src/HOL/GCD.thy
author nipkow
Tue Jul 21 14:08:58 2009 +0200 (2009-07-21)
changeset 32112 6da9c2a49fed
parent 32111 7c39fcfffd61
child 32415 1dddf2f64266
permissions -rw-r--r--
Made dvd/gcd/lcm a complete lattice by introducing Gcd/GCD/Lcm/LCM
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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 = zero + one + dvd +
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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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   340
  by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff_nat)
wenzelm@21256
   341
nipkow@31952
   342
lemma gcd_zero_int [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
huffman@31706
   343
  by (auto simp add: gcd_int_def)
wenzelm@21256
   344
nipkow@31952
   345
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
nipkow@31952
   346
  by (insert gcd_zero_nat [of m n], arith)
wenzelm@21256
   347
nipkow@31952
   348
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
nipkow@31952
   349
  by (insert gcd_zero_int [of m n], insert gcd_ge_0_int [of m n], arith)
huffman@31706
   350
nipkow@31952
   351
lemma gcd_commute_nat: "gcd (m::nat) n = gcd n m"
huffman@31706
   352
  by (rule dvd_anti_sym, auto)
haftmann@23687
   353
nipkow@31952
   354
lemma gcd_commute_int: "gcd (m::int) n = gcd n m"
nipkow@31952
   355
  by (auto simp add: gcd_int_def gcd_commute_nat)
huffman@31706
   356
nipkow@31952
   357
lemma gcd_assoc_nat: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
huffman@31706
   358
  apply (rule dvd_anti_sym)
huffman@31706
   359
  apply (blast intro: dvd_trans)+
huffman@31706
   360
done
wenzelm@21256
   361
nipkow@31952
   362
lemma gcd_assoc_int: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
nipkow@31952
   363
  by (auto simp add: gcd_int_def gcd_assoc_nat)
huffman@31706
   364
nipkow@31952
   365
lemmas gcd_left_commute_nat =
nipkow@31952
   366
  mk_left_commute[of gcd, OF gcd_assoc_nat gcd_commute_nat]
huffman@31706
   367
nipkow@31952
   368
lemmas gcd_left_commute_int =
nipkow@31952
   369
  mk_left_commute[of gcd, OF gcd_assoc_int gcd_commute_int]
huffman@31706
   370
nipkow@31952
   371
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
huffman@31706
   372
  -- {* gcd is an AC-operator *}
wenzelm@21256
   373
nipkow@31952
   374
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
huffman@31706
   375
nipkow@31952
   376
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   377
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   378
  apply auto
huffman@31706
   379
  apply (rule dvd_anti_sym)
nipkow@31952
   380
  apply (erule (1) gcd_greatest_nat)
huffman@31706
   381
  apply auto
huffman@31706
   382
done
wenzelm@21256
   383
nipkow@31952
   384
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   385
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   386
  apply (case_tac "d = 0")
huffman@31706
   387
  apply force
huffman@31706
   388
  apply (rule iffI)
huffman@31706
   389
  apply (rule zdvd_anti_sym)
huffman@31706
   390
  apply arith
nipkow@31952
   391
  apply (subst gcd_pos_int)
huffman@31706
   392
  apply clarsimp
huffman@31706
   393
  apply (drule_tac x = "d + 1" in spec)
huffman@31706
   394
  apply (frule zdvd_imp_le)
nipkow@31952
   395
  apply (auto intro: gcd_greatest_int)
huffman@31706
   396
done
huffman@30082
   397
nipkow@31798
   398
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
nipkow@31952
   399
by (metis dvd.eq_iff gcd_unique_nat)
nipkow@31798
   400
nipkow@31798
   401
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
nipkow@31952
   402
by (metis dvd.eq_iff gcd_unique_nat)
nipkow@31798
   403
nipkow@31798
   404
lemma gcd_proj1_if_dvd_int[simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
nipkow@31952
   405
by (metis abs_dvd_iff abs_eq_0 gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   406
nipkow@31798
   407
lemma gcd_proj2_if_dvd_int[simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
nipkow@31952
   408
by (metis gcd_proj1_if_dvd_int gcd_commute_int)
nipkow@31798
   409
nipkow@31798
   410
wenzelm@21256
   411
text {*
wenzelm@21256
   412
  \medskip Multiplication laws
wenzelm@21256
   413
*}
wenzelm@21256
   414
nipkow@31952
   415
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256
   416
    -- {* \cite[page 27]{davenport92} *}
nipkow@31952
   417
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   418
  apply simp
wenzelm@21256
   419
  apply (case_tac "k = 0")
nipkow@31952
   420
  apply (simp_all add: mod_geq gcd_non_0_nat mod_mult_distrib2)
huffman@31706
   421
done
wenzelm@21256
   422
nipkow@31952
   423
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   424
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   425
  apply (subst (1 2) abs_mult)
nipkow@31952
   426
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   427
  apply auto
huffman@31706
   428
done
wenzelm@21256
   429
nipkow@31952
   430
lemma coprime_dvd_mult_nat: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31952
   431
  apply (insert gcd_mult_distrib_nat [of m k n])
wenzelm@21256
   432
  apply simp
wenzelm@21256
   433
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   434
  apply simp
wenzelm@21256
   435
  done
wenzelm@21256
   436
nipkow@31952
   437
lemma coprime_dvd_mult_int:
nipkow@31813
   438
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31813
   439
apply (subst abs_dvd_iff [symmetric])
nipkow@31813
   440
apply (subst dvd_abs_iff [symmetric])
nipkow@31952
   441
apply (subst (asm) gcd_abs_int)
nipkow@31952
   442
apply (rule coprime_dvd_mult_nat [transferred])
nipkow@31813
   443
    prefer 4 apply assumption
nipkow@31813
   444
   apply auto
nipkow@31813
   445
apply (subst abs_mult [symmetric], auto)
huffman@31706
   446
done
huffman@31706
   447
nipkow@31952
   448
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   449
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   450
  by (auto intro: coprime_dvd_mult_nat)
huffman@31706
   451
nipkow@31952
   452
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   453
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   454
  by (auto intro: coprime_dvd_mult_int)
huffman@31706
   455
nipkow@31952
   456
lemma gcd_mult_cancel_nat: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
wenzelm@21256
   457
  apply (rule dvd_anti_sym)
nipkow@31952
   458
  apply (rule gcd_greatest_nat)
nipkow@31952
   459
  apply (rule_tac n = k in coprime_dvd_mult_nat)
nipkow@31952
   460
  apply (simp add: gcd_assoc_nat)
nipkow@31952
   461
  apply (simp add: gcd_commute_nat)
huffman@31706
   462
  apply (simp_all add: mult_commute)
huffman@31706
   463
done
wenzelm@21256
   464
nipkow@31952
   465
lemma gcd_mult_cancel_int:
nipkow@31813
   466
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
nipkow@31952
   467
apply (subst (1 2) gcd_abs_int)
nipkow@31813
   468
apply (subst abs_mult)
nipkow@31952
   469
apply (rule gcd_mult_cancel_nat [transferred], auto)
huffman@31706
   470
done
wenzelm@21256
   471
wenzelm@21256
   472
text {* \medskip Addition laws *}
wenzelm@21256
   473
nipkow@31952
   474
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   475
  apply (case_tac "n = 0")
nipkow@31952
   476
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   477
done
huffman@31706
   478
nipkow@31952
   479
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
nipkow@31952
   480
  apply (subst (1 2) gcd_commute_nat)
huffman@31706
   481
  apply (subst add_commute)
huffman@31706
   482
  apply simp
huffman@31706
   483
done
huffman@31706
   484
huffman@31706
   485
(* to do: add the other variations? *)
huffman@31706
   486
nipkow@31952
   487
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
nipkow@31952
   488
  by (subst gcd_add1_nat [symmetric], auto)
huffman@31706
   489
nipkow@31952
   490
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
nipkow@31952
   491
  apply (subst gcd_commute_nat)
nipkow@31952
   492
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
   493
  apply auto
nipkow@31952
   494
  apply (subst gcd_commute_nat)
nipkow@31952
   495
  apply (subst gcd_diff1_nat)
huffman@31706
   496
  apply assumption
nipkow@31952
   497
  apply (rule gcd_commute_nat)
huffman@31706
   498
done
huffman@31706
   499
nipkow@31952
   500
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   501
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   502
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
   503
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
   504
    zmod_zminus1_eq_if)
huffman@31706
   505
  apply (frule_tac a = x in pos_mod_bound)
nipkow@31952
   506
  apply (subst (1 2) gcd_commute_nat)
nipkow@31952
   507
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
   508
    nat_le_eq_zle)
huffman@31706
   509
done
wenzelm@21256
   510
nipkow@31952
   511
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   512
  apply (case_tac "y = 0")
huffman@31706
   513
  apply force
huffman@31706
   514
  apply (case_tac "y > 0")
nipkow@31952
   515
  apply (subst gcd_non_0_int, auto)
nipkow@31952
   516
  apply (insert gcd_non_0_int [of "-y" "-x"])
nipkow@31952
   517
  apply (auto simp add: gcd_neg1_int gcd_neg2_int)
huffman@31706
   518
done
huffman@31706
   519
nipkow@31952
   520
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
nipkow@31952
   521
by (metis gcd_red_int mod_add_self1 zadd_commute)
huffman@31706
   522
nipkow@31952
   523
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
nipkow@31952
   524
by (metis gcd_add1_int gcd_commute_int zadd_commute)
wenzelm@21256
   525
nipkow@31952
   526
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31952
   527
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
wenzelm@21256
   528
nipkow@31952
   529
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
nipkow@31952
   530
by (metis gcd_commute_int gcd_red_int mod_mult_self1 zadd_commute)
nipkow@31798
   531
wenzelm@21256
   532
huffman@31706
   533
(* to do: differences, and all variations of addition rules
huffman@31706
   534
    as simplification rules for nat and int *)
huffman@31706
   535
nipkow@31798
   536
(* FIXME remove iff *)
nipkow@31952
   537
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   538
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   539
huffman@31706
   540
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   541
nipkow@31992
   542
lemma finite_divisors_nat[simp]:
nipkow@31992
   543
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
nipkow@31734
   544
proof-
nipkow@31734
   545
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
nipkow@31734
   546
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   547
    by(bestsimp intro!:dvd_imp_le)
nipkow@31734
   548
qed
nipkow@31734
   549
nipkow@31995
   550
lemma finite_divisors_int[simp]:
nipkow@31734
   551
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
   552
proof-
nipkow@31734
   553
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
   554
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
   555
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   556
    by(bestsimp intro!:dvd_imp_le_int)
nipkow@31734
   557
qed
nipkow@31734
   558
nipkow@31995
   559
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
   560
apply(rule antisym)
nipkow@31995
   561
 apply (fastsimp intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
   562
apply simp
nipkow@31995
   563
done
nipkow@31995
   564
nipkow@31995
   565
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
nipkow@31995
   566
apply(rule antisym)
nipkow@31995
