src/HOL/GCD.thy
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(*  Authors:    Christophe Tabacznyj, Lawrence C. Paulson, Amine Chaieb,
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                Thomas M. Rasmussen, Jeremy Avigad, Tobias Nipkow
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This file deals with the functions gcd and lcm.  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 Chaieb.
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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 {* Greatest common divisor and least common multiple *}
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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 and LCM definitions *}
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class gcd = zero + one + dvd +
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  fixes gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
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    and 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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instantiation nat :: gcd
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begin
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fun
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  gcd_nat  :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "gcd_nat x y =
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   (if y = 0 then x else gcd y (x mod y))"
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definition
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  lcm_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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where
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  "lcm_nat x y = x * y div (gcd x y)"
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instance proof qed
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end
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instantiation 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 {* Transfer setup *}
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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 transfer_morphism_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 transfer_morphism_int_nat[transfer add return:
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    transfer_int_nat_gcd transfer_int_nat_gcd_closures]
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subsection {* GCD properties *}
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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 assms, auto, 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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  using assms by (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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   259
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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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   290
lemma gcd_zero_nat [simp]: "(gcd (m::nat) n = 0) = (m = 0 & n = 0)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   291
  by (simp only: dvd_0_left_iff [symmetric] gcd_greatest_iff_nat)
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parents:
diff changeset
   292
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
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parents: 31814
diff changeset
   293
lemma gcd_zero_int [simp]: "(gcd (m::int) n = 0) = (m = 0 & n = 0)"
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diff changeset
   294
  by (auto simp add: gcd_int_def)
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wenzelm
parents:
diff changeset
   295
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   296
lemma gcd_pos_nat [simp]: "(gcd (m::nat) n > 0) = (m ~= 0 | n ~= 0)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   297
  by (insert gcd_zero_nat [of m n], arith)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   298
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   299
lemma gcd_pos_int [simp]: "(gcd (m::int) n > 0) = (m ~= 0 | n ~= 0)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   300
  by (insert gcd_zero_int [of m n], insert gcd_ge_0_int [of m n], arith)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   301
37770
cddb3106adb8 avoid explicit mandatory prefix markers when prefixes are mandatory implicitly
haftmann
parents: 36350
diff changeset
   302
interpretation gcd_nat: abel_semigroup "gcd :: nat \<Rightarrow> nat \<Rightarrow> nat"
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   303
proof
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   304
qed (auto intro: dvd_antisym dvd_trans)
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parents: 30738
diff changeset
   305
37770
cddb3106adb8 avoid explicit mandatory prefix markers when prefixes are mandatory implicitly
haftmann
parents: 36350
diff changeset
   306
interpretation gcd_int: abel_semigroup "gcd :: int \<Rightarrow> int \<Rightarrow> int"
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   307
proof
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   308
qed (simp_all add: gcd_int_def gcd_nat.assoc gcd_nat.commute gcd_nat.left_commute)
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wenzelm
parents:
diff changeset
   309
34973
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   310
lemmas gcd_assoc_nat = gcd_nat.assoc
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   311
lemmas gcd_commute_nat = gcd_nat.commute
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   312
lemmas gcd_left_commute_nat = gcd_nat.left_commute
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   313
lemmas gcd_assoc_int = gcd_int.assoc
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   314
lemmas gcd_commute_int = gcd_int.commute
ae634fad947e dropped mk_left_commute; use interpretation of locale abel_semigroup instead
haftmann
parents: 34915
diff changeset
   315
lemmas gcd_left_commute_int = gcd_int.left_commute
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parents: 30738
diff changeset
   316
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   317
lemmas gcd_ac_nat = gcd_assoc_nat gcd_commute_nat gcd_left_commute_nat
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wenzelm
parents:
diff changeset
   318
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   319
lemmas gcd_ac_int = gcd_assoc_int gcd_commute_int gcd_left_commute_int
31706
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parents: 30738
diff changeset
   320
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   321
lemma gcd_unique_nat: "(d::nat) dvd a \<and> d dvd b \<and>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
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parents: 30738
diff changeset
   322
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   323
  apply auto
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   324
  apply (rule dvd_antisym)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   325
  apply (erule (1) gcd_greatest_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   326
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   327
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   328
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   329
lemma gcd_unique_int: "d >= 0 & (d::int) dvd a \<and> d dvd b \<and>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   330
    (\<forall>e. e dvd a \<and> e dvd b \<longrightarrow> e dvd d) \<longleftrightarrow> d = gcd a b"
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   331
apply (case_tac "d = 0")
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   332
 apply simp
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   333
apply (rule iffI)
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   334
 apply (rule zdvd_antisym_nonneg)
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   335
 apply (auto intro: gcd_greatest_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   336
done
30082
43c5b7bfc791 make more proofs work whether or not One_nat_def is a simp rule
huffman
parents: 30042
diff changeset
   337
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   338
lemma gcd_proj1_if_dvd_nat [simp]: "(x::nat) dvd y \<Longrightarrow> gcd x y = x"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   339
by (metis dvd.eq_iff gcd_unique_nat)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   340
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   341
lemma gcd_proj2_if_dvd_nat [simp]: "(y::nat) dvd x \<Longrightarrow> gcd x y = y"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   342
by (metis dvd.eq_iff gcd_unique_nat)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   343
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   344
lemma gcd_proj1_if_dvd_int[simp]: "x dvd y \<Longrightarrow> gcd (x::int) y = abs x"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   345
by (metis abs_dvd_iff abs_eq_0 gcd_0_left_int gcd_abs_int gcd_unique_int)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   346
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   347
lemma gcd_proj2_if_dvd_int[simp]: "y dvd x \<Longrightarrow> gcd (x::int) y = abs y"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   348
by (metis gcd_proj1_if_dvd_int gcd_commute_int)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   349
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   350
21256
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wenzelm
parents:
diff changeset
   351
text {*
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   352
  \medskip Multiplication laws
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   353
*}
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   354
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   355
lemma gcd_mult_distrib_nat: "(k::nat) * gcd m n = gcd (k * m) (k * n)"
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   356
    -- {* \cite[page 27]{davenport92} *}
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   357
  apply (induct m n rule: gcd_nat_induct)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   358
  apply simp
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   359
  apply (case_tac "k = 0")
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   360
  apply (simp_all add: mod_geq gcd_non_0_nat mod_mult_distrib2)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   361
