src/HOL/GCD.thy
author avigad
Fri Jul 10 10:45:30 2009 -0400 (2009-07-10)
changeset 32036 8a9228872fbd
parent 31952 40501bb2d57c
child 32040 830141c9e528
permissions -rw-r--r--
Moved factorial lemmas from Binomial.thy to Fact.thy and merged.
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(*  Title:      GCD.thy
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    Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with the functions gcd and lcm, and properties of
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primes. Definitions and lemmas are proved uniformly for the natural
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numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by Chiaeb.
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Jeremy Avigad combined all of these, made everything uniform for the
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natural numbers and the integers, and added a number of new theorems.
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Tobias Nipkow cleaned up a lot.
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*)
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header {* GCD *}
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theory GCD
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imports Fact
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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 gcd_nat_induct:
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  fixes m n :: nat
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  assumes "\<And>m. P m 0"
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    and "\<And>m n. 0 < n \<Longrightarrow> P n (m mod n) \<Longrightarrow> P m n"
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  shows "P m n"
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  apply (rule gcd_nat.induct)
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  apply (case_tac "y = 0")
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  using assms apply simp_all
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done
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(* specific to int *)
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lemma gcd_neg1_int [simp]: "gcd (-x::int) y = gcd x y"
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  by (simp add: gcd_int_def)
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lemma gcd_neg2_int [simp]: "gcd (x::int) (-y) = gcd x y"
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  by (simp add: gcd_int_def)
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lemma abs_gcd_int[simp]: "abs(gcd (x::int) y) = gcd x y"
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by(simp add: gcd_int_def)
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lemma gcd_abs_int: "gcd (x::int) y = gcd (abs x) (abs y)"
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by (simp add: gcd_int_def)
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lemma gcd_abs1_int[simp]: "gcd (abs x) (y::int) = gcd x y"
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by (metis abs_idempotent gcd_abs_int)
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lemma gcd_abs2_int[simp]: "gcd x (abs y::int) = gcd x y"
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by (metis abs_idempotent gcd_abs_int)
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lemma gcd_cases_int:
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  fixes x :: int and y
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  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd x y)"
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      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd x (-y))"
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      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (gcd (-x) y)"
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      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (gcd (-x) (-y))"
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  shows "P (gcd x y)"
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by (insert prems, auto simp add: gcd_neg1_int gcd_neg2_int, arith)
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lemma gcd_ge_0_int [simp]: "gcd (x::int) y >= 0"
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  by (simp add: gcd_int_def)
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lemma lcm_neg1_int: "lcm (-x::int) y = lcm x y"
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  by (simp add: lcm_int_def)
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lemma lcm_neg2_int: "lcm (x::int) (-y) = lcm x y"
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  by (simp add: lcm_int_def)
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lemma lcm_abs_int: "lcm (x::int) y = lcm (abs x) (abs y)"
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  by (simp add: lcm_int_def)
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lemma abs_lcm_int [simp]: "abs (lcm i j::int) = lcm i j"
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by(simp add:lcm_int_def)
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lemma lcm_abs1_int[simp]: "lcm (abs x) (y::int) = lcm x y"
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by (metis abs_idempotent lcm_int_def)
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lemma lcm_abs2_int[simp]: "lcm x (abs y::int) = lcm x y"
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by (metis abs_idempotent lcm_int_def)
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lemma lcm_cases_int:
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  fixes x :: int and y
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  assumes "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm x y)"
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      and "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm x (-y))"
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      and "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> P (lcm (-x) y)"
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      and "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> P (lcm (-x) (-y))"
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  shows "P (lcm x y)"
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by (insert prems, auto simp add: lcm_neg1_int lcm_neg2_int, arith)
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lemma lcm_ge_0_int [simp]: "lcm (x::int) y >= 0"
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  by (simp add: lcm_int_def)
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(* was gcd_0, etc. *)
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lemma gcd_0_nat [simp]: "gcd (x::nat) 0 = x"
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  by simp
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(* was igcd_0, etc. *)
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lemma gcd_0_int [simp]: "gcd (x::int) 0 = abs x"
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  by (unfold gcd_int_def, auto)
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lemma gcd_0_left_nat [simp]: "gcd 0 (x::nat) = x"
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  by simp
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lemma gcd_0_left_int [simp]: "gcd 0 (x::int) = abs x"
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  by (unfold gcd_int_def, auto)
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lemma gcd_red_nat: "gcd (x::nat) y = gcd y (x mod y)"
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  by (case_tac "y = 0", auto)
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(* weaker, but useful for the simplifier *)
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lemma gcd_non_0_nat: "y ~= (0::nat) \<Longrightarrow> gcd (x::nat) y = gcd y (x mod y)"
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  by simp
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lemma gcd_1_nat [simp]: "gcd (m::nat) 1 = 1"
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  by simp
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lemma gcd_Suc_0 [simp]: "gcd (m::nat) (Suc 0) = Suc 0"
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  by (simp add: One_nat_def)
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lemma gcd_1_int [simp]: "gcd (m::int) 1 = 1"
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  by (simp add: gcd_int_def)
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lemma gcd_idem_nat: "gcd (x::nat) x = x"
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by simp
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lemma gcd_idem_int: "gcd (x::int) x = abs x"
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by (auto simp add: gcd_int_def)
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declare gcd_nat.simps [simp del]
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text {*
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  \medskip @{term "gcd m n"} divides @{text m} and @{text n}.  The
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  conjunctions don't seem provable separately.
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*}
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lemma gcd_dvd1_nat [iff]: "(gcd (m::nat)) n dvd m"
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  and gcd_dvd2_nat [iff]: "(gcd m n) dvd n"
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  apply (induct m n rule: gcd_nat_induct)
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  apply (simp_all add: gcd_non_0_nat)
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  apply (blast dest: dvd_mod_imp_dvd)
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done
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lemma gcd_dvd1_int [iff]: "gcd (x::int) y dvd x"
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by (metis gcd_int_def int_dvd_iff gcd_dvd1_nat)
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lemma gcd_dvd2_int [iff]: "gcd (x::int) y dvd y"
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by (metis gcd_int_def int_dvd_iff gcd_dvd2_nat)
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lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
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by(metis gcd_dvd1_nat dvd_trans)
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lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
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by(metis gcd_dvd2_nat dvd_trans)
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lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
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by(metis gcd_dvd1_int dvd_trans)
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lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
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by(metis gcd_dvd2_int dvd_trans)
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lemma gcd_le1_nat [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
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  by (rule dvd_imp_le, auto)
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lemma gcd_le2_nat [simp]: "b \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> b"
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  by (rule dvd_imp_le, auto)
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lemma gcd_le1_int [simp]: "a > 0 \<Longrightarrow> gcd (a::int) b \<le> a"
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  by (rule zdvd_imp_le, auto)
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lemma gcd_le2_int [simp]: "b > 0 \<Longrightarrow> gcd (a::int) b \<le> b"
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  by (rule zdvd_imp_le, auto)
