src/HOL/GCD.thy
author haftmann
Thu Oct 29 11:41:36 2009 +0100 (2009-10-29)
changeset 33318 ddd97d9dfbfb
parent 33197 de6285ebcc05
child 33657 a4179bf442d1
permissions -rw-r--r--
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with the functions gcd and lcm.  Definitions and
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lemmas are proved uniformly for the natural numbers and integers.
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This file combines and revises a number of prior developments.
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The original theories "GCD" and "Primes" were by Christophe Tabacznyj
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and Lawrence C. Paulson, based on \cite{davenport92}. They introduced
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gcd, lcm, and prime for the natural numbers.
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The original theory "IntPrimes" was by Thomas M. Rasmussen, and
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extended gcd, lcm, primes to the integers. Amine Chaieb provided
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another extension of the notions to the integers, and added a number
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of results to "Primes" and "GCD". IntPrimes also defined and developed
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the congruence relations on the integers. The notion was extended to
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the natural numbers by 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 Parity
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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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(* 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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(* 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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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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  unfolding gcd_int_def lcm_int_def
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  by auto
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lemma transfer_nat_int_gcd_closures:
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  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> gcd x y >= 0"
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  "x >= (0::int) \<Longrightarrow> y >= 0 \<Longrightarrow> lcm x y >= 0"
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  by (auto simp add: gcd_int_def lcm_int_def)
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declare 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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  by (unfold gcd_int_def lcm_int_def, auto)
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lemma transfer_int_nat_gcd_closures:
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  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> gcd x y >= 0"
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  "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> lcm x y >= 0"
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  by (auto simp add: gcd_int_def lcm_int_def)
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declare 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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  by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff_nat)
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lemma gcd_zero_int [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
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  by (auto simp add: gcd_int_def)
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lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
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  by (insert gcd_zero_nat [of m n], arith)
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lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
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  by (insert gcd_zero_int [of m n], insert gcd_ge_0_int [of m n], arith)
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lemma gcd_commute_nat: "gcd (m::nat) n = gcd n m"
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  by (rule dvd_anti_sym, auto)
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lemma gcd_commute_int: "gcd (m::int) n = gcd n m"
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  by (auto simp add: gcd_int_def gcd_commute_nat)
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lemma gcd_assoc_nat: "gcd (gcd (k::nat) m) n = gcd k (gcd m n)"
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  apply (rule dvd_anti_sym)
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  apply (blast intro: dvd_trans)+
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done
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lemma gcd_assoc_int: "gcd (gcd (k::int) m) n = gcd k (gcd m n)"
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  by (auto simp add: gcd_int_def gcd_assoc_nat)
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nipkow@31952
   328
lemmas gcd_left_commute_nat =
nipkow@31952
   329
  mk_left_commute[of gcd, OF gcd_assoc_nat gcd_commute_nat]
huffman@31706
   330
nipkow@31952
   331
lemmas gcd_left_commute_int =
nipkow@31952
   332
  mk_left_commute[of gcd, OF gcd_assoc_int gcd_commute_int]
huffman@31706
   333
nipkow@31952
   334
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
huffman@31706
   335
  -- {* gcd is an AC-operator *}
wenzelm@21256
   336
nipkow@31952
   337
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
huffman@31706
   338
nipkow@31952
   339
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
huffman@31706
   340
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   341
  apply auto
huffman@31706
   342
  apply (rule dvd_anti_sym)
nipkow@31952
   343
  apply (erule (1) gcd_greatest_nat)
huffman@31706
   344
  apply auto
huffman@31706
   345
done
wenzelm@21256
   346
nipkow@31952
   347
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
huffman@31706
   348
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
huffman@31706
   349
  apply (case_tac "d = 0")
huffman@31706
   350
  apply force
huffman@31706
   351
  apply (rule iffI)
huffman@31706
   352
  apply (rule zdvd_anti_sym)
huffman@31706
   353
  apply arith
nipkow@31952
   354
  apply (subst gcd_pos_int)
huffman@31706
   355
  apply clarsimp
huffman@31706
   356
  apply (drule_tac x = "d + 1" in spec)
huffman@31706
   357
  apply (frule zdvd_imp_le)
nipkow@31952
   358
  apply (auto intro: gcd_greatest_int)
huffman@31706
   359
done
huffman@30082
   360
nipkow@31798
   361
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
nipkow@31952
   362
by (metis dvd.eq_iff gcd_unique_nat)
nipkow@31798
   363
nipkow@31798
   364
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
nipkow@31952
   365
by (metis dvd.eq_iff gcd_unique_nat)
nipkow@31798
   366
nipkow@31798
   367
lemma gcd_proj1_if_dvd_int[simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
nipkow@31952
   368
by (metis abs_dvd_iff abs_eq_0 gcd_0_left_int gcd_abs_int gcd_unique_int)
nipkow@31798
   369
nipkow@31798
   370
lemma gcd_proj2_if_dvd_int[simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
nipkow@31952
   371
by (metis gcd_proj1_if_dvd_int gcd_commute_int)
nipkow@31798
   372
nipkow@31798
   373
wenzelm@21256
   374
text {*
wenzelm@21256
   375
  \medskip Multiplication laws
wenzelm@21256
   376
*}
wenzelm@21256
   377
nipkow@31952
   378
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
wenzelm@21256
   379
    -- {* \cite[page 27]{davenport92} *}
nipkow@31952
   380
  apply (induct m n rule: gcd_nat_induct)
huffman@31706
   381
  apply simp
wenzelm@21256
   382
  apply (case_tac "k = 0")
nipkow@31952
   383
  apply (simp_all add: mod_geq gcd_non_0_nat mod_mult_distrib2)
huffman@31706
   384
done
wenzelm@21256
   385
nipkow@31952
   386
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
nipkow@31952
   387
  apply (subst (1 2) gcd_abs_int)
nipkow@31813
   388
  apply (subst (1 2) abs_mult)
nipkow@31952
   389
  apply (rule gcd_mult_distrib_nat [transferred])
huffman@31706
   390
  apply auto
huffman@31706
   391
done
wenzelm@21256
   392
nipkow@31952
   393
lemma coprime_dvd_mult_nat: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31952
   394
  apply (insert gcd_mult_distrib_nat [of m k n])
wenzelm@21256
   395
  apply simp
wenzelm@21256
   396
  apply (erule_tac t = m in ssubst)
wenzelm@21256
   397
  apply simp
wenzelm@21256
   398
  done
wenzelm@21256
   399
nipkow@31952
   400
lemma coprime_dvd_mult_int:
nipkow@31813
   401
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
nipkow@31813
   402
apply (subst abs_dvd_iff [symmetric])
nipkow@31813
   403
apply (subst dvd_abs_iff [symmetric])
nipkow@31952
   404
apply (subst (asm) gcd_abs_int)
nipkow@31952
   405
apply (rule coprime_dvd_mult_nat [transferred])
nipkow@31813
   406
    prefer 4 apply assumption
nipkow@31813
   407
   apply auto
nipkow@31813
   408
apply (subst abs_mult [symmetric], auto)
huffman@31706
   409
done
huffman@31706
   410
nipkow@31952
   411
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
huffman@31706
   412
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   413
  by (auto intro: coprime_dvd_mult_nat)
huffman@31706
   414
nipkow@31952
   415
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
huffman@31706
   416
    (k dvd m * n) = (k dvd m)"
nipkow@31952
   417
  by (auto intro: coprime_dvd_mult_int)
huffman@31706
   418
nipkow@31952
   419
lemma gcd_mult_cancel_nat: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
wenzelm@21256
   420
  apply (rule dvd_anti_sym)
nipkow@31952
   421
  apply (rule gcd_greatest_nat)
nipkow@31952
   422
  apply (rule_tac n = k in coprime_dvd_mult_nat)
nipkow@31952
   423
  apply (simp add: gcd_assoc_nat)
nipkow@31952
   424
  apply (simp add: gcd_commute_nat)
huffman@31706
   425
  apply (simp_all add: mult_commute)
huffman@31706
   426
done
wenzelm@21256
   427
nipkow@31952
   428
lemma gcd_mult_cancel_int:
nipkow@31813
   429
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
nipkow@31952
   430
