--- a/src/HOL/NewNumberTheory/UniqueFactorization.thy Tue Sep 01 14:13:34 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,967 +0,0 @@
-(* Title: UniqueFactorization.thy
- ID:
- Author: Jeremy Avigad
-
-
- Unique factorization for the natural numbers and the integers.
-
- Note: there were previous Isabelle formalizations of unique
- factorization due to Thomas Marthedal Rasmussen, and, building on
- that, by Jeremy Avigad and David Gray.
-*)
-
-header {* UniqueFactorization *}
-
-theory UniqueFactorization
-imports Cong Multiset
-begin
-
-(* inherited from Multiset *)
-declare One_nat_def [simp del]
-
-(* As a simp or intro rule,
-
- prime p \<Longrightarrow> p > 0
-
- wreaks havoc here. When the premise includes ALL x :# M. prime x, it
- leads to the backchaining
-
- x > 0
- prime x
- x :# M which is, unfortunately,
- count M x > 0
-*)
-
-
-(* useful facts *)
-
-lemma setsum_Un2: "finite (A Un B) \<Longrightarrow>
- setsum f (A Un B) = setsum f (A - B) + setsum f (B - A) +
- setsum f (A Int B)"
- apply (subgoal_tac "A Un B = (A - B) Un (B - A) Un (A Int B)")
- apply (erule ssubst)
- apply (subst setsum_Un_disjoint)
- apply auto
- apply (subst setsum_Un_disjoint)
- apply auto
-done
-
-lemma setprod_Un2: "finite (A Un B) \<Longrightarrow>
- setprod f (A Un B) = setprod f (A - B) * setprod f (B - A) *
- setprod f (A Int B)"
- apply (subgoal_tac "A Un B = (A - B) Un (B - A) Un (A Int B)")
- apply (erule ssubst)
- apply (subst setprod_Un_disjoint)
- apply auto
- apply (subst setprod_Un_disjoint)
- apply auto
-done
-
-(* Should this go in Multiset.thy? *)
-(* TN: No longer an intro-rule; needed only once and might get in the way *)
-lemma multiset_eqI: "[| !!x. count M x = count N x |] ==> M = N"
- by (subst multiset_eq_conv_count_eq, blast)
-
-(* Here is a version of set product for multisets. Is it worth moving
- to multiset.thy? If so, one should similarly define msetsum for abelian
- semirings, using of_nat. Also, is it worth developing bounded quantifiers
- "ALL i :# M. P i"?
-*)
-
-constdefs
- msetprod :: "('a => ('b::{power,comm_monoid_mult})) => 'a multiset => 'b"
- "msetprod f M == setprod (%x. (f x)^(count M x)) (set_of M)"
-
-syntax
- "_msetprod" :: "pttrn => 'a set => 'b => 'b::comm_monoid_mult"
- ("(3PROD _:#_. _)" [0, 51, 10] 10)
-
-translations
- "PROD i :# A. b" == "msetprod (%i. b) A"
-
-lemma msetprod_Un: "msetprod f (A+B) = msetprod f A * msetprod f B"
- apply (simp add: msetprod_def power_add)
- apply (subst setprod_Un2)
- apply auto
- apply (subgoal_tac
- "(PROD x:set_of A - set_of B. f x ^ count A x * f x ^ count B x) =
- (PROD x:set_of A - set_of B. f x ^ count A x)")
- apply (erule ssubst)
- apply (subgoal_tac
- "(PROD x:set_of B - set_of A. f x ^ count A x * f x ^ count B x) =
- (PROD x:set_of B - set_of A. f x ^ count B x)")
- apply (erule ssubst)
- apply (subgoal_tac "(PROD x:set_of A. f x ^ count A x) =
- (PROD x:set_of A - set_of B. f x ^ count A x) *
- (PROD x:set_of A Int set_of B. f x ^ count A x)")
- apply (erule ssubst)
- apply (subgoal_tac "(PROD x:set_of B. f x ^ count B x) =
- (PROD x:set_of B - set_of A. f x ^ count B x) *
- (PROD x:set_of A Int set_of B. f x ^ count B x)")
- apply (erule ssubst)
- apply (subst setprod_timesf)
- apply (force simp add: mult_ac)
- apply (subst setprod_Un_disjoint [symmetric])
- apply (auto intro: setprod_cong)
- apply (subst setprod_Un_disjoint [symmetric])
- apply (auto intro: setprod_cong)
-done
-
-
-subsection {* unique factorization: multiset version *}
-
-lemma multiset_prime_factorization_exists [rule_format]: "n > 0 -->
- (EX M. (ALL (p::nat) : set_of M. prime p) & n = (PROD i :# M. i))"
-proof (rule nat_less_induct, clarify)
- fix n :: nat
- assume ih: "ALL m < n. 0 < m --> (EX M. (ALL p : set_of M. prime p) & m =
- (PROD i :# M. i))"
- assume "(n::nat) > 0"
- then have "n = 1 | (n > 1 & prime n) | (n > 1 & ~ prime n)"
- by arith
- moreover
- {
- assume "n = 1"
- then have "(ALL p : set_of {#}. prime p) & n = (PROD i :# {#}. i)"
- by (auto simp add: msetprod_def)
- }
- moreover
- {
- assume "n > 1" and "prime n"
- then have "(ALL p : set_of {# n #}. prime p) & n = (PROD i :# {# n #}. i)"
- by (auto simp add: msetprod_def)
- }
- moreover
- {
- assume "n > 1" and "~ prime n"
- from prems not_prime_eq_prod_nat
- obtain m k where "n = m * k & 1 < m & m < n & 1 < k & k < n"
- by blast
- with ih obtain Q R where "(ALL p : set_of Q. prime p) & m = (PROD i:#Q. i)"
- and "(ALL p: set_of R. prime p) & k = (PROD i:#R. i)"
- by blast
- hence "(ALL p: set_of (Q + R). prime p) & n = (PROD i :# Q + R. i)"
- by (auto simp add: prems msetprod_Un set_of_union)
- then have "EX M. (ALL p : set_of M. prime p) & n = (PROD i :# M. i)"..