   567
 apply(rule Max_le_iff[THEN iffD2])
nipkow@31995
   568
   apply simp
nipkow@31995
   569
  apply fastsimp
nipkow@31995
   570
 apply (metis Collect_def abs_ge_self dvd_imp_le_int mem_def zle_trans)
nipkow@31995
   571
apply simp
nipkow@31995
   572
done
nipkow@31995
   573
nipkow@31734
   574
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   575
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   576
apply(rule Max_eqI[THEN sym])
nipkow@31995
   577
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   578
 apply simp
nipkow@31952
   579
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
nipkow@31734
   580
apply simp
nipkow@31734
   581
done
nipkow@31734
   582
nipkow@31734
   583
lemma gcd_is_Max_divisors_int:
nipkow@31734
   584
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   585
apply(rule Max_eqI[THEN sym])
nipkow@31995
   586
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
   587
 apply simp
nipkow@31952
   588
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
nipkow@31734
   589
apply simp
nipkow@31734
   590
done
nipkow@31734
   591
chaieb@22027
   592
huffman@31706
   593
subsection {* Coprimality *}
huffman@31706
   594
nipkow@31952
   595
lemma div_gcd_coprime_nat:
huffman@31706
   596
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   597
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   598
proof -
haftmann@27556
   599
  let ?g = "gcd a b"
chaieb@22027
   600
  let ?a' = "a div ?g"
chaieb@22027
   601
  let ?b' = "b div ?g"
haftmann@27556
   602
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   603
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   604
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   605
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   606
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   607
    unfolding dvd_def by blast
huffman@31706
   608
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   609
    by simp_all
wenzelm@22367
   610
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   611
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   612
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
nipkow@31952
   613
  have "?g \<noteq> 0" using nz by (simp add: gcd_zero_nat)
huffman@31706
   614
  then have gp: "?g > 0" by arith
nipkow@31952
   615
  from gcd_greatest_nat [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   616
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   617
qed
chaieb@22027
   618
nipkow@31952
   619
lemma div_gcd_coprime_int:
huffman@31706
   620
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   621
  shows "coprime (a div gcd a b) (b div gcd a b)"
nipkow@31952
   622
apply (subst (1 2 3) gcd_abs_int)
nipkow@31813
   623
apply (subst (1 2) abs_div)
nipkow@31813
   624
  apply simp
nipkow@31813
   625
 apply simp
nipkow@31813
   626
apply(subst (1 2) abs_gcd_int)
nipkow@31952
   627
apply (rule div_gcd_coprime_nat [transferred])
nipkow@31952
   628
using nz apply (auto simp add: gcd_abs_int [symmetric])
huffman@31706
   629
done
huffman@31706
   630
nipkow@31952
   631
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   632
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
   633
nipkow@31952
   634
lemma coprime_Suc_0_nat:
huffman@31706
   635
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
nipkow@31952
   636
  using coprime_nat by (simp add: One_nat_def)
huffman@31706
   637
nipkow@31952
   638
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   639
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   640
  using gcd_unique_int [of 1 a b]
huffman@31706
   641
  apply clarsimp
huffman@31706
   642
  apply (erule subst)
huffman@31706
   643
  apply (rule iffI)
huffman@31706
   644
  apply force
huffman@31706
   645
  apply (drule_tac x = "abs e" in exI)
huffman@31706
   646
  apply (case_tac "e >= 0")
huffman@31706
   647
  apply force
huffman@31706
   648
  apply force
huffman@31706
   649
done
huffman@31706
   650
nipkow@31952
   651
lemma gcd_coprime_nat:
huffman@31706
   652
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   653
    b: "b = b' * gcd a b"
huffman@31706
   654
  shows    "coprime a' b'"
huffman@31706
   655
huffman@31706
   656
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   657
  apply (erule ssubst)
huffman@31706
   658
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   659
  apply (erule ssubst)
nipkow@31952
   660
  apply (rule div_gcd_coprime_nat)
huffman@31706
   661
  using prems
huffman@31706
   662
  apply force
huffman@31706
   663
  apply (subst (1) b)
huffman@31706
   664
  using z apply force
huffman@31706
   665
  apply (subst (1) a)
huffman@31706
   666
  using z apply force
huffman@31706
   667
done
huffman@31706
   668
nipkow@31952
   669
lemma gcd_coprime_int:
huffman@31706
   670
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   671
    b: "b = b' * gcd a b"
huffman@31706
   672
  shows    "coprime a' b'"
huffman@31706
   673
huffman@31706
   674
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   675
  apply (erule ssubst)
huffman@31706
   676
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   677
  apply (erule ssubst)
nipkow@31952
   678
  apply (rule div_gcd_coprime_int)
huffman@31706
   679
  using prems
huffman@31706
   680
  apply force
huffman@31706
   681
  apply (subst (1) b)
huffman@31706
   682
  using z apply force
huffman@31706
   683
  apply (subst (1) a)
huffman@31706
   684
  using z apply force
huffman@31706
   685
done
huffman@31706
   686
nipkow@31952
   687
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   688
    shows "coprime d (a * b)"
nipkow@31952
   689
  apply (subst gcd_commute_nat)
nipkow@31952
   690
  using da apply (subst gcd_mult_cancel_nat)
nipkow@31952
   691
  apply (subst gcd_commute_nat, assumption)
nipkow@31952
   692
  apply (subst gcd_commute_nat, rule db)
huffman@31706
   693
done
huffman@31706
   694
nipkow@31952
   695
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   696
    shows "coprime d (a * b)"
nipkow@31952
   697
  apply (subst gcd_commute_int)
nipkow@31952
   698
  using da apply (subst gcd_mult_cancel_int)
nipkow@31952
   699
  apply (subst gcd_commute_int, assumption)
nipkow@31952
   700
  apply (subst gcd_commute_int, rule db)
huffman@31706
   701
done
huffman@31706
   702
nipkow@31952
   703
lemma coprime_lmult_nat:
huffman@31706
   704
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   705
proof -
huffman@31706
   706
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   707
    by (rule gcd_greatest_nat, auto)
huffman@31706
   708
  with dab show ?thesis
huffman@31706
   709
    by auto
huffman@31706
   710
qed
huffman@31706
   711
nipkow@31952
   712
lemma coprime_lmult_int:
nipkow@31798
   713
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   714
proof -
huffman@31706
   715
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   716
    by (rule gcd_greatest_int, auto)
nipkow@31798
   717
  with assms show ?thesis
huffman@31706
   718
    by auto
huffman@31706
   719
qed
huffman@31706
   720
nipkow@31952
   721
lemma coprime_rmult_nat:
nipkow@31798
   722
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   723
proof -
huffman@31706
   724
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   725
    by (rule gcd_greatest_nat, auto intro: dvd_mult)
nipkow@31798
   726
  with assms show ?thesis
huffman@31706
   727
    by auto
huffman@31706
   728
qed
huffman@31706
   729
nipkow@31952
   730
lemma coprime_rmult_int:
huffman@31706
   731
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   732
proof -
huffman@31706
   733
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   734
    by (rule gcd_greatest_int, auto intro: dvd_mult)
huffman@31706
   735
  with dab show ?thesis
huffman@31706
   736
    by auto
huffman@31706
   737
qed
huffman@31706
   738
nipkow@31952
   739
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   740
    coprime d a \<and>  coprime d b"
nipkow@31952
   741
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
nipkow@31952
   742
    coprime_mult_nat[of d a b]
huffman@31706
   743
  by blast
huffman@31706
   744
nipkow@31952
   745
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   746
    coprime d a \<and>  coprime d b"
nipkow@31952
   747
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
nipkow@31952
   748
    coprime_mult_int[of d a b]
huffman@31706
   749
  by blast
huffman@31706
   750
nipkow@31952
   751
lemma gcd_coprime_exists_nat:
huffman@31706
   752
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   753
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   754
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   755
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   756
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
huffman@31706
   757
done
huffman@31706
   758
nipkow@31952
   759
lemma gcd_coprime_exists_int:
huffman@31706
   760
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   761
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   762
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   763
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   764
  using nz apply (auto simp add: div_gcd_coprime_int dvd_div_mult_self)
huffman@31706
   765
done
huffman@31706
   766
nipkow@31952
   767
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   768
  by (induct n, simp_all add: coprime_mult_nat)
huffman@31706
   769
nipkow@31952
   770
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   771
  by (induct n, simp_all add: coprime_mult_int)
huffman@31706
   772
nipkow@31952
   773
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   774
  apply (rule coprime_exp_nat)
nipkow@31952
   775
  apply (subst gcd_commute_nat)
nipkow@31952
   776
  apply (rule coprime_exp_nat)
nipkow@31952
   777
  apply (subst gcd_commute_nat, assumption)
huffman@31706
   778
done
huffman@31706
   779
nipkow@31952
   780
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   781
  apply (rule coprime_exp_int)
nipkow@31952
   782
  apply (subst gcd_commute_int)
nipkow@31952
   783
  apply (rule coprime_exp_int)
nipkow@31952
   784
  apply (subst gcd_commute_int, assumption)
huffman@31706
   785
done
huffman@31706
   786
nipkow@31952
   787
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   788
proof (cases)
huffman@31706
   789
  assume "a = 0 & b = 0"
huffman@31706
   790
  thus ?thesis by simp
huffman@31706
   791
  next assume "~(a = 0 & b = 0)"
huffman@31706
   792
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
nipkow@31952
   793
    by (auto simp:div_gcd_coprime_nat)
huffman@31706
   794
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   795
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   796
    apply (subst (1 2) mult_commute)
nipkow@31952
   797
    apply (subst gcd_mult_distrib_nat [symmetric])
huffman@31706
   798
    apply simp
huffman@31706
   799
    done
huffman@31706
   800
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
huffman@31706
   801
    apply (subst div_power)
huffman@31706
   802
    apply auto
huffman@31706
   803
    apply (rule dvd_div_mult_self)
huffman@31706
   804
    apply (rule dvd_power_same)
huffman@31706
   805
    apply auto
huffman@31706
   806
    done
huffman@31706
   807
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   808
    apply (subst div_power)
huffman@31706
   809
    apply auto
huffman@31706
   810
    apply (rule dvd_div_mult_self)
huffman@31706
   811
    apply (rule dvd_power_same)
huffman@31706
   812
    apply auto
huffman@31706
   813
    done
huffman@31706
   814
  finally show ?thesis .