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   362
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   363
lemma gcd_mult_distrib_int: "abs (k::int) * gcd m n = gcd (k * m) (k * n)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   364
  apply (subst (1 2) gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   365
  apply (subst (1 2) abs_mult)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   366
  apply (rule gcd_mult_distrib_nat [transferred])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   367
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   368
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   369
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   370
lemma coprime_dvd_mult_nat: "coprime (k::nat) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   371
  apply (insert gcd_mult_distrib_nat [of m k n])
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   372
  apply simp
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   373
  apply (erule_tac t = m in ssubst)
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   374
  apply simp
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   375
  done
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   376
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   377
lemma coprime_dvd_mult_int:
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   378
  "coprime (k::int) n \<Longrightarrow> k dvd m * n \<Longrightarrow> k dvd m"
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   379
apply (subst abs_dvd_iff [symmetric])
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   380
apply (subst dvd_abs_iff [symmetric])
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   381
apply (subst (asm) gcd_abs_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   382
apply (rule coprime_dvd_mult_nat [transferred])
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   383
    prefer 4 apply assumption
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   384
   apply auto
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   385
apply (subst abs_mult [symmetric], auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   386
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   387
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   388
lemma coprime_dvd_mult_iff_nat: "coprime (k::nat) n \<Longrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   389
    (k dvd m * n) = (k dvd m)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   390
  by (auto intro: coprime_dvd_mult_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   391
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   392
lemma coprime_dvd_mult_iff_int: "coprime (k::int) n \<Longrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   393
    (k dvd m * n) = (k dvd m)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   394
  by (auto intro: coprime_dvd_mult_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   395
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   396
lemma gcd_mult_cancel_nat: "coprime k n \<Longrightarrow> gcd ((k::nat) * m) n = gcd m n"
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33318
diff changeset
   397
  apply (rule dvd_antisym)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   398
  apply (rule gcd_greatest_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   399
  apply (rule_tac n = k in coprime_dvd_mult_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   400
  apply (simp add: gcd_assoc_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   401
  apply (simp add: gcd_commute_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   402
  apply (simp_all add: mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   403
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   404
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   405
lemma gcd_mult_cancel_int:
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   406
  "coprime (k::int) n \<Longrightarrow> gcd (k * m) n = gcd m n"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   407
apply (subst (1 2) gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   408
apply (subst abs_mult)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   409
apply (rule gcd_mult_cancel_nat [transferred], auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   410
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   411
35368
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   412
lemma coprime_crossproduct_nat:
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   413
  fixes a b c d :: nat
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   414
  assumes "coprime a d" and "coprime b c"
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   415
  shows "a * c = b * d \<longleftrightarrow> a = b \<and> c = d" (is "?lhs \<longleftrightarrow> ?rhs")
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   416
proof
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   417
  assume ?rhs then show ?lhs by simp
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   418
next
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   419
  assume ?lhs
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   420
  from `?lhs` have "a dvd b * d" by (auto intro: dvdI dest: sym)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   421
  with `coprime a d` have "a dvd b" by (simp add: coprime_dvd_mult_iff_nat)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   422
  from `?lhs` have "b dvd a * c" by (auto intro: dvdI dest: sym)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   423
  with `coprime b c` have "b dvd a" by (simp add: coprime_dvd_mult_iff_nat)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   424
  from `?lhs` have "c dvd d * b" by (auto intro: dvdI dest: sym simp add: mult_commute)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   425
  with `coprime b c` have "c dvd d" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   426
  from `?lhs` have "d dvd c * a" by (auto intro: dvdI dest: sym simp add: mult_commute)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   427
  with `coprime a d` have "d dvd c" by (simp add: coprime_dvd_mult_iff_nat gcd_commute_nat)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   428
  from `a dvd b` `b dvd a` have "a = b" by (rule Nat.dvd.antisym)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   429
  moreover from `c dvd d` `d dvd c` have "c = d" by (rule Nat.dvd.antisym)
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   430
  ultimately show ?rhs ..
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   431
qed
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   432
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   433
lemma coprime_crossproduct_int:
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   434
  fixes a b c d :: int
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   435
  assumes "coprime a d" and "coprime b c"
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   436
  shows "\<bar>a\<bar> * \<bar>c\<bar> = \<bar>b\<bar> * \<bar>d\<bar> \<longleftrightarrow> \<bar>a\<bar> = \<bar>b\<bar> \<and> \<bar>c\<bar> = \<bar>d\<bar>"
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   437
  using assms by (intro coprime_crossproduct_nat [transferred]) auto
19b340c3f1ff crossproduct coprimality lemmas
haftmann
parents: 35216
diff changeset
   438
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   439
text {* \medskip Addition laws *}
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   440
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   441
lemma gcd_add1_nat [simp]: "gcd ((m::nat) + n) n = gcd m n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   442
  apply (case_tac "n = 0")
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   443
  apply (simp_all add: gcd_non_0_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   444
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   445
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   446
lemma gcd_add2_nat [simp]: "gcd (m::nat) (m + n) = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   447
  apply (subst (1 2) gcd_commute_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   448
  apply (subst add_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   449
  apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   450
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   451
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   452
(* to do: add the other variations? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   453
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   454
lemma gcd_diff1_nat: "(m::nat) >= n \<Longrightarrow> gcd (m - n) n = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   455
  by (subst gcd_add1_nat [symmetric], auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   456
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   457
lemma gcd_diff2_nat: "(n::nat) >= m \<Longrightarrow> gcd (n - m) n = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   458
  apply (subst gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   459
  apply (subst gcd_diff1_nat [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   460
  apply auto
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   461
  apply (subst gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   462
  apply (subst gcd_diff1_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   463
  apply assumption
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   464
  apply (rule gcd_commute_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   465
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   466
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   467
lemma gcd_non_0_int: "(y::int) > 0 \<Longrightarrow> gcd x y = gcd y (x mod y)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   468
  apply (frule_tac b = y and a = x in pos_mod_sign)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   469
  apply (simp del: pos_mod_sign add: gcd_int_def abs_if nat_mod_distrib)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   470
  apply (auto simp add: gcd_non_0_nat nat_mod_distrib [symmetric]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   471
    zmod_zminus1_eq_if)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   472
  apply (frule_tac a = x in pos_mod_bound)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   473