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lemma gcd_greatest_nat: "(k::nat) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
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by (induct m n rule: gcd_nat_induct) (simp_all add: gcd_non_0_nat dvd_mod)
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lemma gcd_greatest_int:
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  "(k::int) dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd gcd m n"
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  apply (subst gcd_abs_int)
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  apply (subst abs_dvd_iff [symmetric])
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  apply (rule gcd_greatest_nat [transferred])
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  apply auto
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done
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lemma gcd_greatest_iff_nat [iff]: "(k dvd gcd (m::nat) n) =
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    (k dvd m & k dvd n)"
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  by (blast intro!: gcd_greatest_nat intro: dvd_trans)
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lemma gcd_greatest_iff_int: "((k::int) dvd gcd m n) = (k dvd m & k dvd n)"
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  by (blast intro!: gcd_greatest_int intro: dvd_trans)
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lemma gcd_zero_nat [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
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   341
  by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff_nat)
wenzelm@21256
   342
nipkow@31952
   343
lemma gcd_zero_int [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
huffman@31706
   344
  by (auto simp add: gcd_int_def)
wenzelm@21256
   345
nipkow@31952
   346
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
nipkow@31952
   347
  by (insert gcd_zero_nat [of m n], arith)
wenzelm@21256
   348
nipkow@31952
   349
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
nipkow@31952
   350
  by (insert gcd_zero_int [of m n], insert gcd_ge_0_int [of m n], arith)
huffman@31706
   351
nipkow@31952
   352
lemma gcd_commute_nat: "gcd (m::nat) n = gcd n m"
huffman@31706
   353
  by (rule dvd_anti_sym, auto)
haftmann@23687
   354
nipkow@31952
   355
lemma gcd_commute_int: "gcd (m::int) n = gcd n m"
nipkow@31952
   356
  by (auto simp add: gcd_int_def gcd_commute_nat)
huffman@31706
   357
nipkow@31952
   358
lemma gcd_assoc_nat: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
huffman@31706
   359
  apply (rule dvd_anti_sym)
huffman@31706
   360
  apply (blast intro: dvd_trans)+
huffman@31706
   361
done
wenzelm@21256
   362
nipkow@31952
   363
lemma gcd_assoc_int: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
nipkow@31952
   364
  by (auto simp add: gcd_int_def gcd_assoc_nat)
huffman@31706
   365
nipkow@31952
   366
lemmas gcd_left_commute_nat =
nipkow@31952
   367
  mk_left_commute[of gcd, OF gcd_assoc_nat gcd_commute_nat]
huffman@31706
   368
nipkow@31952
   369
lemmas gcd_left_commute_int =
nipkow@31952
   370
  mk_left_commute[of gcd, OF gcd_assoc_int gcd_commute_int]
huffman@31706
   371
nipkow@31952
   372
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
huffman@31706
   373
  -- {* gcd is an AC-operator *}
wenzelm@21256
   374
nipkow@31952
   375
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
huffman@31706
   376
nipkow@31952
   377
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   378
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   379
  apply auto
huffman@31706
   380
  apply (rule dvd_anti_sym)
nipkow@31952
   381
  apply (erule (1) gcd_greatest_nat)
huffman@31706
   382
  apply auto
huffman@31706
   383
done
wenzelm@21256
   384
nipkow@31952
   385
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   386
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   387
  apply (case_tac "d = 0")
huffman@31706
   388
  apply force
huffman@31706
   389
  apply (rule iffI)
huffman@31706
   390
  apply (rule zdvd_anti_sym)
huffman@31706
   391
  apply arith
nipkow@31952
   392
  apply (subst gcd_pos_int)
huffman@31706
   393
  apply clarsimp
huffman@31706
   394
  apply (drule_tac x = "d + 1" in spec)
huffman@31706
   395
  apply (frule zdvd_imp_le)
nipkow@31952
   396
  apply (auto intro: gcd_greatest_int)
huffman@31706
   397
done
huffman@30082
   398
nipkow@31798
   399
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
nipkow@31952
   400
by (metis dvd.eq_iff gcd_unique_nat)
nipkow@31798
   401
nipkow@31798
   402
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
nipkow@31952
   403
by (metis dvd.eq_iff gcd_unique_nat)
nipkow@31798
   404
nipkow@31798
   405
lemma gcd_proj1_if_dvd_int[simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
nipkow@31952
   406
by (metis abs_dvd_iff abs_eq_0 gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   407
nipkow@31798
   408
lemma gcd_proj2_if_dvd_int[simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
nipkow@31952
   409
by (metis gcd_proj1_if_dvd_int gcd_commute_int)
nipkow@31798
   410
nipkow@31798
   411
wenzelm@21256
   412
text {*
wenzelm@21256
   413
  \medskip Multiplication laws
wenzelm@21256
   414
*}
wenzelm@21256
   415
nipkow@31952
   416
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256
   417
    -- {* \cite[page 27]{davenport92} *}
nipkow@31952
   418
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   419
  apply simp
wenzelm@21256
   420
  apply (case_tac "k = 0")
nipkow@31952
   421
  apply (simp_all add: mod_geq gcd_non_0_nat mod_mult_distrib2)
huffman@31706
   422
done
wenzelm@21256
   423
nipkow@31952
   424
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   425
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   426
  apply (subst (1 2) abs_mult)
nipkow@31952
   427
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   428
  apply auto
huffman@31706
   429
done
wenzelm@21256
   430
nipkow@31952
   431
lemma coprime_dvd_mult_nat: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31952
   432
  apply (insert gcd_mult_distrib_nat [of m k n])
wenzelm@21256
   433
  apply simp
wenzelm@21256
   434
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   435
  apply simp
wenzelm@21256
   436
  done
wenzelm@21256
   437
nipkow@31952
   438
lemma coprime_dvd_mult_int:
nipkow@31813
   439
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31813
   440
apply (subst abs_dvd_iff [symmetric])
nipkow@31813
   441
apply (subst dvd_abs_iff [symmetric])
nipkow@31952
   442
apply (subst (asm) gcd_abs_int)
nipkow@31952
   443
apply (rule coprime_dvd_mult_nat [transferred])
nipkow@31813
   444
    prefer 4 apply assumption
nipkow@31813
   445
   apply auto
nipkow@31813
   446
apply (subst abs_mult [symmetric], auto)
huffman@31706
   447
done
huffman@31706
   448
nipkow@31952
   449
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   450
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   451
  by (auto intro: coprime_dvd_mult_nat)
huffman@31706
   452
nipkow@31952
   453
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   454
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   455
  by (auto intro: coprime_dvd_mult_int)
huffman@31706
   456
nipkow@31952
   457
lemma gcd_mult_cancel_nat: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
wenzelm@21256
   458
  apply (rule dvd_anti_sym)
nipkow@31952
   459
  apply (rule gcd_greatest_nat)
nipkow@31952
   460
  apply (rule_tac n = k in coprime_dvd_mult_nat)
nipkow@31952
   461
  apply (simp add: gcd_assoc_nat)
nipkow@31952
   462
  apply (simp add: gcd_commute_nat)
huffman@31706
   463
  apply (simp_all add: mult_commute)
huffman@31706
   464
done
wenzelm@21256
   465
nipkow@31952
   466
lemma gcd_mult_cancel_int:
nipkow@31813
   467
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
nipkow@31952
   468
apply (subst (1 2) gcd_abs_int)
nipkow@31813
   469
apply (subst abs_mult)
nipkow@31952
   470
apply (rule gcd_mult_cancel_nat [transferred], auto)
huffman@31706
   471
done
wenzelm@21256
   472
wenzelm@21256
   473
text {* \medskip Addition laws *}
wenzelm@21256
   474
nipkow@31952
   475
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   476
  apply (case_tac "n = 0")
nipkow@31952
   477
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   478
done
huffman@31706
   479
nipkow@31952
   480
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
nipkow@31952
   481
  apply (subst (1 2) gcd_commute_nat)
huffman@31706
   482
  apply (subst add_commute)
huffman@31706
   483
  apply simp
huffman@31706
   484
done
huffman@31706
   485
huffman@31706
   486
(* to do: add the other variations? *)
huffman@31706
   487
nipkow@31952
   488
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
nipkow@31952
   489
  by (subst gcd_add1_nat [symmetric], auto)
huffman@31706
   490
nipkow@31952
   491
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
nipkow@31952
   492
  apply (subst gcd_commute_nat)
nipkow@31952
   493
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
   494
  apply auto
nipkow@31952
   495
  apply (subst gcd_commute_nat)
nipkow@31952
   496
  apply (subst gcd_diff1_nat)
huffman@31706
   497
  apply assumption
nipkow@31952
   498
  apply (rule gcd_commute_nat)
huffman@31706
   499
done
huffman@31706
   500
nipkow@31952
   501
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   502
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   503
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
   504
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
   505
    zmod_zminus1_eq_if)
huffman@31706
   506
  apply (frule_tac a = x in pos_mod_bound)
nipkow@31952
   507
  apply (subst (1 2) gcd_commute_nat)
nipkow@31952
   508
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
   509
    nat_le_eq_zle)
huffman@31706
   510
done
wenzelm@21256
   511
nipkow@31952
   512
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   513
  apply (case_tac "y = 0")
huffman@31706
   514
  apply force
huffman@31706
   515
  apply (case_tac "y > 0")
nipkow@31952
   516
  apply (subst gcd_non_0_int, auto)
nipkow@31952
   517
  apply (insert gcd_non_0_int [of "-y" "-x"])
nipkow@31952
   518
  apply (auto simp add: gcd_neg1_int gcd_neg2_int)
huffman@31706
   519
done
huffman@31706
   520
nipkow@31952
   521
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
nipkow@31952
   522
by (metis gcd_red_int mod_add_self1 zadd_commute)
huffman@31706
   523
nipkow@31952
   524
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
nipkow@31952
   525
by (metis gcd_add1_int gcd_commute_int zadd_commute)
wenzelm@21256
   526
nipkow@31952
   527
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31952
   528
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
wenzelm@21256
   529
nipkow@31952
   530
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
nipkow@31952
   531
by (metis gcd_commute_int gcd_red_int mod_mult_self1 zadd_commute)
nipkow@31798
   532
wenzelm@21256
   533
huffman@31706
   534
(* to do: differences, and all variations of addition rules
huffman@31706
   535
    as simplification rules for nat and int *)
huffman@31706
   536
nipkow@31798
   537
(* FIXME remove iff *)
nipkow@31952
   538
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   539
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   540
huffman@31706
   541
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   542
nipkow@31734
   543
lemma finite_divisors_nat:
nipkow@31734
   544
  assumes "(m::nat)~= 0" shows "finite{d. d dvd m}"
nipkow@31734
   545
proof-
nipkow@31734
   546
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
nipkow@31734
   547
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   548
    by(bestsimp intro!:dvd_imp_le)
nipkow@31734
   549
qed