huffman@31706
   815
qed
huffman@31706
   816
nipkow@31952
   817
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
   818
  apply (subst (1 2) gcd_abs_int)
huffman@31706
   819
  apply (subst (1 2) power_abs)
nipkow@31952
   820
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
   821
  apply auto
huffman@31706
   822
done
huffman@31706
   823
nipkow@31952
   824
lemma coprime_divprod_nat: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
nipkow@31952
   825
  using coprime_dvd_mult_iff_nat[of d a b]
huffman@31706
   826
  by (auto simp add: mult_commute)
huffman@31706
   827
nipkow@31952
   828
lemma coprime_divprod_int: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
nipkow@31952
   829
  using coprime_dvd_mult_iff_int[of d a b]
huffman@31706
   830
  by (auto simp add: mult_commute)
huffman@31706
   831
nipkow@31952
   832
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   833
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   834
proof-
huffman@31706
   835
  let ?g = "gcd a b"
huffman@31706
   836
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   837
  moreover
huffman@31706
   838
  {assume z: "?g \<noteq> 0"
nipkow@31952
   839
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   840
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   841
      by blast
huffman@31706
   842
    have thb: "?g dvd b" by auto
huffman@31706
   843
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   844
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   845
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   846
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   847
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   848
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
huffman@31706
   849
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   850
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   851
    with thb thc have ?thesis by blast }
huffman@31706
   852
  ultimately show ?thesis by blast
huffman@31706
   853
qed
huffman@31706
   854
nipkow@31952
   855
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
   856
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   857
proof-
huffman@31706
   858
  let ?g = "gcd a b"
huffman@31706
   859
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   860
  moreover
huffman@31706
   861
  {assume z: "?g \<noteq> 0"
nipkow@31952
   862
    from gcd_coprime_exists_int[OF z]
huffman@31706
   863
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   864
      by blast
huffman@31706
   865
    have thb: "?g dvd b" by auto
huffman@31706
   866
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   867
    with dc have th0: "a' dvd b*c"
huffman@31706
   868
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   869
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   870
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   871
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   872
    from coprime_dvd_mult_int[OF ab'(3)] th_1
huffman@31706
   873
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   874
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   875
    with thb thc have ?thesis by blast }
huffman@31706
   876
  ultimately show ?thesis by blast
chaieb@27669
   877
qed
chaieb@27669
   878
nipkow@31952
   879
lemma pow_divides_pow_nat:
huffman@31706
   880
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   881
  shows "a dvd b"
huffman@31706
   882
proof-
huffman@31706
   883
  let ?g = "gcd a b"
huffman@31706
   884
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   885
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   886
  moreover
huffman@31706
   887
  {assume z: "?g \<noteq> 0"
huffman@31706
   888
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
nipkow@31952
   889
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   890
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   891
      by blast
huffman@31706
   892
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   893
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   894
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   895
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   896
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   897
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   898
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   899
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   900
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
huffman@31706
   901
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   902
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   903
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   904
  ultimately show ?thesis by blast
huffman@31706
   905
qed
huffman@31706
   906
nipkow@31952
   907
lemma pow_divides_pow_int:
huffman@31706
   908
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   909
  shows "a dvd b"
chaieb@27669
   910
proof-
huffman@31706
   911
  let ?g = "gcd a b"
huffman@31706
   912
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   913
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   914
  moreover
huffman@31706
   915
  {assume z: "?g \<noteq> 0"
huffman@31706
   916
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
nipkow@31952
   917
    from gcd_coprime_exists_int[OF z]
huffman@31706
   918
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   919
      by blast
huffman@31706
   920
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   921
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   922
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   923
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   924
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   925
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   926
    with th0 have "a' dvd b'^n"
huffman@31706
   927
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   928
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   929
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
huffman@31706
   930
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   931
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   932
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   933
  ultimately show ?thesis by blast
huffman@31706
   934
qed
huffman@31706
   935
nipkow@31798
   936
(* FIXME move to Divides(?) *)
nipkow@31952
   937
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
   938
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
   939
nipkow@31952
   940
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
   941
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
   942
nipkow@31952
   943
lemma divides_mult_nat:
huffman@31706
   944
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   945
  shows "m * n dvd r"
huffman@31706
   946
proof-
huffman@31706
   947
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   948
    unfolding dvd_def by blast
huffman@31706
   949
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   950
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
huffman@31706
   951
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   952
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   953
qed
huffman@31706
   954
nipkow@31952
   955
lemma divides_mult_int:
huffman@31706
   956
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   957
  shows "m * n dvd r"
huffman@31706
   958
proof-
huffman@31706
   959
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   960
    unfolding dvd_def by blast
huffman@31706
   961
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   962
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
huffman@31706
   963
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   964
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   965
qed
chaieb@27669
   966
nipkow@31952
   967
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   968
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   969
  apply force
nipkow@31952
   970
  apply (rule dvd_diff_nat)
huffman@31706
   971
  apply auto
huffman@31706
   972
done
huffman@31706
   973
nipkow@31952
   974
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
nipkow@31952
   975
  using coprime_plus_one_nat by (simp add: One_nat_def)
huffman@31706
   976
nipkow@31952
   977
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   978
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   979
  apply force
huffman@31706
   980
  apply (rule dvd_diff)
huffman@31706
   981
  apply auto
huffman@31706
   982
done
huffman@31706
   983
nipkow@31952
   984
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
   985
  using coprime_plus_one_nat [of "n - 1"]
nipkow@31952
   986
    gcd_commute_nat [of "n - 1" n] by auto
huffman@31706
   987
nipkow@31952
   988
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
   989
  using coprime_plus_one_int [of "n - 1"]
nipkow@31952
   990
    gcd_commute_int [of "n - 1" n] by auto
huffman@31706
   991
nipkow@31952
   992
lemma setprod_coprime_nat [rule_format]:
huffman@31706
   993
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   994
  apply (case_tac "finite A")
huffman@31706
   995
  apply (induct set: finite)
nipkow@31952
   996
  apply (auto simp add: gcd_mult_cancel_nat)
huffman@31706
   997
done
huffman@31706
   998
nipkow@31952
   999
lemma setprod_coprime_int [rule_format]:
huffman@31706
  1000
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
  1001
  apply (case_tac "finite A")
huffman@31706
  1002
  apply (induct set: finite)
nipkow@31952
  1003
  apply (auto simp add: gcd_mult_cancel_int)
huffman@31706
  1004
done
huffman@31706
  1005
nipkow@31952
  1006
lemma prime_odd_nat: "prime (p::nat) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
  1007
  unfolding prime_nat_def
huffman@31706
  1008
  apply (subst even_mult_two_ex)
huffman@31706
  1009
  apply clarify
huffman@31706
  1010
  apply (drule_tac x = 2 in spec)
huffman@31706
  1011
  apply auto
huffman@31706
  1012
done
huffman@31706
  1013
nipkow@31952
  1014
lemma prime_odd_int: "prime (p::int) \<Longrightarrow> p > 2 \<Longrightarrow> odd p"
huffman@31706
  1015
  unfolding prime_int_def
nipkow@31952
  1016
  apply (frule prime_odd_nat)
huffman@31706
  1017
  apply (auto simp add: even_nat_def)
huffman@31706
  1018
done
huffman@31706
  1019
nipkow@31952
  1020
lemma coprime_common_divisor_nat: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1021
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
  1022
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1023
  apply simp
nipkow@31952
  1024
  apply (erule (1) gcd_greatest_nat)
huffman@31706
  1025
done
huffman@31706
  1026
nipkow@31952
  1027
lemma coprime_common_divisor_int: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
  1028
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
  1029
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
  1030
  apply simp
nipkow@31952
  1031
  apply (erule (1) gcd_greatest_int)
huffman@31706
  1032
done
huffman@31706
  1033
nipkow@31952
  1034
lemma coprime_divisors_nat: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
  1035
    coprime d e"
huffman@31706
  1036
  apply (auto simp add: dvd_def)
nipkow@31952
  1037
  apply (frule coprime_lmult_int)
nipkow@31952
  1038
  apply (subst gcd_commute_int)
nipkow@31952
  1039
  apply (subst (asm) (2) gcd_commute_int)
nipkow@31952
  1040
  apply (erule coprime_lmult_int)
huffman@31706
  1041
done
huffman@31706
  1042
nipkow@31952
  1043
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
  1044
apply (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
huffman@31706
  1045
done
huffman@31706
  1046
nipkow@31952
  1047
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
  1048
apply (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
huffman@31706
  1049
done
huffman@31706
  1050
huffman@31706
  1051
huffman@31706
  1052
subsection {* Bezout's theorem *}
huffman@31706
  1053
huffman@31706
  1054
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1055
   see the theorems that follow the definition. *)
huffman@31706
  1056
fun
huffman@31706
  1057
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1058
where
huffman@31706
  1059
  "bezw x y =
huffman@31706
  1060
  (if y = 0 then (1, 0) else
huffman@31706
  1061
      (snd (bezw y (x mod y)),
huffman@31706
  1062
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1063
huffman@31706
  1064
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1065
huffman@31706
  1066
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1067
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1068
  by simp
huffman@31706
  1069
huffman@31706
  1070
declare bezw.simps [simp del]
huffman@31706
  1071
huffman@31706
  1072
lemma bezw_aux [rule_format]:
huffman@31706
  1073
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1074
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1075
  fix m :: nat
huffman@31706
  1076
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1077
    by auto
huffman@31706
  1078
  next fix m :: nat and n
huffman@31706
  1079
    assume ngt0: "n > 0" and
huffman@31706
  1080
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1081
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1082
        int (gcd n (m mod n))"
huffman@31706
  1083
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1084
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1085
      apply (erule subst)
huffman@31706
  1086
      apply (simp add: ring_simps)
huffman@31706
  1087
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1088
      (* applying simp here undoes the last substitution!