  apply (subst (1 2) gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   474
  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2_nat
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   475
    nat_le_eq_zle)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   476
done
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   477
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   478
lemma gcd_red_int: "gcd (x::int) y = gcd y (x mod y)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   479
  apply (case_tac "y = 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   480
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   481
  apply (case_tac "y > 0")
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   482
  apply (subst gcd_non_0_int, auto)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   483
  apply (insert gcd_non_0_int [of "-y" "-x"])
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   484
  apply auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   485
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   486
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   487
lemma gcd_add1_int [simp]: "gcd ((m::int) + n) n = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   488
by (metis gcd_red_int mod_add_self1 zadd_commute)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   489
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   490
lemma gcd_add2_int [simp]: "gcd m ((m::int) + n) = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   491
by (metis gcd_add1_int gcd_commute_int zadd_commute)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   492
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   493
lemma gcd_add_mult_nat: "gcd (m::nat) (k * m + n) = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   494
by (metis mod_mult_self3 gcd_commute_nat gcd_red_nat)
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   495
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   496
lemma gcd_add_mult_int: "gcd (m::int) (k * m + n) = gcd m n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   497
by (metis gcd_commute_int gcd_red_int mod_mult_self1 zadd_commute)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   498
21256
47195501ecf7 moved theories Parity, GCD, Binomial to Library;
wenzelm
parents:
diff changeset
   499
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   500
(* to do: differences, and all variations of addition rules
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   501
    as simplification rules for nat and int *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   502
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   503
(* FIXME remove iff *)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   504
lemma gcd_dvd_prod_nat [iff]: "gcd (m::nat) n dvd k * n"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
   505
  using mult_dvd_mono [of 1] by auto
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   506
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   507
(* to do: add the three variations of these, and for ints? *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   508
31992
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31952
diff changeset
   509
lemma finite_divisors_nat[simp]:
f8aed98faae7 More about gcd/lcm, and some cleaning up
nipkow
parents: 31952
diff changeset
   510
  assumes "(m::nat) ~= 0" shows "finite{d. d dvd m}"
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   511
proof-
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   512
  have "finite{d. d <= m}" by(blast intro: bounded_nat_set_is_finite)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   513
  from finite_subset[OF _ this] show ?thesis using assms
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   514
    by(bestsimp intro!:dvd_imp_le)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   515
qed
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   516
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   517
lemma finite_divisors_int[simp]:
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   518
  assumes "(i::int) ~= 0" shows "finite{d. d dvd i}"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   519
proof-
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   520
  have "{d. abs d <= abs i} = {- abs i .. abs i}" by(auto simp:abs_if)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   521
  hence "finite{d. abs d <= abs i}" by simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   522
  from finite_subset[OF _ this] show ?thesis using assms
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   523
    by(bestsimp intro!:dvd_imp_le_int)
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   524
qed
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   525
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   526
lemma Max_divisors_self_nat[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::nat. d dvd n} = n"
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   527
apply(rule antisym)
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   528
 apply (fastsimp intro: Max_le_iff[THEN iffD2] simp: dvd_imp_le)
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   529
apply simp
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   530
done
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   531
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   532
lemma Max_divisors_self_int[simp]: "n\<noteq>0 \<Longrightarrow> Max{d::int. d dvd n} = abs n"
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   533
apply(rule antisym)
44278
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 42871
diff changeset
   534
 apply(rule Max_le_iff [THEN iffD2])
1220ecb81e8f observe distinction between sets and predicates more properly
haftmann
parents: 42871
diff changeset
   535
  apply (auto intro: abs_le_D1 dvd_imp_le_int)
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   536
done
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   537
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   538
lemma gcd_is_Max_divisors_nat:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   539
  "m ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> gcd (m::nat) n = (Max {d. d dvd m & d dvd n})"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   540
apply(rule Max_eqI[THEN sym])
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   541
  apply (metis finite_Collect_conjI finite_divisors_nat)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   542
 apply simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   543
 apply(metis Suc_diff_1 Suc_neq_Zero dvd_imp_le gcd_greatest_iff_nat gcd_pos_nat)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   544
apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   545
done
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   546
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   547
lemma gcd_is_Max_divisors_int:
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   548
  "m ~= 0 ==> n ~= 0 ==> gcd (m::int) n = (Max {d. d dvd m & d dvd n})"
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   549
apply(rule Max_eqI[THEN sym])
31995
8f37cf60b885 more gcd/lcm lemmas
nipkow
parents: 31992
diff changeset
   550
  apply (metis finite_Collect_conjI finite_divisors_int)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   551
 apply simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   552
 apply (metis gcd_greatest_iff_int gcd_pos_int zdvd_imp_le)
31734
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   553
apply simp
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   554
done
a4a79836d07b new lemmas
nipkow
parents: 31730
diff changeset
   555
34030
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
   556
lemma gcd_code_int [code]:
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
   557
  "gcd k l = \<bar>if l = (0::int) then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
   558
  by (simp add: gcd_int_def nat_mod_distrib gcd_non_0_nat)
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
   559
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   560
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   561
subsection {* Coprimality *}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   562
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   563
lemma div_gcd_coprime_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   564
  assumes nz: "(a::nat) \<noteq> 0 \<or> b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   565
  shows "coprime (a div gcd a b) (b div gcd a b)"
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   566
proof -
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
   567
  let ?g = "gcd a b"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   568
  let ?a' = "a div ?g"
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   569
  let ?b' = "b div ?g"
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
   570
  let ?g' = "gcd ?a' ?b'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   571
  have dvdg: "?g dvd a" "?g dvd b" by simp_all
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   572
  have dvdg': "?g' dvd ?a'" "?g' dvd ?b'" by simp_all
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   573
  from dvdg dvdg' obtain ka kb ka' kb' where
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   574
      kab: "a = ?g * ka" "b = ?g * kb" "?a' = ?g' * ka'" "?b' = ?g' * kb'"
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   575
    unfolding dvd_def by blast
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   576
  then have "?g * ?a' = (?g * ?g') * ka'" "?g * ?b' = (?g * ?g') * kb'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   577
    by simp_all
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   578
  then have dvdgg':"?g * ?g' dvd a" "?g* ?g' dvd b"
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   579
    by (auto simp add: dvd_mult_div_cancel [OF dvdg(1)]
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   580
      dvd_mult_div_cancel [OF dvdg(2)] dvd_def)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   581
  have "?g \<noteq> 0" using nz by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   582
  then have gp: "?g > 0" by arith
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   583
  from gcd_greatest_nat [OF dvdgg'] have "?g * ?g' dvd ?g" .
