd_mult)
huffman@31706
   440
huffman@31706
   441
lemma int_coprime_dvd_mult_iff: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   442
    (k dvd m * n) = (k dvd m)"
huffman@31706
   443
  by (auto intro: int_coprime_dvd_mult)
huffman@31706
   444
huffman@31706
   445
lemma nat_gcd_mult_cancel: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
wenzelm@21256
   446
  apply (rule dvd_anti_sym)
huffman@31706
   447
  apply (rule nat_gcd_greatest)
huffman@31706
   448
  apply (rule_tac n = k in nat_coprime_dvd_mult)
huffman@31706
   449
  apply (simp add: nat_gcd_assoc)
huffman@31706
   450
  apply (simp add: nat_gcd_commute)
huffman@31706
   451
  apply (simp_all add: mult_commute)
huffman@31706
   452
done
wenzelm@21256
   453
huffman@31706
   454
lemma int_gcd_mult_cancel:
huffman@31706
   455
  assumes "coprime (k::int) n"
huffman@31706
   456
  shows "gcd (k * m) n = gcd m n"
huffman@31706
   457
huffman@31706
   458
  using prems
huffman@31706
   459
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   460
  apply (subst abs_mult)
huffman@31706
   461
  apply (rule nat_gcd_mult_cancel [transferred])
huffman@31706
   462
  apply (auto simp add: int_gcd_abs [symmetric])
huffman@31706
   463
done
wenzelm@21256
   464
wenzelm@21256
   465
text {* \medskip Addition laws *}
wenzelm@21256
   466
huffman@31706
   467
lemma nat_gcd_add1 [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   468
  apply (case_tac "n = 0")
huffman@31706
   469
  apply (simp_all add: nat_gcd_non_0)
huffman@31706
   470
done
huffman@31706
   471
huffman@31706
   472
lemma nat_gcd_add2 [simp]: "gcd (m::nat) (m + n) = gcd m n"
huffman@31706
   473
  apply (subst (1 2) nat_gcd_commute)
huffman@31706
   474
  apply (subst add_commute)
huffman@31706
   475
  apply simp
huffman@31706
   476
done
huffman@31706
   477
huffman@31706
   478
(* to do: add the other variations? *)
huffman@31706
   479
huffman@31706
   480
lemma nat_gcd_diff1: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
huffman@31706
   481
  by (subst nat_gcd_add1 [symmetric], auto)
huffman@31706
   482
huffman@31706
   483
lemma nat_gcd_diff2: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
huffman@31706
   484
  apply (subst nat_gcd_commute)
huffman@31706
   485
  apply (subst nat_gcd_diff1 [symmetric])
huffman@31706
   486
  apply auto
huffman@31706
   487
  apply (subst nat_gcd_commute)
huffman@31706
   488
  apply (subst nat_gcd_diff1)
huffman@31706
   489
  apply assumption
huffman@31706
   490
  apply (rule nat_gcd_commute)
huffman@31706
   491
done
huffman@31706
   492
huffman@31706
   493
lemma int_gcd_non_0: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   494
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   495
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
huffman@31706
   496
  apply (auto simp add: nat_gcd_non_0 nat_mod_distrib [symmetric]
huffman@31706
   497
    zmod_zminus1_eq_if)
huffman@31706
   498
  apply (frule_tac a = x in pos_mod_bound)
huffman@31706
   499
  apply (subst (1 2) nat_gcd_commute)
huffman@31706
   500
  apply (simp del: pos_mod_bound add: nat_diff_distrib nat_gcd_diff2
huffman@31706
   501
    nat_le_eq_zle)
huffman@31706
   502
done
wenzelm@21256
   503
huffman@31706
   504
lemma int_gcd_red: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   505
  apply (case_tac "y = 0")
huffman@31706
   506
  apply force
huffman@31706
   507
  apply (case_tac "y > 0")
huffman@31706
   508
  apply (subst int_gcd_non_0, auto)
huffman@31706
   509
  apply (insert int_gcd_non_0 [of "-y" "-x"])
huffman@31706
   510
  apply (auto simp add: int_gcd_neg1 int_gcd_neg2)
huffman@31706
   511
done
huffman@31706
   512
huffman@31706
   513
lemma int_gcd_add1 [simp]: "gcd ((m::int) + n) n = gcd m n"
huffman@31706
   514
  apply (case_tac "n = 0", force)
huffman@31706
   515
  apply (subst (1 2) int_gcd_red)
huffman@31706
   516
  apply auto
huffman@31706
   517
done
huffman@31706
   518
huffman@31706
   519
lemma int_gcd_add2 [simp]: "gcd m ((m::int) + n) = gcd m n"
huffman@31706
   520
  apply (subst int_gcd_commute)
huffman@31706
   521
  apply (subst add_commute)
huffman@31706
   522
  apply (subst int_gcd_add1)
huffman@31706
   523
  apply (subst int_gcd_commute)
huffman@31706
   524
  apply (rule refl)
huffman@31706
   525
done
wenzelm@21256
   526
huffman@31706
   527
lemma nat_gcd_add_mult: "gcd (m::nat) (k * m + n) = gcd m n"
huffman@31706
   528
  by (induct k, simp_all add: ring_simps)
wenzelm@21256
   529
huffman@31706
   530
lemma int_gcd_add_mult: "gcd (m::int) (k * m + n) = gcd m n"
huffman@31706
   531
  apply (subgoal_tac "ALL s. ALL m. ALL n.
huffman@31706
   532
      gcd m (int (s::nat) * m + n) = gcd m n")
huffman@31706
   533
  apply (case_tac "k >= 0")
huffman@31706
   534
  apply (drule_tac x = "nat k" in spec, force)
huffman@31706
   535
  apply (subst (1 2) int_gcd_neg2 [symmetric])
huffman@31706
   536
  apply (drule_tac x = "nat (- k)" in spec)
huffman@31706
   537
  apply (drule_tac x = "m" in spec)
huffman@31706
   538
  apply (drule_tac x = "-n" in spec)
huffman@31706
   539
  apply auto
huffman@31706
   540
  apply (rule nat_induct)
huffman@31706
   541
  apply auto
huffman@31706
   542
  apply (auto simp add: left_distrib)
huffman@31706
   543
  apply (subst add_assoc)
huffman@31706
   544
  apply simp
huffman@31706
   545
done
wenzelm@21256
   546
huffman@31706
   547
(* to do: differences, and all variations of addition rules
huffman@31706
   548
    as simplification rules for nat and int *)
huffman@31706
   549
huffman@31706
   550
lemma nat_gcd_dvd_prod [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   551
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   552
huffman@31706
   553
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   554
chaieb@22027
   555
huffman@31706
   556
subsection {* Coprimality *}
huffman@31706
   557
huffman@31706
   558
lemma nat_div_gcd_coprime [intro]:
huffman@31706
   559
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   560
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   561
proof -
haftmann@27556
   562
  let ?g = "gcd a b"
chaieb@22027
   563
  let ?a' = "a div ?g"
chaieb@22027
   564
  let ?b' = "b div ?g"
haftmann@27556
   565
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   566
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   567
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   568
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   569
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   570
    unfolding dvd_def by blast
huffman@31706
   571
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   572
    by simp_all
wenzelm@22367
   573
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   574
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   575
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
huffman@31706
   576
  have "?g \<noteq> 0" using nz by (simp add: nat_gcd_zero)
huffman@31706
   577
  then have gp: "?g > 0" by arith
huffman@31706
   578
  from nat_gcd_greatest [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   579
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   580
qed
chaieb@22027
   581
huffman@31706
   582
lemma int_div_gcd_coprime [intro]:
huffman@31706
   583
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   584
  shows "coprime (a div gcd a b) (b div gcd a b)"
chaieb@27669
   585
huffman@31706
   586
  apply (subst (1 2 3) int_gcd_abs)
huffman@31706
   587
  apply (subst (1 2) abs_div)
huffman@31706
   588
  apply auto
huffman@31706
   589
  prefer 3
huffman@31706
   590
  apply (rule nat_div_gcd_coprime [transferred])
huffman@31706
   591
  using nz apply (auto simp add: int_gcd_abs [symmetric])
huffman@31706
   592
done
huffman@31706
   593
huffman@31706
   594
lemma nat_coprime: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
huffman@31706
   595
  using nat_gcd_unique[of 1 a b, simplified] by auto
huffman@31706
   596
huffman@31706
   597
lemma nat_coprime_Suc_0:
huffman@31706
   598
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
huffman@31706
   599
  using nat_coprime by (simp add: One_nat_def)
huffman@31706
   600
huffman@31706
   601
lemma int_coprime: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   602
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
huffman@31706
   603
  using int_gcd_unique [of 1 a b]
huffman@31706
   604
  apply clarsimp
huffman@31706
   605
  apply (erule subst)
huffman@31706
   606
  apply (rule iffI)
huffman@31706
   607
  apply force
huffman@31706
   608
  apply (drule_tac x = "abs e" in exI)
huffman@31706
   609
  apply (case_tac "e >= 0")
huffman@31706
   610
  apply force
huffman@31706
   611
  apply force
huffman@31706
   612
done
huffman@31706
   613
huffman@31706
   614
lemma nat_gcd_coprime:
huffman@31706
   615
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   616
    b: "b = b' * gcd a b"
huffman@31706
   617
  shows    "coprime a' b'"
huffman@31706
   618
huffman@31706
   619
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   620
  apply (erule ssubst)
huffman@31706
   621
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   622
  apply (erule ssubst)
huffman@31706
   623
  apply (rule nat_div_gcd_coprime)
huffman@31706
   624
  using prems
huffman@31706
   625
  apply force
huffman@31706
   626
  apply (subst (1) b)
huffman@31706
   627
  using z apply force
huffman@31706
   628
  apply (subst (1) a)
huffman@31706
   629
  using z apply force
huffman@31706
   630
done
huffman@31706
   631
huffman@31706
   632
lemma int_gcd_coprime:
huffman@31706
   633
  assumes z: "gcd (a::int) 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 int_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 nat_coprime_mult: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   651
    shows "coprime d (a * b)"
huffman@31706
   652
  apply (subst nat_gcd_commute)
huffman@31706
   653
  using da apply (subst nat_gcd_mult_cancel)
huffman@31706
   654
  apply (subst nat_gcd_commute, assumption)
huffman@31706
   655
  apply (subst nat_gcd_commute, rule db)
huffman@31706
   656
done
huffman@31706
   657
huffman@31706
   658
lemma int_coprime_mult: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   659
    shows "coprime d (a * b)"
huffman@31706
   660
  apply (subst int_gcd_commute)
huffman@31706
   661
  using da apply (subst int_gcd_mult_cancel)
huffman@31706
   662
  apply (subst int_gcd_commute, assumption)
huffman@31706
   663
  apply (subst int_gcd_commute, rule db)
huffman@31706
   664
done
huffman@31706
   665
huffman@31706
   666
lemma nat_coprime_lmult:
huffman@31706
   667
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   668
proof -
huffman@31706
   669
  have "gcd d a dvd gcd d (a * b)"
huffman@31706
   670
    by (rule nat_gcd_greatest, auto)
huffman@31706
   671
  with dab show ?thesis
huffman@31706
   672
    by auto
huffman@31706
   673
qed
huffman@31706
   674
huffman@31706
   675
lemma int_coprime_lmult:
huffman@31706
   676
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   677
proof -
huffman@31706
   678
  have "gcd d a dvd gcd d (a * b)"
huffman@31706
   679
    by (rule int_gcd_greatest, auto)
huffman@31706
   680
  with dab show ?thesis
huffman@31706
   681
    by auto
huffman@31706
   682
qed
huffman@31706
   683
huffman@31706
   684
lemma nat_coprime_rmult:
huffman@31706
   685
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   686
proof -
huffman@31706
   687
  have "gcd d b dvd gcd d (a * b)"
huffman@31706
   688
    by (rule nat_gcd_greatest, auto intro: dvd_mult)
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_rmult:
huffman@31706
   694
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   695
proof -
huffman@31706
   696
  have "gcd d b dvd gcd d (a * b)"
huffman@31706
   697
    by (rule int_gcd_greatest, auto intro: dvd_mult)
huffman@31706
   698
  with dab show ?thesis
huffman@31706
   699
    by auto
huffman@31706
   700
qed
huffman@31706
   701
huffman@31706
   702
lemma nat_coprime_mul_eq: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   703
    coprime d a \<and>  coprime d b"
huffman@31706
   704
  using nat_coprime_rmult[of d a b] nat_coprime_lmult[of d a b]
huffman@31706
   705
    nat_coprime_mult[of d a b]
huffman@31706
   706
  by blast
huffman@31706
   707
huffman@31706
   708
lemma int_coprime_mul_eq: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   709
    coprime d a \<and>  coprime d b"
huffman@31706
   710
  using int_coprime_rmult[of d a b] int_coprime_lmult[of d a b]
huffman@31706
   711
    int_coprime_mult[of d a b]
huffman@31706
   712
  by blast
huffman@31706
   713
huffman@31706
   714
lemma nat_gcd_coprime_exists:
huffman@31706
   715
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   716
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   717
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   718
  apply (rule_tac x = "b div gcd a b" in exI)
huffman@31706
   719
  using nz apply (auto simp add: nat_div_gcd_coprime dvd_div_mult)
huffman@31706
   720
done
huffman@31706
   721
huffman@31706
   722
lemma int_gcd_coprime_exists:
huffman@31706
   723
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   724
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   725
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   726
  apply (rule_tac x = "b div gcd a b" in exI)
huffman@31706
   727
  using nz apply (auto simp add: int_div_gcd_coprime dvd_div_mult_self)
huffman@31706
   728
done
huffman@31706
   729
huffman@31706
   730
lemma nat_coprime_exp: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
huffman@31706
   731
  by (induct n, simp_all add: nat_coprime_mult)
huffman@31706
   732
huffman@31706
   733
lemma int_coprime_exp: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
huffman@31706
   734
  by (induct n, simp_all add: int_coprime_mult)
huffman@31706
   735
huffman@31706
   736
lemma nat_coprime_exp2 [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
huffman@31706
   737
  apply (rule nat_coprime_exp)
huffman@31706
   738
  apply (subst nat_gcd_commute)
huffman@31706
   739
  apply (rule nat_coprime_exp)
huffman@31706
   740
  apply (subst nat_gcd_commute, assumption)
huffman@31706
   741
done
huffman@31706
   742
huffman@31706
   743
lemma int_coprime_exp2 [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
huffman@31706
   744
  apply (rule int_coprime_exp)
huffman@31706
   745
  apply (subst int_gcd_commute)
huffman@31706
   746
  apply (rule int_coprime_exp)
huffman@31706
   747
  apply (subst int_gcd_commute, assumption)
huffman@31706
   748
done
huffman@31706
   749
huffman@31706
   750
lemma nat_gcd_exp: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   751
proof (cases)
huffman@31706
   752
  assume "a = 0 & b = 0"
huffman@31706
   753
  thus ?thesis by simp
huffman@31706
   754
  next assume "~(a = 0 & b = 0)"
huffman@31706
   755
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
huffman@31706
   756
    by auto
huffman@31706
   757
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   758
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   759
    apply (subst (1 2) mult_commute)
huffman@31706
   760
    apply (subst nat_gcd_mult_distrib [symmetric])
huffman@31706
   761
    apply simp
huffman@31706
   762
    done
huffman@31706
   763
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
huffman@31706
   764
    apply (subst div_power)
huffman@31706
   765
    apply auto
huffman@31706
   766
    apply (rule dvd_div_mult_self)
huffman@31706
   767
    apply (rule dvd_power_same)
huffman@31706
   768
    apply auto
huffman@31706
   769
    done
huffman@31706
   770
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   771
    apply (subst div_power)
huffman@31706
   772
    apply auto
huffman@31706
   773
    apply (rule dvd_div_mult_self)
huffman@31706
   774
    apply (rule dvd_power_same)
huffman@31706
   775
    apply auto
huffman@31706
   776
    done
huffman@31706
   777
  finally show ?thesis .