huffman@31706
  1089
         what is procedure cancel_div_mod? *)
huffman@31706
  1090
      apply (simp only: ring_simps zadd_int [symmetric]
huffman@31706
  1091
        zmult_int [symmetric])
huffman@31706
  1092
      done
huffman@31706
  1093
qed
huffman@31706
  1094
nipkow@31952
  1095
lemma bezout_int:
huffman@31706
  1096
  fixes x y
huffman@31706
  1097
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1098
proof -
huffman@31706
  1099
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1100
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1101
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1102
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1103
    apply (unfold gcd_int_def)
huffman@31706
  1104
    apply simp
huffman@31706
  1105
    apply (subst bezw_aux [symmetric])
huffman@31706
  1106
    apply auto
huffman@31706
  1107
    done
huffman@31706
  1108
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1109
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1110
    by auto
huffman@31706
  1111
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1112
    by (erule (1) bezout_aux)
huffman@31706
  1113
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1114
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1115
    apply auto
huffman@31706
  1116
    apply (rule_tac x = u in exI)
huffman@31706
  1117
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1118
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1119
    apply auto
huffman@31706
  1120
    done
huffman@31706
  1121
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1122
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1123
    apply auto
huffman@31706
  1124
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1125
    apply (rule_tac x = v in exI)
nipkow@31952
  1126
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1127
    apply auto
huffman@31706
  1128
    done
huffman@31706
  1129
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1130
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1131
    apply auto
huffman@31706
  1132
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1133
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1134
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1135
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1136
    apply auto
huffman@31706
  1137
    done
huffman@31706
  1138
  ultimately show ?thesis by blast
huffman@31706
  1139
qed
huffman@31706
  1140
huffman@31706
  1141
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1142
huffman@31706
  1143
lemma ind_euclid:
huffman@31706
  1144
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1145
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1146
  shows "P a b"
chaieb@27669
  1147
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
chaieb@27669
  1148
  fix n a b
chaieb@27669
  1149
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
chaieb@27669
  1150
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1151
  moreover {assume eq: "a= b"
huffman@31706
  1152
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1153
    by simp}
chaieb@27669
  1154
  moreover
chaieb@27669
  1155
  {assume lt: "a < b"
chaieb@27669
  1156
    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
chaieb@27669
  1157
    moreover
chaieb@27669
  1158
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1159
    moreover
chaieb@27669
  1160
    {assume ab: "a + b - a < n"
chaieb@27669
  1161
      have th0: "a + b - a = a + (b - a)" using lt by arith
chaieb@27669
  1162
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1163
      have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1164
    ultimately have "P a b" by blast}
chaieb@27669
  1165
  moreover
chaieb@27669
  1166
  {assume lt: "a > b"
chaieb@27669
  1167
    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
chaieb@27669
  1168
    moreover
chaieb@27669
  1169
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1170
    moreover
chaieb@27669
  1171
    {assume ab: "b + a - b < n"
chaieb@27669
  1172
      have th0: "b + a - b = b + (a - b)" using lt by arith
chaieb@27669
  1173
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1174
      have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1175
      hence "P a b" using c by blast }
chaieb@27669
  1176
    ultimately have "P a b" by blast}
chaieb@27669
  1177
ultimately  show "P a b" by blast
chaieb@27669
  1178
qed
chaieb@27669
  1179
nipkow@31952
  1180
lemma bezout_lemma_nat:
huffman@31706
  1181
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1182
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1183
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1184
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1185
  using ex
huffman@31706
  1186
  apply clarsimp
huffman@31706
  1187
  apply (rule_tac x="d" in exI, simp add: dvd_add)
huffman@31706
  1188
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1189
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1190
  apply (rule_tac x="y" in exI)
huffman@31706
  1191
  apply algebra
huffman@31706
  1192
  apply (rule_tac x="x" in exI)
huffman@31706
  1193
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1194
  apply algebra
chaieb@27669
  1195
done
chaieb@27669
  1196
nipkow@31952
  1197
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1198
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1199
  apply(induct a b rule: ind_euclid)
huffman@31706
  1200
  apply blast
huffman@31706
  1201
  apply clarify
huffman@31706
  1202
  apply (rule_tac x="a" in exI, simp add: dvd_add)
huffman@31706
  1203
  apply clarsimp
huffman@31706
  1204
  apply (rule_tac x="d" in exI)
huffman@31706
  1205
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
huffman@31706
  1206
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1207
  apply (rule_tac x="y" in exI)
huffman@31706
  1208
  apply algebra
huffman@31706
  1209
  apply (rule_tac x="x" in exI)
huffman@31706
  1210
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1211
  apply algebra
chaieb@27669
  1212
done
chaieb@27669
  1213
nipkow@31952
  1214
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1215
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1216
  using bezout_add_nat[of a b]
huffman@31706
  1217
  apply clarsimp
huffman@31706
  1218
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1219
  apply (rule_tac x="x" in exI)
huffman@31706
  1220
  apply (rule_tac x="y" in exI)
huffman@31706
  1221
  apply auto
chaieb@27669
  1222
done
chaieb@27669
  1223
nipkow@31952
  1224
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1225
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1226
proof-
huffman@31706
  1227
 from nz have ap: "a > 0" by simp
nipkow@31952
  1228
 from bezout_add_nat[of a b]
huffman@31706
  1229
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1230
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1231
 moreover
huffman@31706
  1232
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1233
     from H have ?thesis by blast }
chaieb@27669
  1234
 moreover
chaieb@27669
  1235
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1236
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1237
   moreover
chaieb@27669
  1238
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1239
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1240
       by auto
chaieb@27669
  1241
     moreover
chaieb@27669
  1242
     {assume db: "d=b"
chaieb@27669
  1243
       from prems have ?thesis apply simp
chaieb@27669
  1244
	 apply (rule exI[where x = b], simp)
chaieb@27669
  1245
	 apply (rule exI[where x = b])
chaieb@27669
  1246
	by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1247
    moreover
huffman@31706
  1248
    {assume db: "d < b"
chaieb@27669
  1249
	{assume "x=0" hence ?thesis  using prems by simp }
chaieb@27669
  1250
	moreover
chaieb@27669
  1251
	{assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
chaieb@27669
  1252
	  from db have "d \<le> b - 1" by simp
chaieb@27669
  1253
	  hence "d*b \<le> b*(b - 1)" by simp
chaieb@27669
  1254
	  with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
chaieb@27669
  1255
	  have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
huffman@31706
  1256
	  from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1257
            by simp
huffman@31706
  1258
	  hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
huffman@31706
  1259
	    by (simp only: mult_assoc right_distrib)
huffman@31706
  1260
	  hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1261
            by algebra
chaieb@27669
  1262
	  hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
huffman@31706
  1263
	  hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
chaieb@27669
  1264
	    by (simp only: diff_add_assoc[OF dble, of d, symmetric])
chaieb@27669
  1265
	  hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
chaieb@27669
  1266
	    by (simp only: diff_mult_distrib2 add_commute mult_ac)
chaieb@27669
  1267
	  hence ?thesis using H(1,2)
chaieb@27669
  1268
	    apply -
chaieb@27669
  1269
	    apply (rule exI[where x=d], simp)
chaieb@27669
  1270
	    apply (rule exI[where x="(b - 1) * y"])
chaieb@27669
  1271
	    by (rule exI[where x="x*(b - 1) - d"], simp)}
chaieb@27669
  1272
	ultimately have ?thesis by blast}
chaieb@27669
  1273
    ultimately have ?thesis by blast}
chaieb@27669
  1274
  ultimately have ?thesis by blast}
chaieb@27669
  1275
 ultimately show ?thesis by blast
chaieb@27669
  1276
qed
chaieb@27669
  1277
nipkow@31952
  1278
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1279
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1280
proof-
chaieb@27669
  1281
  let ?g = "gcd a b"
nipkow@31952
  1282
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1283
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1284
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1285
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1286
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1287
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1288
  thus ?thesis by blast
chaieb@27669
  1289
qed
chaieb@27669
  1290
huffman@31706
  1291
huffman@31706
  1292
subsection {* LCM *}
huffman@31706