- }
- ultimately show "EX M. (ALL p : set_of M. prime p) & n = (PROD i::nat:#M. i)"
- by blast
-qed
-
-lemma multiset_prime_factorization_unique_aux:
- fixes a :: nat
- assumes "(ALL p : set_of M. prime p)" and
- "(ALL p : set_of N. prime p)" and
- "(PROD i :# M. i) dvd (PROD i:# N. i)"
- shows
- "count M a <= count N a"
-proof cases
- assume "a : set_of M"
- with prems have a: "prime a"
- by auto
- with prems have "a ^ count M a dvd (PROD i :# M. i)"
- by (auto intro: dvd_setprod simp add: msetprod_def)
- also have "... dvd (PROD i :# N. i)"
- by (rule prems)
- also have "... = (PROD i : (set_of N). i ^ (count N i))"
- by (simp add: msetprod_def)
- also have "... =
- a^(count N a) * (PROD i : (set_of N - {a}). i ^ (count N i))"
- proof (cases)
- assume "a : set_of N"
- hence b: "set_of N = {a} Un (set_of N - {a})"
- by auto
- thus ?thesis
- by (subst (1) b, subst setprod_Un_disjoint, auto)
- next
- assume "a ~: set_of N"
- thus ?thesis
- by auto
- qed
- finally have "a ^ count M a dvd
- a^(count N a) * (PROD i : (set_of N - {a}). i ^ (count N i))".
- moreover have "coprime (a ^ count M a)
- (PROD i : (set_of N - {a}). i ^ (count N i))"
- apply (subst gcd_commute_nat)
- apply (rule setprod_coprime_nat)
- apply (rule primes_imp_powers_coprime_nat)
- apply (insert prems, auto)
- done
- ultimately have "a ^ count M a dvd a^(count N a)"
- by (elim coprime_dvd_mult_nat)
- with a show ?thesis
- by (intro power_dvd_imp_le, auto)
-next
- assume "a ~: set_of M"
- thus ?thesis by auto
-qed
-
-lemma multiset_prime_factorization_unique:
- assumes "(ALL (p::nat) : set_of M. prime p)" and
- "(ALL p : set_of N. prime p)" and
- "(PROD i :# M. i) = (PROD i:# N. i)"
- shows
- "M = N"
-proof -
- {
- fix a
- from prems have "count M a <= count N a"
- by (intro multiset_prime_factorization_unique_aux, auto)
- moreover from prems have "count N a <= count M a"
- by (intro multiset_prime_factorization_unique_aux, auto)
- ultimately have "count M a = count N a"
- by auto
- }
- thus ?thesis by (simp add:multiset_eq_conv_count_eq)
-qed
-
-constdefs
- multiset_prime_factorization :: "nat => nat multiset"
- "multiset_prime_factorization n ==
- if n > 0 then (THE M. ((ALL p : set_of M. prime p) &
- n = (PROD i :# M. i)))
- else {#}"
-
-lemma multiset_prime_factorization: "n > 0 ==>
- (ALL p : set_of (multiset_prime_factorization n). prime p) &
- n = (PROD i :# (multiset_prime_factorization n). i)"
- apply (unfold multiset_prime_factorization_def)
- apply clarsimp
- apply (frule multiset_prime_factorization_exists)
- apply clarify
- apply (rule theI)
- apply (insert multiset_prime_factorization_unique, blast)+
-done
-
-
-subsection {* Prime factors and multiplicity for nats and ints *}
-
-class unique_factorization =
-
-fixes
- multiplicity :: "'a \<Rightarrow> 'a \<Rightarrow> nat" and
- prime_factors :: "'a \<Rightarrow> 'a set"
-
-(* definitions for the natural numbers *)
-
-instantiation nat :: unique_factorization
-
-begin
-
-definition
- multiplicity_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
- "multiplicity_nat p n = count (multiset_prime_factorization n) p"
-
-definition
- prime_factors_nat :: "nat \<Rightarrow> nat set"
-where
- "prime_factors_nat n = set_of (multiset_prime_factorization n)"
-
-instance proof qed
-
-end
-
-(* definitions for the integers *)
-
-instantiation int :: unique_factorization
-
-begin
-
-definition
- multiplicity_int :: "int \<Rightarrow> int \<Rightarrow> nat"
-where
- "multiplicity_int p n = multiplicity (nat p) (nat n)"
-
-definition
- prime_factors_int :: "int \<Rightarrow> int set"
-where
- "prime_factors_int n = int ` (prime_factors (nat n))"
-
-instance proof qed
-
-end
-
-
-subsection {* Set up transfer *}