huffman@31706
   778
qed
huffman@31706
   779
huffman@31706
   780
lemma int_gcd_exp: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
huffman@31706
   781
  apply (subst (1 2) int_gcd_abs)
huffman@31706
   782
  apply (subst (1 2) power_abs)
huffman@31706
   783
  apply (rule nat_gcd_exp [where n = n, transferred])
huffman@31706
   784
  apply auto
huffman@31706
   785
done
huffman@31706
   786
huffman@31706
   787
lemma nat_coprime_divprod: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
huffman@31706
   788
  using nat_coprime_dvd_mult_iff[of d a b]
huffman@31706
   789
  by (auto simp add: mult_commute)
huffman@31706
   790
huffman@31706
   791
lemma int_coprime_divprod: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
huffman@31706
   792
  using int_coprime_dvd_mult_iff[of d a b]
huffman@31706
   793
  by (auto simp add: mult_commute)
huffman@31706
   794
huffman@31706
   795
lemma nat_division_decomp: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   796
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   797
proof-
huffman@31706
   798
  let ?g = "gcd a b"
huffman@31706
   799
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   800
  moreover
huffman@31706
   801
  {assume z: "?g \<noteq> 0"
huffman@31706
   802
    from nat_gcd_coprime_exists[OF z]
huffman@31706
   803
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   804
      by blast
huffman@31706
   805
    have thb: "?g dvd b" by auto
huffman@31706
   806
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   807
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   808
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   809
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   810
    with z have th_1: "a' dvd b' * c" by auto
huffman@31706
   811
    from nat_coprime_dvd_mult[OF ab'(3)] th_1
huffman@31706
   812
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   813
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   814
    with thb thc have ?thesis by blast }
huffman@31706
   815
  ultimately show ?thesis by blast
huffman@31706
   816
qed
huffman@31706
   817
huffman@31706
   818
lemma int_division_decomp: assumes dc: "(a::int) dvd b * c"
huffman@31706
   819
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   820
proof-
huffman@31706
   821
  let ?g = "gcd a b"
huffman@31706
   822
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   823
  moreover
huffman@31706
   824
  {assume z: "?g \<noteq> 0"
huffman@31706
   825
    from int_gcd_coprime_exists[OF z]
huffman@31706
   826
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   827
      by blast
huffman@31706
   828
    have thb: "?g dvd b" by auto
huffman@31706
   829
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   830
    with dc have th0: "a' dvd b*c"
huffman@31706
   831
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   832
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   833
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   834
    with z have th_1: "a' dvd b' * c" by auto
huffman@31706
   835
    from int_coprime_dvd_mult[OF ab'(3)] th_1
huffman@31706
   836
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   837
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   838
    with thb thc have ?thesis by blast }
huffman@31706
   839
  ultimately show ?thesis by blast
chaieb@27669
   840
qed
chaieb@27669
   841
huffman@31706
   842
lemma nat_pow_divides_pow:
huffman@31706
   843
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   844
  shows "a dvd b"
huffman@31706
   845
proof-
huffman@31706
   846
  let ?g = "gcd a b"
huffman@31706
   847
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   848
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   849
  moreover
huffman@31706
   850
  {assume z: "?g \<noteq> 0"
huffman@31706
   851
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
huffman@31706
   852
    from nat_gcd_coprime_exists[OF z]
huffman@31706
   853
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   854
      by blast
huffman@31706
   855
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   856
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   857
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   858
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   859
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   860
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   861
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   862
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
huffman@31706
   863
    from nat_coprime_dvd_mult[OF nat_coprime_exp [OF ab'(3), of m]] th1
huffman@31706
   864
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   865
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   866
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   867
  ultimately show ?thesis by blast
huffman@31706
   868
qed
huffman@31706
   869
huffman@31706
   870
lemma int_pow_divides_pow:
huffman@31706
   871
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   872
  shows "a dvd b"
chaieb@27669
   873
proof-
huffman@31706
   874
  let ?g = "gcd a b"
huffman@31706
   875
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   876
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   877
  moreover
huffman@31706
   878
  {assume z: "?g \<noteq> 0"
huffman@31706
   879
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
huffman@31706
   880
    from int_gcd_coprime_exists[OF z]
huffman@31706
   881
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   882
      by blast
huffman@31706
   883
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   884
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   885
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   886
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   887
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   888
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   889
    with th0 have "a' dvd b'^n"
huffman@31706
   890
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   891
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
huffman@31706
   892
    from int_coprime_dvd_mult[OF int_coprime_exp [OF ab'(3), of m]] th1
huffman@31706
   893
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   894
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   895
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   896
  ultimately show ?thesis by blast
huffman@31706
   897
qed
huffman@31706
   898
huffman@31706
   899
lemma nat_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
huffman@31706
   900
  by (auto intro: nat_pow_divides_pow dvd_power_same)
huffman@31706
   901
huffman@31706
   902
lemma int_pow_divides_eq [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
huffman@31706
   903
  by (auto intro: int_pow_divides_pow dvd_power_same)
huffman@31706
   904
huffman@31706
   905
lemma nat_divides_mult:
huffman@31706
   906
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   907
  shows "m * n dvd r"
huffman@31706
   908
proof-
huffman@31706
   909
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   910
    unfolding dvd_def by blast
huffman@31706
   911
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
huffman@31706
   912
  hence "m dvd n'" using nat_coprime_dvd_mult_iff[OF mn] by simp
huffman@31706
   913
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   914
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   915
qed
huffman@31706
   916
huffman@31706
   917
lemma int_divides_mult:
huffman@31706
   918
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   919
  shows "m * n dvd r"
huffman@31706
   920
proof-
huffman@31706
   921
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   922
    unfolding dvd_def by blast
huffman@31706
   923
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
huffman@31706
   924
  hence "m dvd n'" using int_coprime_dvd_mult_iff[OF mn] by simp
huffman@31706
   925
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   926
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   927
qed
chaieb@27669
   928
huffman@31706
   929
lemma nat_coprime_plus_one [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   930
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   931