  1293
nipkow@31952
  1294
lemma lcm_altdef_int: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1295
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@31706
  1296
    zmult_int [symmetric] gcd_int_def)
huffman@31706
  1297
nipkow@31952
  1298
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1299
  unfolding lcm_nat_def
nipkow@31952
  1300
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1301
nipkow@31952
  1302
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1303
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1304
  apply (subst int_mult [symmetric])
nipkow@31952
  1305
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1306
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1307
  apply (simp, simp add: abs_mult)
huffman@31706
  1308
done
huffman@31706
  1309
nipkow@31952
  1310
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1311
  unfolding lcm_nat_def by simp
huffman@31706
  1312
nipkow@31952
  1313
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1314
  unfolding lcm_int_def by simp
huffman@31706
  1315
nipkow@31952
  1316
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1317
  unfolding lcm_nat_def by simp
chaieb@27669
  1318
nipkow@31952
  1319
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1320
  unfolding lcm_int_def by simp
huffman@31706
  1321
nipkow@31952
  1322
lemma lcm_commute_nat: "lcm (m::nat) n = lcm n m"
nipkow@31952
  1323
  unfolding lcm_nat_def by (simp add: gcd_commute_nat ring_simps)
huffman@31706
  1324
nipkow@31952
  1325
lemma lcm_commute_int: "lcm (m::int) n = lcm n m"
nipkow@31952
  1326
  unfolding lcm_int_def by (subst lcm_commute_nat, rule refl)
huffman@31706
  1327
huffman@31706
  1328
nipkow@31952
  1329
lemma lcm_pos_nat:
nipkow@31798
  1330
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1331
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1332
nipkow@31952
  1333
lemma lcm_pos_int:
nipkow@31798
  1334
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1335
  apply (subst lcm_abs_int)
nipkow@31952
  1336
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1337
  apply auto
huffman@31706
  1338
done
haftmann@23687
  1339
nipkow@31952
  1340
lemma dvd_pos_nat:
haftmann@23687
  1341
  fixes n m :: nat
haftmann@23687
  1342
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1343
  shows "m > 0"
haftmann@23687
  1344
using assms by (cases m) auto
haftmann@23687
  1345
nipkow@31952
  1346
lemma lcm_least_nat:
huffman@31706
  1347
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1348
  shows "lcm m n dvd k"
haftmann@23687
  1349
proof (cases k)
haftmann@23687
  1350
  case 0 then show ?thesis by auto
haftmann@23687
  1351
next
haftmann@23687
  1352
  case (Suc _) then have pos_k: "k > 0" by auto
nipkow@31952
  1353
  from assms dvd_pos_nat [OF this] have pos_mn: "m > 0" "n > 0" by auto
nipkow@31952
  1354
  with gcd_zero_nat [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1355
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1356
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1357
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1358
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1359
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
nipkow@31952
  1360
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1361
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1362
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1363
  also have "\<dots> = k * p * q * gcd m n"
nipkow@31952
  1364
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1365
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1366
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1367
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1368
    by (simp add: mult_ac)
haftmann@27556
  1369
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1370
    by simp
nipkow@31952
  1371
  with prod_gcd_lcm_nat [of m n]
haftmann@27556
  1372
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1373
    by (simp add: mult_ac)
huffman@31706
  1374
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1375
  then show ?thesis using dvd_def by auto
haftmann@23687
  1376
qed
haftmann@23687
  1377
nipkow@31952
  1378
lemma lcm_least_int:
nipkow@31798
  1379
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
nipkow@31952
  1380
apply (subst lcm_abs_int)
nipkow@31798
  1381
apply (rule dvd_trans)
nipkow@31952
  1382
apply (rule lcm_least_nat [transferred, of _ "abs k" _])
nipkow@31798
  1383
apply auto
huffman@31706
  1384
done
huffman@31706
  1385
nipkow@31952
  1386
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
haftmann@23687
  1387
proof (cases m)
haftmann@23687
  1388
  case 0 then show ?thesis by simp
haftmann@23687
  1389
next
haftmann@23687
  1390
  case (Suc _)
haftmann@23687
  1391
  then have mpos: "m > 0" by simp
haftmann@23687
  1392
  show ?thesis
haftmann@23687
  1393
  proof (cases n)
haftmann@23687
  1394
    case 0 then show ?thesis by simp
haftmann@23687
  1395
  next
haftmann@23687
  1396
    case (Suc _)
haftmann@23687
  1397
    then have npos: "n > 0" by simp
haftmann@27556
  1398
    have "gcd m n dvd n" by simp
haftmann@27556
  1399
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1400
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1401
      by (simp add: mult_ac)
nipkow@31952
  1402
    also have "\<dots> = m * k" using mpos npos gcd_zero_nat by simp
huffman@31706
  1403
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1404
  qed
haftmann@23687
  1405
qed
haftmann@23687
  1406
nipkow@31952
  1407
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
nipkow@31952
  1408
  apply (subst lcm_abs_int)
huffman@31706
  1409
  apply (rule dvd_trans)
huffman@31706
  1410
  prefer 2
nipkow@31952
  1411
  apply (rule lcm_dvd1_nat [transferred])
huffman@31706
  1412
  apply auto
huffman@31706
  1413
done
huffman@31706
  1414
nipkow@31952
  1415
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
nipkow@31952
  1416
  by (subst lcm_commute_nat, rule lcm_dvd1_nat)
huffman@31706
  1417
nipkow@31952
  1418
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
nipkow@31952
  1419
  by (subst lcm_commute_int, rule lcm_dvd1_int)
huffman@31706
  1420
nipkow@31730
  1421
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1422
by(metis lcm_dvd1_nat dvd_trans)
nipkow@31729
  1423
nipkow@31730
  1424
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1425
by(metis lcm_dvd2_nat dvd_trans)
nipkow@31729
  1426
nipkow@31730
  1427
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1428
by(metis lcm_dvd1_int dvd_trans)
nipkow@31729
  1429
nipkow@31730
  1430
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1431
by(metis lcm_dvd2_int dvd_trans)
nipkow@31729
  1432
nipkow@31952
  1433
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1434
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31952
  1435
  by (auto intro: dvd_anti_sym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
chaieb@27568
  1436
nipkow@31952
  1437
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1438
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31952
  1439
  by (auto intro: dvd_anti_sym [transferred] lcm_least_int)
huffman@31706
  1440
nipkow@31798
  1441
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1442
  apply (rule sym)
nipkow@31952
  1443
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1444
  apply auto
huffman@31706
  1445
done
huffman@31706
  1446
nipkow@31798
  1447
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1448
  apply (rule sym)
nipkow@31952
  1449
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1450
  apply auto
huffman@31706
  1451
done
huffman@31706
  1452
nipkow@31798
  1453
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31952
  1454
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1455
nipkow@31798
  1456
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31952
  1457
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
huffman@31706
  1458
nipkow@31992
  1459
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1460
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1461
nipkow@31992
  1462
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1463
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1464
nipkow@31992
  1465
lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) \<longleftrightarrow> n dvd m"
nipkow@31992
  1466
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1467
nipkow@31992
  1468
lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) \<longleftrightarrow> m dvd n"
nipkow@31992
  1469
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1470
nipkow@31766
  1471
lemma lcm_assoc_nat: "lcm (lcm n m) (p::nat) = lcm n (lcm m p)"
nipkow@31992
  1472
by(rule lcm_unique_nat[THEN iffD1])(metis dvd.order_trans lcm_unique_nat)
nipkow@31766
  1473
nipkow@31766
  1474
lemma lcm_assoc_int: "lcm (lcm n m) (p::int) = lcm n (lcm m p)"
nipkow@31992
  1475
by(rule lcm_unique_int[THEN iffD1])(metis dvd_trans lcm_unique_int)
nipkow@31766
  1476
nipkow@31992
  1477
lemmas lcm_left_commute_nat = mk_left_commute[of lcm, OF lcm_assoc_nat lcm_commute_nat]
nipkow@31992
  1478
lemmas lcm_left_commute_int = mk_left_commute[of lcm, OF lcm_assoc_int lcm_commute_int]
nipkow@31766
  1479
nipkow@31952
  1480
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
nipkow@31952
  1481
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
nipkow@31766
  1482
nipkow@31992
  1483
lemma fun_left_comm_idem_gcd_nat: "fun_left_comm_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1484
proof qed (auto simp add: gcd_ac_nat)
nipkow@31992
  1485
nipkow@31992
  1486
lemma fun_left_comm_idem_gcd_int: "fun_left_comm_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1487
proof qed (auto simp add: gcd_ac_int)
nipkow@31992
  1488
nipkow@31992
  1489
lemma fun_left_comm_idem_lcm_nat: "fun_left_comm_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1490
proof qed (auto simp add: lcm_ac_nat)
nipkow@31992
  1491
nipkow@31992
  1492
lemma fun_left_comm_idem_lcm_int: "fun_left_comm_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1493
proof qed (auto simp add: lcm_ac_int)
nipkow@31992
  1494
haftmann@23687
  1495
nipkow@31995
  1496
(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)