-
-lemma transfer_nat_int_prime_factors:
- "prime_factors (nat n) = nat ` prime_factors n"
- unfolding prime_factors_int_def apply auto
- by (subst transfer_int_nat_set_return_embed, assumption)
-
-lemma transfer_nat_int_prime_factors_closure: "n >= 0 \<Longrightarrow>
- nat_set (prime_factors n)"
- by (auto simp add: nat_set_def prime_factors_int_def)
-
-lemma transfer_nat_int_multiplicity: "p >= 0 \<Longrightarrow> n >= 0 \<Longrightarrow>
- multiplicity (nat p) (nat n) = multiplicity p n"
- by (auto simp add: multiplicity_int_def)
-
-declare TransferMorphism_nat_int[transfer add return:
- transfer_nat_int_prime_factors transfer_nat_int_prime_factors_closure
- transfer_nat_int_multiplicity]
-
-
-lemma transfer_int_nat_prime_factors:
- "prime_factors (int n) = int ` prime_factors n"
- unfolding prime_factors_int_def by auto
-
-lemma transfer_int_nat_prime_factors_closure: "is_nat n \<Longrightarrow>
- nat_set (prime_factors n)"
- by (simp only: transfer_nat_int_prime_factors_closure is_nat_def)
-
-lemma transfer_int_nat_multiplicity:
- "multiplicity (int p) (int n) = multiplicity p n"
- by (auto simp add: multiplicity_int_def)
-
-declare TransferMorphism_int_nat[transfer add return:
- transfer_int_nat_prime_factors transfer_int_nat_prime_factors_closure
- transfer_int_nat_multiplicity]
-
-
-subsection {* Properties of prime factors and multiplicity for nats and ints *}
-
-lemma prime_factors_ge_0_int [elim]: "p : prime_factors (n::int) \<Longrightarrow> p >= 0"
- by (unfold prime_factors_int_def, auto)
-
-lemma prime_factors_prime_nat [intro]: "p : prime_factors (n::nat) \<Longrightarrow> prime p"
- apply (case_tac "n = 0")
- apply (simp add: prime_factors_nat_def multiset_prime_factorization_def)
- apply (auto simp add: prime_factors_nat_def multiset_prime_factorization)
-done
-
-lemma prime_factors_prime_int [intro]:
- assumes "n >= 0" and "p : prime_factors (n::int)"
- shows "prime p"
-
- apply (rule prime_factors_prime_nat [transferred, of n p])
- using prems apply auto
-done
-
-lemma prime_factors_gt_0_nat [elim]: "p : prime_factors x \<Longrightarrow> p > (0::nat)"
- by (frule prime_factors_prime_nat, auto)
-
-lemma prime_factors_gt_0_int [elim]: "x >= 0 \<Longrightarrow> p : prime_factors x \<Longrightarrow>
- p > (0::int)"
- by (frule (1) prime_factors_prime_int, auto)
-
-lemma prime_factors_finite_nat [iff]: "finite (prime_factors (n::nat))"
- by (unfold prime_factors_nat_def, auto)
-
-lemma prime_factors_finite_int [iff]: "finite (prime_factors (n::int))"
- by (unfold prime_factors_int_def, auto)
-
-lemma prime_factors_altdef_nat: "prime_factors (n::nat) =
- {p. multiplicity p n > 0}"
- by (force simp add: prime_factors_nat_def multiplicity_nat_def)
-
-lemma prime_factors_altdef_int: "prime_factors (n::int) =
- {p. p >= 0 & multiplicity p n > 0}"
- apply (unfold prime_factors_int_def multiplicity_int_def)
- apply (subst prime_factors_altdef_nat)
- apply (auto simp add: image_def)
-done
-
-lemma prime_factorization_nat: "(n::nat) > 0 \<Longrightarrow>
- n = (PROD p : prime_factors n. p^(multiplicity p n))"
- by (frule multiset_prime_factorization,
- simp add: prime_factors_nat_def multiplicity_nat_def msetprod_def)
-
-thm prime_factorization_nat [transferred]
-
-lemma prime_factorization_int:
- assumes "(n::int) > 0"
- shows "n = (PROD p : prime_factors n. p^(multiplicity p n))"
-
- apply (rule prime_factorization_nat [transferred, of n])
- using prems apply auto
-done
-
-lemma neq_zero_eq_gt_zero_nat: "((x::nat) ~= 0) = (x > 0)"
- by auto
-
-lemma prime_factorization_unique_nat:
- "S = { (p::nat) . f p > 0} \<Longrightarrow> finite S \<Longrightarrow> (ALL p : S. prime p) \<Longrightarrow>
- n = (PROD p : S. p^(f p)) \<Longrightarrow>