apply (subst (1 2) gcd_abs_int)
nipkow@31813
   431
apply (subst abs_mult)
nipkow@31952
   432
apply (rule gcd_mult_cancel_nat [transferred], auto)
huffman@31706
   433
done
wenzelm@21256
   434
wenzelm@21256
   435
text {* \medskip Addition laws *}
wenzelm@21256
   436
nipkow@31952
   437
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
huffman@31706
   438
  apply (case_tac "n = 0")
nipkow@31952
   439
  apply (simp_all add: gcd_non_0_nat)
huffman@31706
   440
done
huffman@31706
   441
nipkow@31952
   442
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
nipkow@31952
   443
  apply (subst (1 2) gcd_commute_nat)
huffman@31706
   444
  apply (subst add_commute)
huffman@31706
   445
  apply simp
huffman@31706
   446
done
huffman@31706
   447
huffman@31706
   448
(* to do: add the other variations? *)
huffman@31706
   449
nipkow@31952
   450
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
nipkow@31952
   451
  by (subst gcd_add1_nat [symmetric], auto)
huffman@31706
   452
nipkow@31952
   453
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
nipkow@31952
   454
  apply (subst gcd_commute_nat)
nipkow@31952
   455
  apply (subst gcd_diff1_nat [symmetric])
huffman@31706
   456
  apply auto
nipkow@31952
   457
  apply (subst gcd_commute_nat)
nipkow@31952
   458
  apply (subst gcd_diff1_nat)
huffman@31706
   459
  apply assumption
nipkow@31952
   460
  apply (rule gcd_commute_nat)
huffman@31706
   461
done
huffman@31706
   462
nipkow@31952
   463
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
huffman@31706
   464
  apply (frule_tac b = y and a = x in pos_mod_sign)
huffman@31706
   465
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
nipkow@31952
   466
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
huffman@31706
   467
    zmod_zminus1_eq_if)
huffman@31706
   468
  apply (frule_tac a = x in pos_mod_bound)
nipkow@31952
   469
  apply (subst (1 2) gcd_commute_nat)
nipkow@31952
   470
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
huffman@31706
   471
    nat_le_eq_zle)
huffman@31706
   472
done
wenzelm@21256
   473
nipkow@31952
   474
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
huffman@31706
   475
  apply (case_tac "y = 0")
huffman@31706
   476
  apply force
huffman@31706
   477
  apply (case_tac "y > 0")
nipkow@31952
   478
  apply (subst gcd_non_0_int, auto)
nipkow@31952
   479
  apply (insert gcd_non_0_int [of "-y" "-x"])
nipkow@31952
   480
  apply (auto simp add: gcd_neg1_int gcd_neg2_int)
huffman@31706
   481
done
huffman@31706
   482
nipkow@31952
   483
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
nipkow@31952
   484
by (metis gcd_red_int mod_add_self1 zadd_commute)
huffman@31706
   485
nipkow@31952
   486
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
nipkow@31952
   487
by (metis gcd_add1_int gcd_commute_int zadd_commute)
wenzelm@21256
   488
nipkow@31952
   489
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
nipkow@31952
   490
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
wenzelm@21256
   491
nipkow@31952
   492
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
nipkow@31952
   493
by (metis gcd_commute_int gcd_red_int mod_mult_self1 zadd_commute)
nipkow@31798
   494
wenzelm@21256
   495
huffman@31706
   496
(* to do: differences, and all variations of addition rules
huffman@31706
   497
    as simplification rules for nat and int *)
huffman@31706
   498
nipkow@31798
   499
(* FIXME remove iff *)
nipkow@31952
   500
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
haftmann@23687
   501
  using mult_dvd_mono [of 1] by auto
chaieb@22027
   502
huffman@31706
   503
(* to do: add the three variations of these, and for ints? *)
huffman@31706
   504
nipkow@31992
   505
lemma finite_divisors_nat[simp]:
nipkow@31992
   506
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
nipkow@31734
   507
proof-
nipkow@31734
   508
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
nipkow@31734
   509
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   510
    by(bestsimp intro!:dvd_imp_le)
nipkow@31734
   511
qed
nipkow@31734
   512
nipkow@31995
   513
lemma finite_divisors_int[simp]:
nipkow@31734
   514
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
nipkow@31734
   515
proof-
nipkow@31734
   516
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
nipkow@31734
   517
  hence "finite{d. abs d <= abs i}" by simp
nipkow@31734
   518
  from finite_subset[OF _ this] show ?thesis using assms
nipkow@31734
   519
    by(bestsimp intro!:dvd_imp_le_int)
nipkow@31734
   520
qed
nipkow@31734
   521
nipkow@31995
   522
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
nipkow@31995
   523
apply(rule antisym)
nipkow@31995
   524
 apply (fastsimp intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
nipkow@31995
   525
apply simp
nipkow@31995
   526
done
nipkow@31995
   527
nipkow@31995
   528
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
nipkow@31995
   529
apply(rule antisym)
nipkow@31995
   530
 apply(rule Max_le_iff[THEN iffD2])
nipkow@31995
   531
   apply simp
nipkow@31995
   532
  apply fastsimp
nipkow@31995
   533
 apply (metis Collect_def abs_ge_self dvd_imp_le_int mem_def zle_trans)
nipkow@31995
   534
apply simp
nipkow@31995
   535
done
nipkow@31995
   536
nipkow@31734
   537
lemma gcd_is_Max_divisors_nat:
nipkow@31734
   538
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   539
apply(rule Max_eqI[THEN sym])
nipkow@31995
   540
  apply (metis finite_Collect_conjI finite_divisors_nat)
nipkow@31734
   541
 apply simp
nipkow@31952
   542
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
nipkow@31734
   543
apply simp
nipkow@31734
   544
done
nipkow@31734
   545
nipkow@31734
   546
lemma gcd_is_Max_divisors_int:
nipkow@31734
   547
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
nipkow@31734
   548
apply(rule Max_eqI[THEN sym])
nipkow@31995
   549
  apply (metis finite_Collect_conjI finite_divisors_int)
nipkow@31734
   550
 apply simp
nipkow@31952
   551
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
nipkow@31734
   552
apply simp
nipkow@31734
   553
done
nipkow@31734
   554
chaieb@22027
   555
huffman@31706
   556
subsection {* Coprimality *}
huffman@31706
   557
nipkow@31952
   558
lemma div_gcd_coprime_nat:
huffman@31706
   559
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   560
  shows "coprime (a div gcd a b) (b div gcd a b)"
wenzelm@22367
   561
proof -
haftmann@27556
   562
  let ?g = "gcd a b"
chaieb@22027
   563
  let ?a' = "a div ?g"
chaieb@22027
   564
  let ?b' = "b div ?g"
haftmann@27556
   565
  let ?g' = "gcd ?a' ?b'"
chaieb@22027
   566
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
chaieb@22027
   567
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
wenzelm@22367
   568
  from dvdg dvdg' obtain ka kb ka' kb' where
wenzelm@22367
   569
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
chaieb@22027
   570
    unfolding dvd_def by blast
huffman@31706
   571
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
huffman@31706
   572
    by simp_all
wenzelm@22367
   573
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
wenzelm@22367
   574
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
wenzelm@22367
   575
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
nipkow@31952
   576
  have "?g \<noteq> 0" using nz by (simp add: gcd_zero_nat)
huffman@31706
   577
  then have gp: "?g > 0" by arith
nipkow@31952
   578
  from gcd_greatest_nat [OF dvdgg'] have "?g * ?g' dvd ?g" .
wenzelm@22367
   579
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
chaieb@22027
   580
qed
chaieb@22027
   581
nipkow@31952
   582
lemma div_gcd_coprime_int:
huffman@31706
   583
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
huffman@31706
   584
  shows "coprime (a div gcd a b) (b div gcd a b)"
nipkow@31952
   585
apply (subst (1 2 3) gcd_abs_int)
nipkow@31813
   586
apply (subst (1 2) abs_div)
nipkow@31813
   587
  apply simp
nipkow@31813
   588
 apply simp
nipkow@31813
   589
apply(subst (1 2) abs_gcd_int)
nipkow@31952
   590
apply (rule div_gcd_coprime_nat [transferred])
nipkow@31952
   591
using nz apply (auto simp add: gcd_abs_int [symmetric])
huffman@31706
   592
done
huffman@31706
   593
nipkow@31952
   594
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   595
  using gcd_unique_nat[of 1 a b, simplified] by auto
huffman@31706
   596
nipkow@31952
   597
lemma coprime_Suc_0_nat:
huffman@31706
   598
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
nipkow@31952
   599
  using coprime_nat by (simp add: One_nat_def)
huffman@31706
   600
nipkow@31952
   601
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
huffman@31706
   602
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
nipkow@31952
   603
  using gcd_unique_int [of 1 a b]
huffman@31706
   604
  apply clarsimp
huffman@31706
   605
  apply (erule subst)
huffman@31706
   606
  apply (rule iffI)
huffman@31706
   607
  apply force
huffman@31706
   608
  apply (drule_tac x = "abs e" in exI)
huffman@31706
   609
  apply (case_tac "e >= 0")
huffman@31706
   610
  apply force
huffman@31706
   611
  apply force
huffman@31706
   612
done
huffman@31706
   613
nipkow@31952
   614
lemma gcd_coprime_nat:
huffman@31706
   615
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   616
    b: "b = b' * gcd a b"
huffman@31706
   617
  shows    "coprime a' b'"
huffman@31706
   618
huffman@31706
   619
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   620
  apply (erule ssubst)
huffman@31706
   621
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   622
  apply (erule ssubst)
nipkow@31952
   623
  apply (rule div_gcd_coprime_nat)
huffman@31706
   624
  using prems
huffman@31706
   625
  apply force
huffman@31706
   626
  apply (subst (1) b)
huffman@31706
   627
  using z apply force
huffman@31706
   628
  apply (subst (1) a)
huffman@31706
   629
  using z apply force
huffman@31706
   630
done
huffman@31706
   631
nipkow@31952
   632
lemma gcd_coprime_int:
huffman@31706
   633
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
huffman@31706
   634
    b: "b = b' * gcd a b"
huffman@31706
   635
  shows    "coprime a' b'"
huffman@31706
   636
huffman@31706
   637
  apply (subgoal_tac "a' = a div gcd a b")
huffman@31706
   638
  apply (erule ssubst)
huffman@31706
   639