nipkow@31734
   550
nipkow@31734
   551
lemma finite_divisors_int:
nipkow@31734
   552
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
   553
proof-
nipkow@31734
   554
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
   555
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
   556
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   557
    by(bestsimp intro!:dvd_imp_le_int)
nipkow@31734
   558
qed
nipkow@31734
   559
nipkow@31734
   560
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   561
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   562
apply(rule Max_eqI[THEN sym])
nipkow@31734
   563
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   564
 apply simp
nipkow@31952
   565
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
nipkow@31734
   566
apply simp
nipkow@31734
   567
done
nipkow@31734
   568
nipkow@31734
   569
lemma gcd_is_Max_divisors_int:
nipkow@31734
   570
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   571
apply(rule Max_eqI[THEN sym])
nipkow@31734
   572
  apply (metis dvd.eq_iff finite_Collect_conjI finite_divisors_int)
nipkow@31734
   573
 apply simp
nipkow@31952
   574
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
nipkow@31734
   575
apply simp
nipkow@31734
   576
done
nipkow@31734
   577
chaieb@22027
   578
huffman@31706
   579
subsection {* Coprimality *}
huffman@31706
   580
nipkow@31952
   581
lemma div_gcd_coprime_nat:
huffman@31706
   582
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   583
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   584
proof -
haftmann@27556
   585
  let ?g = "gcd a b"
chaieb@22027
   586
  let ?a' = "a div ?g"
chaieb@22027
   587
  let ?b' = "b div ?g"
haftmann@27556
   588
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   589
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   590
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   591
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   592
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   593
    unfolding dvd_def by blast
huffman@31706
   594
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   595
    by simp_all
wenzelm@22367
   596
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   597
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   598
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
nipkow@31952
   599
  have "?g \<noteq> 0" using nz by (simp add: gcd_zero_nat)
huffman@31706
   600
  then have gp: "?g > 0" by arith
nipkow@31952
   601
  from gcd_greatest_nat [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   602
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   603
qed
chaieb@22027
   604
nipkow@31952
   605
lemma div_gcd_coprime_int:
huffman@31706
   606
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   607
  shows "coprime (a div gcd a b) (b div gcd a b)"
nipkow@31952
   608
apply (subst (1 2 3) gcd_abs_int)
nipkow@31813
   609
apply (subst (1 2) abs_div)
nipkow@31813
   610
  apply simp
nipkow@31813
   611
 apply simp
nipkow@31813
   612
apply(subst (1 2) abs_gcd_int)
nipkow@31952
   613
apply (rule div_gcd_coprime_nat [transferred])
nipkow@31952
   614
using nz apply (auto simp add: gcd_abs_int [symmetric])
huffman@31706
   615
done
huffman@31706
   616
nipkow@31952
   617
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   618
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
   619
nipkow@31952
   620
lemma coprime_Suc_0_nat:
huffman@31706
   621
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
nipkow@31952
   622
  using coprime_nat by (simp add: One_nat_def)
huffman@31706
   623
nipkow@31952
   624
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   625
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   626
  using gcd_unique_int [of 1 a b]
huffman@31706
   627
  apply clarsimp
huffman@31706
   628
  apply (erule subst)
huffman@31706
   629
  apply (rule iffI)
huffman@31706
   630
  apply force
huffman@31706
   631
  apply (drule_tac x = "abs e" in exI)
huffman@31706
   632
  apply (case_tac "e >= 0")
huffman@31706
   633
  apply force
huffman@31706
   634
  apply force
huffman@31706
   635
done
huffman@31706
   636
nipkow@31952
   637
lemma gcd_coprime_nat:
huffman@31706
   638
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   639
    b: "b = b' * gcd a b"
huffman@31706
   640
  shows    "coprime a' b'"
huffman@31706
   641
huffman@31706
   642
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   643
  apply (erule ssubst)
huffman@31706
   644
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   645
  apply (erule ssubst)
nipkow@31952
   646
  apply (rule div_gcd_coprime_nat)
huffman@31706
   647
  using prems
huffman@31706
   648
  apply force
huffman@31706
   649
  apply (subst (1) b)
huffman@31706
   650
  using z apply force
huffman@31706
   651
  apply (subst (1) a)
huffman@31706
   652
  using z apply force
huffman@31706
   653
done
huffman@31706
   654
nipkow@31952
   655
lemma gcd_coprime_int:
huffman@31706
   656
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   657
    b: "b = b' * gcd a b"
huffman@31706
   658
  shows    "coprime a' b'"
huffman@31706
   659
huffman@31706
   660
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   661
  apply (erule ssubst)
huffman@31706
   662
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   663
  apply (erule ssubst)
nipkow@31952
   664
  apply (rule div_gcd_coprime_int)
huffman@31706
   665
  using prems
huffman@31706
   666
  apply force
huffman@31706
   667
  apply (subst (1) b)
huffman@31706
   668
  using z apply force
huffman@31706
   669
  apply (subst (1) a)
huffman@31706
   670
  using z apply force
huffman@31706
   671
done
huffman@31706
   672
nipkow@31952
   673
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   674
    shows "coprime d (a * b)"
nipkow@31952
   675
  apply (subst gcd_commute_nat)
nipkow@31952
   676
  using da apply (subst gcd_mult_cancel_nat)
nipkow@31952
   677
  apply (subst gcd_commute_nat, assumption)
nipkow@31952
   678
  apply (subst gcd_commute_nat, rule db)
huffman@31706
   679
done
huffman@31706
   680
nipkow@31952
   681
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   682
    shows "coprime d (a * b)"
nipkow@31952
   683
  apply (subst gcd_commute_int)
nipkow@31952
   684
  using da apply (subst gcd_mult_cancel_int)
nipkow@31952
   685
  apply (subst gcd_commute_int, assumption)
nipkow@31952
   686
  apply (subst gcd_commute_int, rule db)
huffman@31706
   687
done
huffman@31706
   688
nipkow@31952
   689
lemma coprime_lmult_nat:
huffman@31706
   690
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   691
proof -
huffman@31706
   692
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   693
    by (rule gcd_greatest_nat, auto)
huffman@31706
   694
  with dab show ?thesis
huffman@31706
   695
    by auto
huffman@31706
   696
qed
huffman@31706
   697
nipkow@31952
   698
lemma coprime_lmult_int:
nipkow@31798
   699
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   700
proof -
huffman@31706
   701
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   702
    by (rule gcd_greatest_int, auto)
nipkow@31798
   703
  with assms show ?thesis
huffman@31706
   704
    by auto
huffman@31706
   705
qed
huffman@31706
   706
nipkow@31952
   707
lemma coprime_rmult_nat:
nipkow@31798
   708
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   709
proof -
huffman@31706
   710
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   711
    by (rule gcd_greatest_nat, auto intro: dvd_mult)
nipkow@31798
   712
  with assms show ?thesis
huffman@31706
   713
    by auto
huffman@31706
   714
qed
huffman@31706
   715
nipkow@31952
   716
lemma coprime_rmult_int:
huffman@31706
   717
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   718
proof -
huffman@31706
   719
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   720
    by (rule gcd_greatest_int, auto intro: dvd_mult)
huffman@31706
   721
  with dab show ?thesis
huffman@31706
   722
    by auto
huffman@31706
   723
qed
huffman@31706
   724
nipkow@31952
   725
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   726
    coprime d a \<and>  coprime d b"
nipkow@31952
   727
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
nipkow@31952
   728
    coprime_mult_nat[of d a b]
huffman@31706
   729
  by blast
huffman@31706
   730
nipkow@31952
   731
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   732
    coprime d a \<and>  coprime d b"
nipkow@31952
   733
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
nipkow@31952
   734
    coprime_mult_int[of d a b]
huffman@31706
   735
  by blast
huffman@31706
   736
nipkow@31952
   737
lemma gcd_coprime_exists_nat:
huffman@31706
   738
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   739
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   740
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   741
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   742
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
huffman@31706
   743
done
huffman@31706
   744
nipkow@31952
   745
lemma gcd_coprime_exists_int:
huffman@31706
   746
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   747
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   748
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   749
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   750
  using nz apply (auto simp add: div_gcd_coprime_int dvd_div_mult_self)
huffman@31706
   751
done
huffman@31706
   752
nipkow@31952
   753
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   754
  by (induct n, simp_all add: coprime_mult_nat)
huffman@31706
   755
nipkow@31952
   756
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   757
  by (induct n, simp_all add: coprime_mult_int)
huffman@31706
   758
nipkow@31952
   759
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   760
  apply (rule coprime_exp_nat)
nipkow@31952
   761
  apply (subst gcd_commute_nat)
nipkow@31952
   762
  apply (rule coprime_exp_nat)
nipkow@31952
   763
  apply (subst gcd_commute_nat, assumption)
huffman@31706
   764
done
huffman@31706
   765
nipkow@31952
   766
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   767
  apply (rule coprime_exp_int)
nipkow@31952
   768
  apply (subst gcd_commute_int)
nipkow@31952
   769
  apply (rule coprime_exp_int)
nipkow@31952
   770
  apply (subst gcd_commute_int, assumption)
huffman@31706
   771
done
huffman@31706
   772
nipkow@31952
   773
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   774
proof (cases)
huffman@31706
   775
  assume "a = 0 & b = 0"
huffman@31706
   776
  thus ?thesis by simp
huffman@31706
   777
  next assume "~(a = 0 & b = 0)"
huffman@31706
   778
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
nipkow@31952
   779
    by (auto simp:div_gcd_coprime_nat)
huffman@31706
   780
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   781
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   782
    apply (subst (1 2) mult_commute)
nipkow@31952
   783
    apply (subst gcd_mult_distrib_nat [symmetric])
huffman@31706
   784
    apply simp
huffman@31706
   785
    done
huffman@31706
   786
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
huffman@31706
   787
    apply (subst div_power)
huffman@31706
   788
    apply auto
huffman@31706
   789
    apply (rule dvd_div_mult_self)
huffman@31706
   790
    apply (rule dvd_power_same)
huffman@31706
   791
    apply auto
huffman@31706
   792
    done
huffman@31706
   793
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   794
    apply (subst div_power)
huffman@31706
   795
    apply auto
huffman@31706
   796
    apply (rule dvd_div_mult_self)
huffman@31706
   797
    apply (rule dvd_power_same)
huffman@31706
   798
    apply auto
huffman@31706
   799
    done
huffman@31706
   800
  finally show ?thesis .