- S = prime_factors n & (ALL p. f p = multiplicity p n)"
- apply (subgoal_tac "multiset_prime_factorization n = Abs_multiset
- f")
- apply (unfold prime_factors_nat_def multiplicity_nat_def)
- apply (simp add: set_of_def count_def Abs_multiset_inverse multiset_def)
- apply (unfold multiset_prime_factorization_def)
- apply (subgoal_tac "n > 0")
- prefer 2
- apply force
- apply (subst if_P, assumption)
- apply (rule the1_equality)
- apply (rule ex_ex1I)
- apply (rule multiset_prime_factorization_exists, assumption)
- apply (rule multiset_prime_factorization_unique)
- apply force
- apply force
- apply force
- unfolding set_of_def count_def msetprod_def
- apply (subgoal_tac "f : multiset")
- apply (auto simp only: Abs_multiset_inverse)
- unfolding multiset_def apply force
-done
-
-lemma prime_factors_characterization_nat: "S = {p. 0 < f (p::nat)} \<Longrightarrow>
- finite S \<Longrightarrow> (ALL p:S. prime p) \<Longrightarrow> n = (PROD p:S. p ^ f p) \<Longrightarrow>
- prime_factors n = S"
- by (rule prime_factorization_unique_nat [THEN conjunct1, symmetric],
- assumption+)
-
-lemma prime_factors_characterization'_nat:
- "finite {p. 0 < f (p::nat)} \<Longrightarrow>
- (ALL p. 0 < f p \<longrightarrow> prime p) \<Longrightarrow>
- prime_factors (PROD p | 0 < f p . p ^ f p) = {p. 0 < f p}"
- apply (rule prime_factors_characterization_nat)
- apply auto
-done
-
-(* A minor glitch:*)
-
-thm prime_factors_characterization'_nat
- [where f = "%x. f (int (x::nat))",
- transferred direction: nat "op <= (0::int)", rule_format]
-
-(*
- Transfer isn't smart enough to know that the "0 < f p" should
- remain a comparison between nats. But the transfer still works.
-*)
-
-lemma primes_characterization'_int [rule_format]:
- "finite {p. p >= 0 & 0 < f (p::int)} \<Longrightarrow>
- (ALL p. 0 < f p \<longrightarrow> prime p) \<Longrightarrow>
- prime_factors (PROD p | p >=0 & 0 < f p . p ^ f p) =
- {p. p >= 0 & 0 < f p}"
-
- apply (insert prime_factors_characterization'_nat
- [where f = "%x. f (int (x::nat))",
- transferred direction: nat "op <= (0::int)"])
- apply auto
-done
-
-lemma prime_factors_characterization_int: "S = {p. 0 < f (p::int)} \<Longrightarrow>
- finite S \<Longrightarrow> (ALL p:S. prime p) \<Longrightarrow> n = (PROD p:S. p ^ f p) \<Longrightarrow>
- prime_factors n = S"
- apply simp
- apply (subgoal_tac "{p. 0 < f p} = {p. 0 <= p & 0 < f p}")
- apply (simp only:)
- apply (subst primes_characterization'_int)
- apply auto
- apply (auto simp add: prime_ge_0_int)
-done
-
-lemma multiplicity_characterization_nat: "S = {p. 0 < f (p::nat)} \<Longrightarrow>
- finite S \<Longrightarrow> (ALL p:S. prime p) \<Longrightarrow> n = (PROD p:S. p ^ f p) \<Longrightarrow>
- multiplicity p n = f p"
- by (frule prime_factorization_unique_nat [THEN conjunct2, rule_format,
- symmetric], auto)
-
-lemma multiplicity_characterization'_nat: "finite {p. 0 < f (p::nat)} \<longrightarrow>
- (ALL p. 0 < f p \<longrightarrow> prime p) \<longrightarrow>
- multiplicity p (PROD p | 0 < f p . p ^ f p) = f p"
- apply (rule impI)+
- apply (rule multiplicity_characterization_nat)
- apply auto
-done
-
-lemma multiplicity_characterization'_int [rule_format]:
- "finite {p. p >= 0 & 0 < f (p::int)} \<Longrightarrow>
- (ALL p. 0 < f p \<longrightarrow> prime p) \<Longrightarrow> p >= 0 \<Longrightarrow>
- multiplicity p (PROD p | p >= 0 & 0 < f p . p ^ f p) = f p"
-
- apply (insert multiplicity_characterization'_nat
- [where f = "%x. f (int (x::nat))",
- transferred direction: nat "op <= (0::int)", rule_format])
- apply auto
-done
-
-lemma multiplicity_characterization_int: "S = {p. 0 < f (p::int)} \<Longrightarrow>