  apply force
huffman@31706
   932
  apply (rule nat_dvd_diff)
huffman@31706
   933
  apply auto
huffman@31706
   934
done
huffman@31706
   935
huffman@31706
   936
lemma nat_coprime_Suc [simp]: "coprime (Suc n) n"
huffman@31706
   937
  using nat_coprime_plus_one by (simp add: One_nat_def)
huffman@31706
   938
huffman@31706
   939
lemma int_coprime_plus_one [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   940
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   941
  apply force
huffman@31706
   942
  apply (rule dvd_diff)
huffman@31706
   943
  apply auto
huffman@31706
   944
done
huffman@31706
   945
huffman@31706
   946
lemma nat_coprime_minus_one: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
huffman@31706
   947
  using nat_coprime_plus_one [of "n - 1"]
huffman@31706
   948
    nat_gcd_commute [of "n - 1" n] by auto
huffman@31706
   949
huffman@31706
   950
lemma int_coprime_minus_one: "coprime ((n::int) - 1) n"
huffman@31706
   951
  using int_coprime_plus_one [of "n - 1"]
huffman@31706
   952
    int_gcd_commute [of "n - 1" n] by auto
huffman@31706
   953
huffman@31706
   954
lemma nat_setprod_coprime [rule_format]:
huffman@31706
   955
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   956
  apply (case_tac "finite A")
huffman@31706
   957
  apply (induct set: finite)
huffman@31706
   958
  apply (auto simp add: nat_gcd_mult_cancel)
huffman@31706
   959
done
huffman@31706
   960
huffman@31706
   961
lemma int_setprod_coprime [rule_format]:
huffman@31706
   962
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
   963
  apply (case_tac "finite A")
huffman@31706
   964
  apply (induct set: finite)
huffman@31706
   965
  apply (auto simp add: int_gcd_mult_cancel)
huffman@31706
   966
done
huffman@31706
   967
huffman@31706
   968
lemma nat_prime_odd: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
   969
  unfolding prime_nat_def
huffman@31706
   970
  apply (subst even_mult_two_ex)
huffman@31706
   971
  apply clarify
huffman@31706
   972
  apply (drule_tac x = 2 in spec)
huffman@31706
   973
  apply auto
huffman@31706
   974
done
huffman@31706
   975
huffman@31706
   976
lemma int_prime_odd: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
   977
  unfolding prime_int_def
huffman@31706
   978
  apply (frule nat_prime_odd)
huffman@31706
   979
  apply (auto simp add: even_nat_def)
huffman@31706
   980
done
huffman@31706
   981
huffman@31706
   982
lemma nat_coprime_common_divisor: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
   983
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
   984
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
   985
  apply simp
huffman@31706
   986
  apply (erule (1) nat_gcd_greatest)
huffman@31706
   987
done
huffman@31706
   988
huffman@31706
   989
lemma int_coprime_common_divisor: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
   990
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
   991
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
   992
  apply simp
huffman@31706
   993
  apply (erule (1) int_gcd_greatest)
huffman@31706
   994
done
huffman@31706
   995
huffman@31706
   996
lemma nat_coprime_divisors: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
   997
    coprime d e"
huffman@31706
   998
  apply (auto simp add: dvd_def)
huffman@31706
   999
  apply (frule int_coprime_lmult)
huffman@31706
  1000
  apply (subst int_gcd_commute)
huffman@31706
  1001
  apply (subst (asm) (2) int_gcd_commute)
huffman@31706
  1002
  apply (erule int_coprime_lmult)
huffman@31706
  1003
done
huffman@31706
  1004
huffman@31706
  1005
lemma nat_invertible_coprime: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
huffman@31706
  1006
apply (metis nat_coprime_lmult nat_gcd_1 nat_gcd_commute nat_gcd_red)
huffman@31706
  1007
done
huffman@31706
  1008
huffman@31706
  1009
lemma int_invertible_coprime: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
huffman@31706
  1010
apply (metis int_coprime_lmult int_gcd_1 int_gcd_commute int_gcd_red)
huffman@31706
  1011
done
huffman@31706
  1012
huffman@31706
  1013
huffman@31706
  1014
subsection {* Bezout's theorem *}
huffman@31706
  1015
huffman@31706
  1016
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1017
   see the theorems that follow the definition. *)
huffman@31706
  1018
fun
huffman@31706
  1019
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1020
where
huffman@31706
  1021
  "bezw x y =
huffman@31706
  1022
  (if y = 0 then (1, 0) else
huffman@31706
  1023
      (snd (bezw y (x mod y)),
huffman@31706
  1024
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1025
huffman@31706
  1026
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1027
huffman@31706
  1028
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1029
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1030
  by simp
huffman@31706
  1031
huffman@31706
  1032
declare bezw.simps [simp del]
huffman@31706
  1033
huffman@31706
  1034
lemma bezw_aux [rule_format]:
huffman@31706
  1035
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
huffman@31706
  1036
proof (induct x y rule: nat_gcd_induct)
huffman@31706
  1037
  fix m :: nat
huffman@31706
  1038
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1039
    by auto
huffman@31706
  1040
  next fix m :: nat and n
huffman@31706
  1041
    assume ngt0: "n > 0" and
huffman@31706
  1042
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1043
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1044
        int (gcd n (m mod n))"
huffman@31706
  1045
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
huffman@31706
  1046
      apply (simp add: bezw_non_0 nat_gcd_non_0)
huffman@31706
  1047
      apply (erule subst)
huffman@31706
  1048
      apply (simp add: ring_simps)
huffman@31706
  1049
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1050
      (* applying simp here undoes the last substitution!
huffman@31706
  1051
         what is procedure cancel_div_mod? *)
huffman@31706
  1052
      apply (simp only: ring_simps zadd_int [symmetric]
huffman@31706
  1053
        zmult_int [symmetric])
huffman@31706
  1054
      done
huffman@31706
  1055
qed
huffman@31706
  1056
huffman@31706
  1057
lemma int_bezout:
huffman@31706
  1058
  fixes x y
huffman@31706
  1059
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1060
proof -
huffman@31706
  1061
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1062
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1063
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1064
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1065
    apply (unfold gcd_int_def)
huffman@31706
  1066
    apply simp
huffman@31706
  1067
    apply (subst bezw_aux [symmetric])
huffman@31706
  1068
    apply auto
huffman@31706
  1069
    done
huffman@31706
  1070
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1071
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1072
    by auto
huffman@31706
  1073
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1074
    by (erule (1) bezout_aux)
huffman@31706
  1075
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1076
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1077
    apply auto
huffman@31706
  1078
    apply (rule_tac x = u in exI)
huffman@31706
  1079
    apply (rule_tac x = "-v" in exI)
huffman@31706
  1080
    apply (subst int_gcd_neg2 [symmetric])
huffman@31706
  1081
    apply auto
huffman@31706
  1082
    done
huffman@31706
  1083
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1084
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1085
    apply auto
huffman@31706
  1086
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1087
    apply (rule_tac x = v in exI)
huffman@31706
  1088
    apply (subst int_gcd_neg1 [symmetric])
huffman@31706
  1089
    apply auto
huffman@31706
  1090
    done
huffman@31706
  1091
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1092
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1093
    apply auto
huffman@31706
  1094
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1095
    apply (rule_tac x = "-v" in exI)
huffman@31706
  1096
    apply (subst int_gcd_neg1 [symmetric])
huffman@31706
  1097
    apply (subst int_gcd_neg2 [symmetric])