huffman@31706
   801
qed
huffman@31706
   802
nipkow@31952
   803
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
   804
  apply (subst (1 2) gcd_abs_int)
huffman@31706
   805
  apply (subst (1 2) power_abs)
nipkow@31952
   806
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
   807
  apply auto
huffman@31706
   808
done
huffman@31706
   809
nipkow@31952
   810
lemma coprime_divprod_nat: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
nipkow@31952
   811
  using coprime_dvd_mult_iff_nat[of d a b]
huffman@31706
   812
  by (auto simp add: mult_commute)
huffman@31706
   813
nipkow@31952
   814
lemma coprime_divprod_int: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
nipkow@31952
   815
  using coprime_dvd_mult_iff_int[of d a b]
huffman@31706
   816
  by (auto simp add: mult_commute)
huffman@31706
   817
nipkow@31952
   818
lemma division_decomp_nat: assumes dc: "(a::nat) 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"
nipkow@31952
   825
    from gcd_coprime_exists_nat[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" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   831
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   832
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   833
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   834
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
huffman@31706
   835
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   836
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   837
    with thb thc have ?thesis by blast }
huffman@31706
   838
  ultimately show ?thesis by blast
huffman@31706
   839
qed
huffman@31706
   840
nipkow@31952
   841
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
   842
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   843
proof-
huffman@31706
   844
  let ?g = "gcd a b"
huffman@31706
   845
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   846
  moreover
huffman@31706
   847
  {assume z: "?g \<noteq> 0"
nipkow@31952
   848
    from gcd_coprime_exists_int[OF z]
huffman@31706
   849
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   850
      by blast
huffman@31706
   851
    have thb: "?g dvd b" by auto
huffman@31706
   852
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   853
    with dc have th0: "a' dvd b*c"
huffman@31706
   854
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   855
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   856
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   857
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   858
    from coprime_dvd_mult_int[OF ab'(3)] th_1
huffman@31706
   859
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   860
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   861
    with thb thc have ?thesis by blast }
huffman@31706
   862
  ultimately show ?thesis by blast
chaieb@27669
   863
qed
chaieb@27669
   864
nipkow@31952
   865
lemma pow_divides_pow_nat:
huffman@31706
   866
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   867
  shows "a dvd b"
huffman@31706
   868
proof-
huffman@31706
   869
  let ?g = "gcd a b"
huffman@31706
   870
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   871
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   872
  moreover
huffman@31706
   873
  {assume z: "?g \<noteq> 0"
huffman@31706
   874
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
nipkow@31952
   875
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   876
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   877
      by blast
huffman@31706
   878
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   879
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   880
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   881
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   882
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   883
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   884
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   885
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   886
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
huffman@31706
   887
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   888
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   889
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   890
  ultimately show ?thesis by blast
huffman@31706
   891
qed
huffman@31706
   892
nipkow@31952
   893
lemma pow_divides_pow_int:
huffman@31706
   894
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   895
  shows "a dvd b"
chaieb@27669
   896
proof-
huffman@31706
   897
  let ?g = "gcd a b"
huffman@31706
   898
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   899
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   900
  moreover
huffman@31706
   901
  {assume z: "?g \<noteq> 0"
huffman@31706
   902
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
nipkow@31952
   903
    from gcd_coprime_exists_int[OF z]
huffman@31706
   904
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   905
      by blast
huffman@31706
   906
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   907
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   908
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   909
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   910
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   911
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   912
    with th0 have "a' dvd b'^n"
huffman@31706
   913
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   914
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   915
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
huffman@31706
   916
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   917
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   918
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   919
  ultimately show ?thesis by blast
huffman@31706
   920
qed
huffman@31706
   921
nipkow@31798
   922
(* FIXME move to Divides(?) *)
nipkow@31952
   923
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
   924
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
   925
nipkow@31952
   926
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
   927
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
   928
nipkow@31952
   929
lemma divides_mult_nat:
huffman@31706
   930
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   931
  shows "m * n dvd r"
huffman@31706
   932
proof-
huffman@31706
   933
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   934
    unfolding dvd_def by blast
huffman@31706
   935
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   936
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
huffman@31706
   937
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   938
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   939
qed
huffman@31706
   940
nipkow@31952
   941
lemma divides_mult_int:
huffman@31706
   942
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   943
  shows "m * n dvd r"
huffman@31706
   944
proof-
huffman@31706
   945
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   946
    unfolding dvd_def by blast
huffman@31706
   947
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   948
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
huffman@31706
   949
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   950
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   951
qed
chaieb@27669
   952
nipkow@31952
   953
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   954
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   955
  apply force
nipkow@31952
   956
  apply (rule dvd_diff_nat)
huffman@31706
   957
  apply auto
huffman@31706
   958
done
huffman@31706
   959
nipkow@31952
   960
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
nipkow@31952
   961
  using coprime_plus_one_nat by (simp add: One_nat_def)
huffman@31706
   962
nipkow@31952
   963
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   964
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   965
  apply force
huffman@31706
   966
  apply (rule dvd_diff)
huffman@31706
   967
  apply auto
huffman@31706
   968
done
huffman@31706
   969
nipkow@31952
   970
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
   971
  using coprime_plus_one_nat [of "n - 1"]
nipkow@31952
   972
    gcd_commute_nat [of "n - 1" n] by auto
huffman@31706
   973
nipkow@31952
   974
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
   975
  using coprime_plus_one_int [of "n - 1"]
nipkow@31952
   976
    gcd_commute_int [of "n - 1" n] by auto
huffman@31706
   977
nipkow@31952
   978
lemma setprod_coprime_nat [rule_format]:
huffman@31706
   979
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   980
  apply (case_tac "finite A")
huffman@31706
   981
  apply (induct set: finite)
nipkow@31952
   982
  apply (auto simp add: gcd_mult_cancel_nat)
huffman@31706
   983
done
huffman@31706
   984
nipkow@31952
   985
lemma setprod_coprime_int [rule_format]:
huffman@31706
   986
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
   987
  apply (case_tac "finite A")
huffman@31706
   988
  apply (induct set: finite)
nipkow@31952
   989
  apply (auto simp add: gcd_mult_cancel_int)
huffman@31706
   990
done
huffman@31706
   991
nipkow@31952
   992
lemma prime_odd_nat: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
   993
  unfolding prime_nat_def
huffman@31706
   994
  apply (subst even_mult_two_ex)
huffman@31706
   995
  apply clarify
huffman@31706
   996
  apply (drule_tac x = 2 in spec)
huffman@31706
   997
  apply auto
huffman@31706
   998
done
huffman@31706
   999
nipkow@31952
  1000
lemma prime_odd_int: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
  1001
  unfolding prime_int_def
nipkow@31952
  1002
  apply (frule prime_odd_nat)
huffman@31706
  1003
  apply (auto simp add: even_nat_def)
huffman@31706
  1004
done
huffman@31706
  1005
nipkow@31952
  1006
lemma coprime_common_divisor_nat: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1007
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
  1008
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1009
  apply simp
nipkow@31952
  1010
  apply (erule (1) gcd_greatest_nat)
huffman@31706
  1011
done
huffman@31706
  1012
nipkow@31952
  1013
lemma coprime_common_divisor_int: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1014
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
  1015
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1016
  apply simp
nipkow@31952
  1017
  apply (erule (1) gcd_greatest_int)
huffman@31706
  1018
done
huffman@31706
  1019
nipkow@31952
  1020
lemma coprime_divisors_nat: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
  1021
    coprime d e"
huffman@31706
  1022
  apply (auto simp add: dvd_def)
nipkow@31952
  1023
  apply (frule coprime_lmult_int)
nipkow@31952
  1024
  apply (subst gcd_commute_int)
nipkow@31952
  1025
  apply (subst (asm) (2) gcd_commute_int)
nipkow@31952
  1026
  apply (erule coprime_lmult_int)
huffman@31706
  1027
done
huffman@31706
  1028
nipkow@31952
  1029
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
  1030
apply (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
huffman@31706
  1031
done
huffman@31706
  1032
nipkow@31952
  1033
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
  1034
apply (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
huffman@31706
  1035
done
huffman@31706
  1036
huffman@31706
  1037
huffman@31706
  1038
subsection {* Bezout's theorem *}
huffman@31706
  1039
huffman@31706
  1040
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1041
   see the theorems that follow the definition. *)
huffman@31706
  1042
fun
huffman@31706
  1043
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1044
where
huffman@31706
  1045
  "bezw x y =
huffman@31706
  1046
  (if y = 0 then (1, 0) else
huffman@31706
  1047
      (snd (bezw y (x mod y)),
huffman@31706
  1048
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1049
huffman@31706
  1050
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1051
huffman@31706
  1052
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1053
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1054
  by simp
huffman@31706
  1055
huffman@31706
  1056
declare bezw.simps [simp del]
huffman@31706
  1057
huffman@31706
  1058
lemma bezw_aux [rule_format]:
huffman@31706
  1059
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1060
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1061
  fix m :: nat
huffman@31706
  1062
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1063
    by auto
huffman@31706
  1064
  next fix m :: nat and n
huffman@31706
  1065
    assume ngt0: "n > 0" and
huffman@31706
  1066
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1067
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1068
        int (gcd n (m mod n))"
huffman@31706
  1069
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1070
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1071
      apply (erule subst)
huffman@31706
  1072
      apply (simp add: ring_simps)
huffman@31706
  1073
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1074
      (* applying simp here undoes the last substitution!