- finite S \<Longrightarrow> (ALL p:S. prime p) \<Longrightarrow> n = (PROD p:S. p ^ f p) \<Longrightarrow>
- p >= 0 \<Longrightarrow> multiplicity p n = f p"
- apply simp
- apply (subgoal_tac "{p. 0 < f p} = {p. 0 <= p & 0 < f p}")
- apply (simp only:)
- apply (subst multiplicity_characterization'_int)
- apply auto
- apply (auto simp add: prime_ge_0_int)
-done
-
-lemma multiplicity_zero_nat [simp]: "multiplicity (p::nat) 0 = 0"
- by (simp add: multiplicity_nat_def multiset_prime_factorization_def)
-
-lemma multiplicity_zero_int [simp]: "multiplicity (p::int) 0 = 0"
- by (simp add: multiplicity_int_def)
-
-lemma multiplicity_one_nat [simp]: "multiplicity p (1::nat) = 0"
- by (subst multiplicity_characterization_nat [where f = "%x. 0"], auto)
-
-lemma multiplicity_one_int [simp]: "multiplicity p (1::int) = 0"
- by (simp add: multiplicity_int_def)
-
-lemma multiplicity_prime_nat [simp]: "prime (p::nat) \<Longrightarrow> multiplicity p p = 1"
- apply (subst multiplicity_characterization_nat
- [where f = "(%q. if q = p then 1 else 0)"])
- apply auto
- apply (case_tac "x = p")
- apply auto
-done
-
-lemma multiplicity_prime_int [simp]: "prime (p::int) \<Longrightarrow> multiplicity p p = 1"
- unfolding prime_int_def multiplicity_int_def by auto
-
-lemma multiplicity_prime_power_nat [simp]: "prime (p::nat) \<Longrightarrow>
- multiplicity p (p^n) = n"
- apply (case_tac "n = 0")
- apply auto
- apply (subst multiplicity_characterization_nat
- [where f = "(%q. if q = p then n else 0)"])
- apply auto
- apply (case_tac "x = p")
- apply auto
-done
-
-lemma multiplicity_prime_power_int [simp]: "prime (p::int) \<Longrightarrow>
- multiplicity p (p^n) = n"
- apply (frule prime_ge_0_int)
- apply (auto simp add: prime_int_def multiplicity_int_def nat_power_eq)
-done
-
-lemma multiplicity_nonprime_nat [simp]: "~ prime (p::nat) \<Longrightarrow>
- multiplicity p n = 0"
- apply (case_tac "n = 0")
- apply auto
- apply (frule multiset_prime_factorization)
- apply (auto simp add: set_of_def multiplicity_nat_def)
-done
-
-lemma multiplicity_nonprime_int [simp]: "~ prime (p::int) \<Longrightarrow> multiplicity p n = 0"
- by (unfold multiplicity_int_def prime_int_def, auto)
-
-lemma multiplicity_not_factor_nat [simp]:
- "p ~: prime_factors (n::nat) \<Longrightarrow> multiplicity p n = 0"
- by (subst (asm) prime_factors_altdef_nat, auto)
-
-lemma multiplicity_not_factor_int [simp]:
- "p >= 0 \<Longrightarrow> p ~: prime_factors (n::int) \<Longrightarrow> multiplicity p n = 0"
- by (subst (asm) prime_factors_altdef_int, auto)
-
-lemma multiplicity_product_aux_nat: "(k::nat) > 0 \<Longrightarrow> l > 0 \<Longrightarrow>
- (prime_factors k) Un (prime_factors l) = prime_factors (k * l) &
- (ALL p. multiplicity p k + multiplicity p l = multiplicity p (k * l))"
- apply (rule prime_factorization_unique_nat)
- apply (simp only: prime_factors_altdef_nat)
- apply auto
- apply (subst power_add)
- apply (subst setprod_timesf)
- apply (rule arg_cong2)back back
- apply (subgoal_tac "prime_factors k Un prime_factors l = prime_factors k Un
- (prime_factors l - prime_factors k)")
- apply (erule ssubst)
- apply (subst setprod_Un_disjoint)
- apply auto
- apply (subgoal_tac "(\<Prod>p\<in>prime_factors l - prime_factors k. p ^ multiplicity p k) =
- (\<Prod>p\<in>prime_factors l - prime_factors k. 1)")
- apply (erule ssubst)
- apply (simp add: setprod_1)
- apply (erule prime_factorization_nat)
- apply (rule setprod_cong, auto)
- apply (subgoal_tac "prime_factors k Un prime_factors l = prime_factors l Un
- (prime_factors k - prime_factors l)")
- apply (erule ssubst)
- apply (subst setprod_Un_disjoint)
- apply auto
- apply (subgoal_tac "(\<Prod>p\<in>prime_factors k - prime_factors l. p ^ multiplicity p l) =