22367
6860f09242bf tuned document;
wenzelm
parents: 22027
diff changeset
   584
  with dvd_mult_cancel1 [OF gp] show "?g' = 1" by simp
22027
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   585
qed
e4a08629c4bd A few lemmas about relative primes when dividing trough gcd
chaieb
parents: 21404
diff changeset
   586
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   587
lemma div_gcd_coprime_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   588
  assumes nz: "(a::int) \<noteq> 0 \<or> b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   589
  shows "coprime (a div gcd a b) (b div gcd a b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   590
apply (subst (1 2 3) gcd_abs_int)
31813
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   591
apply (subst (1 2) abs_div)
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   592
  apply simp
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   593
 apply simp
4df828bbc411 gcd abs lemmas
nipkow
parents: 31798
diff changeset
   594
apply(subst (1 2) abs_gcd_int)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   595
apply (rule div_gcd_coprime_nat [transferred])
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   596
using nz apply (auto simp add: gcd_abs_int [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   597
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   598
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   599
lemma coprime_nat: "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   600
  using gcd_unique_nat[of 1 a b, simplified] by auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   601
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   602
lemma coprime_Suc_0_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   603
    "coprime (a::nat) b \<longleftrightarrow> (\<forall>d. d dvd a \<and> d dvd b \<longleftrightarrow> d = Suc 0)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   604
  using coprime_nat by (simp add: One_nat_def)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   605
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   606
lemma coprime_int: "coprime (a::int) b \<longleftrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   607
    (\<forall>d. d >= 0 \<and> d dvd a \<and> d dvd b \<longleftrightarrow> d = 1)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   608
  using gcd_unique_int [of 1 a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   609
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   610
  apply (erule subst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   611
  apply (rule iffI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   612
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   613
  apply (drule_tac x = "abs e" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   614
  apply (case_tac "e >= 0")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   615
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   616
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   617
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   618
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   619
lemma gcd_coprime_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   620
  assumes z: "gcd (a::nat) b \<noteq> 0" and a: "a = a' * gcd a b" and
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   621
    b: "b = b' * gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   622
  shows    "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   623
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   624
  apply (subgoal_tac "a' = a div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   625
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   626
  apply (subgoal_tac "b' = b div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   627
  apply (erule ssubst)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   628
  apply (rule div_gcd_coprime_nat)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
   629
  using z apply force
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   630
  apply (subst (1) b)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   631
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   632
  apply (subst (1) a)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   633
  using z apply force
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
   634
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   635
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   636
lemma gcd_coprime_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   637
  assumes z: "gcd (a::int) b \<noteq> 0" and a: "a = a' * gcd a b" and
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   638
    b: "b = b' * gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   639
  shows    "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   640
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   641
  apply (subgoal_tac "a' = a div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   642
  apply (erule ssubst)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   643
  apply (subgoal_tac "b' = b div gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   644
  apply (erule ssubst)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   645
  apply (rule div_gcd_coprime_int)
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
   646
  using z apply force
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   647
  apply (subst (1) b)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   648
  using z apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   649
  apply (subst (1) a)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   650
  using z apply force
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
   651
  done
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   652
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   653
lemma coprime_mult_nat: assumes da: "coprime (d::nat) a" and db: "coprime d b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   654
    shows "coprime d (a * b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   655
  apply (subst gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   656
  using da apply (subst gcd_mult_cancel_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   657
  apply (subst gcd_commute_nat, assumption)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   658
  apply (subst gcd_commute_nat, rule db)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   659
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   660
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   661
lemma coprime_mult_int: assumes da: "coprime (d::int) a" and db: "coprime d b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   662
    shows "coprime d (a * b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   663
  apply (subst gcd_commute_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   664
  using da apply (subst gcd_mult_cancel_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   665
  apply (subst gcd_commute_int, assumption)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   666
  apply (subst gcd_commute_int, rule db)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   667
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   668
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   669
lemma coprime_lmult_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   670
  assumes dab: "coprime (d::nat) (a * b)" shows "coprime d a"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   671
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   672
  have "gcd d a dvd gcd d (a * b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   673
    by (rule gcd_greatest_nat, auto)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   674
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   675
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   676
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   677
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   678
lemma coprime_lmult_int:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   679
  assumes "coprime (d::int) (a * b)" shows "coprime d a"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   680
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   681
  have "gcd d a dvd gcd d (a * b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   682
    by (rule gcd_greatest_int, auto)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   683
  with assms show ?thesis
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   684
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   685
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   686
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   687
lemma coprime_rmult_nat:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   688
  assumes "coprime (d::nat) (a * b)" shows "coprime d b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   689
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   690
  have "gcd d b dvd gcd d (a * b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   691
    by (rule gcd_greatest_nat, auto intro: dvd_mult)
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
   692
  with assms show ?thesis
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   693
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   694
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   695
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   696
lemma coprime_rmult_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   697
  assumes dab: "coprime (d::int) (a * b)" shows "coprime d b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   698
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   699
  have "gcd d b dvd gcd d (a * b)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   700
    by (rule gcd_greatest_int, auto intro: dvd_mult)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   701
  with dab show ?thesis
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   702
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   703
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   704
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   705
lemma coprime_mul_eq_nat: "coprime (d::nat) (a * b) \<longleftrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   706
    coprime d a \<and>  coprime d b"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   707
  using coprime_rmult_nat[of d a b] coprime_lmult_nat[of d a b]
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   708
    coprime_mult_nat[of d a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   709
  by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   710
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   711
lemma coprime_mul_eq_int: "coprime (d::int) (a * b) \<longleftrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   712
    coprime d a \<and>  coprime d b"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   713
  using coprime_rmult_int[of d a b] coprime_lmult_int[of d a b]
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   714
    coprime_mult_int[of d a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   715
  by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   716
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   717
lemma gcd_coprime_exists_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   718
    assumes nz: "gcd (a::nat) b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   719
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   720
  apply (rule_tac x = "a div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   721
  apply (rule_tac x = "b div gcd a b" in exI)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   722
  using nz apply (auto simp add: div_gcd_coprime_nat dvd_div_mult)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   723
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   724
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   725
lemma gcd_coprime_exists_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   726
    assumes nz: "gcd (a::int) b \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   727
    shows "\<exists>a' b'. a = a' * gcd a b \<and> b = b' * gcd a b \<and> coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   728
  apply (rule_tac x = "a div gcd a b" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   729
  apply (rule_tac x = "b div gcd a b" in exI)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   730
  using nz apply (auto simp add: div_gcd_coprime_int dvd_div_mult_self)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   731
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   732
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   733
lemma coprime_exp_nat: "coprime (d::nat) a \<Longrightarrow> coprime d (a^n)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   734
  by (induct n, simp_all add: coprime_mult_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   735
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   736
lemma coprime_exp_int: "coprime (d::int) a \<Longrightarrow> coprime d (a^n)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   737
  by (induct n, simp_all add: coprime_mult_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   738
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   739
lemma coprime_exp2_nat [intro]: "coprime (a::nat) b \<Longrightarrow> coprime (a^n) (b^m)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   740
  apply (rule coprime_exp_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   741
  apply (subst gcd_commute_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   742
  apply (rule coprime_exp_nat)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   743
  apply (subst gcd_commute_nat, assumption)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   744
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   745
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   746
lemma coprime_exp2_int [intro]: "coprime (a::int) b \<Longrightarrow> coprime (a^n) (b^m)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   747
  apply (rule coprime_exp_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   748
  apply (subst gcd_commute_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   749
  apply (rule coprime_exp_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   750
  apply (subst gcd_commute_int, assumption)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   751
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   752
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   753
lemma gcd_exp_nat: "gcd ((a::nat)^n) (b^n) = (gcd a b)^n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   754
proof (cases)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   755
  assume "a = 0 & b = 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   756
  thus ?thesis by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   757
  next assume "~(a = 0 & b = 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   758
  hence "coprime ((a div gcd a b)^n) ((b div gcd a b)^n)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   759
    by (auto simp:div_gcd_coprime_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   760
  hence "gcd ((a div gcd a b)^n * (gcd a b)^n)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   761
      ((b div gcd a b)^n * (gcd a b)^n) = (gcd a b)^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   762
    apply (subst (1 2) mult_commute)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   763
    apply (subst gcd_mult_distrib_nat [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   764
    apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   765
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   766
  also have "(a div gcd a b)^n * (gcd a b)^n = a^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   767
    apply (subst div_power)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   768
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   769
    apply (rule dvd_div_mult_self)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   770
    apply (rule dvd_power_same)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   771
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   772
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   773
  also have "(b div gcd a b)^n * (gcd a b)^n = b^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   774
    apply (subst div_power)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   775
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   776
    apply (rule dvd_div_mult_self)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   777
    apply (rule dvd_power_same)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   778
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   779
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   780
  finally show ?thesis .