huffman@31706
  1075
         what is procedure cancel_div_mod? *)
huffman@31706
  1076
      apply (simp only: ring_simps zadd_int [symmetric]
huffman@31706
  1077
        zmult_int [symmetric])
huffman@31706
  1078
      done
huffman@31706
  1079
qed
huffman@31706
  1080
nipkow@31952
  1081
lemma bezout_int:
huffman@31706
  1082
  fixes x y
huffman@31706
  1083
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1084
proof -
huffman@31706
  1085
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1086
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1087
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1088
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1089
    apply (unfold gcd_int_def)
huffman@31706
  1090
    apply simp
huffman@31706
  1091
    apply (subst bezw_aux [symmetric])
huffman@31706
  1092
    apply auto
huffman@31706
  1093
    done
huffman@31706
  1094
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1095
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1096
    by auto
huffman@31706
  1097
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1098
    by (erule (1) bezout_aux)
huffman@31706
  1099
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1100
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1101
    apply auto
huffman@31706
  1102
    apply (rule_tac x = u in exI)
huffman@31706
  1103
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1104
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1105
    apply auto
huffman@31706
  1106
    done
huffman@31706
  1107
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1108
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1109
    apply auto
huffman@31706
  1110
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1111
    apply (rule_tac x = v in exI)
nipkow@31952
  1112
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1113
    apply auto
huffman@31706
  1114
    done
huffman@31706
  1115
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1116
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1117
    apply auto
huffman@31706
  1118
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1119
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1120
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1121
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1122
    apply auto
huffman@31706
  1123
    done
huffman@31706
  1124
  ultimately show ?thesis by blast
huffman@31706
  1125
qed
huffman@31706
  1126
huffman@31706
  1127
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1128
huffman@31706
  1129
lemma ind_euclid:
huffman@31706
  1130
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1131
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1132
  shows "P a b"
chaieb@27669
  1133
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
chaieb@27669
  1134
  fix n a b
chaieb@27669
  1135
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
chaieb@27669
  1136
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1137
  moreover {assume eq: "a= b"
huffman@31706
  1138
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1139
    by simp}
chaieb@27669
  1140
  moreover
chaieb@27669
  1141
  {assume lt: "a < b"
chaieb@27669
  1142
    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
chaieb@27669
  1143
    moreover
chaieb@27669
  1144
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1145
    moreover
chaieb@27669
  1146
    {assume ab: "a + b - a < n"
chaieb@27669
  1147
      have th0: "a + b - a = a + (b - a)" using lt by arith
chaieb@27669
  1148
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1149
      have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1150
    ultimately have "P a b" by blast}
chaieb@27669
  1151
  moreover
chaieb@27669
  1152
  {assume lt: "a > b"
chaieb@27669
  1153
    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
chaieb@27669
  1154
    moreover
chaieb@27669
  1155
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1156
    moreover
chaieb@27669
  1157
    {assume ab: "b + a - b < n"
chaieb@27669
  1158
      have th0: "b + a - b = b + (a - b)" using lt by arith
chaieb@27669
  1159
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1160
      have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1161
      hence "P a b" using c by blast }
chaieb@27669
  1162
    ultimately have "P a b" by blast}
chaieb@27669
  1163
ultimately  show "P a b" by blast
chaieb@27669
  1164
qed
chaieb@27669
  1165
nipkow@31952
  1166
lemma bezout_lemma_nat:
huffman@31706
  1167
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1168
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1169
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1170
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1171
  using ex
huffman@31706
  1172
  apply clarsimp
huffman@31706
  1173
  apply (rule_tac x="d" in exI, simp add: dvd_add)
huffman@31706
  1174
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1175
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1176
  apply (rule_tac x="y" in exI)
huffman@31706
  1177
  apply algebra
huffman@31706
  1178
  apply (rule_tac x="x" in exI)
huffman@31706
  1179
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1180
  apply algebra
chaieb@27669
  1181
done
chaieb@27669
  1182
nipkow@31952
  1183
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1184
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1185
  apply(induct a b rule: ind_euclid)
huffman@31706
  1186
  apply blast
huffman@31706
  1187
  apply clarify
huffman@31706
  1188
  apply (rule_tac x="a" in exI, simp add: dvd_add)
huffman@31706
  1189
  apply clarsimp
huffman@31706
  1190
  apply (rule_tac x="d" in exI)
huffman@31706
  1191
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
huffman@31706
  1192
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1193
  apply (rule_tac x="y" in exI)
huffman@31706
  1194
  apply algebra
huffman@31706
  1195
  apply (rule_tac x="x" in exI)
huffman@31706
  1196
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1197
  apply algebra
chaieb@27669
  1198
done
chaieb@27669
  1199
nipkow@31952
  1200
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1201
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1202
  using bezout_add_nat[of a b]
huffman@31706
  1203
  apply clarsimp
huffman@31706
  1204
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1205
  apply (rule_tac x="x" in exI)
huffman@31706
  1206
  apply (rule_tac x="y" in exI)
huffman@31706
  1207
  apply auto
chaieb@27669
  1208
done
chaieb@27669
  1209
nipkow@31952
  1210
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1211
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1212
proof-
huffman@31706
  1213
 from nz have ap: "a > 0" by simp
nipkow@31952
  1214
 from bezout_add_nat[of a b]
huffman@31706
  1215
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1216
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1217
 moreover
huffman@31706
  1218
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1219
     from H have ?thesis by blast }
chaieb@27669
  1220
 moreover
chaieb@27669
  1221
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1222
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1223
   moreover
chaieb@27669
  1224
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1225
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1226
       by auto
chaieb@27669
  1227
     moreover
chaieb@27669
  1228
     {assume db: "d=b"
chaieb@27669
  1229
       from prems have ?thesis apply simp
chaieb@27669
  1230
	 apply (rule exI[where x = b], simp)
chaieb@27669
  1231
	 apply (rule exI[where x = b])
chaieb@27669
  1232
	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1233
    moreover
huffman@31706
  1234
    {assume db: "d < b"
chaieb@27669
  1235
	{assume "x=0" hence ?thesis  using prems by simp }
chaieb@27669
  1236
	moreover
chaieb@27669
  1237
	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
chaieb@27669
  1238
	  from db have "d \<le> b - 1" by simp
chaieb@27669
  1239
	  hence "d*b \<le> b*(b - 1)" by simp
chaieb@27669
  1240
	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
chaieb@27669
  1241
	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
huffman@31706
  1242
	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1243
            by simp
huffman@31706
  1244
	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
huffman@31706
  1245
	    by (simp only: mult_assoc right_distrib)
huffman@31706
  1246
	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1247
            by algebra
chaieb@27669
  1248
	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
huffman@31706
  1249
	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
chaieb@27669
  1250
	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
chaieb@27669
  1251