  apply (subgoal_tac "b' = b div gcd a b")
huffman@31706
   640
  apply (erule ssubst)
nipkow@31952
   641
  apply (rule div_gcd_coprime_int)
huffman@31706
   642
  using prems
huffman@31706
   643
  apply force
huffman@31706
   644
  apply (subst (1) b)
huffman@31706
   645
  using z apply force
huffman@31706
   646
  apply (subst (1) a)
huffman@31706
   647
  using z apply force
huffman@31706
   648
done
huffman@31706
   649
nipkow@31952
   650
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
huffman@31706
   651
    shows "coprime d (a * b)"
nipkow@31952
   652
  apply (subst gcd_commute_nat)
nipkow@31952
   653
  using da apply (subst gcd_mult_cancel_nat)
nipkow@31952
   654
  apply (subst gcd_commute_nat, assumption)
nipkow@31952
   655
  apply (subst gcd_commute_nat, rule db)
huffman@31706
   656
done
huffman@31706
   657
nipkow@31952
   658
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
huffman@31706
   659
    shows "coprime d (a * b)"
nipkow@31952
   660
  apply (subst gcd_commute_int)
nipkow@31952
   661
  using da apply (subst gcd_mult_cancel_int)
nipkow@31952
   662
  apply (subst gcd_commute_int, assumption)
nipkow@31952
   663
  apply (subst gcd_commute_int, rule db)
huffman@31706
   664
done
huffman@31706
   665
nipkow@31952
   666
lemma coprime_lmult_nat:
huffman@31706
   667
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
huffman@31706
   668
proof -
huffman@31706
   669
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   670
    by (rule gcd_greatest_nat, auto)
huffman@31706
   671
  with dab show ?thesis
huffman@31706
   672
    by auto
huffman@31706
   673
qed
huffman@31706
   674
nipkow@31952
   675
lemma coprime_lmult_int:
nipkow@31798
   676
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
huffman@31706
   677
proof -
huffman@31706
   678
  have "gcd d a dvd gcd d (a * b)"
nipkow@31952
   679
    by (rule gcd_greatest_int, auto)
nipkow@31798
   680
  with assms show ?thesis
huffman@31706
   681
    by auto
huffman@31706
   682
qed
huffman@31706
   683
nipkow@31952
   684
lemma coprime_rmult_nat:
nipkow@31798
   685
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
huffman@31706
   686
proof -
huffman@31706
   687
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   688
    by (rule gcd_greatest_nat, auto intro: dvd_mult)
nipkow@31798
   689
  with assms show ?thesis
huffman@31706
   690
    by auto
huffman@31706
   691
qed
huffman@31706
   692
nipkow@31952
   693
lemma coprime_rmult_int:
huffman@31706
   694
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
huffman@31706
   695
proof -
huffman@31706
   696
  have "gcd d b dvd gcd d (a * b)"
nipkow@31952
   697
    by (rule gcd_greatest_int, auto intro: dvd_mult)
huffman@31706
   698
  with dab show ?thesis
huffman@31706
   699
    by auto
huffman@31706
   700
qed
huffman@31706
   701
nipkow@31952
   702
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
huffman@31706
   703
    coprime d a \<and>  coprime d b"
nipkow@31952
   704
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
nipkow@31952
   705
    coprime_mult_nat[of d a b]
huffman@31706
   706
  by blast
huffman@31706
   707
nipkow@31952
   708
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
huffman@31706
   709
    coprime d a \<and>  coprime d b"
nipkow@31952
   710
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
nipkow@31952
   711
    coprime_mult_int[of d a b]
huffman@31706
   712
  by blast
huffman@31706
   713
nipkow@31952
   714
lemma gcd_coprime_exists_nat:
huffman@31706
   715
    assumes nz: "gcd (a::nat) b \<noteq> 0"
huffman@31706
   716
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   717
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   718
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   719
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
huffman@31706
   720
done
huffman@31706
   721
nipkow@31952
   722
lemma gcd_coprime_exists_int:
huffman@31706
   723
    assumes nz: "gcd (a::int) b \<noteq> 0"
huffman@31706
   724
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
huffman@31706
   725
  apply (rule_tac x = "a div gcd a b" in exI)
huffman@31706
   726
  apply (rule_tac x = "b div gcd a b" in exI)
nipkow@31952
   727
  using nz apply (auto simp add: div_gcd_coprime_int dvd_div_mult_self)
huffman@31706
   728
done
huffman@31706
   729
nipkow@31952
   730
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   731
  by (induct n, simp_all add: coprime_mult_nat)
huffman@31706
   732
nipkow@31952
   733
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
nipkow@31952
   734
  by (induct n, simp_all add: coprime_mult_int)
huffman@31706
   735
nipkow@31952
   736
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   737
  apply (rule coprime_exp_nat)
nipkow@31952
   738
  apply (subst gcd_commute_nat)
nipkow@31952
   739
  apply (rule coprime_exp_nat)
nipkow@31952
   740
  apply (subst gcd_commute_nat, assumption)
huffman@31706
   741
done
huffman@31706
   742
nipkow@31952
   743
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
nipkow@31952
   744
  apply (rule coprime_exp_int)
nipkow@31952
   745
  apply (subst gcd_commute_int)
nipkow@31952
   746
  apply (rule coprime_exp_int)
nipkow@31952
   747
  apply (subst gcd_commute_int, assumption)
huffman@31706
   748
done
huffman@31706
   749
nipkow@31952
   750
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
huffman@31706
   751
proof (cases)
huffman@31706
   752
  assume "a = 0 & b = 0"
huffman@31706
   753
  thus ?thesis by simp
huffman@31706
   754
  next assume "~(a = 0 & b = 0)"
huffman@31706
   755
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
nipkow@31952
   756
    by (auto simp:div_gcd_coprime_nat)
huffman@31706
   757
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
huffman@31706
   758
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
huffman@31706
   759
    apply (subst (1 2) mult_commute)
nipkow@31952
   760
    apply (subst gcd_mult_distrib_nat [symmetric])
huffman@31706
   761
    apply simp
huffman@31706
   762
    done
huffman@31706
   763
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
huffman@31706
   764
    apply (subst div_power)
huffman@31706
   765
    apply auto
huffman@31706
   766
    apply (rule dvd_div_mult_self)
huffman@31706
   767
    apply (rule dvd_power_same)
huffman@31706
   768
    apply auto
huffman@31706
   769
    done
huffman@31706
   770
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
huffman@31706
   771
    apply (subst div_power)
huffman@31706
   772
    apply auto
huffman@31706
   773
    apply (rule dvd_div_mult_self)
huffman@31706
   774
    apply (rule dvd_power_same)
huffman@31706
   775
    apply auto
huffman@31706
   776
    done
huffman@31706
   777
  finally show ?thesis .
huffman@31706
   778
qed
huffman@31706
   779
nipkow@31952
   780
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
nipkow@31952
   781
  apply (subst (1 2) gcd_abs_int)
huffman@31706
   782
  apply (subst (1 2) power_abs)
nipkow@31952
   783
  apply (rule gcd_exp_nat [where n = n, transferred])
huffman@31706
   784
  apply auto
huffman@31706
   785
done
huffman@31706
   786
nipkow@31952
   787
lemma coprime_divprod_nat: "(d::nat) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
nipkow@31952
   788
  using coprime_dvd_mult_iff_nat[of d a b]
huffman@31706
   789
  by (auto simp add: mult_commute)
huffman@31706
   790
nipkow@31952
   791
lemma coprime_divprod_int: "(d::int) dvd a * b  \<Longrightarrow> coprime d a \<Longrightarrow> d dvd b"
nipkow@31952
   792
  using coprime_dvd_mult_iff_int[of d a b]
huffman@31706
   793
  by (auto simp add: mult_commute)
huffman@31706
   794
nipkow@31952
   795
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
huffman@31706
   796
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   797
proof-
huffman@31706
   798
  let ?g = "gcd a b"
huffman@31706
   799
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   800
  moreover
huffman@31706
   801
  {assume z: "?g \<noteq> 0"
nipkow@31952
   802
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   803
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   804
      by blast
huffman@31706
   805
    have thb: "?g dvd b" by auto
huffman@31706
   806
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   807
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   808
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   809
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   810
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   811
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
huffman@31706
   812
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   813
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   814
    with thb thc have ?thesis by blast }
huffman@31706
   815
  ultimately show ?thesis by blast
huffman@31706
   816
qed
huffman@31706
   817
nipkow@31952
   818
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
huffman@31706
   819
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
huffman@31706
   820
proof-
huffman@31706
   821
  let ?g = "gcd a b"
huffman@31706
   822
  {assume "?g = 0" with dc have ?thesis by auto}
huffman@31706
   823
  moreover
huffman@31706
   824
  {assume z: "?g \<noteq> 0"
nipkow@31952
   825
    from gcd_coprime_exists_int[OF z]
huffman@31706
   826
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   827
      by blast
huffman@31706
   828
    have thb: "?g dvd b" by auto
huffman@31706
   829
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
huffman@31706
   830
    with dc have th0: "a' dvd b*c"
huffman@31706
   831
      using dvd_trans[of a' a "b*c"] by simp
huffman@31706
   832
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
huffman@31706
   833
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
huffman@31706
   834
    with z have th_1: "a' dvd b' * c" by auto
nipkow@31952
   835
    from coprime_dvd_mult_int[OF ab'(3)] th_1
huffman@31706
   836
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
huffman@31706
   837
    from ab' have "a = ?g*a'" by algebra
huffman@31706
   838
    with thb thc have ?thesis by blast }
huffman@31706
   839
  ultimately show ?thesis by blast
chaieb@27669
   840
qed
chaieb@27669
   841
nipkow@31952
   842
lemma pow_divides_pow_nat:
huffman@31706
   843
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   844
  shows "a dvd b"
huffman@31706
   845
proof-
huffman@31706
   846
  let ?g = "gcd a b"
huffman@31706
   847
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   848
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   849
  moreover
huffman@31706
   850
  {assume z: "?g \<noteq> 0"
huffman@31706
   851
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
nipkow@31952
   852
    from gcd_coprime_exists_nat[OF z]
huffman@31706
   853
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   854
      by blast
huffman@31706
   855
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   856
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   857
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   858
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   859
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   860
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   861
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   862
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   863
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
huffman@31706
   864
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   865
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   866
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   867
  ultimately show ?thesis by blast
huffman@31706
   868
qed
huffman@31706
   869
nipkow@31952
   870
lemma pow_divides_pow_int:
huffman@31706
   871
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
huffman@31706
   872
  shows "a dvd b"
chaieb@27669
   873
proof-
huffman@31706
   874
  let ?g = "gcd a b"
huffman@31706
   875
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
huffman@31706
   876
  {assume "?g = 0" with ab n have ?thesis by auto }
huffman@31706
   877
  moreover
huffman@31706
   878
  {assume z: "?g \<noteq> 0"
huffman@31706
   879
    hence zn: "?g ^ n \<noteq> 0" using n by (simp add: neq0_conv)
nipkow@31952
   880
    from gcd_coprime_exists_int[OF z]
huffman@31706
   881
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
huffman@31706
   882
      by blast
huffman@31706
   883
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
huffman@31706
   884
      by (simp add: ab'(1,2)[symmetric])
huffman@31706
   885
    hence "?g^n*a'^n dvd ?g^n *b'^n"
huffman@31706
   886
      by (simp only: power_mult_distrib mult_commute)
huffman@31706
   887
    with zn z n have th0:"a'^n dvd b'^n" by auto
huffman@31706
   888
    have "a' dvd a'^n" by (simp add: m)
huffman@31706
   889
    with th0 have "a' dvd b'^n"
huffman@31706
   890
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
huffman@31706
   891
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
nipkow@31952
   892
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
huffman@31706
   893
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
huffman@31706
   894
    hence "a'*?g dvd b'*?g" by simp
huffman@31706
   895
    with ab'(1,2)  have ?thesis by simp }
huffman@31706
   896
  ultimately show ?thesis by blast
huffman@31706
   897
qed
huffman@31706
   898
nipkow@31798
   899
(* FIXME move to Divides(?) *)
nipkow@31952
   900
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
nipkow@31952
   901
  by (auto intro: pow_divides_pow_nat dvd_power_same)
huffman@31706
   902
nipkow@31952
   903
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
nipkow@31952
   904
  by (auto intro: pow_divides_pow_int dvd_power_same)
huffman@31706
   905
nipkow@31952
   906
lemma divides_mult_nat:
huffman@31706
   907
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   908
  shows "m * n dvd r"
huffman@31706
   909
proof-
huffman@31706
   910
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   911
    unfolding dvd_def by blast
huffman@31706
   912
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   913
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
huffman@31706
   914
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   915
  from n' k show ?thesis unfolding dvd_def by auto
huffman@31706
   916
qed
huffman@31706
   917
nipkow@31952
   918
lemma divides_mult_int:
huffman@31706
   919
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
huffman@31706
   920
  shows "m * n dvd r"
huffman@31706
   921
proof-
huffman@31706
   922
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
huffman@31706
   923
    unfolding dvd_def by blast
huffman@31706
   924
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
nipkow@31952
   925
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
huffman@31706
   926
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
huffman@31706
   927
  from n' k show ?thesis unfolding dvd_def by auto
chaieb@27669
   928
qed
chaieb@27669
   929
nipkow@31952
   930
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
huffman@31706
   931
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   932
  apply force
nipkow@31952
   933
  apply (rule dvd_diff_nat)
huffman@31706
   934
  apply auto
huffman@31706
   935
done
huffman@31706
   936
nipkow@31952
   937
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
nipkow@31952
   938
  using coprime_plus_one_nat by (simp add: One_nat_def)
huffman@31706
   939
nipkow@31952
   940
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
huffman@31706
   941
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
huffman@31706
   942
  apply force
huffman@31706
   943
  apply (rule dvd_diff)
huffman@31706
   944
  apply auto
huffman@31706
   945
done
huffman@31706
   946
nipkow@31952
   947
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
nipkow@31952
   948
  using coprime_plus_one_nat [of "n - 1"]
nipkow@31952
   949
    gcd_commute_nat [of "n - 1" n] by auto
huffman@31706
   950
nipkow@31952
   951
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
nipkow@31952
   952
  using coprime_plus_one_int [of "n - 1"]
nipkow@31952
   953
    gcd_commute_int [of "n - 1" n] by auto
huffman@31706
   954
nipkow@31952
   955
lemma setprod_coprime_nat [rule_format]:
huffman@31706
   956
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
huffman@31706
   957
  apply (case_tac "finite A")
huffman@31706
   958
  apply (induct set: finite)
nipkow@31952
   959
  apply (auto simp add: gcd_mult_cancel_nat)
huffman@31706
   960
done
huffman@31706
   961
nipkow@31952
   962
lemma setprod_coprime_int [rule_format]:
huffman@31706
   963
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
huffman@31706
   964
  apply (case_tac "finite A")
huffman@31706
   965
  apply (induct set: finite)
nipkow@31952
   966
  apply (auto simp add: gcd_mult_cancel_int)
huffman@31706
   967
done
huffman@31706
   968
nipkow@31952
   969
lemma coprime_common_divisor_nat: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
   970
    x dvd b \<Longrightarrow> x = 1"
huffman@31706
   971
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
   972
  apply simp
nipkow@31952
   973
  apply (erule (1) gcd_greatest_nat)
huffman@31706
   974
done
huffman@31706
   975
nipkow@31952
   976
lemma coprime_common_divisor_int: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
huffman@31706
   977
    x dvd b \<Longrightarrow> abs x = 1"
huffman@31706
   978
  apply (subgoal_tac "x dvd gcd a b")
huffman@31706
   979
  apply simp
nipkow@31952
   980
  apply (erule (1) gcd_greatest_int)
huffman@31706
   981
done
huffman@31706
   982
nipkow@31952
   983
lemma coprime_divisors_nat: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
huffman@31706
   984
    coprime d e"
huffman@31706
   985
  apply (auto simp add: dvd_def)
nipkow@31952
   986
  apply (frule coprime_lmult_int)
nipkow@31952
   987
  apply (subst gcd_commute_int)
nipkow@31952
   988
  apply (subst (asm) (2) gcd_commute_int)
nipkow@31952
   989
  apply (erule coprime_lmult_int)
huffman@31706
   990
done
huffman@31706
   991
nipkow@31952
   992
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
   993
apply (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
huffman@31706
   994
done
huffman@31706
   995
nipkow@31952
   996
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
nipkow@31952
   997
apply (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
huffman@31706
   998
done
huffman@31706
   999
huffman@31706
  1000
huffman@31706
  1001
subsection {* Bezout's theorem *}
huffman@31706
  1002
huffman@31706
  1003
(* Function bezw returns a pair of witnesses to Bezout's theorem --
huffman@31706
  1004
   see the theorems that follow the definition. *)
huffman@31706
  1005
fun
huffman@31706
  1006
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
huffman@31706
  1007
where
huffman@31706
  1008
  "bezw x y =
huffman@31706
  1009
  (if y = 0 then (1, 0) else
huffman@31706
  1010
      (snd (bezw y (x mod y)),
huffman@31706
  1011
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
huffman@31706
  1012
huffman@31706
  1013
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
huffman@31706
  1014
huffman@31706
  1015
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
huffman@31706
  1016
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
huffman@31706
  1017
  by simp
huffman@31706
  1018
huffman@31706
  1019
declare bezw.simps [simp del]
huffman@31706
  1020
huffman@31706
  1021
lemma bezw_aux [rule_format]:
huffman@31706
  1022
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
nipkow@31952
  1023
proof (induct x y rule: gcd_nat_induct)
huffman@31706
  1024
  fix m :: nat
huffman@31706
  1025
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
huffman@31706
  1026
    by auto
huffman@31706
  1027
  next fix m :: nat and n
huffman@31706
  1028
    assume ngt0: "n > 0" and
huffman@31706
  1029
      ih: "fst (bezw n (m mod n)) * int n +
huffman@31706
  1030
        snd (bezw n (m mod n)) * int (m mod n) =
huffman@31706
  1031
        int (gcd n (m mod n))"
huffman@31706
  1032
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
nipkow@31952
  1033
      apply (simp add: bezw_non_0 gcd_non_0_nat)
huffman@31706
  1034
      apply (erule subst)
huffman@31706
  1035
      apply (simp add: ring_simps)
huffman@31706
  1036
      apply (subst mod_div_equality [of m n, symmetric])
huffman@31706
  1037
      (* applying simp here undoes the last substitution!