huffman@31706
  1098
    apply auto
huffman@31706
  1099
    done
huffman@31706
  1100
  ultimately show ?thesis by blast
huffman@31706
  1101
qed
huffman@31706
  1102
huffman@31706
  1103
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1104
huffman@31706
  1105
lemma ind_euclid:
huffman@31706
  1106
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1107
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1108
  shows "P a b"
chaieb@27669
  1109
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
chaieb@27669
  1110
  fix n a b
chaieb@27669
  1111
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
chaieb@27669
  1112
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1113
  moreover {assume eq: "a= b"
huffman@31706
  1114
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1115
    by simp}
chaieb@27669
  1116
  moreover
chaieb@27669
  1117
  {assume lt: "a < b"
chaieb@27669
  1118
    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
chaieb@27669
  1119
    moreover
chaieb@27669
  1120
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1121
    moreover
chaieb@27669
  1122
    {assume ab: "a + b - a < n"
chaieb@27669
  1123
      have th0: "a + b - a = a + (b - a)" using lt by arith
chaieb@27669
  1124
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1125
      have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1126
    ultimately have "P a b" by blast}
chaieb@27669
  1127
  moreover
chaieb@27669
  1128
  {assume lt: "a > b"
chaieb@27669
  1129
    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
chaieb@27669
  1130
    moreover
chaieb@27669
  1131
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1132
    moreover
chaieb@27669
  1133
    {assume ab: "b + a - b < n"
chaieb@27669
  1134
      have th0: "b + a - b = b + (a - b)" using lt by arith
chaieb@27669
  1135
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1136
      have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1137
      hence "P a b" using c by blast }
chaieb@27669
  1138
    ultimately have "P a b" by blast}
chaieb@27669
  1139
ultimately  show "P a b" by blast
chaieb@27669
  1140
qed
chaieb@27669
  1141
huffman@31706
  1142
lemma nat_bezout_lemma:
huffman@31706
  1143
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1144
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1145
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1146
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1147
  using ex
huffman@31706
  1148
  apply clarsimp
huffman@31706
  1149
  apply (rule_tac x="d" in exI, simp add: dvd_add)
huffman@31706
  1150
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1151
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1152
  apply (rule_tac x="y" in exI)
huffman@31706
  1153
  apply algebra
huffman@31706
  1154
  apply (rule_tac x="x" in exI)
huffman@31706
  1155
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1156
  apply algebra
chaieb@27669
  1157
done
chaieb@27669
  1158
huffman@31706
  1159
lemma nat_bezout_add: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1160
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1161
  apply(induct a b rule: ind_euclid)
huffman@31706
  1162
  apply blast
huffman@31706
  1163
  apply clarify
huffman@31706
  1164
  apply (rule_tac x="a" in exI, simp add: dvd_add)
huffman@31706
  1165
  apply clarsimp
huffman@31706
  1166
  apply (rule_tac x="d" in exI)
huffman@31706
  1167
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
huffman@31706
  1168
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1169
  apply (rule_tac x="y" in exI)
huffman@31706
  1170
  apply algebra
huffman@31706
  1171
  apply (rule_tac x="x" in exI)
huffman@31706
  1172
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1173
  apply algebra
chaieb@27669
  1174
done
chaieb@27669
  1175
huffman@31706
  1176
lemma nat_bezout1: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1177
    (a * x - b * y = d \<or> b * x - a * y = d)"
huffman@31706
  1178
  using nat_bezout_add[of a b]
huffman@31706
  1179
  apply clarsimp
huffman@31706
  1180
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1181
  apply (rule_tac x="x" in exI)
huffman@31706
  1182
  apply (rule_tac x="y" in exI)
huffman@31706
  1183
  apply auto
chaieb@27669
  1184
done
chaieb@27669
  1185
huffman@31706
  1186
lemma nat_bezout_add_strong: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1187
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1188
proof-
huffman@31706
  1189
 from nz have ap: "a > 0" by simp
huffman@31706
  1190
 from nat_bezout_add[of a b]
huffman@31706
  1191
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1192
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1193
 moreover
huffman@31706
  1194
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1195
     from H have ?thesis by blast }
chaieb@27669
  1196
 moreover
chaieb@27669
  1197
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1198
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1199
   moreover
chaieb@27669
  1200
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1201
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1202
       by auto
chaieb@27669
  1203
     moreover
chaieb@27669
  1204
     {assume db: "d=b"
chaieb@27669
  1205
       from prems have ?thesis apply simp
chaieb@27669
  1206
	 apply (rule exI[where x = b], simp)
chaieb@27669
  1207
	 apply (rule exI[where x = b])
chaieb@27669
  1208
	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1209
    moreover
huffman@31706
  1210
    {assume db: "d < b"
chaieb@27669
  1211
	{assume "x=0" hence ?thesis  using prems by simp }
chaieb@27669
  1212
	moreover
chaieb@27669
  1213
	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
chaieb@27669
  1214
	  from db have "d \<le> b - 1" by simp
chaieb@27669
  1215
	  hence "d*b \<le> b*(b - 1)" by simp
chaieb@27669
  1216
	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
chaieb@27669
  1217
	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
huffman@31706
  1218
	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1219
            by simp
huffman@31706
  1220
	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
huffman@31706
  1221
	    by (simp only: mult_assoc right_distrib)
huffman@31706
  1222
	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1223
            by algebra
chaieb@27669
  1224
	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
huffman@31706
  1225
	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
chaieb@27669
  1226
	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
chaieb@27669
  1227
	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
chaieb@27669
  1228
	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
chaieb@27669
  1229
	  hence ?thesis using H(1,2)
chaieb@27669
  1230
	    apply -
chaieb@27669
  1231
	    apply (rule exI[where x=d], simp)
chaieb@27669
  1232
	    apply (rule exI[where x="(b - 1) * y"])
chaieb@27669
  1233
	    by (rule exI[where x="x*(b - 1) - d"], simp)}
chaieb@27669
  1234
	ultimately have ?thesis by blast}
chaieb@27669
  1235
    ultimately have ?thesis by blast}
chaieb@27669
  1236
  ultimately have ?thesis by blast}
chaieb@27669
  1237
 ultimately show ?thesis by blast
chaieb@27669
  1238
qed
chaieb@27669
  1239
huffman@31706
  1240
lemma nat_bezout: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1241
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1242
proof-
chaieb@27669
  1243
  let ?g = "gcd a b"
huffman@31706
  1244
  from nat_bezout_add_strong[OF a, of b]
chaieb@27669
  1245
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1246
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1247
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1248
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1249
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1250
  thus ?thesis by blast
chaieb@27669
  1251
qed
chaieb@27669
  1252
huffman@31706
  1253
huffman@31706
  1254
subsection {* LCM *}
huffman@31706
  1255
huffman@31706
  1256
lemma int_lcm_altdef: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1257
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@31706
  1258
    zmult_int [symmetric] gcd_int_def)
huffman@31706