nipkow@31995
  1497
nipkow@31995
  1498
lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1499
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
nipkow@31995
  1500
nipkow@31995
  1501
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1502
by (metis lcm_0_int lcm_0_left_int lcm_pos_int zless_le)
nipkow@31995
  1503
nipkow@31995
  1504
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
nipkow@31995
  1505
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
nipkow@31995
  1506
nipkow@31995
  1507
lemma lcm_1_iff_int[simp]: "lcm (m::int) n = 1 \<longleftrightarrow> (m=1 \<or> m = -1) \<and> (n=1 \<or> n = -1)"
berghofe@31996
  1508
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
nipkow@31995
  1509
nipkow@32112
  1510
subsubsection {* The complete divisibility lattice *}
nipkow@32112
  1511
nipkow@32112
  1512
nipkow@32112
  1513
interpretation gcd_semilattice_nat: lower_semilattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd
nipkow@32112
  1514
proof
nipkow@32112
  1515
  case goal3 thus ?case by(metis gcd_unique_nat)
nipkow@32112
  1516
qed auto
nipkow@32112
  1517
nipkow@32112
  1518
interpretation lcm_semilattice_nat: upper_semilattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm
nipkow@32112
  1519
proof
nipkow@32112
  1520
  case goal3 thus ?case by(metis lcm_unique_nat)
nipkow@32112
  1521
qed auto
nipkow@32112
  1522
nipkow@32112
  1523
interpretation gcd_lcm_lattice_nat: lattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd lcm ..
nipkow@32112
  1524
nipkow@32112
  1525
text{* Lifting gcd and lcm to finite (Gcd/Lcm) and infinite sets (GCD/LCM).
nipkow@32112
  1526
GCD is defined via LCM to facilitate the proof that we have a complete lattice.
nipkow@32112
  1527
Later on we show that GCD and Gcd coincide on finite sets.
nipkow@32112
  1528
*}
nipkow@32112
  1529
context gcd
nipkow@32112
  1530
begin
nipkow@32112
  1531
nipkow@32112
  1532
definition Gcd :: "'a set \<Rightarrow> 'a"
nipkow@32112
  1533
where "Gcd = fold gcd 0"
nipkow@32112
  1534
nipkow@32112
  1535
definition Lcm :: "'a set \<Rightarrow> 'a"
nipkow@32112
  1536
where "Lcm = fold lcm 1"
nipkow@32112
  1537
nipkow@32112
  1538
definition LCM :: "'a set \<Rightarrow> 'a" where
nipkow@32112
  1539
"LCM M = (if finite M then Lcm M else 0)"
nipkow@32112
  1540
nipkow@32112
  1541
definition GCD :: "'a set \<Rightarrow> 'a" where
nipkow@32112
  1542
"GCD M = LCM(INT m:M. {d. d dvd m})"
nipkow@32112
  1543
nipkow@32112
  1544
end
nipkow@32112
  1545
nipkow@32112
  1546
lemma Gcd_empty[simp]: "Gcd {} = 0"
nipkow@32112
  1547
by(simp add:Gcd_def)
nipkow@32112
  1548
nipkow@32112
  1549
lemma Lcm_empty[simp]: "Lcm {} = 1"
nipkow@32112
  1550
by(simp add:Lcm_def)
nipkow@32112
  1551
nipkow@32112
  1552
lemma GCD_empty_nat[simp]: "GCD {} = (0::nat)"
nipkow@32112
  1553
by(simp add:GCD_def LCM_def)
nipkow@32112
  1554
nipkow@32112
  1555
lemma LCM_eq_Lcm[simp]: "finite M \<Longrightarrow> LCM M = Lcm M"
nipkow@32112
  1556
by(simp add:LCM_def)
nipkow@32112
  1557
nipkow@32112
  1558
lemma Lcm_insert_nat [simp]:
nipkow@32112
  1559
  assumes "finite N"
nipkow@32112
  1560
  shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
nipkow@32112
  1561
proof -
nipkow@32112
  1562
  interpret fun_left_comm_idem "lcm::nat\<Rightarrow>nat\<Rightarrow>nat"
nipkow@32112
  1563
    by (rule fun_left_comm_idem_lcm_nat)
nipkow@32112
  1564
  from assms show ?thesis by(simp add: Lcm_def)
nipkow@32112
  1565
qed
nipkow@32112
  1566
nipkow@32112
  1567
lemma Lcm_insert_int [simp]:
nipkow@32112
  1568
  assumes "finite N"
nipkow@32112
  1569
  shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"
nipkow@32112
  1570
proof -
nipkow@32112
  1571
  interpret fun_left_comm_idem "lcm::int\<Rightarrow>int\<Rightarrow>int"
nipkow@32112
  1572
    by (rule fun_left_comm_idem_lcm_int)
nipkow@32112
  1573
  from assms show ?thesis by(simp add: Lcm_def)
nipkow@32112
  1574
qed
nipkow@32112
  1575
nipkow@32112
  1576
lemma Gcd_insert_nat [simp]:
nipkow@32112
  1577
  assumes "finite N"
nipkow@32112
  1578
  shows "Gcd (insert (n::nat) N) = gcd n (Gcd N)"
nipkow@32112
  1579
proof -
nipkow@32112
  1580
  interpret fun_left_comm_idem "gcd::nat\<Rightarrow>nat\<Rightarrow>nat"
nipkow@32112
  1581
    by (rule fun_left_comm_idem_gcd_nat)
nipkow@32112
  1582
  from assms show ?thesis by(simp add: Gcd_def)
nipkow@32112
  1583
qed
nipkow@32112
  1584
nipkow@32112
  1585
lemma Gcd_insert_int [simp]:
nipkow@32112
  1586
  assumes "finite N"
nipkow@32112
  1587
  shows "Gcd (insert (n::int) N) = gcd n (Gcd N)"
nipkow@32112
  1588
proof -
nipkow@32112
  1589
  interpret fun_left_comm_idem "gcd::int\<Rightarrow>int\<Rightarrow>int"
nipkow@32112
  1590
    by (rule fun_left_comm_idem_gcd_int)
nipkow@32112
  1591
  from assms show ?thesis by(simp add: Gcd_def)
nipkow@32112
  1592
qed
nipkow@32112
  1593
nipkow@32112
  1594
lemma Lcm0_iff[simp]: "finite (M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> Lcm M = 0 \<longleftrightarrow> 0 : M"
nipkow@32112
  1595
by(induct rule:finite_ne_induct) auto
nipkow@32112
  1596
nipkow@32112
  1597
lemma Lcm_eq_0[simp]: "finite (M::nat set) \<Longrightarrow> 0 : M \<Longrightarrow> Lcm M = 0"
nipkow@32112
  1598
by (metis Lcm0_iff empty_iff)
nipkow@32112
  1599
nipkow@32112
  1600
lemma Gcd_dvd_nat [simp]:
nipkow@32112
  1601
  assumes "finite M" and "(m::nat) \<in> M"
nipkow@32112
  1602
  shows "Gcd M dvd m"
nipkow@32112
  1603
proof -
nipkow@32112
  1604
  show ?thesis using gcd_semilattice_nat.fold_inf_le_inf[OF assms, of 0] by (simp add: Gcd_def)
nipkow@32112
  1605
qed
nipkow@32112
  1606
nipkow@32112
  1607
lemma dvd_Gcd_nat[simp]:
nipkow@32112
  1608
  assumes "finite M" and "ALL (m::nat) : M. n dvd m"
nipkow@32112
  1609
  shows "n dvd Gcd M"
nipkow@32112
  1610
proof -
nipkow@32112
  1611
  show ?thesis using gcd_semilattice_nat.inf_le_fold_inf[OF assms, of 0] by (simp add: Gcd_def)
nipkow@32112
  1612
qed
nipkow@32112
  1613
nipkow@32112
  1614
lemma dvd_Lcm_nat [simp]:
nipkow@32112
  1615
  assumes "finite M" and "(m::nat) \<in> M"
nipkow@32112
  1616
  shows "m dvd Lcm M"
nipkow@32112
  1617
proof -
nipkow@32112
  1618
  show ?thesis using lcm_semilattice_nat.sup_le_fold_sup[OF assms, of 1] by (simp add: Lcm_def)
nipkow@32112
  1619
qed
nipkow@32112
  1620
nipkow@32112
  1621
lemma Lcm_dvd_nat[simp]:
nipkow@32112
  1622
  assumes "finite M" and "ALL (m::nat) : M. m dvd n"
nipkow@32112
  1623
  shows "Lcm M dvd n"
nipkow@32112
  1624
proof -
nipkow@32112
  1625
  show ?thesis using lcm_semilattice_nat.fold_sup_le_sup[OF assms, of 1] by (simp add: Lcm_def)
nipkow@32112
  1626
qed
nipkow@32112
  1627
nipkow@32112
  1628
interpretation gcd_lcm_complete_lattice_nat:
nipkow@32112
  1629
  complete_lattice "op dvd" "%m n::nat. m dvd n & ~ n dvd m" gcd lcm 1 0 GCD LCM
nipkow@32112
  1630
proof
nipkow@32112
  1631
  case goal1 show ?case by simp
nipkow@32112
  1632
next
nipkow@32112
  1633
  case goal2 show ?case by simp
nipkow@32112
  1634
next
nipkow@32112
  1635
  case goal5 thus ?case by (auto simp: LCM_def)
nipkow@32112
  1636
next
nipkow@32112
  1637
  case goal6 thus ?case
nipkow@32112
  1638
    by(auto simp: LCM_def)(metis finite_nat_set_iff_bounded_le gcd_proj2_if_dvd_nat gcd_le1_nat)
nipkow@32112
  1639
next
nipkow@32112
  1640
  case goal3 thus ?case by (auto simp: GCD_def LCM_def)(metis finite_INT finite_divisors_nat)
nipkow@32112
  1641
next
nipkow@32112
  1642
  case goal4 thus ?case by(auto simp: LCM_def GCD_def)
nipkow@32112
  1643
qed
nipkow@32112
  1644
nipkow@32112
  1645
text{* Alternative characterizations of Gcd and GCD: *}
nipkow@32112
  1646
nipkow@32112
  1647
lemma Gcd_eq_Max: "finite(M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> Gcd M = Max(\<Inter>m\<in>M. {d. d dvd m})"
nipkow@32112
  1648
apply(rule antisym)
nipkow@32112
  1649
 apply(rule Max_ge)
nipkow@32112
  1650
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@32112
  1651
 apply simp
nipkow@32112
  1652
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  1653
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@32112
  1654
 apply fastsimp
nipkow@32112
  1655
apply clarsimp
nipkow@32112
  1656
apply (metis Gcd_dvd_nat Max_in dvd_0_left dvd_Gcd_nat dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  1657
done
nipkow@32112
  1658
nipkow@32112
  1659
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  1660
apply(induct pred:finite)
nipkow@32112
  1661
 apply simp
nipkow@32112
  1662
apply(case_tac "x=0")
nipkow@32112
  1663
 apply simp
nipkow@32112
  1664
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  1665
 apply simp
nipkow@32112
  1666
apply blast
nipkow@32112
  1667
done
nipkow@32112
  1668
nipkow@32112
  1669
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  1670
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> Lcm M : M"
nipkow@32112
  1671
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  1672
 apply simp
nipkow@32112
  1673
apply simp
nipkow@32112
  1674
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  1675
 apply simp
nipkow@32112
  1676
 apply(case_tac "A={}")
nipkow@32112
  1677
  apply simp
nipkow@32112
  1678
 apply simp
nipkow@32112
  1679
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  1680
done
nipkow@32112
  1681
nipkow@32112
  1682
lemma Lcm_eq_Max_nat:
nipkow@32112
  1683
  "finite M \<Longrightarrow> M \<noteq> {} \<Longrightarrow> 0 \<notin> M \<Longrightarrow> ALL m n :: nat. m:M \<longrightarrow> n:M \<longrightarrow> lcm m n : M \<Longrightarrow> Lcm M = Max M"
nipkow@32112
  1684
apply(rule antisym)
nipkow@32112
  1685
 apply(rule Max_ge, assumption)
nipkow@32112
  1686
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
nipkow@32112
  1687
apply clarsimp
nipkow@32112
  1688
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
nipkow@32112
  1689
done
nipkow@32112
  1690
nipkow@32112
  1691
text{* Finally GCD is Gcd: *}
nipkow@32112
  1692
nipkow@32112
  1693
lemma GCD_eq_Gcd[simp]: assumes "finite(M::nat set)" shows "GCD M = Gcd M"
nipkow@32112
  1694
proof-
nipkow@32112
  1695
  have divisors_remove0_nat: "(\<Inter>m\<in>M. {d::nat. d dvd m}) = (\<Inter>m\<in>M-{0}. {d::nat. d dvd m})" by auto
nipkow@32112
  1696
  show ?thesis
nipkow@32112
  1697
  proof cases