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   781
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   782
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   783
lemma gcd_exp_int: "gcd ((a::int)^n) (b^n) = (gcd a b)^n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   784
  apply (subst (1 2) gcd_abs_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   785
  apply (subst (1 2) power_abs)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   786
  apply (rule gcd_exp_nat [where n = n, transferred])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   787
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   788
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   789
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   790
lemma division_decomp_nat: assumes dc: "(a::nat) dvd b * c"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   791
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   792
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   793
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   794
  {assume "?g = 0" with dc have ?thesis by auto}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   795
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   796
  {assume z: "?g \<noteq> 0"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   797
    from gcd_coprime_exists_nat[OF z]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   798
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   799
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   800
    have thb: "?g dvd b" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   801
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   802
    with dc have th0: "a' dvd b*c" using dvd_trans[of a' a "b*c"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   803
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   804
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   805
    with z have th_1: "a' dvd b' * c" by auto
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   806
    from coprime_dvd_mult_nat[OF ab'(3)] th_1
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   807
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   808
    from ab' have "a = ?g*a'" by algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   809
    with thb thc have ?thesis by blast }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   810
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   811
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   812
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   813
lemma division_decomp_int: assumes dc: "(a::int) dvd b * c"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   814
  shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   815
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   816
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   817
  {assume "?g = 0" with dc have ?thesis by auto}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   818
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   819
  {assume z: "?g \<noteq> 0"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   820
    from gcd_coprime_exists_int[OF z]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   821
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   822
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   823
    have thb: "?g dvd b" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   824
    from ab'(1) have "a' dvd a"  unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   825
    with dc have th0: "a' dvd b*c"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   826
      using dvd_trans[of a' a "b*c"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   827
    from dc ab'(1,2) have "a'*?g dvd (b'*?g) *c" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   828
    hence "?g*a' dvd ?g * (b' * c)" by (simp add: mult_assoc)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   829
    with z have th_1: "a' dvd b' * c" by auto
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   830
    from coprime_dvd_mult_int[OF ab'(3)] th_1
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   831
    have thc: "a' dvd c" by (subst (asm) mult_commute, blast)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   832
    from ab' have "a = ?g*a'" by algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   833
    with thb thc have ?thesis by blast }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   834
  ultimately show ?thesis by blast
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   835
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   836
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   837
lemma pow_divides_pow_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   838
  assumes ab: "(a::nat) ^ n dvd b ^n" and n:"n \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   839
  shows "a dvd b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   840
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   841
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   842
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   843
  {assume "?g = 0" with ab n have ?thesis by auto }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   844
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   845
  {assume z: "?g \<noteq> 0"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   846
    hence zn: "?g ^ n \<noteq> 0" using n by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   847
    from gcd_coprime_exists_nat[OF z]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   848
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   849
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   850
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   851
      by (simp add: ab'(1,2)[symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   852
    hence "?g^n*a'^n dvd ?g^n *b'^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   853
      by (simp only: power_mult_distrib mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   854
    with zn z n have th0:"a'^n dvd b'^n" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   855
    have "a' dvd a'^n" by (simp add: m)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   856
    with th0 have "a' dvd b'^n" using dvd_trans[of a' "a'^n" "b'^n"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   857
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   858
    from coprime_dvd_mult_nat[OF coprime_exp_nat [OF ab'(3), of m]] th1
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   859
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   860
    hence "a'*?g dvd b'*?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   861
    with ab'(1,2)  have ?thesis by simp }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   862
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   863
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   864
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   865
lemma pow_divides_pow_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   866
  assumes ab: "(a::int) ^ n dvd b ^n" and n:"n \<noteq> 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   867
  shows "a dvd b"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   868
proof-
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   869
  let ?g = "gcd a b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   870
  from n obtain m where m: "n = Suc m" by (cases n, simp_all)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   871
  {assume "?g = 0" with ab n have ?thesis by auto }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   872
  moreover
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   873
  {assume z: "?g \<noteq> 0"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
   874
    hence zn: "?g ^ n \<noteq> 0" using n by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   875
    from gcd_coprime_exists_int[OF z]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   876
    obtain a' b' where ab': "a = a' * ?g" "b = b' * ?g" "coprime a' b'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   877
      by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   878
    from ab have "(a' * ?g) ^ n dvd (b' * ?g)^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   879
      by (simp add: ab'(1,2)[symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   880
    hence "?g^n*a'^n dvd ?g^n *b'^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   881
      by (simp only: power_mult_distrib mult_commute)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   882
    with zn z n have th0:"a'^n dvd b'^n" by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   883
    have "a' dvd a'^n" by (simp add: m)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   884
    with th0 have "a' dvd b'^n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   885
      using dvd_trans[of a' "a'^n" "b'^n"] by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   886
    hence th1: "a' dvd b'^m * b'" by (simp add: m mult_commute)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   887
    from coprime_dvd_mult_int[OF coprime_exp_int [OF ab'(3), of m]] th1
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   888
    have "a' dvd b'" by (subst (asm) mult_commute, blast)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   889
    hence "a'*?g dvd b'*?g" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   890
    with ab'(1,2)  have ?thesis by simp }
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   891
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   892
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   893
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   894
lemma pow_divides_eq_nat [simp]: "n ~= 0 \<Longrightarrow> ((a::nat)^n dvd b^n) = (a dvd b)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   895
  by (auto intro: pow_divides_pow_nat dvd_power_same)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   896
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   897
lemma pow_divides_eq_int [simp]: "n ~= 0 \<Longrightarrow> ((a::int)^n dvd b^n) = (a dvd b)"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   898
  by (auto intro: pow_divides_pow_int dvd_power_same)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   899
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   900
lemma divides_mult_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   901
  assumes mr: "(m::nat) dvd r" and nr: "n dvd r" and mn:"coprime m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   902
  shows "m * n dvd r"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   903
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   904
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   905
    unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   906
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   907
  hence "m dvd n'" using coprime_dvd_mult_iff_nat[OF mn] by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   908
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   909
  from n' k show ?thesis unfolding dvd_def by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   910
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   911
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   912
lemma divides_mult_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   913
  assumes mr: "(m::int) dvd r" and nr: "n dvd r" and mn:"coprime m n"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   914
  shows "m * n dvd r"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   915
proof-
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   916
  from mr nr obtain m' n' where m': "r = m*m'" and n': "r = n*n'"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   917
    unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   918
  from mr n' have "m dvd n'*n" by (simp add: mult_commute)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   919
  hence "m dvd n'" using coprime_dvd_mult_iff_int[OF mn] by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   920
  then obtain k where k: "n' = m*k" unfolding dvd_def by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   921
  from n' k show ?thesis unfolding dvd_def by auto
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   922
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
   923
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   924
lemma coprime_plus_one_nat [simp]: "coprime ((n::nat) + 1) n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   925
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   926
  apply force
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   927
  apply (rule dvd_diff_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   928
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   929
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   930
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   931
lemma coprime_Suc_nat [simp]: "coprime (Suc n) n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   932
  using coprime_plus_one_nat by (simp add: One_nat_def)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   933
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   934
lemma coprime_plus_one_int [simp]: "coprime ((n::int) + 1) n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   935
  apply (subgoal_tac "gcd (n + 1) n dvd (n + 1 - n)")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   936
  apply force
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   937
  apply (rule dvd_diff)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   938
  apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   939
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   940
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   941
lemma coprime_minus_one_nat: "(n::nat) \<noteq> 0 \<Longrightarrow> coprime (n - 1) n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   942
  using coprime_plus_one_nat [of "n - 1"]
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   943
    gcd_commute_nat [of "n - 1" n] by auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   944