  1259
huffman@31706
  1260
lemma nat_prod_gcd_lcm: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1261
  unfolding lcm_nat_def
huffman@31706
  1262
  by (simp add: dvd_mult_div_cancel [OF nat_gcd_dvd_prod])
huffman@31706
  1263
huffman@31706
  1264
lemma int_prod_gcd_lcm: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1265
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1266
  apply (subst int_mult [symmetric])
huffman@31706
  1267
  apply (subst nat_prod_gcd_lcm [symmetric])
huffman@31706
  1268
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1269
  apply (simp, simp add: abs_mult)
huffman@31706
  1270
done
huffman@31706
  1271
huffman@31706
  1272
lemma nat_lcm_0 [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1273
  unfolding lcm_nat_def by simp
huffman@31706
  1274
huffman@31706
  1275
lemma int_lcm_0 [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1276
  unfolding lcm_int_def by simp
huffman@31706
  1277
huffman@31706
  1278
lemma nat_lcm_1 [simp]: "lcm (m::nat) 1 = m"
huffman@31706
  1279
  unfolding lcm_nat_def by simp
huffman@31706
  1280
huffman@31706
  1281
lemma nat_lcm_Suc_0 [simp]: "lcm (m::nat) (Suc 0) = m"
huffman@31706
  1282
  unfolding lcm_nat_def by (simp add: One_nat_def)
huffman@31706
  1283
huffman@31706
  1284
lemma int_lcm_1 [simp]: "lcm (m::int) 1 = abs m"
huffman@31706
  1285
  unfolding lcm_int_def by simp
huffman@31706
  1286
huffman@31706
  1287
lemma nat_lcm_0_left [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1288
  unfolding lcm_nat_def by simp
chaieb@27669
  1289
huffman@31706
  1290
lemma int_lcm_0_left [simp]: "lcm (0::int) n = 0"
huffman@31706
  1291
  unfolding lcm_int_def by simp
huffman@31706
  1292
huffman@31706
  1293
lemma nat_lcm_1_left [simp]: "lcm (1::nat) m = m"
huffman@31706
  1294
  unfolding lcm_nat_def by simp
huffman@31706
  1295
huffman@31706
  1296
lemma nat_lcm_Suc_0_left [simp]: "lcm (Suc 0) m = m"
huffman@31706
  1297
  unfolding lcm_nat_def by (simp add: One_nat_def)
huffman@31706
  1298
huffman@31706
  1299
lemma int_lcm_1_left [simp]: "lcm (1::int) m = abs m"
huffman@31706
  1300
  unfolding lcm_int_def by simp
huffman@31706
  1301
huffman@31706
  1302
lemma nat_lcm_commute: "lcm (m::nat) n = lcm n m"
huffman@31706
  1303
  unfolding lcm_nat_def by (simp add: nat_gcd_commute ring_simps)
huffman@31706
  1304
huffman@31706
  1305
lemma int_lcm_commute: "lcm (m::int) n = lcm n m"
huffman@31706
  1306
  unfolding lcm_int_def by (subst nat_lcm_commute, rule refl)
huffman@31706
  1307
huffman@31706
  1308
(* to do: show lcm is associative, and then declare ac simps *)
huffman@31706
  1309
huffman@31706
  1310
lemma nat_lcm_pos:
huffman@31706
  1311
  assumes mpos: "(m::nat) > 0"
huffman@31706
  1312
  and npos: "n>0"
huffman@31706
  1313
  shows "lcm m n > 0"
huffman@31706
  1314
proof(rule ccontr, simp add: lcm_nat_def nat_gcd_zero)
huffman@31706
  1315
  assume h:"m*n div gcd m n = 0"
huffman@31706
  1316
  from mpos npos have "gcd m n \<noteq> 0" using nat_gcd_zero by simp
huffman@31706
  1317
  hence gcdp: "gcd m n > 0" by simp
huffman@31706
  1318
  with h
huffman@31706
  1319
  have "m*n < gcd m n"
huffman@31706
  1320
    by (cases "m * n < gcd m n")
huffman@31706
  1321
       (auto simp add: div_if[OF gcdp, where m="m*n"])
chaieb@27669
  1322
  moreover
huffman@31706
  1323
  have "gcd m n dvd m" by simp
huffman@31706
  1324
  with mpos dvd_imp_le have t1:"gcd m n \<le> m" by simp
huffman@31706
  1325
  with npos have t1:"gcd m n*n \<le> m*n" by simp
huffman@31706
  1326
  have "gcd m n \<le> gcd m n*n" using npos by simp
huffman@31706
  1327
  with t1 have "gcd m n \<le> m*n" by arith
huffman@31706
  1328
  ultimately show "False" by simp
chaieb@27669
  1329
qed
chaieb@27669
  1330
huffman@31706
  1331
lemma int_lcm_pos:
huffman@31706
  1332
  assumes mneq0: "(m::int) ~= 0"
huffman@31706
  1333
  and npos: "n ~= 0"
huffman@31706
  1334
  shows "lcm m n > 0"
chaieb@27669
  1335
huffman@31706
  1336
  apply (subst int_lcm_abs)
huffman@31706
  1337
  apply (rule nat_lcm_pos [transferred])
huffman@31706
  1338
  using prems apply auto
huffman@31706
  1339
done
haftmann@23687
  1340
huffman@31706
  1341
lemma nat_dvd_pos:
haftmann@23687
  1342
  fixes n m :: nat
haftmann@23687
  1343
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1344
  shows "m > 0"
haftmann@23687
  1345
using assms by (cases m) auto
haftmann@23687
  1346
huffman@31706
  1347
lemma nat_lcm_least:
huffman@31706
  1348
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1349
  shows "lcm m n dvd k"
haftmann@23687
  1350
proof (cases k)
haftmann@23687
  1351
  case 0 then show ?thesis by auto
haftmann@23687
  1352
next
haftmann@23687
  1353
  case (Suc _) then have pos_k: "k > 0" by auto
huffman@31706
  1354
  from assms nat_dvd_pos [OF this] have pos_mn: "m > 0" "n > 0" by auto
huffman@31706
  1355
  with nat_gcd_zero [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1356
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1357
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1358
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1359
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1360
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
huffman@31706
  1361
    by (simp add: mult_ac nat_gcd_mult_distrib)
haftmann@27556
  1362
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1363
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1364
  also have "\<dots> = k * p * q * gcd m n"
huffman@31706
  1365
    by (simp add: mult_ac nat_gcd_mult_distrib)
haftmann@27556
  1366
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1367
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1368
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1369
    by (simp add: mult_ac)
haftmann@27556
  1370
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1371
    by simp
huffman@31706
  1372
  with nat_prod_gcd_lcm [of m n]
haftmann@27556
  1373
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1374
    by (simp add: mult_ac)
huffman@31706
  1375
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1376
  then show ?thesis using dvd_def by auto
haftmann@23687
  1377
qed
haftmann@23687
  1378
huffman@31706
  1379
lemma int_lcm_least:
huffman@31706
  1380
  assumes "(m::int) dvd k" and "n dvd k"
huffman@31706
  1381
  shows "lcm m n dvd k"
huffman@31706
  1382
huffman@31706
  1383
  apply (subst int_lcm_abs)
huffman@31706
  1384
  apply (rule dvd_trans)
huffman@31706
  1385
  apply (rule nat_lcm_least [transferred, of _ "abs k" _])
huffman@31706
  1386
  using prems apply auto
huffman@31706
  1387
done
huffman@31706
  1388
huffman@31706
  1389
lemma nat_lcm_dvd1 [iff]: "(m::nat) dvd lcm m n"
haftmann@23687
  1390
proof (cases m)
haftmann@23687
  1391
  case 0 then show ?thesis by simp
haftmann@23687
  1392
next
haftmann@23687
  1393
  case (Suc _)
haftmann@23687
  1394
  then have mpos: "m > 0" by simp
haftmann@23687
  1395
  show ?thesis
haftmann@23687
  1396
  proof (cases n)
haftmann@23687
  1397
    case 0 then show ?thesis by simp
haftmann@23687
  1398
  next
haftmann@23687
  1399
    case (Suc _)
haftmann@23687
  1400
    then have npos: "n > 0" by simp
haftmann@27556
  1401
    have "gcd m n dvd n" by simp
haftmann@27556
  1402
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1403
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1404
      by (simp add: mult_ac)
huffman@31706
  1405
    also have "\<dots> = m * k" using mpos npos nat_gcd_zero by simp
huffman@31706
  1406
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1407
  qed
haftmann@23687
  1408
qed
haftmann@23687
  1409
huffman@31706
  1410
lemma int_lcm_dvd1 [iff]: "(m::int) dvd lcm m n"
huffman@31706
  1411
  apply (subst int_lcm_abs)
huffman@31706
  1412
  apply (rule dvd_trans)
huffman@31706
  1413
  prefer 2
huffman@31706
  1414
  apply (rule nat_lcm_dvd1 [transferred])
huffman@31706
  1415
  apply auto
huffman@31706
  1416
done
huffman@31706
  1417
huffman@31706
  1418
lemma nat_lcm_dvd2 [iff]: "(n::nat) dvd lcm m n"
huffman@31706
  1419
  by (subst nat_lcm_commute, rule nat_lcm_dvd1)
huffman@31706
  1420
huffman@31706
  1421
lemma int_lcm_dvd2 [iff]: "(n::int) dvd lcm m n"
huffman@31706
  1422