	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
chaieb@27669
  1252
	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
chaieb@27669
  1253
	  hence ?thesis using H(1,2)
chaieb@27669
  1254
	    apply -
chaieb@27669
  1255
	    apply (rule exI[where x=d], simp)
chaieb@27669
  1256
	    apply (rule exI[where x="(b - 1) * y"])
chaieb@27669
  1257
	    by (rule exI[where x="x*(b - 1) - d"], simp)}
chaieb@27669
  1258
	ultimately have ?thesis by blast}
chaieb@27669
  1259
    ultimately have ?thesis by blast}
chaieb@27669
  1260
  ultimately have ?thesis by blast}
chaieb@27669
  1261
 ultimately show ?thesis by blast
chaieb@27669
  1262
qed
chaieb@27669
  1263
nipkow@31952
  1264
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1265
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1266
proof-
chaieb@27669
  1267
  let ?g = "gcd a b"
nipkow@31952
  1268
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1269
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1270
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1271
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1272
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1273
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1274
  thus ?thesis by blast
chaieb@27669
  1275
qed
chaieb@27669
  1276
huffman@31706
  1277
huffman@31706
  1278
subsection {* LCM *}
huffman@31706
  1279
nipkow@31952
  1280
lemma lcm_altdef_int: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1281
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@31706
  1282
    zmult_int [symmetric] gcd_int_def)
huffman@31706
  1283
nipkow@31952
  1284
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1285
  unfolding lcm_nat_def
nipkow@31952
  1286
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1287
nipkow@31952
  1288
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1289
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1290
  apply (subst int_mult [symmetric])
nipkow@31952
  1291
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1292
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1293
  apply (simp, simp add: abs_mult)
huffman@31706
  1294
done
huffman@31706
  1295
nipkow@31952
  1296
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1297
  unfolding lcm_nat_def by simp
huffman@31706
  1298
nipkow@31952
  1299
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1300
  unfolding lcm_int_def by simp
huffman@31706
  1301
nipkow@31952
  1302
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1303
  unfolding lcm_nat_def by simp
chaieb@27669
  1304
nipkow@31952
  1305
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1306
  unfolding lcm_int_def by simp
huffman@31706
  1307
nipkow@31952
  1308
lemma lcm_commute_nat: "lcm (m::nat) n = lcm n m"
nipkow@31952
  1309
  unfolding lcm_nat_def by (simp add: gcd_commute_nat ring_simps)
huffman@31706
  1310
nipkow@31952
  1311
lemma lcm_commute_int: "lcm (m::int) n = lcm n m"
nipkow@31952
  1312
  unfolding lcm_int_def by (subst lcm_commute_nat, rule refl)
huffman@31706
  1313
huffman@31706
  1314
nipkow@31952
  1315
lemma lcm_pos_nat:
nipkow@31798
  1316
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1317
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1318
nipkow@31952
  1319
lemma lcm_pos_int:
nipkow@31798
  1320
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1321
  apply (subst lcm_abs_int)
nipkow@31952
  1322
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1323
  apply auto
huffman@31706
  1324
done
haftmann@23687
  1325
nipkow@31952
  1326
lemma dvd_pos_nat:
haftmann@23687
  1327
  fixes n m :: nat
haftmann@23687
  1328
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1329
  shows "m > 0"
haftmann@23687
  1330
using assms by (cases m) auto
haftmann@23687
  1331
nipkow@31952
  1332
lemma lcm_least_nat:
huffman@31706
  1333
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1334
  shows "lcm m n dvd k"
haftmann@23687
  1335
proof (cases k)
haftmann@23687
  1336
  case 0 then show ?thesis by auto
haftmann@23687
  1337
next
haftmann@23687
  1338
  case (Suc _) then have pos_k: "k > 0" by auto
nipkow@31952
  1339
  from assms dvd_pos_nat [OF this] have pos_mn: "m > 0" "n > 0" by auto
nipkow@31952
  1340
  with gcd_zero_nat [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1341
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1342
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1343
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1344
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1345
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
nipkow@31952
  1346
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1347
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1348
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1349
  also have "\<dots> = k * p * q * gcd m n"
nipkow@31952
  1350
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1351
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1352
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1353
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1354
    by (simp add: mult_ac)
haftmann@27556
  1355
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1356
    by simp
nipkow@31952
  1357
  with prod_gcd_lcm_nat [of m n]
haftmann@27556
  1358
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1359
    by (simp add: mult_ac)
huffman@31706
  1360
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1361
  then show ?thesis using dvd_def by auto
haftmann@23687
  1362
qed
haftmann@23687
  1363
nipkow@31952
  1364
lemma lcm_least_int:
nipkow@31798
  1365
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
nipkow@31952
  1366
apply (subst lcm_abs_int)
nipkow@31798
  1367
apply (rule dvd_trans)
nipkow@31952
  1368
apply (rule lcm_least_nat [transferred, of _ "abs k" _])
nipkow@31798
  1369
apply auto
huffman@31706
  1370
done
huffman@31706
  1371
nipkow@31952
  1372
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
haftmann@23687
  1373
proof (cases m)
haftmann@23687
  1374
  case 0 then show ?thesis by simp
haftmann@23687
  1375
next
haftmann@23687
  1376
  case (Suc _)
haftmann@23687
  1377
  then have mpos: "m > 0" by simp
haftmann@23687
  1378
  show ?thesis
haftmann@23687
  1379
  proof (cases n)
haftmann@23687
  1380
    case 0 then show ?thesis by simp
haftmann@23687
  1381
  next
haftmann@23687
  1382
    case (Suc _)
haftmann@23687
  1383
    then have npos: "n > 0" by simp
haftmann@27556
  1384
    have "gcd m n dvd n" by simp
haftmann@27556
  1385
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1386
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1387
      by (simp add: mult_ac)
nipkow@31952
  1388
    also have "\<dots> = m * k" using mpos npos gcd_zero_nat by simp
huffman@31706
  1389
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1390
  qed
haftmann@23687
  1391
qed
haftmann@23687
  1392
nipkow@31952
  1393
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
nipkow@31952
  1394
  apply (subst lcm_abs_int)
huffman@31706
  1395
  apply (rule dvd_trans)
huffman@31706
  1396
  prefer 2
nipkow@31952
  1397
  apply (rule lcm_dvd1_nat [transferred])
huffman@31706
  1398
  apply auto
huffman@31706
  1399
done
huffman@31706
  1400
nipkow@31952
  1401
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
nipkow@31952
  1402
  by (subst lcm_commute_nat, rule lcm_dvd1_nat)
huffman@31706
  1403
nipkow@31952
  1404
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
nipkow@31952
  1405
  by (subst lcm_commute_int, rule lcm_dvd1_int)
huffman@31706
  1406
nipkow@31730
  1407
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1408
by(metis lcm_dvd1_nat dvd_trans)
nipkow@31729
  1409
nipkow@31730
  1410
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1411
by(metis lcm_dvd2_nat dvd_trans)
nipkow@31729
  1412
nipkow@31730
  1413
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1414
by(metis lcm_dvd1_int dvd_trans)
nipkow@31729
  1415
nipkow@31730
  1416
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1417
by(metis lcm_dvd2_int dvd_trans)
nipkow@31729
  1418
nipkow@31952
  1419
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1420
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31952
  1421
  by (auto intro: dvd_anti_sym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
chaieb@27568
  1422
nipkow@31952
  1423
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1424
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31952
  1425
  by (auto intro: dvd_anti_sym [transferred] lcm_least_int)
huffman@31706
  1426
nipkow@31798
  1427
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1428
  apply (rule sym)
nipkow@31952
  1429
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1430
  apply auto
huffman@31706
  1431
done
huffman@31706
  1432
nipkow@31798
  1433
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1434