- (\<Prod>p\<in>prime_factors k - prime_factors l. 1)")
- apply (erule ssubst)
- apply (simp add: setprod_1)
- apply (erule prime_factorization_nat)
- apply (rule setprod_cong, auto)
-done
-
-(* transfer doesn't have the same problem here with the right
- choice of rules. *)
-
-lemma multiplicity_product_aux_int:
- assumes "(k::int) > 0" and "l > 0"
- shows
- "(prime_factors k) Un (prime_factors l) = prime_factors (k * l) &
- (ALL p >= 0. multiplicity p k + multiplicity p l = multiplicity p (k * l))"
-
- apply (rule multiplicity_product_aux_nat [transferred, of l k])
- using prems apply auto
-done
-
-lemma prime_factors_product_nat: "(k::nat) > 0 \<Longrightarrow> l > 0 \<Longrightarrow> prime_factors (k * l) =
- prime_factors k Un prime_factors l"
- by (rule multiplicity_product_aux_nat [THEN conjunct1, symmetric])
-
-lemma prime_factors_product_int: "(k::int) > 0 \<Longrightarrow> l > 0 \<Longrightarrow> prime_factors (k * l) =
- prime_factors k Un prime_factors l"
- by (rule multiplicity_product_aux_int [THEN conjunct1, symmetric])
-
-lemma multiplicity_product_nat: "(k::nat) > 0 \<Longrightarrow> l > 0 \<Longrightarrow> multiplicity p (k * l) =
- multiplicity p k + multiplicity p l"
- by (rule multiplicity_product_aux_nat [THEN conjunct2, rule_format,
- symmetric])
-
-lemma multiplicity_product_int: "(k::int) > 0 \<Longrightarrow> l > 0 \<Longrightarrow> p >= 0 \<Longrightarrow>
- multiplicity p (k * l) = multiplicity p k + multiplicity p l"
- by (rule multiplicity_product_aux_int [THEN conjunct2, rule_format,
- symmetric])
-
-lemma multiplicity_setprod_nat: "finite S \<Longrightarrow> (ALL x : S. f x > 0) \<Longrightarrow>
- multiplicity (p::nat) (PROD x : S. f x) =
- (SUM x : S. multiplicity p (f x))"
- apply (induct set: finite)
- apply auto
- apply (subst multiplicity_product_nat)
- apply auto
-done
-
-(* Transfer is delicate here for two reasons: first, because there is
- an implicit quantifier over functions (f), and, second, because the
- product over the multiplicity should not be translated to an integer
- product.
-
- The way to handle the first is to use quantifier rules for functions.
- The way to handle the second is to turn off the offending rule.
-*)
-
-lemma transfer_nat_int_sum_prod_closure3:
- "(SUM x : A. int (f x)) >= 0"
- "(PROD x : A. int (f x)) >= 0"
- apply (rule setsum_nonneg, auto)
- apply (rule setprod_nonneg, auto)
-done
-
-declare TransferMorphism_nat_int[transfer
- add return: transfer_nat_int_sum_prod_closure3
- del: transfer_nat_int_sum_prod2 (1)]
-
-lemma multiplicity_setprod_int: "p >= 0 \<Longrightarrow> finite S \<Longrightarrow>
- (ALL x : S. f x > 0) \<Longrightarrow>
- multiplicity (p::int) (PROD x : S. f x) =
- (SUM x : S. multiplicity p (f x))"
-
- apply (frule multiplicity_setprod_nat
- [where f = "%x. nat(int(nat(f x)))",
- transferred direction: nat "op <= (0::int)"])
- apply auto
- apply (subst (asm) setprod_cong)
- apply (rule refl)
- apply (rule if_P)
- apply auto
- apply (rule setsum_cong)
- apply auto
-done
-
-declare TransferMorphism_nat_int[transfer
- add return: transfer_nat_int_sum_prod2 (1)]
-
-lemma multiplicity_prod_prime_powers_nat:
- "finite S \<Longrightarrow> (ALL p : S. prime (p::nat)) \<Longrightarrow>
- multiplicity p (PROD p : S. p ^ f p) = (if p : S then f p else 0)"
- apply (subgoal_tac "(PROD p : S. p ^ f p) =
- (PROD p : S. p ^ (%x. if x : S then f x else 0) p)")
- apply (erule ssubst)
- apply (subst multiplicity_characterization_nat)
- prefer 5 apply (rule refl)
- apply (rule refl)
- apply auto
- apply (subst setprod_mono_one_right)
- apply assumption
- prefer 3
- apply (rule setprod_cong)
- apply (rule refl)
- apply auto
-done
-
-(* Here the issue with transfer is the implicit quantifier over S *)