nipkow@32112
  1698
    assume "M={}" thus ?thesis by simp
nipkow@32112
  1699
  next
nipkow@32112
  1700
    assume "M\<noteq>{}"
nipkow@32112
  1701
    show ?thesis
nipkow@32112
  1702
    proof cases
nipkow@32112
  1703
      assume "M={0}" thus ?thesis by(simp add:GCD_def LCM_def)
nipkow@32112
  1704
    next
nipkow@32112
  1705
      assume "M\<noteq>{0}"
nipkow@32112
  1706
      with `M\<noteq>{}` assms show ?thesis
nipkow@32112
  1707
	apply(subst Gcd_remove0_nat[OF assms])
nipkow@32112
  1708
	apply(simp add:GCD_def)
nipkow@32112
  1709
	apply(subst divisors_remove0_nat)
nipkow@32112
  1710
	apply(simp add:LCM_def)
nipkow@32112
  1711
	apply rule
nipkow@32112
  1712
	 apply rule
nipkow@32112
  1713
	 apply(subst Gcd_eq_Max)
nipkow@32112
  1714
	    apply simp
nipkow@32112
  1715
	   apply blast
nipkow@32112
  1716
	  apply blast
nipkow@32112
  1717
	 apply(rule Lcm_eq_Max_nat)
nipkow@32112
  1718
	    apply simp
nipkow@32112
  1719
	   apply blast
nipkow@32112
  1720
	  apply fastsimp
nipkow@32112
  1721
	 apply clarsimp
nipkow@32112
  1722
	apply(fastsimp intro: finite_divisors_nat intro!: finite_INT)
nipkow@32112
  1723
	done
nipkow@32112
  1724
    qed
nipkow@32112
  1725
  qed
nipkow@32112
  1726
qed
nipkow@32112
  1727
nipkow@32112
  1728
lemma Lcm_set_nat [code_unfold]:
nipkow@32112
  1729
  "Lcm (set ns) = foldl lcm (1::nat) ns"
nipkow@32112
  1730
proof -
nipkow@32112
  1731
  interpret fun_left_comm_idem "lcm::nat\<Rightarrow>nat\<Rightarrow>nat" by (rule fun_left_comm_idem_lcm_nat)
nipkow@32112
  1732
  show ?thesis by(simp add: Lcm_def fold_set lcm_commute_nat)
nipkow@32112
  1733
qed
nipkow@32112
  1734
nipkow@32112
  1735
lemma Lcm_set_int [code_unfold]:
nipkow@32112
  1736
  "Lcm (set is) = foldl lcm (1::int) is"
nipkow@32112
  1737
proof -
nipkow@32112
  1738
  interpret fun_left_comm_idem "lcm::int\<Rightarrow>int\<Rightarrow>int" by (rule fun_left_comm_idem_lcm_int)
nipkow@32112
  1739
  show ?thesis by(simp add: Lcm_def fold_set lcm_commute_int)
nipkow@32112
  1740
qed
nipkow@32112
  1741
nipkow@32112
  1742
lemma Gcd_set_nat [code_unfold]:
nipkow@32112
  1743
  "Gcd (set ns) = foldl gcd (0::nat) ns"
nipkow@32112
  1744
proof -
nipkow@32112
  1745
  interpret fun_left_comm_idem "gcd::nat\<Rightarrow>nat\<Rightarrow>nat" by (rule fun_left_comm_idem_gcd_nat)
nipkow@32112
  1746
  show ?thesis by(simp add: Gcd_def fold_set gcd_commute_nat)
nipkow@32112
  1747
qed
nipkow@32112
  1748
nipkow@32112
  1749
lemma Gcd_set_int [code_unfold]:
nipkow@32112
  1750
  "Gcd (set ns) = foldl gcd (0::int) ns"
nipkow@32112
  1751
proof -
nipkow@32112
  1752
  interpret fun_left_comm_idem "gcd::int\<Rightarrow>int\<Rightarrow>int" by (rule fun_left_comm_idem_gcd_int)
nipkow@32112
  1753
  show ?thesis by(simp add: Gcd_def fold_set gcd_commute_int)
nipkow@32112
  1754
qed
nipkow@32112
  1755
nipkow@31995
  1756
huffman@31706
  1757
subsection {* Primes *}
wenzelm@22367
  1758
nipkow@31992
  1759
(* FIXME Is there a better way to handle these, rather than making them elim rules? *)
chaieb@22027
  1760
nipkow@31952
  1761
lemma prime_ge_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 0"
huffman@31706
  1762
  by (unfold prime_nat_def, auto)
chaieb@22027
  1763
nipkow@31952
  1764
lemma prime_gt_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p > 0"
huffman@31706
  1765
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1766
nipkow@31952
  1767
lemma prime_ge_1_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 1"
huffman@31706
  1768
  by (unfold prime_nat_def, auto)
chaieb@22027
  1769
nipkow@31952
  1770
lemma prime_gt_1_nat [elim]: "prime (p::nat) \<Longrightarrow> p > 1"
huffman@31706
  1771
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1772
nipkow@31952
  1773
lemma prime_ge_Suc_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= Suc 0"
huffman@31706
  1774
  by (unfold prime_nat_def, auto)
wenzelm@22367
  1775
nipkow@31952
  1776
lemma prime_gt_Suc_0_nat [elim]: "prime (p::nat) \<Longrightarrow> p > Suc 0"
huffman@31706
  1777
  by (unfold prime_nat_def, auto)
huffman@31706
  1778
nipkow@31952
  1779
lemma prime_ge_2_nat [elim]: "prime (p::nat) \<Longrightarrow> p >= 2"
huffman@31706
  1780
  by (unfold prime_nat_def, auto)
huffman@31706
  1781
nipkow@31952
  1782
lemma prime_ge_0_int [elim]: "prime (p::int) \<Longrightarrow> p >= 0"
nipkow@31992
  1783
  by (unfold prime_int_def prime_nat_def) auto
wenzelm@22367
  1784
nipkow@31952
  1785
lemma prime_gt_0_int [elim]: "prime (p::int) \<Longrightarrow> p > 0"
huffman@31706
  1786
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1787
nipkow@31952
  1788
lemma prime_ge_1_int [elim]: "prime (p::int) \<Longrightarrow> p >= 1"
huffman@31706
  1789
  by (unfold prime_int_def prime_nat_def, auto)
chaieb@22027
  1790
nipkow@31952
  1791
lemma prime_gt_1_int [elim]: "prime (p::int) \<Longrightarrow> p > 1"
huffman@31706
  1792
  by (unfold prime_int_def prime_nat_def, auto)
huffman@31706
  1793
nipkow@31952
  1794
lemma prime_ge_2_int [elim]: "prime (p::int) \<Longrightarrow> p >= 2"
huffman@31706
  1795
  by (unfold prime_int_def prime_nat_def, auto)
wenzelm@22367
  1796
huffman@31706
  1797
huffman@31706
  1798
lemma prime_int_altdef: "prime (p::int) = (1 < p \<and> (\<forall>m \<ge> 0. m dvd p \<longrightarrow>
huffman@31706
  1799
    m = 1 \<or> m = p))"
huffman@31706
  1800
  using prime_nat_def [transferred]
huffman@31706
  1801
    apply (case_tac "p >= 0")
nipkow@31952
  1802
    by (blast, auto simp add: prime_ge_0_int)
huffman@31706
  1803
nipkow@31952
  1804
lemma prime_imp_coprime_nat: "prime (p::nat) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1805
  apply (unfold prime_nat_def)
nipkow@31952
  1806
  apply (metis gcd_dvd1_nat gcd_dvd2_nat)
huffman@31706
  1807
  done
huffman@31706
  1808
nipkow@31952
  1809
lemma prime_imp_coprime_int: "prime (p::int) \<Longrightarrow> \<not> p dvd n \<Longrightarrow> coprime p n"
huffman@31706
  1810
  apply (unfold prime_int_altdef)
nipkow@31952
  1811
  apply (metis gcd_dvd1_int gcd_dvd2_int gcd_ge_0_int)
chaieb@27568
  1812
  done
chaieb@27568
  1813
nipkow@31952
  1814
lemma prime_dvd_mult_nat: "prime (p::nat) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
nipkow@31952
  1815
  by (blast intro: coprime_dvd_mult_nat prime_imp_coprime_nat)
huffman@31706
  1816
nipkow@31952
  1817
lemma prime_dvd_mult_int: "prime (p::int) \<Longrightarrow> p dvd m * n \<Longrightarrow> p dvd m \<or> p dvd n"
nipkow@31952
  1818
  by (blast intro: coprime_dvd_mult_int prime_imp_coprime_int)
huffman@31706
  1819
nipkow@31952
  1820
lemma prime_dvd_mult_eq_nat [simp]: "prime (p::nat) \<Longrightarrow>
huffman@31706
  1821
    p dvd m * n = (p dvd m \<or> p dvd n)"
nipkow@31952
  1822
  by (rule iffI, rule prime_dvd_mult_nat, auto)
chaieb@27568
  1823
nipkow@31952
  1824
lemma prime_dvd_mult_eq_int [simp]: "prime (p::int) \<Longrightarrow>
huffman@31706
  1825
    p dvd m * n = (p dvd m \<or> p dvd n)"
nipkow@31952
  1826
  by (rule iffI, rule prime_dvd_mult_int, auto)
chaieb@27568
  1827
nipkow@31952
  1828
lemma not_prime_eq_prod_nat: "(n::nat) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1829
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1830
  unfolding prime_nat_def dvd_def apply auto
nipkow@31992
  1831
  by(metis mult_commute linorder_neq_iff linorder_not_le mult_1 n_less_n_mult_m one_le_mult_iff less_imp_le_nat)
chaieb@27568
  1832
nipkow@31952
  1833
lemma not_prime_eq_prod_int: "(n::int) > 1 \<Longrightarrow> ~ prime n \<Longrightarrow>
huffman@31706
  1834
    EX m k. n = m * k & 1 < m & m < n & 1 < k & k < n"
huffman@31706
  1835
  unfolding prime_int_altdef dvd_def
huffman@31706
  1836
  apply auto
nipkow@31992
  1837
  by(metis div_mult_self1_is_id div_mult_self2_is_id int_div_less_self int_one_le_iff_zero_less zero_less_mult_pos zless_le)
chaieb@27568
  1838
nipkow@31952
  1839
lemma prime_dvd_power_nat [rule_format]: "prime (p::nat) -->
huffman@31706
  1840
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1841
  by (induct n rule: nat_induct, auto)
chaieb@27568
  1842
nipkow@31952
  1843
lemma prime_dvd_power_int [rule_format]: "prime (p::int) -->
huffman@31706
  1844
    n > 0 --> (p dvd x^n --> p dvd x)"
huffman@31706
  1845
  apply (induct n rule: nat_induct, auto)
nipkow@31952
  1846
  apply (frule prime_ge_0_int)
huffman@31706
  1847
  apply auto
huffman@31706
  1848
done
huffman@31706
  1849
nipkow@32007
  1850
subsubsection{* Make prime naively executable *}
nipkow@32007
  1851
nipkow@32007
  1852
lemma zero_not_prime_nat [simp]: "~prime (0::nat)"
nipkow@32007
  1853
  by (simp add: prime_nat_def)
nipkow@32007
  1854
nipkow@32007
  1855
lemma zero_not_prime_int [simp]: "~prime (0::int)"
nipkow@32007
  1856
  by (simp add: prime_int_def)
nipkow@32007
  1857
nipkow@32007
  1858
lemma one_not_prime_nat [simp]: "~prime (1::nat)"
nipkow@32007
  1859
  by (simp add: prime_nat_def)
nipkow@32007
  1860
nipkow@32007
  1861
lemma Suc_0_not_prime_nat [simp]: "~prime (Suc 0)"
nipkow@32007
  1862
  by (simp add: prime_nat_def One_nat_def)
nipkow@32007
  1863
nipkow@32007
  1864
lemma one_not_prime_int [simp]: "~prime (1::int)"
nipkow@32007
  1865
  by (simp add: prime_int_def)
nipkow@32007
  1866
nipkow@32007
  1867
lemma prime_nat_code[code]:
nipkow@32007
  1868
 "prime(p::nat) = (p > 1 & (ALL n : {1<..<p}. ~(n dvd p)))"
nipkow@32007
  1869
apply(simp add: Ball_def)
nipkow@32007
  1870
apply (metis less_not_refl prime_nat_def dvd_triv_right not_prime_eq_prod_nat)
nipkow@32007
  1871
done
nipkow@32007
  1872
nipkow@32007
  1873
lemma prime_nat_simp:
nipkow@32007
  1874
 "prime(p::nat) = (p > 1 & (list_all (%n. ~ n dvd p) [2..<p]))"
nipkow@32007
  1875
apply(simp only:prime_nat_code list_ball_code greaterThanLessThan_upt)
nipkow@32007
  1876
apply(simp add:nat_number One_nat_def)
nipkow@32007
  1877
done
nipkow@32007
  1878
nipkow@32007
  1879
lemmas prime_nat_simp_number_of[simp] = prime_nat_simp[of "number_of m", standard]
nipkow@32007
  1880
nipkow@32007
  1881
lemma prime_int_code[code]:
nipkow@32007
  1882
  "prime(p::int) = (p > 1 & (ALL n : {1<..<p}. ~(n dvd p)))" (is "?L = ?R")
nipkow@32007
  1883
proof
nipkow@32007
  1884
  assume "?L" thus "?R"
nipkow@32007
  1885