  by (subst int_lcm_commute, rule int_lcm_dvd1)
huffman@31706
  1423
nipkow@31729
  1424
lemma dvd_lcm_if_dvd1_nat: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31729
  1425
by(metis nat_lcm_dvd1 dvd_trans)
nipkow@31729
  1426
nipkow@31729
  1427
lemma dvd_lcm_if_dvd2_nat: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31729
  1428
by(metis nat_lcm_dvd2 dvd_trans)
nipkow@31729
  1429
nipkow@31729
  1430
lemma dvd_lcm_if_dvd1_int: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31729
  1431
by(metis int_lcm_dvd1 dvd_trans)
nipkow@31729
  1432
nipkow@31729
  1433
lemma dvd_lcm_if_dvd2_int: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31729
  1434
by(metis int_lcm_dvd2 dvd_trans)
nipkow@31729
  1435
huffman@31706
  1436
lemma nat_lcm_unique: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1437
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
huffman@31706
  1438
  by (auto intro: dvd_anti_sym nat_lcm_least)
chaieb@27568
  1439
huffman@31706
  1440
lemma int_lcm_unique: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1441
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
huffman@31706
  1442
  by (auto intro: dvd_anti_sym [transferred] int_lcm_least)
huffman@31706
  1443
huffman@31706
  1444
lemma nat_lcm_dvd_eq [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1445
  apply (rule sym)
huffman@31706
  1446
  apply (subst nat_lcm_unique [symmetric])
huffman@31706
  1447
  apply auto
huffman@31706
  1448
done
huffman@31706
  1449
huffman@31706
  1450
lemma int_lcm_dvd_eq [simp]: "0 <= y \<Longrightarrow> (x::int) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1451
  apply (rule sym)
huffman@31706
  1452
  apply (subst int_lcm_unique [symmetric])
huffman@31706
  1453
  apply auto
huffman@31706
  1454
done
huffman@31706
  1455
huffman@31706
  1456
lemma nat_lcm_dvd_eq' [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
huffman@31706
  1457
  by (subst nat_lcm_commute, erule nat_lcm_dvd_eq)
huffman@31706
  1458
huffman@31706
  1459
lemma int_lcm_dvd_eq' [simp]: "y >= 0 \<Longrightarrow> (x::int) dvd y \<Longrightarrow> lcm y x = y"
huffman@31706
  1460
  by (subst int_lcm_commute, erule (1) int_lcm_dvd_eq)
huffman@31706
  1461
chaieb@27568
  1462
haftmann@23687
  1463
huffman@31706
  1464
subsection {* Primes *}
wenzelm@22367
  1465
huffman@31706
  1466
(* Is there a better way to handle these, rather than making them
huffman@31706
  1467
   elim rules? *)
chaieb@22027
  1468
huffman@31706
  1469
lemma nat_prime_ge_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
huffman@31706
  1470
  by (unfold prime_nat_def, auto)
chaieb@22027
  1471
huffman@31706
  1472
lemma nat_prime_gt_0 [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
huffman@31706
  1473
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1474
huffman@31706
  1475
lemma nat_prime_ge_1 [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
huffman@31706
  1476
  by (unfold prime_nat_def, auto)
chaieb@22027
  1477
huffman@31706
  1478
lemma nat_prime_gt_1 [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
huffman@31706
  1479
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1480
huffman@31706
  1481
lemma nat_prime_ge_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
huffman@31706
  1482
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1483
huffman@31706
  1484
lemma nat_prime_gt_Suc_0 [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
huffman@31706
  1485
  by (unfold prime_nat_def, auto)
huffman@31706
  1486
huffman@31706
  1487
lemma nat_prime_ge_2 [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
huffman@31706
  1488
  by (unfold prime_nat_def, auto)
huffman@31706
  1489
huffman@31706
  1490
lemma int_prime_ge_0 [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
huffman@31706
  1491
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1492
huffman@31706
  1493
lemma int_prime_gt_0 [elim]: "prime (p::int) \<Longrightarrow> p > 0"
huffman@31706
  1494
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1495
huffman@31706
  1496
lemma int_prime_ge_1 [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
huffman@31706
  1497
  by (unfold prime_int_def prime_nat_def, auto)
chaieb@22027
  1498
huffman@31706
  1499
lemma int_prime_gt_1 [elim]: "prime (p::int) \<Longrightarrow> p > 1"
huffman@31706
  1500
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1501
huffman@31706
  1502
lemma int_prime_ge_2 [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
huffman@31706
  1503
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1504
huffman@31706
  1505
thm prime_nat_def;
huffman@31706
  1506
thm prime_nat_def [transferred];
huffman@31706
  1507
huffman@31706
  1508
lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
huffman@31706
  1509
    m = 1 \<or> m = p))"
huffman@31706
  1510
  using prime_nat_def [transferred]
huffman@31706
  1511
    apply (case_tac "p >= 0")
huffman@31706
  1512
    by (blast, auto simp add: int_prime_ge_0)
huffman@31706
  1513
huffman@31706
  1514
(* To do: determine primality of any numeral *)
huffman@31706
  1515
huffman@31706
  1516
lemma nat_zero_not_prime [simp]: "~prime (0::nat)"
huffman@31706
  1517
  by (simp add: prime_nat_def)
huffman@31706
  1518
huffman@31706
  1519
lemma int_zero_not_prime [simp]: "~prime (0::int)"
huffman@31706
  1520
  by (simp add: prime_int_def)
huffman@31706
  1521
huffman@31706
  1522
lemma nat_one_not_prime [simp]: "~prime (1::nat)"
huffman@31706
  1523
  by (simp add: prime_nat_def)
chaieb@22027
  1524
huffman@31706
  1525
lemma nat_Suc_0_not_prime [simp]: "~prime (Suc 0)"
huffman@31706
  1526
  by (simp add: prime_nat_def One_nat_def)
huffman@31706
  1527
huffman@31706
  1528
lemma int_one_not_prime [simp]: "~prime (1::int)"
huffman@31706
  1529
  by (simp add: prime_int_def)
huffman@31706
  1530
huffman@31706
  1531
lemma nat_two_is_prime [simp]: "prime (2::nat)"
huffman@31706
  1532
  apply (auto simp add: prime_nat_def)
huffman@31706
  1533
  apply (case_tac m)
huffman@31706
  1534
  apply (auto dest!: dvd_imp_le)
huffman@31706
  1535
  done
chaieb@22027
  1536
huffman@31706
  1537
lemma int_two_is_prime [simp]: "prime (2::int)"
huffman@31706
  1538
  by (rule nat_two_is_prime [transferred direction: nat "op <= (0::int)"])
chaieb@27568
  1539
huffman@31706
  1540
lemma nat_prime_imp_coprime: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1541
  apply (unfold prime_nat_def)
huffman@31706
  1542
  apply (metis nat_gcd_dvd1 nat_gcd_dvd2)
huffman@31706
  1543
  done
huffman@31706
  1544
huffman@31706
  1545
lemma int_prime_imp_coprime: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1546
  apply (unfold prime_int_altdef)
huffman@31706
  1547
  apply (metis int_gcd_dvd1 int_gcd_dvd2 int_gcd_ge_0)
chaieb@27568
  1548
  done
chaieb@27568
  1549
huffman@31706
  1550
lemma nat_prime_dvd_mult: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
huffman@31706
  1551
  by (blast intro: nat_coprime_dvd_mult nat_prime_imp_coprime)
huffman@31706
  1552
huffman@31706
  1553
lemma int_prime_dvd_mult: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
huffman@31706
  1554
  by (blast intro: int_coprime_dvd_mult int_prime_imp_coprime)
huffman@31706
  1555
huffman@31706
  1556
lemma nat_prime_dvd_mult_eq [simp]: "prime (p::nat) \<Longrightarrow>
huffman@31706
  1557
    p dvd m * n = (p dvd m \<or> p dvd n)"
huffman@31706
  1558
  by (rule iffI, rule nat_prime_dvd_mult, auto)
chaieb@27568
  1559
huffman@31706
  1560
lemma int_prime_dvd_mult_eq [simp]: "prime (p::int) \<Longrightarrow>
huffman@31706
  1561
    p dvd m * n = (p dvd m \<or> p dvd n)"
huffman@31706
  1562
  by (rule iffI, rule int_prime_dvd_mult, auto)
chaieb@27568
  1563
huffman@31706
  1564
lemma nat_not_prime_eq_prod: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1565
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1566
  unfolding prime_nat_def dvd_def apply auto
huffman@31706
  1567
  apply (subgoal_tac "k > 1")
huffman@31706
  1568
  apply force
huffman@31706
  1569
  apply (subgoal_tac "k ~= 0")
huffman@31706
  1570
  apply force
huffman@31706
  1571
  apply (rule notI)
huffman@31706
  1572
  apply force
huffman@31706
  1573
done
chaieb@27568
  1574
huffman@31706
  1575
(* there's a lot of messing around with signs of products here --