-
-lemma multiplicity_prod_prime_powers_int:
- "(p::int) >= 0 \<Longrightarrow> finite S \<Longrightarrow> (ALL p : S. prime p) \<Longrightarrow>
- multiplicity p (PROD p : S. p ^ f p) = (if p : S then f p else 0)"
-
- apply (subgoal_tac "int ` nat ` S = S")
- apply (frule multiplicity_prod_prime_powers_nat [where f = "%x. f(int x)"
- and S = "nat ` S", transferred])
- apply auto
- apply (subst prime_int_def [symmetric])
- apply auto
- apply (subgoal_tac "xb >= 0")
- apply force
- apply (rule prime_ge_0_int)
- apply force
- apply (subst transfer_nat_int_set_return_embed)
- apply (unfold nat_set_def, auto)
-done
-
-lemma multiplicity_distinct_prime_power_nat: "prime (p::nat) \<Longrightarrow> prime q \<Longrightarrow>
- p ~= q \<Longrightarrow> multiplicity p (q^n) = 0"
- apply (subgoal_tac "q^n = setprod (%x. x^n) {q}")
- apply (erule ssubst)
- apply (subst multiplicity_prod_prime_powers_nat)
- apply auto
-done
-
-lemma multiplicity_distinct_prime_power_int: "prime (p::int) \<Longrightarrow> prime q \<Longrightarrow>
- p ~= q \<Longrightarrow> multiplicity p (q^n) = 0"
- apply (frule prime_ge_0_int [of q])
- apply (frule multiplicity_distinct_prime_power_nat [transferred leaving: n])
- prefer 4
- apply assumption
- apply auto
-done
-
-lemma dvd_multiplicity_nat:
- "(0::nat) < y \<Longrightarrow> x dvd y \<Longrightarrow> multiplicity p x <= multiplicity p y"
- apply (case_tac "x = 0")
- apply (auto simp add: dvd_def multiplicity_product_nat)
-done
-
-lemma dvd_multiplicity_int:
- "(0::int) < y \<Longrightarrow> 0 <= x \<Longrightarrow> x dvd y \<Longrightarrow> p >= 0 \<Longrightarrow>
- multiplicity p x <= multiplicity p y"
- apply (case_tac "x = 0")
- apply (auto simp add: dvd_def)
- apply (subgoal_tac "0 < k")
- apply (auto simp add: multiplicity_product_int)
- apply (erule zero_less_mult_pos)
- apply arith
-done
-
-lemma dvd_prime_factors_nat [intro]:
- "0 < (y::nat) \<Longrightarrow> x dvd y \<Longrightarrow> prime_factors x <= prime_factors y"
- apply (simp only: prime_factors_altdef_nat)
- apply auto
- apply (frule dvd_multiplicity_nat)
- apply auto
-(* It is a shame that auto and arith don't get this. *)
- apply (erule order_less_le_trans)back
- apply assumption
-done
-
-lemma dvd_prime_factors_int [intro]:
- "0 < (y::int) \<Longrightarrow> 0 <= x \<Longrightarrow> x dvd y \<Longrightarrow> prime_factors x <= prime_factors y"
- apply (auto simp add: prime_factors_altdef_int)
- apply (erule order_less_le_trans)
- apply (rule dvd_multiplicity_int)
- apply auto
-done
-
-lemma multiplicity_dvd_nat: "0 < (x::nat) \<Longrightarrow> 0 < y \<Longrightarrow>
- ALL p. multiplicity p x <= multiplicity p y \<Longrightarrow>
- x dvd y"
- apply (subst prime_factorization_nat [of x], assumption)
- apply (subst prime_factorization_nat [of y], assumption)
- apply (rule setprod_dvd_setprod_subset2)
- apply force
- apply (subst prime_factors_altdef_nat)+
- apply auto
-(* Again, a shame that auto and arith don't get this. *)
- apply (drule_tac x = xa in spec, auto)
- apply (rule le_imp_power_dvd)
- apply blast
-done
-
-lemma multiplicity_dvd_int: "0 < (x::int) \<Longrightarrow> 0 < y \<Longrightarrow>
- ALL p >= 0. multiplicity p x <= multiplicity p y \<Longrightarrow>
- x dvd y"
- apply (subst prime_factorization_int [of x], assumption)
- apply (subst prime_factorization_int [of y], assumption)
- apply (rule setprod_dvd_setprod_subset2)
- apply force
- apply (subst prime_factors_altdef_int)+
- apply auto
- apply (rule dvd_power_le)
- apply auto
- apply (drule_tac x = xa in spec)
- apply (erule impE)
- apply auto
-done
-
-lemma multiplicity_dvd'_nat: "(0::nat) < x \<Longrightarrow>