huffman@31706
  1038
         what is procedure cancel_div_mod? *)
huffman@31706
  1039
      apply (simp only: ring_simps zadd_int [symmetric]
huffman@31706
  1040
        zmult_int [symmetric])
huffman@31706
  1041
      done
huffman@31706
  1042
qed
huffman@31706
  1043
nipkow@31952
  1044
lemma bezout_int:
huffman@31706
  1045
  fixes x y
huffman@31706
  1046
  shows "EX u v. u * (x::int) + v * y = gcd x y"
huffman@31706
  1047
proof -
huffman@31706
  1048
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
huffman@31706
  1049
      EX u v. u * x + v * y = gcd x y"
huffman@31706
  1050
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
huffman@31706
  1051
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
huffman@31706
  1052
    apply (unfold gcd_int_def)
huffman@31706
  1053
    apply simp
huffman@31706
  1054
    apply (subst bezw_aux [symmetric])
huffman@31706
  1055
    apply auto
huffman@31706
  1056
    done
huffman@31706
  1057
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
huffman@31706
  1058
      (x \<le> 0 \<and> y \<le> 0)"
huffman@31706
  1059
    by auto
huffman@31706
  1060
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
huffman@31706
  1061
    by (erule (1) bezout_aux)
huffman@31706
  1062
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1063
    apply (insert bezout_aux [of x "-y"])
huffman@31706
  1064
    apply auto
huffman@31706
  1065
    apply (rule_tac x = u in exI)
huffman@31706
  1066
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1067
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1068
    apply auto
huffman@31706
  1069
    done
huffman@31706
  1070
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1071
    apply (insert bezout_aux [of "-x" y])
huffman@31706
  1072
    apply auto
huffman@31706
  1073
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1074
    apply (rule_tac x = v in exI)
nipkow@31952
  1075
    apply (subst gcd_neg1_int [symmetric])
huffman@31706
  1076
    apply auto
huffman@31706
  1077
    done
huffman@31706
  1078
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
huffman@31706
  1079
    apply (insert bezout_aux [of "-x" "-y"])
huffman@31706
  1080
    apply auto
huffman@31706
  1081
    apply (rule_tac x = "-u" in exI)
huffman@31706
  1082
    apply (rule_tac x = "-v" in exI)
nipkow@31952
  1083
    apply (subst gcd_neg1_int [symmetric])
nipkow@31952
  1084
    apply (subst gcd_neg2_int [symmetric])
huffman@31706
  1085
    apply auto
huffman@31706
  1086
    done
huffman@31706
  1087
  ultimately show ?thesis by blast
huffman@31706
  1088
qed
huffman@31706
  1089
huffman@31706
  1090
text {* versions of Bezout for nat, by Amine Chaieb *}
huffman@31706
  1091
huffman@31706
  1092
lemma ind_euclid:
huffman@31706
  1093
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
huffman@31706
  1094
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
chaieb@27669
  1095
  shows "P a b"
chaieb@27669
  1096
proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
chaieb@27669
  1097
  fix n a b
chaieb@27669
  1098
  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
chaieb@27669
  1099
  have "a = b \<or> a < b \<or> b < a" by arith
chaieb@27669
  1100
  moreover {assume eq: "a= b"
huffman@31706
  1101
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
huffman@31706
  1102
    by simp}
chaieb@27669
  1103
  moreover
chaieb@27669
  1104
  {assume lt: "a < b"
chaieb@27669
  1105
    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
chaieb@27669
  1106
    moreover
chaieb@27669
  1107
    {assume "a =0" with z c have "P a b" by blast }
chaieb@27669
  1108
    moreover
chaieb@27669
  1109
    {assume ab: "a + b - a < n"
chaieb@27669
  1110
      have th0: "a + b - a = a + (b - a)" using lt by arith
chaieb@27669
  1111
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1112
      have "P a b" by (simp add: th0[symmetric])}
chaieb@27669
  1113
    ultimately have "P a b" by blast}
chaieb@27669
  1114
  moreover
chaieb@27669
  1115
  {assume lt: "a > b"
chaieb@27669
  1116
    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
chaieb@27669
  1117
    moreover
chaieb@27669
  1118
    {assume "b =0" with z c have "P a b" by blast }
chaieb@27669
  1119
    moreover
chaieb@27669
  1120
    {assume ab: "b + a - b < n"
chaieb@27669
  1121
      have th0: "b + a - b = b + (a - b)" using lt by arith
chaieb@27669
  1122
      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
chaieb@27669
  1123
      have "P b a" by (simp add: th0[symmetric])
chaieb@27669
  1124
      hence "P a b" using c by blast }
chaieb@27669
  1125
    ultimately have "P a b" by blast}
chaieb@27669
  1126
ultimately  show "P a b" by blast
chaieb@27669
  1127
qed
chaieb@27669
  1128
nipkow@31952
  1129
lemma bezout_lemma_nat:
huffman@31706
  1130
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1131
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1132
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
huffman@31706
  1133
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
huffman@31706
  1134
  using ex
huffman@31706
  1135
  apply clarsimp
huffman@31706
  1136
  apply (rule_tac x="d" in exI, simp add: dvd_add)
huffman@31706
  1137
  apply (case_tac "a * x = b * y + d" , simp_all)
huffman@31706
  1138
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1139
  apply (rule_tac x="y" in exI)
huffman@31706
  1140
  apply algebra
huffman@31706
  1141
  apply (rule_tac x="x" in exI)
huffman@31706
  1142
  apply (rule_tac x="x + y" in exI)
huffman@31706
  1143
  apply algebra
chaieb@27669
  1144
done
chaieb@27669
  1145
nipkow@31952
  1146
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1147
    (a * x = b * y + d \<or> b * x = a * y + d)"
huffman@31706
  1148
  apply(induct a b rule: ind_euclid)
huffman@31706
  1149
  apply blast
huffman@31706
  1150
  apply clarify
huffman@31706
  1151
  apply (rule_tac x="a" in exI, simp add: dvd_add)
huffman@31706
  1152
  apply clarsimp
huffman@31706
  1153
  apply (rule_tac x="d" in exI)
huffman@31706
  1154
  apply (case_tac "a * x = b * y + d", simp_all add: dvd_add)
huffman@31706
  1155
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1156
  apply (rule_tac x="y" in exI)
huffman@31706
  1157
  apply algebra
huffman@31706
  1158
  apply (rule_tac x="x" in exI)
huffman@31706
  1159
  apply (rule_tac x="x+y" in exI)
huffman@31706
  1160
  apply algebra
chaieb@27669
  1161
done
chaieb@27669
  1162
nipkow@31952
  1163
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
huffman@31706
  1164
    (a * x - b * y = d \<or> b * x - a * y = d)"
nipkow@31952
  1165
  using bezout_add_nat[of a b]
huffman@31706
  1166
  apply clarsimp
huffman@31706
  1167
  apply (rule_tac x="d" in exI, simp)
huffman@31706
  1168
  apply (rule_tac x="x" in exI)
huffman@31706
  1169
  apply (rule_tac x="y" in exI)
huffman@31706
  1170
  apply auto
chaieb@27669
  1171
done
chaieb@27669
  1172
nipkow@31952
  1173
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
chaieb@27669
  1174
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
chaieb@27669
  1175
proof-
huffman@31706
  1176
 from nz have ap: "a > 0" by simp
nipkow@31952
  1177
 from bezout_add_nat[of a b]
huffman@31706
  1178
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
huffman@31706
  1179
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
chaieb@27669
  1180
 moreover
huffman@31706
  1181
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
huffman@31706
  1182
     from H have ?thesis by blast }
chaieb@27669
  1183
 moreover
chaieb@27669
  1184
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
chaieb@27669
  1185
   {assume b0: "b = 0" with H  have ?thesis by simp}
huffman@31706
  1186
   moreover
chaieb@27669
  1187
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
huffman@31706
  1188
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
huffman@31706
  1189
       by auto
chaieb@27669
  1190
     moreover
chaieb@27669
  1191
     {assume db: "d=b"
chaieb@27669
  1192
       from prems have ?thesis apply simp
wenzelm@32960
  1193
         apply (rule exI[where x = b], simp)
wenzelm@32960
  1194
         apply (rule exI[where x = b])
wenzelm@32960
  1195
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
chaieb@27669
  1196
    moreover
huffman@31706
  1197
    {assume db: "d < b"
wenzelm@32960
  1198
        {assume "x=0" hence ?thesis  using prems by simp }
wenzelm@32960
  1199
        moreover
wenzelm@32960
  1200
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
wenzelm@32960
  1201
          from db have "d \<le> b - 1" by simp
wenzelm@32960
  1202
          hence "d*b \<le> b*(b - 1)" by simp
wenzelm@32960
  1203
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
wenzelm@32960
  1204
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
wenzelm@32960
  1205
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
huffman@31706
  1206
            by simp
wenzelm@32960
  1207