huffman@31706
  1576
   could this be made more automatic? *)
huffman@31706
  1577
lemma int_not_prime_eq_prod: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1578
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1579
  unfolding prime_int_altdef dvd_def
huffman@31706
  1580
  apply auto
huffman@31706
  1581
  apply (rule_tac x = m in exI)
huffman@31706
  1582
  apply (rule_tac x = k in exI)
huffman@31706
  1583
  apply (auto simp add: mult_compare_simps)
huffman@31706
  1584
  apply (subgoal_tac "k > 0")
huffman@31706
  1585
  apply arith
huffman@31706
  1586
  apply (case_tac "k <= 0")
huffman@31706
  1587
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1588
  apply force
huffman@31706
  1589
  apply (subst zero_compare_simps(8))
huffman@31706
  1590
  apply auto
huffman@31706
  1591
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1592
  apply force
huffman@31706
  1593
  apply (subst zero_compare_simps(8))
huffman@31706
  1594
  apply auto
huffman@31706
  1595
done
chaieb@27568
  1596
huffman@31706
  1597
lemma nat_prime_dvd_power [rule_format]: "prime (p::nat) -->
huffman@31706
  1598
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1599
  by (induct n rule: nat_induct, auto)
chaieb@27568
  1600
huffman@31706
  1601
lemma int_prime_dvd_power [rule_format]: "prime (p::int) -->
huffman@31706
  1602
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1603
  apply (induct n rule: nat_induct, auto)
huffman@31706
  1604
  apply (frule int_prime_ge_0)
huffman@31706
  1605
  apply auto
huffman@31706
  1606
done
huffman@31706
  1607
huffman@31706
  1608
lemma nat_prime_imp_power_coprime: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1609
    coprime a (p^m)"
huffman@31706
  1610
  apply (rule nat_coprime_exp)
huffman@31706
  1611
  apply (subst nat_gcd_commute)
huffman@31706
  1612
  apply (erule (1) nat_prime_imp_coprime)
huffman@31706
  1613
done
chaieb@27568
  1614
huffman@31706
  1615
lemma int_prime_imp_power_coprime: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1616
    coprime a (p^m)"
huffman@31706
  1617
  apply (rule int_coprime_exp)
huffman@31706
  1618
  apply (subst int_gcd_commute)
huffman@31706
  1619
  apply (erule (1) int_prime_imp_coprime)
huffman@31706
  1620
done
chaieb@27568
  1621
huffman@31706
  1622
lemma nat_primes_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
huffman@31706
  1623
  apply (rule nat_prime_imp_coprime, assumption)
huffman@31706
  1624
  apply (unfold prime_nat_def, auto)
huffman@31706
  1625
done
chaieb@27568
  1626
huffman@31706
  1627
lemma int_primes_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
huffman@31706
  1628
  apply (rule int_prime_imp_coprime, assumption)
huffman@31706
  1629
  apply (unfold prime_int_altdef, clarify)
huffman@31706
  1630
  apply (drule_tac x = q in spec)
huffman@31706
  1631
  apply (drule_tac x = p in spec)
huffman@31706
  1632
  apply auto
huffman@31706
  1633
done
chaieb@27568
  1634
huffman@31706
  1635
lemma nat_primes_imp_powers_coprime: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1636
    coprime (p^m) (q^n)"
huffman@31706
  1637
  by (rule nat_coprime_exp2, rule nat_primes_coprime)
chaieb@27568
  1638
huffman@31706
  1639
lemma int_primes_imp_powers_coprime: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1640
    coprime (p^m) (q^n)"
huffman@31706
  1641
  by (rule int_coprime_exp2, rule int_primes_coprime)
chaieb@27568
  1642
huffman@31706
  1643
lemma nat_prime_factor: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
huffman@31706
  1644
  apply (induct n rule: nat_less_induct)
huffman@31706
  1645
  apply (case_tac "n = 0")
huffman@31706
  1646
  using nat_two_is_prime apply blast
huffman@31706
  1647
  apply (case_tac "prime n")
huffman@31706
  1648
  apply blast
huffman@31706
  1649
  apply (subgoal_tac "n > 1")
huffman@31706
  1650
  apply (frule (1) nat_not_prime_eq_prod)
huffman@31706
  1651
  apply (auto intro: dvd_mult dvd_mult2)
huffman@31706
  1652
done
chaieb@23244
  1653
huffman@31706
  1654
(* An Isar version:
huffman@31706
  1655
huffman@31706
  1656
lemma nat_prime_factor_b:
huffman@31706
  1657
  fixes n :: nat
huffman@31706
  1658
  assumes "n \<noteq> 1"
huffman@31706
  1659
  shows "\<exists>p. prime p \<and> p dvd n"
nipkow@23983
  1660
huffman@31706
  1661
using `n ~= 1`
huffman@31706
  1662
proof (induct n rule: nat_less_induct)
huffman@31706
  1663
  fix n :: nat
huffman@31706
  1664
  assume "n ~= 1" and
huffman@31706
  1665
    ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
huffman@31706
  1666
  thus "\<exists>p. prime p \<and> p dvd n"
huffman@31706
  1667
  proof -
huffman@31706
  1668
  {
huffman@31706
  1669
    assume "n = 0"
huffman@31706
  1670
    moreover note nat_two_is_prime
huffman@31706
  1671
    ultimately have ?thesis
huffman@31706
  1672
      by (auto simp del: nat_two_is_prime)
huffman@31706
  1673
  }
huffman@31706
  1674
  moreover
huffman@31706
  1675
  {
huffman@31706
  1676
    assume "prime n"
huffman@31706
  1677
    hence ?thesis by auto
huffman@31706
  1678
  }
huffman@31706
  1679
  moreover
huffman@31706
  1680
  {
huffman@31706
  1681
    assume "n ~= 0" and "~ prime n"
huffman@31706
  1682
    with `n ~= 1` have "n > 1" by auto
huffman@31706
  1683
    with `~ prime n` and nat_not_prime_eq_prod obtain m k where
huffman@31706
  1684
      "n = m * k" and "1 < m" and "m < n" by blast
huffman@31706
  1685
    with ih obtain p where "prime p" and "p dvd m" by blast
huffman@31706
  1686
    with `n = m * k` have ?thesis by auto
huffman@31706
  1687
  }
huffman@31706
  1688
  ultimately show ?thesis by blast
huffman@31706
  1689
  qed
nipkow@23983
  1690
qed
nipkow@23983
  1691
huffman@31706
  1692
*)
huffman@31706
  1693
huffman@31706
  1694
text {* One property of coprimality is easier to prove via prime factors. *}
huffman@31706
  1695
huffman@31706
  1696
lemma nat_prime_divprod_pow:
huffman@31706
  1697
  assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
huffman@31706
  1698
  shows "p^n dvd a \<or> p^n dvd b"
huffman@31706
  1699
proof-
huffman@31706
  1700
  {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
huffman@31706
  1701
      apply (cases "n=0", simp_all)
huffman@31706
  1702
      apply (cases "a=1", simp_all) done}
huffman@31706
  1703
  moreover
huffman@31706
  1704
  {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
huffman@31706
  1705
    then obtain m where m: "n = Suc m" by (cases n, auto)
huffman@31706
  1706
    from n have "p dvd p^n" by (intro dvd_power, auto)
huffman@31706
  1707
    also note pab
huffman@31706
  1708
    finally have pab': "p dvd a * b".
huffman@31706
  1709
    from nat_prime_dvd_mult[OF p pab']
huffman@31706
  1710
    have "p dvd a \<or> p dvd b" .
huffman@31706
  1711
    moreover
huffman@31706
  1712
    {assume pa: "p dvd a"
huffman@31706
  1713
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
huffman@31706
  1714
      from nat_coprime_common_divisor [OF ab, OF pa] p have "\<not> p dvd b" by auto
huffman@31706
  1715
      with p have "coprime b p"
huffman@31706
  1716
        by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
huffman@31706
  1717
      hence pnb: "coprime (p^n) b"
huffman@31706
  1718
        by (subst nat_gcd_commute, rule nat_coprime_exp)
huffman@31706
  1719
      from nat_coprime_divprod[OF pnba pnb] have ?thesis by blast }
huffman@31706
  1720
    moreover
huffman@31706
  1721
    {assume pb: "p dvd b"
huffman@31706
  1722
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
huffman@31706
  1723
      from nat_coprime_common_divisor [OF ab, of p] pb p have "\<not> p dvd a"
huffman@31706
  1724
        by auto
huffman@31706
  1725
      with p have "coprime a p"
huffman@31706
  1726
        by (subst nat_gcd_commute, intro nat_prime_imp_coprime)
huffman@31706
  1727
      hence pna: "coprime (p^n) a"
huffman@31706
  1728
        by (subst nat_gcd_commute, rule nat_coprime_exp)
huffman@31706
  1729
      from nat_coprime_divprod[OF pab pna] have ?thesis by blast }
huffman@31706
  1730
    ultimately have ?thesis by blast}
huffman@31706
  1731
  ultimately show ?thesis by blast
nipkow@23983
  1732
qed
nipkow@23983
  1733
wenzelm@21256
  1734
end