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   945
lemma coprime_minus_one_int: "coprime ((n::int) - 1) n"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   946
  using coprime_plus_one_int [of "n - 1"]
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   947
    gcd_commute_int [of "n - 1" n] by auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   948
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   949
lemma setprod_coprime_nat [rule_format]:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   950
    "(ALL i: A. coprime (f i) (x::nat)) --> coprime (PROD i:A. f i) x"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   951
  apply (case_tac "finite A")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   952
  apply (induct set: finite)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   953
  apply (auto simp add: gcd_mult_cancel_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   954
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   955
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   956
lemma setprod_coprime_int [rule_format]:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   957
    "(ALL i: A. coprime (f i) (x::int)) --> coprime (PROD i:A. f i) x"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   958
  apply (case_tac "finite A")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   959
  apply (induct set: finite)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   960
  apply (auto simp add: gcd_mult_cancel_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   961
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   962
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   963
lemma coprime_common_divisor_nat: "coprime (a::nat) b \<Longrightarrow> x dvd a \<Longrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   964
    x dvd b \<Longrightarrow> x = 1"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   965
  apply (subgoal_tac "x dvd gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   966
  apply simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   967
  apply (erule (1) gcd_greatest_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   968
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   969
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   970
lemma coprime_common_divisor_int: "coprime (a::int) b \<Longrightarrow> x dvd a \<Longrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   971
    x dvd b \<Longrightarrow> abs x = 1"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   972
  apply (subgoal_tac "x dvd gcd a b")
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   973
  apply simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   974
  apply (erule (1) gcd_greatest_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   975
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   976
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   977
lemma coprime_divisors_nat: "(d::int) dvd a \<Longrightarrow> e dvd b \<Longrightarrow> coprime a b \<Longrightarrow>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   978
    coprime d e"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   979
  apply (auto simp add: dvd_def)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   980
  apply (frule coprime_lmult_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   981
  apply (subst gcd_commute_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   982
  apply (subst (asm) (2) gcd_commute_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   983
  apply (erule coprime_lmult_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   984
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   985
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   986
lemma invertible_coprime_nat: "(x::nat) * y mod m = 1 \<Longrightarrow> coprime x m"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   987
apply (metis coprime_lmult_nat gcd_1_nat gcd_commute_nat gcd_red_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   988
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   989
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   990
lemma invertible_coprime_int: "(x::int) * y mod m = 1 \<Longrightarrow> coprime x m"
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
   991
apply (metis coprime_lmult_int gcd_1_int gcd_commute_int gcd_red_int)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   992
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   993
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   994
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   995
subsection {* Bezout's theorem *}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   996
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   997
(* Function bezw returns a pair of witnesses to Bezout's theorem --
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   998
   see the theorems that follow the definition. *)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
   999
fun
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1000
  bezw  :: "nat \<Rightarrow> nat \<Rightarrow> int * int"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1001
where
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1002
  "bezw x y =
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1003
  (if y = 0 then (1, 0) else
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1004
      (snd (bezw y (x mod y)),
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1005
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y)))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1006
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1007
lemma bezw_0 [simp]: "bezw x 0 = (1, 0)" by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1008
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1009
lemma bezw_non_0: "y > 0 \<Longrightarrow> bezw x y = (snd (bezw y (x mod y)),
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1010
       fst (bezw y (x mod y)) - snd (bezw y (x mod y)) * int(x div y))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1011
  by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1012
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1013
declare bezw.simps [simp del]
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1014
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1015
lemma bezw_aux [rule_format]:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1016
    "fst (bezw x y) * int x + snd (bezw x y) * int y = int (gcd x y)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1017
proof (induct x y rule: gcd_nat_induct)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1018
  fix m :: nat
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1019
  show "fst (bezw m 0) * int m + snd (bezw m 0) * int 0 = int (gcd m 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1020
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1021
  next fix m :: nat and n
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1022
    assume ngt0: "n > 0" and
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1023
      ih: "fst (bezw n (m mod n)) * int n +
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1024
        snd (bezw n (m mod n)) * int (m mod n) =
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1025
        int (gcd n (m mod n))"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1026
    thus "fst (bezw m n) * int m + snd (bezw m n) * int n = int (gcd m n)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1027
      apply (simp add: bezw_non_0 gcd_non_0_nat)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1028
      apply (erule subst)
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 35726
diff changeset
  1029
      apply (simp add: field_simps)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1030
      apply (subst mod_div_equality [of m n, symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1031
      (* applying simp here undoes the last substitution!
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1032
         what is procedure cancel_div_mod? *)
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 35726
diff changeset
  1033
      apply (simp only: field_simps zadd_int [symmetric]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1034
        zmult_int [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1035
      done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1036
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1037
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1038
lemma bezout_int:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1039
  fixes x y
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1040
  shows "EX u v. u * (x::int) + v * y = gcd x y"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1041
proof -
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1042
  have bezout_aux: "!!x y. x \<ge> (0::int) \<Longrightarrow> y \<ge> 0 \<Longrightarrow>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1043
      EX u v. u * x + v * y = gcd x y"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1044
    apply (rule_tac x = "fst (bezw (nat x) (nat y))" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1045
    apply (rule_tac x = "snd (bezw (nat x) (nat y))" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1046
    apply (unfold gcd_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1047
    apply simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1048
    apply (subst bezw_aux [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1049
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1050
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1051
  have "(x \<ge> 0 \<and> y \<ge> 0) | (x \<ge> 0 \<and> y \<le> 0) | (x \<le> 0 \<and> y \<ge> 0) |
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1052
      (x \<le> 0 \<and> y \<le> 0)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1053
    by auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1054
  moreover have "x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> ?thesis"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1055
    by (erule (1) bezout_aux)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1056
  moreover have "x >= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1057
    apply (insert bezout_aux [of x "-y"])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1058
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1059
    apply (rule_tac x = u in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1060
    apply (rule_tac x = "-v" in exI)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1061
    apply (subst gcd_neg2_int [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1062
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1063
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1064
  moreover have "x <= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> ?thesis"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1065
    apply (insert bezout_aux [of "-x" y])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1066
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1067
    apply (rule_tac x = "-u" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1068
    apply (rule_tac x = v in exI)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1069
    apply (subst gcd_neg1_int [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1070
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1071
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1072
  moreover have "x <= 0 \<Longrightarrow> y <= 0 \<Longrightarrow> ?thesis"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1073
    apply (insert bezout_aux [of "-x" "-y"])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1074
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1075
    apply (rule_tac x = "-u" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1076
    apply (rule_tac x = "-v" in exI)
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1077
    apply (subst gcd_neg1_int [symmetric])
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1078
    apply (subst gcd_neg2_int [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1079
    apply auto
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1080
    done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1081
  ultimately show ?thesis by blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1082
qed
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1083
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1084
text {* versions of Bezout for nat, by Amine Chaieb *}
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1085
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1086
lemma ind_euclid:
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1087
  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1088
  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1089
  shows "P a b"
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1090
proof(induct "a + b" arbitrary: a b rule: less_induct)
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1091
  case less
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1092
  have "a = b \<or> a < b \<or> b < a" by arith
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1093
  moreover {assume eq: "a= b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1094
    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1095
    by simp}
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1096
  moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1097
  {assume lt: "a < b"
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1098
    hence "a + b - a < a + b \<or> a = 0" by arith
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1099
    moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1100
    {assume "a =0" with z c have "P a b" by blast }
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1101
    moreover
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1102
    {assume "a + b - a < a + b"
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1103
      also have th0: "a + b - a = a + (b - a)" using lt by arith
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1104
      finally have "a + (b - a) < a + b" .