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
wenzelm@32960
  1208
            by (simp only: mult_assoc right_distrib)
wenzelm@32960
  1209
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
huffman@31706
  1210
            by algebra
wenzelm@32960
  1211
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
wenzelm@32960
  1212
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
wenzelm@32960
  1213
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
wenzelm@32960
  1214
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
wenzelm@32960
  1215
            by (simp only: diff_mult_distrib2 add_commute mult_ac)
wenzelm@32960
  1216
          hence ?thesis using H(1,2)
wenzelm@32960
  1217
            apply -
wenzelm@32960
  1218
            apply (rule exI[where x=d], simp)
wenzelm@32960
  1219
            apply (rule exI[where x="(b - 1) * y"])
wenzelm@32960
  1220
            by (rule exI[where x="x*(b - 1) - d"], simp)}
wenzelm@32960
  1221
        ultimately have ?thesis by blast}
chaieb@27669
  1222
    ultimately have ?thesis by blast}
chaieb@27669
  1223
  ultimately have ?thesis by blast}
chaieb@27669
  1224
 ultimately show ?thesis by blast
chaieb@27669
  1225
qed
chaieb@27669
  1226
nipkow@31952
  1227
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
chaieb@27669
  1228
  shows "\<exists>x y. a * x = b * y + gcd a b"
chaieb@27669
  1229
proof-
chaieb@27669
  1230
  let ?g = "gcd a b"
nipkow@31952
  1231
  from bezout_add_strong_nat[OF a, of b]
chaieb@27669
  1232
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
chaieb@27669
  1233
  from d(1,2) have "d dvd ?g" by simp
chaieb@27669
  1234
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
huffman@31706
  1235
  from d(3) have "a * x * k = (b * y + d) *k " by auto
chaieb@27669
  1236
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
chaieb@27669
  1237
  thus ?thesis by blast
chaieb@27669
  1238
qed
chaieb@27669
  1239
huffman@31706
  1240
huffman@31706
  1241
subsection {* LCM *}
huffman@31706
  1242
nipkow@31952
  1243
lemma lcm_altdef_int: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
huffman@31706
  1244
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
huffman@31706
  1245
    zmult_int [symmetric] gcd_int_def)
huffman@31706
  1246
nipkow@31952
  1247
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
huffman@31706
  1248
  unfolding lcm_nat_def
nipkow@31952
  1249
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
huffman@31706
  1250
nipkow@31952
  1251
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
huffman@31706
  1252
  unfolding lcm_int_def gcd_int_def
huffman@31706
  1253
  apply (subst int_mult [symmetric])
nipkow@31952
  1254
  apply (subst prod_gcd_lcm_nat [symmetric])
huffman@31706
  1255
  apply (subst nat_abs_mult_distrib [symmetric])
huffman@31706
  1256
  apply (simp, simp add: abs_mult)
huffman@31706
  1257
done
huffman@31706
  1258
nipkow@31952
  1259
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
huffman@31706
  1260
  unfolding lcm_nat_def by simp
huffman@31706
  1261
nipkow@31952
  1262
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
huffman@31706
  1263
  unfolding lcm_int_def by simp
huffman@31706
  1264
nipkow@31952
  1265
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
huffman@31706
  1266
  unfolding lcm_nat_def by simp
chaieb@27669
  1267
nipkow@31952
  1268
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
huffman@31706
  1269
  unfolding lcm_int_def by simp
huffman@31706
  1270
nipkow@31952
  1271
lemma lcm_commute_nat: "lcm (m::nat) n = lcm n m"
nipkow@31952
  1272
  unfolding lcm_nat_def by (simp add: gcd_commute_nat ring_simps)
huffman@31706
  1273
nipkow@31952
  1274
lemma lcm_commute_int: "lcm (m::int) n = lcm n m"
nipkow@31952
  1275
  unfolding lcm_int_def by (subst lcm_commute_nat, rule refl)
huffman@31706
  1276
huffman@31706
  1277
nipkow@31952
  1278
lemma lcm_pos_nat:
nipkow@31798
  1279
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1280
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
chaieb@27669
  1281
nipkow@31952
  1282
lemma lcm_pos_int:
nipkow@31798
  1283
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
nipkow@31952
  1284
  apply (subst lcm_abs_int)
nipkow@31952
  1285
  apply (rule lcm_pos_nat [transferred])
nipkow@31798
  1286
  apply auto
huffman@31706
  1287
done
haftmann@23687
  1288
nipkow@31952
  1289
lemma dvd_pos_nat:
haftmann@23687
  1290
  fixes n m :: nat
haftmann@23687
  1291
  assumes "n > 0" and "m dvd n"
haftmann@23687
  1292
  shows "m > 0"
haftmann@23687
  1293
using assms by (cases m) auto
haftmann@23687
  1294
nipkow@31952
  1295
lemma lcm_least_nat:
huffman@31706
  1296
  assumes "(m::nat) dvd k" and "n dvd k"
haftmann@27556
  1297
  shows "lcm m n dvd k"
haftmann@23687
  1298
proof (cases k)
haftmann@23687
  1299
  case 0 then show ?thesis by auto
haftmann@23687
  1300
next
haftmann@23687
  1301
  case (Suc _) then have pos_k: "k > 0" by auto
nipkow@31952
  1302
  from assms dvd_pos_nat [OF this] have pos_mn: "m > 0" "n > 0" by auto
nipkow@31952
  1303
  with gcd_zero_nat [of m n] have pos_gcd: "gcd m n > 0" by simp
haftmann@23687
  1304
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
haftmann@23687
  1305
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
haftmann@23687
  1306
  from pos_k k_m have pos_p: "p > 0" by auto
haftmann@23687
  1307
  from pos_k k_n have pos_q: "q > 0" by auto
haftmann@27556
  1308
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
nipkow@31952
  1309
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1310
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
haftmann@23687
  1311
    by (simp add: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1312
  also have "\<dots> = k * p * q * gcd m n"
nipkow@31952
  1313
    by (simp add: mult_ac gcd_mult_distrib_nat)
haftmann@27556
  1314
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
haftmann@23687
  1315
    by (simp only: k_m [symmetric] k_n [symmetric])
haftmann@27556
  1316
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
haftmann@23687
  1317
    by (simp add: mult_ac)
haftmann@27556
  1318
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
haftmann@23687
  1319
    by simp
nipkow@31952
  1320
  with prod_gcd_lcm_nat [of m n]
haftmann@27556
  1321
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
haftmann@23687
  1322
    by (simp add: mult_ac)
huffman@31706
  1323
  with pos_gcd have "lcm m n * gcd q p = k" by auto
haftmann@23687
  1324
  then show ?thesis using dvd_def by auto
haftmann@23687
  1325
qed
haftmann@23687
  1326
nipkow@31952
  1327
lemma lcm_least_int:
nipkow@31798
  1328
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"
nipkow@31952
  1329
apply (subst lcm_abs_int)
nipkow@31798
  1330
apply (rule dvd_trans)
nipkow@31952
  1331
apply (rule lcm_least_nat [transferred, of _ "abs k" _])
nipkow@31798
  1332
apply auto
huffman@31706
  1333
done
huffman@31706
  1334
nipkow@31952
  1335
lemma lcm_dvd1_nat: "(m::nat) dvd lcm m n"
haftmann@23687
  1336
proof (cases m)
haftmann@23687
  1337
  case 0 then show ?thesis by simp
haftmann@23687
  1338
next
haftmann@23687
  1339
  case (Suc _)
haftmann@23687
  1340
  then have mpos: "m > 0" by simp
haftmann@23687
  1341
  show ?thesis
haftmann@23687
  1342
  proof (cases n)
haftmann@23687
  1343
    case 0 then show ?thesis by simp
haftmann@23687
  1344
  next
haftmann@23687
  1345
    case (Suc _)
haftmann@23687
  1346
    then have npos: "n > 0" by simp
haftmann@27556
  1347
    have "gcd m n dvd n" by simp
haftmann@27556
  1348
    then obtain k where "n = gcd m n * k" using dvd_def by auto
huffman@31706
  1349
    then have "m * n div gcd m n = m * (gcd m n * k) div gcd m n"
huffman@31706
  1350
      by (simp add: mult_ac)
nipkow@31952
  1351
    also have "\<dots> = m * k" using mpos npos gcd_zero_nat by simp
huffman@31706
  1352
    finally show ?thesis by (simp add: lcm_nat_def)
haftmann@23687
  1353
  qed
haftmann@23687
  1354
qed
haftmann@23687
  1355
nipkow@31952
  1356
lemma lcm_dvd1_int: "(m::int) dvd lcm m n"
nipkow@31952
  1357
  apply (subst lcm_abs_int)
huffman@31706
  1358
  apply (rule dvd_trans)
huffman@31706
  1359
  prefer 2
nipkow@31952
  1360
  apply (rule lcm_dvd1_nat [transferred])
huffman@31706
  1361
  apply auto
huffman@31706
  1362
done
huffman@31706
  1363
nipkow@31952
  1364
lemma lcm_dvd2_nat: "(n::nat) dvd lcm m n"
nipkow@31952
  1365
  by (subst lcm_commute_nat, rule lcm_dvd1_nat)
huffman@31706
  1366
nipkow@31952
  1367
lemma lcm_dvd2_int: "(n::int) dvd lcm m n"
nipkow@31952
  1368
  by (subst lcm_commute_int, rule lcm_dvd1_int)
huffman@31706
  1369
nipkow@31730
  1370
lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1371
by(metis lcm_dvd1_nat dvd_trans)
nipkow@31729
  1372
nipkow@31730
  1373
lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
nipkow@31952
  1374
by(metis lcm_dvd2_nat dvd_trans)
nipkow@31729
  1375
nipkow@31730
  1376
lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1377