  apply (rule sym)
nipkow@31952
  1435
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1436
  apply auto
huffman@31706
  1437
done
huffman@31706
  1438
nipkow@31798
  1439
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31952
  1440
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1441
nipkow@31798
  1442
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31952
  1443
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
huffman@31706
  1444
chaieb@27568
  1445
nipkow@31766
  1446
lemma lcm_assoc_nat: "lcm (lcm n m) (p::nat) = lcm n (lcm m p)"
nipkow@31952
  1447
apply(rule lcm_unique_nat[THEN iffD1])
nipkow@31952
  1448
apply (metis dvd.order_trans lcm_unique_nat)
nipkow@31766
  1449
done
nipkow@31766
  1450
nipkow@31766
  1451
lemma lcm_assoc_int: "lcm (lcm n m) (p::int) = lcm n (lcm m p)"
nipkow@31952
  1452
apply(rule lcm_unique_int[THEN iffD1])
nipkow@31952
  1453
apply (metis dvd_trans lcm_unique_int)
nipkow@31766
  1454
done
nipkow@31766
  1455
nipkow@31766
  1456
lemmas lcm_left_commute_nat =
nipkow@31952
  1457
  mk_left_commute[of lcm, OF lcm_assoc_nat lcm_commute_nat]
nipkow@31766
  1458
nipkow@31766
  1459
lemmas lcm_left_commute_int =
nipkow@31952
  1460
  mk_left_commute[of lcm, OF lcm_assoc_int lcm_commute_int]
nipkow@31766
  1461
nipkow@31952
  1462
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
nipkow@31952
  1463
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
nipkow@31766
  1464
haftmann@23687
  1465
huffman@31706
  1466
subsection {* Primes *}
wenzelm@22367
  1467
huffman@31706
  1468
(* Is there a better way to handle these, rather than making them
huffman@31706
  1469
   elim rules? *)
chaieb@22027
  1470
nipkow@31952
  1471
lemma prime_ge_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
huffman@31706
  1472
  by (unfold prime_nat_def, auto)
chaieb@22027
  1473
nipkow@31952
  1474
lemma prime_gt_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
huffman@31706
  1475
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1476
nipkow@31952
  1477
lemma prime_ge_1_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
huffman@31706
  1478
  by (unfold prime_nat_def, auto)
chaieb@22027
  1479
nipkow@31952
  1480
lemma prime_gt_1_nat [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
huffman@31706
  1481
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1482
nipkow@31952
  1483
lemma prime_ge_Suc_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
huffman@31706
  1484
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1485
nipkow@31952
  1486
lemma prime_gt_Suc_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
huffman@31706
  1487
  by (unfold prime_nat_def, auto)
huffman@31706
  1488
nipkow@31952
  1489
lemma prime_ge_2_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
huffman@31706
  1490
  by (unfold prime_nat_def, auto)
huffman@31706
  1491
nipkow@31952
  1492
lemma prime_ge_0_int [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
huffman@31706
  1493
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1494
nipkow@31952
  1495
lemma prime_gt_0_int [elim]: "prime (p::int) \<Longrightarrow> p > 0"
huffman@31706
  1496
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1497
nipkow@31952
  1498
lemma prime_ge_1_int [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
huffman@31706
  1499
  by (unfold prime_int_def prime_nat_def, auto)
chaieb@22027
  1500
nipkow@31952
  1501
lemma prime_gt_1_int [elim]: "prime (p::int) \<Longrightarrow> p > 1"
huffman@31706
  1502
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1503
nipkow@31952
  1504
lemma prime_ge_2_int [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
huffman@31706
  1505
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1506
huffman@31706
  1507
thm prime_nat_def;
huffman@31706
  1508
thm prime_nat_def [transferred];
huffman@31706
  1509
huffman@31706
  1510
lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
huffman@31706
  1511
    m = 1 \<or> m = p))"
huffman@31706
  1512
  using prime_nat_def [transferred]
huffman@31706
  1513
    apply (case_tac "p >= 0")
nipkow@31952
  1514
    by (blast, auto simp add: prime_ge_0_int)
huffman@31706
  1515
huffman@31706
  1516
(* To do: determine primality of any numeral *)
huffman@31706
  1517
nipkow@31952
  1518
lemma zero_not_prime_nat [simp]: "~prime (0::nat)"
huffman@31706
  1519
  by (simp add: prime_nat_def)
huffman@31706
  1520
nipkow@31952
  1521
lemma zero_not_prime_int [simp]: "~prime (0::int)"
huffman@31706
  1522
  by (simp add: prime_int_def)
huffman@31706
  1523
nipkow@31952
  1524
lemma one_not_prime_nat [simp]: "~prime (1::nat)"
huffman@31706
  1525
  by (simp add: prime_nat_def)
chaieb@22027
  1526
nipkow@31952
  1527
lemma Suc_0_not_prime_nat [simp]: "~prime (Suc 0)"
huffman@31706
  1528
  by (simp add: prime_nat_def One_nat_def)
huffman@31706
  1529
nipkow@31952
  1530
lemma one_not_prime_int [simp]: "~prime (1::int)"
huffman@31706
  1531
  by (simp add: prime_int_def)
huffman@31706
  1532
nipkow@31952
  1533
lemma two_is_prime_nat [simp]: "prime (2::nat)"
huffman@31706
  1534
  apply (auto simp add: prime_nat_def)
huffman@31706
  1535
  apply (case_tac m)
huffman@31706
  1536
  apply (auto dest!: dvd_imp_le)
huffman@31706
  1537
  done
chaieb@22027
  1538
nipkow@31952
  1539
lemma two_is_prime_int [simp]: "prime (2::int)"
nipkow@31952
  1540
  by (rule two_is_prime_nat [transferred direction: nat "op <= (0::int)"])
chaieb@27568
  1541
nipkow@31952
  1542
lemma prime_imp_coprime_nat: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1543
  apply (unfold prime_nat_def)
nipkow@31952
  1544
  apply (metis gcd_dvd1_nat gcd_dvd2_nat)
huffman@31706
  1545
  done
huffman@31706
  1546
nipkow@31952
  1547
lemma prime_imp_coprime_int: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1548
  apply (unfold prime_int_altdef)
nipkow@31952
  1549
  apply (metis gcd_dvd1_int gcd_dvd2_int gcd_ge_0_int)
chaieb@27568
  1550
  done
chaieb@27568
  1551
nipkow@31952
  1552
lemma prime_dvd_mult_nat: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
nipkow@31952
  1553
  by (blast intro: coprime_dvd_mult_nat prime_imp_coprime_nat)
huffman@31706
  1554
nipkow@31952
  1555
lemma prime_dvd_mult_int: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
nipkow@31952
  1556
  by (blast intro: coprime_dvd_mult_int prime_imp_coprime_int)
huffman@31706
  1557
nipkow@31952
  1558
lemma prime_dvd_mult_eq_nat [simp]: "prime (p::nat) \<Longrightarrow>
huffman@31706
  1559
    p dvd m * n = (p dvd m \<or> p dvd n)"
nipkow@31952
  1560
  by (rule iffI, rule prime_dvd_mult_nat, auto)
chaieb@27568
  1561
nipkow@31952
  1562
lemma prime_dvd_mult_eq_int [simp]: "prime (p::int) \<Longrightarrow>
huffman@31706
  1563
    p dvd m * n = (p dvd m \<or> p dvd n)"
nipkow@31952
  1564
  by (rule iffI, rule prime_dvd_mult_int, auto)
chaieb@27568
  1565
nipkow@31952
  1566
lemma not_prime_eq_prod_nat: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1567
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1568
  unfolding prime_nat_def dvd_def apply auto
huffman@31706
  1569
  apply (subgoal_tac "k > 1")
huffman@31706
  1570
  apply force
huffman@31706
  1571
  apply (subgoal_tac "k ~= 0")
huffman@31706
  1572
  apply force
huffman@31706
  1573
  apply (rule notI)
huffman@31706
  1574
  apply force
huffman@31706
  1575
done
chaieb@27568
  1576
huffman@31706
  1577
(* there's a lot of messing around with signs of products here --
huffman@31706
  1578
   could this be made more automatic? *)
nipkow@31952
  1579
lemma not_prime_eq_prod_int: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1580
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1581
  unfolding prime_int_altdef dvd_def
huffman@31706
  1582
  apply auto
huffman@31706
  1583
  apply (rule_tac x = m in exI)
huffman@31706
  1584
  apply (rule_tac x = k in exI)
huffman@31706
  1585
  apply (auto simp add: mult_compare_simps)
huffman@31706
  1586
  apply (subgoal_tac "k > 0")
huffman@31706
  1587
  apply arith
huffman@31706
  1588
  apply (case_tac "k <= 0")
huffman@31706
  1589
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1590
  apply force
huffman@31706
  1591
  apply (subst zero_compare_simps(8))
huffman@31706
  1592
  apply auto
huffman@31706
  1593
  apply (subgoal_tac "m * k <= 0")
huffman@31706
  1594
  apply force
huffman@31706
  1595
  apply (subst zero_compare_simps(8))
huffman@31706
  1596
  apply auto
huffman@31706
  1597
done
chaieb@27568
  1598
nipkow@31952
  1599
lemma prime_dvd_power_nat [rule_format]: "prime (p::nat) -->
huffman@31706
  1600
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1601
  by (induct n rule: nat_induct, auto)
chaieb@27568
  1602
nipkow@31952
  1603
lemma prime_dvd_power_int [rule_format]: "prime (p::int) -->
huffman@31706
  1604
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1605