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1105
      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1106
      then have "P a b" by (simp add: th0[symmetric])}
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1107
    ultimately have "P a b" by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1108
  moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1109
  {assume lt: "a > b"
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1110
    hence "b + a - b < a + b \<or> b = 0" by arith
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1111
    moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1112
    {assume "b =0" with z c have "P a b" by blast }
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1113
    moreover
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1114
    {assume "b + a - b < a + b"
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1115
      also have th0: "b + a - b = b + (a - b)" using lt by arith
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1116
      finally have "b + (a - b) < a + b" .
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1117
      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34223
diff changeset
  1118
      then have "P b a" by (simp add: th0[symmetric])
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1119
      hence "P a b" using c by blast }
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1120
    ultimately have "P a b" by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1121
ultimately  show "P a b" by blast
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1122
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1123
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1124
lemma bezout_lemma_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1125
  assumes ex: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1126
    (a * x = b * y + d \<or> b * x = a * y + d)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1127
  shows "\<exists>d x y. d dvd a \<and> d dvd a + b \<and>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1128
    (a * x = (a + b) * y + d \<or> (a + b) * x = a * y + d)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1129
  using ex
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1130
  apply clarsimp
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
  1131
  apply (rule_tac x="d" in exI, simp)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1132
  apply (case_tac "a * x = b * y + d" , simp_all)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1133
  apply (rule_tac x="x + y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1134
  apply (rule_tac x="y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1135
  apply algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1136
  apply (rule_tac x="x" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1137
  apply (rule_tac x="x + y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1138
  apply algebra
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1139
done
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1140
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1141
lemma bezout_add_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1142
    (a * x = b * y + d \<or> b * x = a * y + d)"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1143
  apply(induct a b rule: ind_euclid)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1144
  apply blast
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1145
  apply clarify
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
  1146
  apply (rule_tac x="a" in exI, simp)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1147
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1148
  apply (rule_tac x="d" in exI)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35028
diff changeset
  1149
  apply (case_tac "a * x = b * y + d", simp_all)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1150
  apply (rule_tac x="x+y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1151
  apply (rule_tac x="y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1152
  apply algebra
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1153
  apply (rule_tac x="x" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1154
  apply (rule_tac x="x+y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1155
  apply algebra
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1156
done
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1157
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1158
lemma bezout1_nat: "\<exists>(d::nat) x y. d dvd a \<and> d dvd b \<and>
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1159
    (a * x - b * y = d \<or> b * x - a * y = d)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1160
  using bezout_add_nat[of a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1161
  apply clarsimp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1162
  apply (rule_tac x="d" in exI, simp)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1163
  apply (rule_tac x="x" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1164
  apply (rule_tac x="y" in exI)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1165
  apply auto
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1166
done
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1167
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1168
lemma bezout_add_strong_nat: assumes nz: "a \<noteq> (0::nat)"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1169
  shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1170
proof-
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1171
 from nz have ap: "a > 0" by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1172
 from bezout_add_nat[of a b]
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1173
 have "(\<exists>d x y. d dvd a \<and> d dvd b \<and> a * x = b * y + d) \<or>
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1174
   (\<exists>d x y. d dvd a \<and> d dvd b \<and> b * x = a * y + d)" by blast
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1175
 moreover
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1176
    {fix d x y assume H: "d dvd a" "d dvd b" "a * x = b * y + d"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1177
     from H have ?thesis by blast }
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1178
 moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1179
 {fix d x y assume H: "d dvd a" "d dvd b" "b * x = a * y + d"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1180
   {assume b0: "b = 0" with H  have ?thesis by simp}
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1181
   moreover
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1182
   {assume b: "b \<noteq> 0" hence bp: "b > 0" by simp
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1183
     from b dvd_imp_le [OF H(2)] have "d < b \<or> d = b"
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1184
       by auto
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1185
     moreover
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1186
     {assume db: "d=b"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1187
       with nz H have ?thesis apply simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1188
         apply (rule exI[where x = b], simp)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1189
         apply (rule exI[where x = b])
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1190
        by (rule exI[where x = "a - 1"], simp add: diff_mult_distrib2)}
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1191
    moreover
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1192
    {assume db: "d < b"
41550
efa734d9b221 eliminated global prems;
wenzelm
parents: 37770
diff changeset
  1193
        {assume "x=0" hence ?thesis using nz H by simp }
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1194
        moreover
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1195
        {assume x0: "x \<noteq> 0" hence xp: "x > 0" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1196
          from db have "d \<le> b - 1" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1197
          hence "d*b \<le> b*(b - 1)" by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1198
          with xp mult_mono[of "1" "x" "d*b" "b*(b - 1)"]
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1199
          have dble: "d*b \<le> x*b*(b - 1)" using bp by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1200
          from H (3) have "d + (b - 1) * (b*x) = d + (b - 1) * (a*y + d)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1201
            by simp
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1202
          hence "d + (b - 1) * a * y + (b - 1) * d = d + (b - 1) * b * x"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1203
            by (simp only: mult_assoc right_distrib)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1204
          hence "a * ((b - 1) * y) + d * (b - 1 + 1) = d + x*b*(b - 1)"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1205
            by algebra
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1206
          hence "a * ((b - 1) * y) = d + x*b*(b - 1) - d*b" using bp by simp
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1207
          hence "a * ((b - 1) * y) = d + (x*b*(b - 1) - d*b)"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1208
            by (simp only: diff_add_assoc[OF dble, of d, symmetric])
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1209
          hence "a * ((b - 1) * y) = b*(x*(b - 1) - d) + d"
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1210
            by (simp only: diff_mult_distrib2 add_commute mult_ac)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1211
          hence ?thesis using H(1,2)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1212
            apply -