by(metis lcm_dvd1_int dvd_trans)
nipkow@31729
  1378
nipkow@31730
  1379
lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
nipkow@31952
  1380
by(metis lcm_dvd2_int dvd_trans)
nipkow@31729
  1381
nipkow@31952
  1382
lemma lcm_unique_nat: "(a::nat) dvd d \<and> b dvd d \<and>
huffman@31706
  1383
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31952
  1384
  by (auto intro: dvd_anti_sym lcm_least_nat lcm_dvd1_nat lcm_dvd2_nat)
chaieb@27568
  1385
nipkow@31952
  1386
lemma lcm_unique_int: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
huffman@31706
  1387
    (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
nipkow@31952
  1388
  by (auto intro: dvd_anti_sym [transferred] lcm_least_int)
huffman@31706
  1389
nipkow@31798
  1390
lemma lcm_proj2_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm x y = y"
huffman@31706
  1391
  apply (rule sym)
nipkow@31952
  1392
  apply (subst lcm_unique_nat [symmetric])
huffman@31706
  1393
  apply auto
huffman@31706
  1394
done
huffman@31706
  1395
nipkow@31798
  1396
lemma lcm_proj2_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm x y = abs y"
huffman@31706
  1397
  apply (rule sym)
nipkow@31952
  1398
  apply (subst lcm_unique_int [symmetric])
huffman@31706
  1399
  apply auto
huffman@31706
  1400
done
huffman@31706
  1401
nipkow@31798
  1402
lemma lcm_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> lcm y x = y"
nipkow@31952
  1403
by (subst lcm_commute_nat, erule lcm_proj2_if_dvd_nat)
huffman@31706
  1404
nipkow@31798
  1405
lemma lcm_proj1_if_dvd_int [simp]: "(x::int) dvd y \<Longrightarrow> lcm y x = abs y"
nipkow@31952
  1406
by (subst lcm_commute_int, erule lcm_proj2_if_dvd_int)
huffman@31706
  1407
nipkow@31992
  1408
lemma lcm_proj1_iff_nat[simp]: "lcm m n = (m::nat) \<longleftrightarrow> n dvd m"
nipkow@31992
  1409
by (metis lcm_proj1_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1410
nipkow@31992
  1411
lemma lcm_proj2_iff_nat[simp]: "lcm m n = (n::nat) \<longleftrightarrow> m dvd n"
nipkow@31992
  1412
by (metis lcm_proj2_if_dvd_nat lcm_unique_nat)
nipkow@31992
  1413
nipkow@31992
  1414
lemma lcm_proj1_iff_int[simp]: "lcm m n = abs(m::int) \<longleftrightarrow> n dvd m"
nipkow@31992
  1415
by (metis dvd_abs_iff lcm_proj1_if_dvd_int lcm_unique_int)
nipkow@31992
  1416
nipkow@31992
  1417
lemma lcm_proj2_iff_int[simp]: "lcm m n = abs(n::int) \<longleftrightarrow> m dvd n"
nipkow@31992
  1418
by (metis dvd_abs_iff lcm_proj2_if_dvd_int lcm_unique_int)
chaieb@27568
  1419
nipkow@31766
  1420
lemma lcm_assoc_nat: "lcm (lcm n m) (p::nat) = lcm n (lcm m p)"
nipkow@31992
  1421
by(rule lcm_unique_nat[THEN iffD1])(metis dvd.order_trans lcm_unique_nat)
nipkow@31766
  1422
nipkow@31766
  1423
lemma lcm_assoc_int: "lcm (lcm n m) (p::int) = lcm n (lcm m p)"
nipkow@31992
  1424
by(rule lcm_unique_int[THEN iffD1])(metis dvd_trans lcm_unique_int)
nipkow@31766
  1425
nipkow@31992
  1426
lemmas lcm_left_commute_nat = mk_left_commute[of lcm, OF lcm_assoc_nat lcm_commute_nat]
nipkow@31992
  1427
lemmas lcm_left_commute_int = mk_left_commute[of lcm, OF lcm_assoc_int lcm_commute_int]
nipkow@31766
  1428
nipkow@31952
  1429
lemmas lcm_ac_nat = lcm_assoc_nat lcm_commute_nat lcm_left_commute_nat
nipkow@31952
  1430
lemmas lcm_ac_int = lcm_assoc_int lcm_commute_int lcm_left_commute_int
nipkow@31766
  1431
nipkow@31992
  1432
lemma fun_left_comm_idem_gcd_nat: "fun_left_comm_idem (gcd :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1433
proof qed (auto simp add: gcd_ac_nat)
nipkow@31992
  1434
nipkow@31992
  1435
lemma fun_left_comm_idem_gcd_int: "fun_left_comm_idem (gcd :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1436
proof qed (auto simp add: gcd_ac_int)
nipkow@31992
  1437
nipkow@31992
  1438
lemma fun_left_comm_idem_lcm_nat: "fun_left_comm_idem (lcm :: nat\<Rightarrow>nat\<Rightarrow>nat)"
nipkow@31992
  1439
proof qed (auto simp add: lcm_ac_nat)
nipkow@31992
  1440
nipkow@31992
  1441
lemma fun_left_comm_idem_lcm_int: "fun_left_comm_idem (lcm :: int\<Rightarrow>int\<Rightarrow>int)"
nipkow@31992
  1442
proof qed (auto simp add: lcm_ac_int)
nipkow@31992
  1443
haftmann@23687
  1444
nipkow@31995
  1445
(* FIXME introduce selimattice_bot/top and derive the following lemmas in there: *)
nipkow@31995
  1446
nipkow@31995
  1447
lemma lcm_0_iff_nat[simp]: "lcm (m::nat) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1448
by (metis lcm_0_left_nat lcm_0_nat mult_is_0 prod_gcd_lcm_nat)
nipkow@31995
  1449
nipkow@31995
  1450
lemma lcm_0_iff_int[simp]: "lcm (m::int) n = 0 \<longleftrightarrow> m=0 \<or> n=0"
nipkow@31995
  1451
by (metis lcm_0_int lcm_0_left_int lcm_pos_int zless_le)
nipkow@31995
  1452
nipkow@31995
  1453
lemma lcm_1_iff_nat[simp]: "lcm (m::nat) n = 1 \<longleftrightarrow> m=1 \<and> n=1"
nipkow@31995
  1454
by (metis gcd_1_nat lcm_unique_nat nat_mult_1 prod_gcd_lcm_nat)
nipkow@31995
  1455
nipkow@31995
  1456
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
  1457
by (auto simp add: abs_mult_self trans [OF lcm_unique_int eq_commute, symmetric] zmult_eq_1_iff)
nipkow@31995
  1458
nipkow@32112
  1459
subsubsection {* The complete divisibility lattice *}
nipkow@32112
  1460
nipkow@32112
  1461
nipkow@32112
  1462
interpretation gcd_semilattice_nat: lower_semilattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd
nipkow@32112
  1463
proof
nipkow@32112
  1464
  case goal3 thus ?case by(metis gcd_unique_nat)
nipkow@32112
  1465
qed auto
nipkow@32112
  1466
nipkow@32112
  1467
interpretation lcm_semilattice_nat: upper_semilattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" lcm
nipkow@32112
  1468
proof
nipkow@32112
  1469
  case goal3 thus ?case by(metis lcm_unique_nat)
nipkow@32112
  1470
qed auto
nipkow@32112
  1471
nipkow@32112
  1472
interpretation gcd_lcm_lattice_nat: lattice "op dvd" "(%m n::nat. m dvd n & ~ n dvd m)" gcd lcm ..
nipkow@32112
  1473
nipkow@32112
  1474
text{* Lifting gcd and lcm to finite (Gcd/Lcm) and infinite sets (GCD/LCM).
nipkow@32112
  1475
GCD is defined via LCM to facilitate the proof that we have a complete lattice.
nipkow@32112
  1476
Later on we show that GCD and Gcd coincide on finite sets.
nipkow@32112
  1477
*}
nipkow@32112
  1478
context gcd
nipkow@32112
  1479
begin
nipkow@32112
  1480
nipkow@32112
  1481
definition Gcd :: "'a set \<Rightarrow> 'a"
nipkow@32112
  1482
where "Gcd = fold gcd 0"
nipkow@32112
  1483
nipkow@32112
  1484
definition Lcm :: "'a set \<Rightarrow> 'a"
nipkow@32112
  1485
where "Lcm = fold lcm 1"
nipkow@32112
  1486
nipkow@32112
  1487
definition LCM :: "'a set \<Rightarrow> 'a" where
nipkow@32112
  1488
"LCM M = (if finite M then Lcm M else 0)"
nipkow@32112
  1489
nipkow@32112
  1490
definition GCD :: "'a set \<Rightarrow> 'a" where
nipkow@32112
  1491
"GCD M = LCM(INT m:M. {d. d dvd m})"
nipkow@32112
  1492
nipkow@32112
  1493
end
nipkow@32112
  1494
nipkow@32112
  1495
lemma Gcd_empty[simp]: "Gcd {} = 0"
nipkow@32112
  1496
by(simp add:Gcd_def)
nipkow@32112
  1497
nipkow@32112
  1498
lemma Lcm_empty[simp]: "Lcm {} = 1"
nipkow@32112
  1499
by(simp add:Lcm_def)
nipkow@32112
  1500
nipkow@32112
  1501
lemma GCD_empty_nat[simp]: "GCD {} = (0::nat)"
nipkow@32112
  1502
by(simp add:GCD_def LCM_def)
nipkow@32112
  1503
nipkow@32112
  1504
lemma LCM_eq_Lcm[simp]: "finite M \<Longrightarrow> LCM M = Lcm M"
nipkow@32112
  1505
by(simp add:LCM_def)
nipkow@32112
  1506
nipkow@32112
  1507
lemma Lcm_insert_nat [simp]:
nipkow@32112
  1508
  assumes "finite N"
nipkow@32112
  1509
  shows "Lcm (insert (n::nat) N) = lcm n (Lcm N)"
nipkow@32112
  1510
proof -
nipkow@32112
  1511
  interpret fun_left_comm_idem "lcm::nat\<Rightarrow>nat\<Rightarrow>nat"
nipkow@32112
  1512
    by (rule fun_left_comm_idem_lcm_nat)
nipkow@32112
  1513
  from assms show ?thesis by(simp add: Lcm_def)
nipkow@32112
  1514
qed
nipkow@32112
  1515
nipkow@32112
  1516
lemma Lcm_insert_int [simp]:
nipkow@32112
  1517
  assumes "finite N"
nipkow@32112
  1518
  shows "Lcm (insert (n::int) N) = lcm n (Lcm N)"
nipkow@32112
  1519
proof -
nipkow@32112
  1520
  interpret fun_left_comm_idem "lcm::int\<Rightarrow>int\<Rightarrow>int"
nipkow@32112
  1521
    by (rule fun_left_comm_idem_lcm_int)
nipkow@32112
  1522
  from assms show ?thesis by(simp add: Lcm_def)
nipkow@32112
  1523
qed
nipkow@32112
  1524
nipkow@32112
  1525
lemma Gcd_insert_nat [simp]:
nipkow@32112
  1526
  assumes "finite N"
nipkow@32112
  1527
  shows "Gcd (insert (n::nat) N) = gcd n (Gcd N)"
nipkow@32112
  1528
proof -
nipkow@32112
  1529
  interpret fun_left_comm_idem "gcd::nat\<Rightarrow>nat\<Rightarrow>nat"
nipkow@32112
  1530
    by (rule fun_left_comm_idem_gcd_nat)
nipkow@32112
  1531
  from assms show ?thesis by(simp add: Gcd_def)
nipkow@32112
  1532
qed
nipkow@32112
  1533
nipkow@32112
  1534
lemma Gcd_insert_int [simp]:
nipkow@32112
  1535
  assumes "finite N"
nipkow@32112
  1536
  shows "Gcd (insert (n::int) N) = gcd n (Gcd N)"
nipkow@32112
  1537
proof -
nipkow@32112
  1538