  apply (induct n rule: nat_induct, auto)
nipkow@31952
  1606
  apply (frule prime_ge_0_int)
huffman@31706
  1607
  apply auto
huffman@31706
  1608
done
huffman@31706
  1609
nipkow@31952
  1610
lemma prime_imp_power_coprime_nat: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1611
    coprime a (p^m)"
nipkow@31952
  1612
  apply (rule coprime_exp_nat)
nipkow@31952
  1613
  apply (subst gcd_commute_nat)
nipkow@31952
  1614
  apply (erule (1) prime_imp_coprime_nat)
huffman@31706
  1615
done
chaieb@27568
  1616
nipkow@31952
  1617
lemma prime_imp_power_coprime_int: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow>
huffman@31706
  1618
    coprime a (p^m)"
nipkow@31952
  1619
  apply (rule coprime_exp_int)
nipkow@31952
  1620
  apply (subst gcd_commute_int)
nipkow@31952
  1621
  apply (erule (1) prime_imp_coprime_int)
huffman@31706
  1622
done
chaieb@27568
  1623
nipkow@31952
  1624
lemma primes_coprime_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
nipkow@31952
  1625
  apply (rule prime_imp_coprime_nat, assumption)
huffman@31706
  1626
  apply (unfold prime_nat_def, auto)
huffman@31706
  1627
done
chaieb@27568
  1628
nipkow@31952
  1629
lemma primes_coprime_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
nipkow@31952
  1630
  apply (rule prime_imp_coprime_int, assumption)
huffman@31706
  1631
  apply (unfold prime_int_altdef, clarify)
huffman@31706
  1632
  apply (drule_tac x = q in spec)
huffman@31706
  1633
  apply (drule_tac x = p in spec)
huffman@31706
  1634
  apply auto
huffman@31706
  1635
done
chaieb@27568
  1636
nipkow@31952
  1637
lemma primes_imp_powers_coprime_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1638
    coprime (p^m) (q^n)"
nipkow@31952
  1639
  by (rule coprime_exp2_nat, rule primes_coprime_nat)
chaieb@27568
  1640
nipkow@31952
  1641
lemma primes_imp_powers_coprime_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow>
huffman@31706
  1642
    coprime (p^m) (q^n)"
nipkow@31952
  1643
  by (rule coprime_exp2_int, rule primes_coprime_int)
chaieb@27568
  1644
nipkow@31952
  1645
lemma prime_factor_nat: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
huffman@31706
  1646
  apply (induct n rule: nat_less_induct)
huffman@31706
  1647
  apply (case_tac "n = 0")
nipkow@31952
  1648
  using two_is_prime_nat apply blast
huffman@31706
  1649
  apply (case_tac "prime n")
huffman@31706
  1650
  apply blast
huffman@31706
  1651
  apply (subgoal_tac "n > 1")
nipkow@31952
  1652
  apply (frule (1) not_prime_eq_prod_nat)
huffman@31706
  1653
  apply (auto intro: dvd_mult dvd_mult2)
huffman@31706
  1654
done
chaieb@23244
  1655
huffman@31706
  1656
(* An Isar version:
huffman@31706
  1657
nipkow@31952
  1658
lemma prime_factor_b_nat:
huffman@31706
  1659
  fixes n :: nat
huffman@31706
  1660
  assumes "n \<noteq> 1"
huffman@31706
  1661
  shows "\<exists>p. prime p \<and> p dvd n"
nipkow@23983
  1662
huffman@31706
  1663
using `n ~= 1`
nipkow@31952
  1664
proof (induct n rule: less_induct_nat)
huffman@31706
  1665
  fix n :: nat
huffman@31706
  1666
  assume "n ~= 1" and
huffman@31706
  1667
    ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
huffman@31706
  1668
  thus "\<exists>p. prime p \<and> p dvd n"
huffman@31706
  1669
  proof -
huffman@31706
  1670
  {
huffman@31706
  1671
    assume "n = 0"
nipkow@31952
  1672
    moreover note two_is_prime_nat
huffman@31706
  1673
    ultimately have ?thesis
nipkow@31952
  1674
      by (auto simp del: two_is_prime_nat)
huffman@31706
  1675
  }
huffman@31706
  1676
  moreover
huffman@31706
  1677
  {
huffman@31706
  1678
    assume "prime n"
huffman@31706
  1679
    hence ?thesis by auto
huffman@31706
  1680
  }
huffman@31706
  1681
  moreover
huffman@31706
  1682
  {
huffman@31706
  1683
    assume "n ~= 0" and "~ prime n"
huffman@31706
  1684
    with `n ~= 1` have "n > 1" by auto
nipkow@31952
  1685
    with `~ prime n` and not_prime_eq_prod_nat obtain m k where
huffman@31706
  1686
      "n = m * k" and "1 < m" and "m < n" by blast
huffman@31706
  1687
    with ih obtain p where "prime p" and "p dvd m" by blast
huffman@31706
  1688
    with `n = m * k` have ?thesis by auto
huffman@31706
  1689
  }
huffman@31706
  1690
  ultimately show ?thesis by blast
huffman@31706
  1691
  qed
nipkow@23983
  1692
qed
nipkow@23983
  1693
huffman@31706
  1694
*)
huffman@31706
  1695
huffman@31706
  1696
text {* One property of coprimality is easier to prove via prime factors. *}
huffman@31706
  1697
nipkow@31952
  1698
lemma prime_divprod_pow_nat:
huffman@31706
  1699
  assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
huffman@31706
  1700
  shows "p^n dvd a \<or> p^n dvd b"
huffman@31706
  1701
proof-
huffman@31706
  1702
  {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
huffman@31706
  1703
      apply (cases "n=0", simp_all)
huffman@31706
  1704
      apply (cases "a=1", simp_all) done}
huffman@31706
  1705
  moreover
huffman@31706
  1706
  {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
huffman@31706
  1707
    then obtain m where m: "n = Suc m" by (cases n, auto)
huffman@31706
  1708
    from n have "p dvd p^n" by (intro dvd_power, auto)
huffman@31706
  1709
    also note pab
huffman@31706
  1710
    finally have pab': "p dvd a * b".
nipkow@31952
  1711
    from prime_dvd_mult_nat[OF p pab']
huffman@31706
  1712
    have "p dvd a \<or> p dvd b" .
huffman@31706
  1713
    moreover
huffman@31706
  1714
    {assume pa: "p dvd a"
huffman@31706
  1715
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
nipkow@31952
  1716
      from coprime_common_divisor_nat [OF ab, OF pa] p have "\<not> p dvd b" by auto
huffman@31706
  1717
      with p have "coprime b p"
nipkow@31952
  1718
        by (subst gcd_commute_nat, intro prime_imp_coprime_nat)
huffman@31706
  1719
      hence pnb: "coprime (p^n) b"
nipkow@31952
  1720
        by (subst gcd_commute_nat, rule coprime_exp_nat)
nipkow@31952
  1721
      from coprime_divprod_nat[OF pnba pnb] have ?thesis by blast }
huffman@31706
  1722
    moreover
huffman@31706
  1723
    {assume pb: "p dvd b"
huffman@31706
  1724
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
nipkow@31952
  1725
      from coprime_common_divisor_nat [OF ab, of p] pb p have "\<not> p dvd a"
huffman@31706
  1726
        by auto
huffman@31706
  1727
      with p have "coprime a p"
nipkow@31952
  1728
        by (subst gcd_commute_nat, intro prime_imp_coprime_nat)
huffman@31706
  1729
      hence pna: "coprime (p^n) a"
nipkow@31952
  1730
        by (subst gcd_commute_nat, rule coprime_exp_nat)
nipkow@31952
  1731
      from coprime_divprod_nat[OF pab pna] have ?thesis by blast }
huffman@31706
  1732
    ultimately have ?thesis by blast}
huffman@31706
  1733
  ultimately show ?thesis by blast
nipkow@23983
  1734
qed
nipkow@23983
  1735
avigad@32036
  1736
subsection {* Infinitely many primes *}
avigad@32036
  1737
avigad@32036
  1738
lemma next_prime_bound: "\<exists>(p::nat). prime p \<and> n < p \<and> p <= fact n + 1"
avigad@32036
  1739
proof-
avigad@32036
  1740
  have f1: "fact n + 1 \<noteq> 1" using fact_ge_one_nat [of n] by arith 
avigad@32036
  1741
  from prime_factor_nat [OF f1]
avigad@32036
  1742
      obtain p where "prime p" and "p dvd fact n + 1" by auto
avigad@32036
  1743
  hence "p \<le> fact n + 1" 
avigad@32036
  1744
    by (intro dvd_imp_le, auto)
avigad@32036
  1745
  {assume "p \<le> n"
avigad@32036
  1746
    from `prime p` have "p \<ge> 1" 
avigad@32036
  1747
      by (cases p, simp_all)
avigad@32036
  1748
    with `p <= n` have "p dvd fact n" 
avigad@32036
  1749
      by (intro dvd_fact_nat)
avigad@32036
  1750
    with `p dvd fact n + 1` have "p dvd fact n + 1 - fact n"
avigad@32036
  1751
      by (rule dvd_diff_nat)
avigad@32036
  1752
    hence "p dvd 1" by simp
avigad@32036
  1753
    hence "p <= 1" by auto
avigad@32036
  1754
    moreover from `prime p` have "p > 1" by auto
avigad@32036
  1755
    ultimately have False by auto}
avigad@32036
  1756
  hence "n < p" by arith
avigad@32036
  1757
  with `prime p` and `p <= fact n + 1` show ?thesis by auto
avigad@32036
  1758
qed
avigad@32036
  1759
avigad@32036
  1760
lemma bigger_prime: "\<exists>p. prime p \<and> p > (n::nat)" 
avigad@32036
  1761
using next_prime_bound by auto
avigad@32036
  1762
avigad@32036
  1763
lemma primes_infinite: "\<not> (finite {(p::nat). prime p})"
avigad@32036
  1764
proof
avigad@32036
  1765
  assume "finite {(p::nat). prime p}"
avigad@32036
  1766
  with Max_ge have "(EX b. (ALL x : {(p::nat). prime p}. x <= b))"
avigad@32036
  1767
    by auto
avigad@32036
  1768
  then obtain b where "ALL (x::nat). prime x \<longrightarrow> x <= b"
avigad@32036
  1769
    by auto
avigad@32036
  1770
  with bigger_prime [of b] show False by auto
avigad@32036
  1771
qed
avigad@32036
  1772
avigad@32036
  1773
wenzelm@21256
  1774
end