    by (clarsimp simp: prime_gt_1_int) (metis int_one_le_iff_zero_less prime_int_altdef zless_le)
nipkow@32007
  1886
next
nipkow@32007
  1887
    assume "?R" thus "?L" by (clarsimp simp:Ball_def) (metis dvdI not_prime_eq_prod_int)
nipkow@32007
  1888
qed
nipkow@32007
  1889
nipkow@32007
  1890
lemma prime_int_simp:
nipkow@32007
  1891
  "prime(p::int) = (p > 1 & (list_all (%n. ~ n dvd p) [2..p - 1]))"
nipkow@32007
  1892
apply(simp only:prime_int_code list_ball_code greaterThanLessThan_upto)
nipkow@32007
  1893
apply simp
nipkow@32007
  1894
done
nipkow@32007
  1895
nipkow@32007
  1896
lemmas prime_int_simp_number_of[simp] = prime_int_simp[of "number_of m", standard]
nipkow@32007
  1897
nipkow@32007
  1898
declare successor_int_def[simp]
nipkow@32007
  1899
nipkow@32007
  1900
lemma two_is_prime_nat [simp]: "prime (2::nat)"
nipkow@32007
  1901
by simp
nipkow@32007
  1902
nipkow@32007
  1903
lemma two_is_prime_int [simp]: "prime (2::int)"
nipkow@32007
  1904
by simp
nipkow@32007
  1905
nipkow@32111
  1906
text{* A bit of regression testing: *}
nipkow@32111
  1907
nipkow@32111
  1908
lemma "prime(97::nat)"
nipkow@32111
  1909
by simp
nipkow@32111
  1910
nipkow@32111
  1911
lemma "prime(97::int)"
nipkow@32111
  1912
by simp
nipkow@32111
  1913
nipkow@32111
  1914
lemma "prime(997::nat)"
nipkow@32111
  1915
by eval
nipkow@32111
  1916
nipkow@32111
  1917
lemma "prime(997::int)"
nipkow@32111
  1918
by eval
nipkow@32111
  1919
nipkow@32007
  1920
nipkow@32007
  1921
lemma prime_imp_power_coprime_nat: "prime (p::nat) \<Longrightarrow> ~ p dvd a \<Longrightarrow> coprime a (p^m)"
nipkow@31952
  1922
  apply (rule coprime_exp_nat)
nipkow@31952
  1923
  apply (subst gcd_commute_nat)
nipkow@31952
  1924
  apply (erule (1) prime_imp_coprime_nat)
huffman@31706
  1925
done
chaieb@27568
  1926
nipkow@32007
  1927
lemma prime_imp_power_coprime_int: "prime (p::int) \<Longrightarrow> ~ p dvd a \<Longrightarrow> coprime a (p^m)"
nipkow@31952
  1928
  apply (rule coprime_exp_int)
nipkow@31952
  1929
  apply (subst gcd_commute_int)
nipkow@31952
  1930
  apply (erule (1) prime_imp_coprime_int)
huffman@31706
  1931
done
chaieb@27568
  1932
nipkow@31952
  1933
lemma primes_coprime_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
nipkow@31952
  1934
  apply (rule prime_imp_coprime_nat, assumption)
huffman@31706
  1935
  apply (unfold prime_nat_def, auto)
huffman@31706
  1936
done
chaieb@27568
  1937
nipkow@31952
  1938
lemma primes_coprime_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p \<noteq> q \<Longrightarrow> coprime p q"
nipkow@31952
  1939
  apply (rule prime_imp_coprime_int, assumption)
huffman@31706
  1940
  apply (unfold prime_int_altdef, clarify)
huffman@31706
  1941
  apply (drule_tac x = q in spec)
huffman@31706
  1942
  apply (drule_tac x = p in spec)
huffman@31706
  1943
  apply auto
huffman@31706
  1944
done
chaieb@27568
  1945
nipkow@32007
  1946
lemma primes_imp_powers_coprime_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow> coprime (p^m) (q^n)"
nipkow@31952
  1947
  by (rule coprime_exp2_nat, rule primes_coprime_nat)
chaieb@27568
  1948
nipkow@32007
  1949
lemma primes_imp_powers_coprime_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow> p ~= q \<Longrightarrow> coprime (p^m) (q^n)"
nipkow@31952
  1950
  by (rule coprime_exp2_int, rule primes_coprime_int)
chaieb@27568
  1951
nipkow@31952
  1952
lemma prime_factor_nat: "n \<noteq> (1::nat) \<Longrightarrow> \<exists> p. prime p \<and> p dvd n"
huffman@31706
  1953
  apply (induct n rule: nat_less_induct)
huffman@31706
  1954
  apply (case_tac "n = 0")
nipkow@31952
  1955
  using two_is_prime_nat apply blast
huffman@31706
  1956
  apply (case_tac "prime n")
huffman@31706
  1957
  apply blast
huffman@31706
  1958
  apply (subgoal_tac "n > 1")
nipkow@31952
  1959
  apply (frule (1) not_prime_eq_prod_nat)
huffman@31706
  1960
  apply (auto intro: dvd_mult dvd_mult2)
huffman@31706
  1961
done
chaieb@23244
  1962
huffman@31706
  1963
(* An Isar version:
huffman@31706
  1964
nipkow@31952
  1965
lemma prime_factor_b_nat:
huffman@31706
  1966
  fixes n :: nat
huffman@31706
  1967
  assumes "n \<noteq> 1"
huffman@31706
  1968
  shows "\<exists>p. prime p \<and> p dvd n"
nipkow@23983
  1969
huffman@31706
  1970
using `n ~= 1`
nipkow@31952
  1971
proof (induct n rule: less_induct_nat)
huffman@31706
  1972
  fix n :: nat
huffman@31706
  1973
  assume "n ~= 1" and
huffman@31706
  1974
    ih: "\<forall>m<n. m \<noteq> 1 \<longrightarrow> (\<exists>p. prime p \<and> p dvd m)"
huffman@31706
  1975
  thus "\<exists>p. prime p \<and> p dvd n"
huffman@31706
  1976
  proof -
huffman@31706
  1977
  {
huffman@31706
  1978
    assume "n = 0"
nipkow@31952
  1979
    moreover note two_is_prime_nat
huffman@31706
  1980
    ultimately have ?thesis
nipkow@31952
  1981
      by (auto simp del: two_is_prime_nat)
huffman@31706
  1982
  }
huffman@31706
  1983
  moreover
huffman@31706
  1984
  {
huffman@31706
  1985
    assume "prime n"
huffman@31706
  1986
    hence ?thesis by auto
huffman@31706
  1987
  }
huffman@31706
  1988
  moreover
huffman@31706
  1989
  {
huffman@31706
  1990
    assume "n ~= 0" and "~ prime n"
huffman@31706
  1991
    with `n ~= 1` have "n > 1" by auto
nipkow@31952
  1992
    with `~ prime n` and not_prime_eq_prod_nat obtain m k where
huffman@31706
  1993
      "n = m * k" and "1 < m" and "m < n" by blast
huffman@31706
  1994
    with ih obtain p where "prime p" and "p dvd m" by blast
huffman@31706
  1995
    with `n = m * k` have ?thesis by auto
huffman@31706
  1996
  }
huffman@31706
  1997
  ultimately show ?thesis by blast
huffman@31706
  1998
  qed
nipkow@23983
  1999
qed
nipkow@23983
  2000
huffman@31706
  2001
*)
huffman@31706
  2002
huffman@31706
  2003
text {* One property of coprimality is easier to prove via prime factors. *}
huffman@31706
  2004
nipkow@31952
  2005
lemma prime_divprod_pow_nat:
huffman@31706
  2006
  assumes p: "prime (p::nat)" and ab: "coprime a b" and pab: "p^n dvd a * b"
huffman@31706
  2007
  shows "p^n dvd a \<or> p^n dvd b"
huffman@31706
  2008
proof-
huffman@31706
  2009
  {assume "n = 0 \<or> a = 1 \<or> b = 1" with pab have ?thesis
huffman@31706
  2010
      apply (cases "n=0", simp_all)
huffman@31706
  2011
      apply (cases "a=1", simp_all) done}
huffman@31706
  2012
  moreover
huffman@31706
  2013
  {assume n: "n \<noteq> 0" and a: "a\<noteq>1" and b: "b\<noteq>1"
huffman@31706
  2014
    then obtain m where m: "n = Suc m" by (cases n, auto)
huffman@31706
  2015
    from n have "p dvd p^n" by (intro dvd_power, auto)
huffman@31706
  2016
    also note pab
huffman@31706
  2017
    finally have pab': "p dvd a * b".
nipkow@31952
  2018
    from prime_dvd_mult_nat[OF p pab']
huffman@31706
  2019
    have "p dvd a \<or> p dvd b" .
huffman@31706
  2020
    moreover
huffman@31706
  2021
    {assume pa: "p dvd a"
huffman@31706
  2022
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
nipkow@31952
  2023
      from coprime_common_divisor_nat [OF ab, OF pa] p have "\<not> p dvd b" by auto
huffman@31706
  2024
      with p have "coprime b p"
nipkow@31952
  2025
        by (subst gcd_commute_nat, intro prime_imp_coprime_nat)
huffman@31706
  2026
      hence pnb: "coprime (p^n) b"
nipkow@31952
  2027
        by (subst gcd_commute_nat, rule coprime_exp_nat)
nipkow@31952
  2028
      from coprime_divprod_nat[OF pnba pnb] have ?thesis by blast }
huffman@31706
  2029
    moreover
huffman@31706
  2030
    {assume pb: "p dvd b"
huffman@31706
  2031
      have pnba: "p^n dvd b*a" using pab by (simp add: mult_commute)
nipkow@31952
  2032
      from coprime_common_divisor_nat [OF ab, of p] pb p have "\<not> p dvd a"
huffman@31706
  2033
        by auto
huffman@31706
  2034
      with p have "coprime a p"
nipkow@31952
  2035
        by (subst gcd_commute_nat, intro prime_imp_coprime_nat)
huffman@31706
  2036
      hence pna: "coprime (p^n) a"
nipkow@31952
  2037
        by (subst gcd_commute_nat, rule coprime_exp_nat)
nipkow@31952
  2038
      from coprime_divprod_nat[OF pab pna] have ?thesis by blast }
huffman@31706
  2039
    ultimately have ?thesis by blast}
huffman@31706
  2040
  ultimately show ?thesis by blast
nipkow@23983
  2041
qed
nipkow@23983
  2042
avigad@32036
  2043
subsection {* Infinitely many primes *}
avigad@32036
  2044
avigad@32036
  2045
lemma next_prime_bound: "\<exists>(p::nat). prime p \<and> n < p \<and> p <= fact n + 1"
avigad@32036
  2046
proof-
avigad@32036
  2047
  have f1: "fact n + 1 \<noteq> 1" using fact_ge_one_nat [of n] by arith 
avigad@32036
  2048
  from prime_factor_nat [OF f1]
avigad@32036
  2049
      obtain p where "prime p" and "p dvd fact n + 1" by auto
avigad@32036
  2050
  hence "p \<le> fact n + 1" 
avigad@32036
  2051
    by (intro dvd_imp_le, auto)
avigad@32036
  2052
  {assume "p \<le> n"
avigad@32036
  2053
    from `prime p` have "p \<ge> 1" 
avigad@32036
  2054
      by (cases p, simp_all)
avigad@32036
  2055
    with `p <= n` have "p dvd fact n" 
avigad@32036
  2056
      by (intro dvd_fact_nat)
avigad@32036
  2057
    with `p dvd fact n + 1` have "p dvd fact n + 1 - fact n"
avigad@32036
  2058
      by (rule dvd_diff_nat)
avigad@32036
  2059
    hence "p dvd 1" by simp
avigad@32036
  2060
    hence "p <= 1" by auto
avigad@32036
  2061
    moreover from `prime p` have "p > 1" by auto
avigad@32036
  2062
    ultimately have False by auto}
avigad@32036
  2063
  hence "n < p" by arith
avigad@32036
  2064
  with `prime p` and `p <= fact n + 1` show ?thesis by auto
avigad@32036
  2065
qed
avigad@32036
  2066
avigad@32036
  2067
lemma bigger_prime: "\<exists>p. prime p \<and> p > (n::nat)" 
avigad@32036
  2068
using next_prime_bound by auto
avigad@32036
  2069
avigad@32036
  2070
lemma primes_infinite: "\<not> (finite {(p::nat). prime p})"
avigad@32036
  2071
proof
avigad@32036
  2072
  assume "finite {(p::nat). prime p}"
avigad@32036
  2073
  with Max_ge have "(EX b. (ALL x : {(p::nat). prime p}. x <= b))"
avigad@32036
  2074
    by auto
avigad@32036
  2075
  then obtain b where "ALL (x::nat). prime x \<longrightarrow> x <= b"
avigad@32036
  2076
    by auto
avigad@32036
  2077
  with bigger_prime [of b] show False by auto
avigad@32036
  2078
qed
avigad@32036
  2079
avigad@32036
  2080
wenzelm@21256
  2081
end