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1213
            apply (rule exI[where x=d], simp)
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1214
            apply (rule exI[where x="(b - 1) * y"])
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1215
            by (rule exI[where x="x*(b - 1) - d"], simp)}
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32879
diff changeset
  1216
        ultimately have ?thesis by blast}
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1217
    ultimately have ?thesis by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1218
  ultimately have ?thesis by blast}
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1219
 ultimately show ?thesis by blast
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1220
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1221
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1222
lemma bezout_nat: assumes a: "(a::nat) \<noteq> 0"
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1223
  shows "\<exists>x y. a * x = b * y + gcd a b"
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1224
proof-
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1225
  let ?g = "gcd a b"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1226
  from bezout_add_strong_nat[OF a, of b]
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1227
  obtain d x y where d: "d dvd a" "d dvd b" "a * x = b * y + d" by blast
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1228
  from d(1,2) have "d dvd ?g" by simp
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1229
  then obtain k where k: "?g = d*k" unfolding dvd_def by blast
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1230
  from d(3) have "a * x * k = (b * y + d) *k " by auto
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1231
  hence "a * (x * k) = b * (y*k) + ?g" by (algebra add: k)
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1232
  thus ?thesis by blast
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1233
qed
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1234
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1235
34030
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
  1236
subsection {* LCM properties *}
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1237
34030
829eb528b226 resorted code equations from "old" number theory version
haftmann
parents: 33946
diff changeset
  1238
lemma lcm_altdef_int [code]: "lcm (a::int) b = (abs a) * (abs b) div gcd a b"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1239
  by (simp add: lcm_int_def lcm_nat_def zdiv_int
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1240
    zmult_int [symmetric] gcd_int_def)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1241
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1242
lemma prod_gcd_lcm_nat: "(m::nat) * n = gcd m n * lcm m n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1243
  unfolding lcm_nat_def
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1244
  by (simp add: dvd_mult_div_cancel [OF gcd_dvd_prod_nat])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1245
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1246
lemma prod_gcd_lcm_int: "abs(m::int) * abs n = gcd m n * lcm m n"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1247
  unfolding lcm_int_def gcd_int_def
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1248
  apply (subst int_mult [symmetric])
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1249
  apply (subst prod_gcd_lcm_nat [symmetric])
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1250
  apply (subst nat_abs_mult_distrib [symmetric])
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1251
  apply (simp, simp add: abs_mult)
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1252
done
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1253
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1254
lemma lcm_0_nat [simp]: "lcm (m::nat) 0 = 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1255
  unfolding lcm_nat_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1256
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1257
lemma lcm_0_int [simp]: "lcm (m::int) 0 = 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1258
  unfolding lcm_int_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1259
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1260
lemma lcm_0_left_nat [simp]: "lcm (0::nat) n = 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1261
  unfolding lcm_nat_def by simp
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1262
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1263
lemma lcm_0_left_int [simp]: "lcm (0::int) n = 0"
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1264
  unfolding lcm_int_def by simp
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1265
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1266
lemma lcm_pos_nat:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1267
  "(m::nat) > 0 \<Longrightarrow> n>0 \<Longrightarrow> lcm m n > 0"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1268
by (metis gr0I mult_is_0 prod_gcd_lcm_nat)
27669
4b1642284dd7 Tuned and simplified proofs; Rules added to presburger's and algebra's context; moved Bezout theorems from Primes.thy
chaieb
parents: 27651
diff changeset
  1269
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1270
lemma lcm_pos_int:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1271
  "(m::int) ~= 0 \<Longrightarrow> n ~= 0 \<Longrightarrow> lcm m n > 0"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1272
  apply (subst lcm_abs_int)
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1273
  apply (rule lcm_pos_nat [transferred])
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1274
  apply auto
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1275
done
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1276
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1277
lemma dvd_pos_nat:
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1278
  fixes n m :: nat
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1279
  assumes "n > 0" and "m dvd n"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1280
  shows "m > 0"
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1281
using assms by (cases m) auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1282
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1283
lemma lcm_least_nat:
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1284
  assumes "(m::nat) dvd k" and "n dvd k"
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1285
  shows "lcm m n dvd k"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1286
proof (cases k)
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1287
  case 0 then show ?thesis by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1288
next
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1289
  case (Suc _) then have pos_k: "k > 0" by auto
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1290
  from assms dvd_pos_nat [OF this] have pos_mn: "m > 0" "n > 0" by auto
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1291
  with gcd_zero_nat [of m n] have pos_gcd: "gcd m n > 0" by simp
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1292
  from assms obtain p where k_m: "k = m * p" using dvd_def by blast
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1293
  from assms obtain q where k_n: "k = n * q" using dvd_def by blast
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1294
  from pos_k k_m have pos_p: "p > 0" by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1295
  from pos_k k_n have pos_q: "q > 0" by auto
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1296
  have "k * k * gcd q p = k * gcd (k * q) (k * p)"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1297
    by (simp add: mult_ac gcd_mult_distrib_nat)
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1298
  also have "\<dots> = k * gcd (m * p * q) (n * q * p)"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1299
    by (simp add: k_m [symmetric] k_n [symmetric])
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1300
  also have "\<dots> = k * p * q * gcd m n"
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1301
    by (simp add: mult_ac gcd_mult_distrib_nat)
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1302
  finally have "(m * p) * (n * q) * gcd q p = k * p * q * gcd m n"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1303
    by (simp only: k_m [symmetric] k_n [symmetric])
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1304
  then have "p * q * m * n * gcd q p = p * q * k * gcd m n"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1305
    by (simp add: mult_ac)
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1306
  with pos_p pos_q have "m * n * gcd q p = k * gcd m n"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1307
    by simp
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1308
  with prod_gcd_lcm_nat [of m n]
27556
292098f2efdf unified curried gcd, lcm, zgcd, zlcm
haftmann
parents: 27487
diff changeset
  1309
  have "lcm m n * gcd q p * gcd m n = k * gcd m n"
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1310
    by (simp add: mult_ac)
31706
1db0c8f235fb new GCD library, courtesy of Jeremy Avigad
huffman
parents: 30738
diff changeset
  1311
  with pos_gcd have "lcm m n * gcd q p = k" by auto
23687
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1312
  then show ?thesis using dvd_def by auto
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1313
qed
06884f7ffb18 extended - convers now basic lcm properties also
haftmann
parents: 23431
diff changeset
  1314
31952
40501bb2d57c renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents: 31814
diff changeset
  1315
lemma lcm_least_int:
31798
fe9a3043d36c Cleaned up GCD
nipkow
parents: 31766
diff changeset
  1316
  "(m::int) dvd k \<Longrightarrow> n dvd k \<Longrightarrow> lcm m n dvd k"