  interpret fun_left_comm_idem "gcd::int\<Rightarrow>int\<Rightarrow>int"
nipkow@32112
  1539
    by (rule fun_left_comm_idem_gcd_int)
nipkow@32112
  1540
  from assms show ?thesis by(simp add: Gcd_def)
nipkow@32112
  1541
qed
nipkow@32112
  1542
nipkow@32112
  1543
lemma Lcm0_iff[simp]: "finite (M::nat set) \<Longrightarrow> M \<noteq> {} \<Longrightarrow> Lcm M = 0 \<longleftrightarrow> 0 : M"
nipkow@32112
  1544
by(induct rule:finite_ne_induct) auto
nipkow@32112
  1545
nipkow@32112
  1546
lemma Lcm_eq_0[simp]: "finite (M::nat set) \<Longrightarrow> 0 : M \<Longrightarrow> Lcm M = 0"
nipkow@32112
  1547
by (metis Lcm0_iff empty_iff)
nipkow@32112
  1548
nipkow@32112
  1549
lemma Gcd_dvd_nat [simp]:
nipkow@32112
  1550
  assumes "finite M" and "(m::nat) \<in> M"
nipkow@32112
  1551
  shows "Gcd M dvd m"
nipkow@32112
  1552
proof -
nipkow@32112
  1553
  show ?thesis using gcd_semilattice_nat.fold_inf_le_inf[OF assms, of 0] by (simp add: Gcd_def)
nipkow@32112
  1554
qed
nipkow@32112
  1555
nipkow@32112
  1556
lemma dvd_Gcd_nat[simp]:
nipkow@32112
  1557
  assumes "finite M" and "ALL (m::nat) : M. n dvd m"
nipkow@32112
  1558
  shows "n dvd Gcd M"
nipkow@32112
  1559
proof -
nipkow@32112
  1560
  show ?thesis using gcd_semilattice_nat.inf_le_fold_inf[OF assms, of 0] by (simp add: Gcd_def)
nipkow@32112
  1561
qed
nipkow@32112
  1562
nipkow@32112
  1563
lemma dvd_Lcm_nat [simp]:
nipkow@32112
  1564
  assumes "finite M" and "(m::nat) \<in> M"
nipkow@32112
  1565
  shows "m dvd Lcm M"
nipkow@32112
  1566
proof -
nipkow@32112
  1567
  show ?thesis using lcm_semilattice_nat.sup_le_fold_sup[OF assms, of 1] by (simp add: Lcm_def)
nipkow@32112
  1568
qed
nipkow@32112
  1569
nipkow@32112
  1570
lemma Lcm_dvd_nat[simp]:
nipkow@32112
  1571
  assumes "finite M" and "ALL (m::nat) : M. m dvd n"
nipkow@32112
  1572
  shows "Lcm M dvd n"
nipkow@32112
  1573
proof -
nipkow@32112
  1574
  show ?thesis using lcm_semilattice_nat.fold_sup_le_sup[OF assms, of 1] by (simp add: Lcm_def)
nipkow@32112
  1575
qed
nipkow@32112
  1576
nipkow@32112
  1577
interpretation gcd_lcm_complete_lattice_nat:
haftmann@32879
  1578
  complete_lattice GCD LCM "op dvd" "%m n::nat. m dvd n & ~ n dvd m" gcd lcm 1 0
nipkow@32112
  1579
proof
nipkow@32112
  1580
  case goal1 show ?case by simp
nipkow@32112
  1581
next
nipkow@32112
  1582
  case goal2 show ?case by simp
nipkow@32112
  1583
next
nipkow@32112
  1584
  case goal5 thus ?case by (auto simp: LCM_def)
nipkow@32112
  1585
next
nipkow@32112
  1586
  case goal6 thus ?case
nipkow@32112
  1587
    by(auto simp: LCM_def)(metis finite_nat_set_iff_bounded_le gcd_proj2_if_dvd_nat gcd_le1_nat)
nipkow@32112
  1588
next
nipkow@32112
  1589
  case goal3 thus ?case by (auto simp: GCD_def LCM_def)(metis finite_INT finite_divisors_nat)
nipkow@32112
  1590
next
nipkow@32112
  1591
  case goal4 thus ?case by(auto simp: LCM_def GCD_def)
nipkow@32112
  1592
qed
nipkow@32112
  1593
nipkow@32112
  1594
text{* Alternative characterizations of Gcd and GCD: *}
nipkow@32112
  1595
nipkow@32112
  1596
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
  1597
apply(rule antisym)
nipkow@32112
  1598
 apply(rule Max_ge)
nipkow@32112
  1599
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@32112
  1600
 apply simp
nipkow@32112
  1601
apply (rule Max_le_iff[THEN iffD2])
nipkow@32112
  1602
  apply (metis all_not_in_conv finite_divisors_nat finite_INT)
nipkow@32112
  1603
 apply fastsimp
nipkow@32112
  1604
apply clarsimp
nipkow@32112
  1605
apply (metis Gcd_dvd_nat Max_in dvd_0_left dvd_Gcd_nat dvd_imp_le linorder_antisym_conv3 not_less0)
nipkow@32112
  1606
done
nipkow@32112
  1607
nipkow@32112
  1608
lemma Gcd_remove0_nat: "finite M \<Longrightarrow> Gcd M = Gcd (M - {0::nat})"
nipkow@32112
  1609
apply(induct pred:finite)
nipkow@32112
  1610
 apply simp
nipkow@32112
  1611
apply(case_tac "x=0")
nipkow@32112
  1612
 apply simp
nipkow@32112
  1613
apply(subgoal_tac "insert x F - {0} = insert x (F - {0})")
nipkow@32112
  1614
 apply simp
nipkow@32112
  1615
apply blast
nipkow@32112
  1616
done
nipkow@32112
  1617
nipkow@32112
  1618
lemma Lcm_in_lcm_closed_set_nat:
nipkow@32112
  1619
  "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
  1620
apply(induct rule:finite_linorder_min_induct)
nipkow@32112
  1621
 apply simp
nipkow@32112
  1622
apply simp
nipkow@32112
  1623
apply(subgoal_tac "ALL m n :: nat. m:A \<longrightarrow> n:A \<longrightarrow> lcm m n : A")
nipkow@32112
  1624
 apply simp
nipkow@32112
  1625
 apply(case_tac "A={}")
nipkow@32112
  1626
  apply simp
nipkow@32112
  1627
 apply simp
nipkow@32112
  1628
apply (metis lcm_pos_nat lcm_unique_nat linorder_neq_iff nat_dvd_not_less not_less0)
nipkow@32112
  1629
done
nipkow@32112
  1630
nipkow@32112
  1631
lemma Lcm_eq_Max_nat:
nipkow@32112
  1632
  "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
  1633
apply(rule antisym)
nipkow@32112
  1634
 apply(rule Max_ge, assumption)
nipkow@32112
  1635
 apply(erule (2) Lcm_in_lcm_closed_set_nat)
nipkow@32112
  1636
apply clarsimp
nipkow@32112
  1637
apply (metis Lcm0_iff dvd_Lcm_nat dvd_imp_le neq0_conv)
nipkow@32112
  1638
done
nipkow@32112
  1639
nipkow@32112
  1640
text{* Finally GCD is Gcd: *}
nipkow@32112
  1641
nipkow@32112
  1642
lemma GCD_eq_Gcd[simp]: assumes "finite(M::nat set)" shows "GCD M = Gcd M"
nipkow@32112
  1643
proof-
nipkow@32112
  1644
  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
  1645
  show ?thesis
nipkow@32112
  1646
  proof cases
nipkow@32112
  1647
    assume "M={}" thus ?thesis by simp
nipkow@32112
  1648
  next
nipkow@32112
  1649
    assume "M\<noteq>{}"
nipkow@32112
  1650
    show ?thesis
nipkow@32112
  1651
    proof cases
nipkow@32112
  1652
      assume "M={0}" thus ?thesis by(simp add:GCD_def LCM_def)
nipkow@32112
  1653
    next
nipkow@32112
  1654
      assume "M\<noteq>{0}"
nipkow@32112
  1655
      with `M\<noteq>{}` assms show ?thesis
wenzelm@32960
  1656
        apply(subst Gcd_remove0_nat[OF assms])
wenzelm@32960
  1657
        apply(simp add:GCD_def)
wenzelm@32960
  1658
        apply(subst divisors_remove0_nat)
wenzelm@32960
  1659
        apply(simp add:LCM_def)
wenzelm@32960
  1660
        apply rule
wenzelm@32960
  1661
         apply rule
wenzelm@32960
  1662
         apply(subst Gcd_eq_Max)
wenzelm@32960
  1663
            apply simp
wenzelm@32960
  1664
           apply blast
wenzelm@32960
  1665
          apply blast
wenzelm@32960
  1666
         apply(rule Lcm_eq_Max_nat)
wenzelm@32960
  1667
            apply simp
wenzelm@32960
  1668
           apply blast
wenzelm@32960
  1669
          apply fastsimp
wenzelm@32960
  1670
         apply clarsimp
wenzelm@32960
  1671
        apply(fastsimp intro: finite_divisors_nat intro!: finite_INT)
wenzelm@32960
  1672
        done
nipkow@32112
  1673
    qed
nipkow@32112
  1674
  qed
nipkow@32112
  1675
qed
nipkow@32112
  1676
nipkow@32112
  1677
lemma Lcm_set_nat [code_unfold]:
nipkow@32112
  1678
  "Lcm (set ns) = foldl lcm (1::nat) ns"
nipkow@32112
  1679
proof -
nipkow@32112
  1680
  interpret fun_left_comm_idem "lcm::nat\<Rightarrow>nat\<Rightarrow>nat" by (rule fun_left_comm_idem_lcm_nat)
nipkow@32112
  1681
  show ?thesis by(simp add: Lcm_def fold_set lcm_commute_nat)
nipkow@32112
  1682
qed
nipkow@32112
  1683
nipkow@32112
  1684
lemma Lcm_set_int [code_unfold]:
nipkow@32112
  1685
  "Lcm (set is) = foldl lcm (1::int) is"
nipkow@32112
  1686
proof -
nipkow@32112
  1687
  interpret fun_left_comm_idem "lcm::int\<Rightarrow>int\<Rightarrow>int" by (rule fun_left_comm_idem_lcm_int)
nipkow@32112
  1688
  show ?thesis by(simp add: Lcm_def fold_set lcm_commute_int)
nipkow@32112
  1689
qed
nipkow@32112
  1690
nipkow@32112
  1691
lemma Gcd_set_nat [code_unfold]:
nipkow@32112
  1692
  "Gcd (set ns) = foldl gcd (0::nat) ns"
nipkow@32112
  1693
proof -
nipkow@32112
  1694
  interpret fun_left_comm_idem "gcd::nat\<Rightarrow>nat\<Rightarrow>nat" by (rule fun_left_comm_idem_gcd_nat)
nipkow@32112
  1695
  show ?thesis by(simp add: Gcd_def fold_set gcd_commute_nat)
nipkow@32112
  1696
qed
nipkow@32112
  1697
nipkow@32112
  1698
lemma Gcd_set_int [code_unfold]:
nipkow@32112
  1699
  "Gcd (set ns) = foldl gcd (0::int) ns"
nipkow@32112
  1700
proof -
nipkow@32112
  1701
  interpret fun_left_comm_idem "gcd::int\<Rightarrow>int\<Rightarrow>int" by (rule fun_left_comm_idem_gcd_int)
nipkow@32112
  1702
  show ?thesis by(simp add: Gcd_def fold_set gcd_commute_int)
nipkow@32112
  1703
qed
nipkow@32112
  1704
blanchet@33197
  1705
lemma gcd_eq_nitpick_gcd [nitpick_def]: "gcd x y \<equiv> Nitpick.nat_gcd x y"
blanchet@33197
  1706
apply (rule eq_reflection)
blanchet@33197
  1707
apply (induct x y rule: nat_gcd.induct)
blanchet@33197
  1708
by (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
blanchet@33197
  1709
blanchet@33197
  1710
lemma lcm_eq_nitpick_lcm [nitpick_def]: "lcm x y \<equiv> Nitpick.nat_lcm x y"
blanchet@33197
  1711
by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
blanchet@33197
  1712
wenzelm@21256
  1713
end