- \<forall>p. prime p \<longrightarrow> multiplicity p x \<le> multiplicity p y \<Longrightarrow> x dvd y"
- apply (cases "y = 0")
- apply auto
- apply (rule multiplicity_dvd_nat, auto)
- apply (case_tac "prime p")
- apply auto
-done
-
-lemma multiplicity_dvd'_int: "(0::int) < x \<Longrightarrow> 0 <= y \<Longrightarrow>
- \<forall>p. prime p \<longrightarrow> multiplicity p x \<le> multiplicity p y \<Longrightarrow> x dvd y"
- apply (cases "y = 0")
- apply auto
- apply (rule multiplicity_dvd_int, auto)
- apply (case_tac "prime p")
- apply auto
-done
-
-lemma dvd_multiplicity_eq_nat: "0 < (x::nat) \<Longrightarrow> 0 < y \<Longrightarrow>
- (x dvd y) = (ALL p. multiplicity p x <= multiplicity p y)"
- by (auto intro: dvd_multiplicity_nat multiplicity_dvd_nat)
-
-lemma dvd_multiplicity_eq_int: "0 < (x::int) \<Longrightarrow> 0 < y \<Longrightarrow>
- (x dvd y) = (ALL p >= 0. multiplicity p x <= multiplicity p y)"
- by (auto intro: dvd_multiplicity_int multiplicity_dvd_int)
-
-lemma prime_factors_altdef2_nat: "(n::nat) > 0 \<Longrightarrow>
- (p : prime_factors n) = (prime p & p dvd n)"
- apply (case_tac "prime p")
- apply auto
- apply (subst prime_factorization_nat [where n = n], assumption)
- apply (rule dvd_trans)
- apply (rule dvd_power [where x = p and n = "multiplicity p n"])
- apply (subst (asm) prime_factors_altdef_nat, force)
- apply (rule dvd_setprod)
- apply auto
- apply (subst prime_factors_altdef_nat)
- apply (subst (asm) dvd_multiplicity_eq_nat)
- apply auto
- apply (drule spec [where x = p])
- apply auto
-done
-
-lemma prime_factors_altdef2_int:
- assumes "(n::int) > 0"
- shows "(p : prime_factors n) = (prime p & p dvd n)"
-
- apply (case_tac "p >= 0")
- apply (rule prime_factors_altdef2_nat [transferred])
- using prems apply auto
- apply (auto simp add: prime_ge_0_int prime_factors_ge_0_int)
-done
-
-lemma multiplicity_eq_nat:
- fixes x and y::nat
- assumes [arith]: "x > 0" "y > 0" and
- mult_eq [simp]: "!!p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y"
- shows "x = y"
-
- apply (rule dvd_anti_sym)
- apply (auto intro: multiplicity_dvd'_nat)
-done
-
-lemma multiplicity_eq_int:
- fixes x and y::int
- assumes [arith]: "x > 0" "y > 0" and
- mult_eq [simp]: "!!p. prime p \<Longrightarrow> multiplicity p x = multiplicity p y"
- shows "x = y"
-
- apply (rule dvd_anti_sym [transferred])
- apply (auto intro: multiplicity_dvd'_int)
-done
-
-
-subsection {* An application *}
-
-lemma gcd_eq_nat:
- assumes pos [arith]: "x > 0" "y > 0"
- shows "gcd (x::nat) y =
- (PROD p: prime_factors x Un prime_factors y.
- p ^ (min (multiplicity p x) (multiplicity p y)))"
-proof -
- def z == "(PROD p: prime_factors (x::nat) Un prime_factors y.
- p ^ (min (multiplicity p x) (multiplicity p y)))"
- have [arith]: "z > 0"
- unfolding z_def by (rule setprod_pos_nat, auto)
- have aux: "!!p. prime p \<Longrightarrow> multiplicity p z =
- min (multiplicity p x) (multiplicity p y)"
- unfolding z_def
- apply (subst multiplicity_prod_prime_powers_nat)
- apply (auto simp add: multiplicity_not_factor_nat)
- done
- have "z dvd x"
- by (intro multiplicity_dvd'_nat, auto simp add: aux)
- moreover have "z dvd y"
- by (intro multiplicity_dvd'_nat, auto simp add: aux)
- moreover have "ALL w. w dvd x & w dvd y \<longrightarrow> w dvd z"
- apply auto
- apply (case_tac "w = 0", auto)
- apply (erule multiplicity_dvd'_nat)
- apply (auto intro: dvd_multiplicity_nat simp add: aux)
- done
- ultimately have "z = gcd x y"
- by (subst gcd_unique_nat [symmetric], blast)
- thus ?thesis
- unfolding z_def by auto
-qed
-
-lemma lcm_eq_nat:
- assumes pos [arith]: "x > 0" "y > 0"
- shows "lcm (x::nat) y =
- (PROD p: prime_factors x Un prime_factors y.
- p ^ (max (multiplicity p x) (multiplicity p y)))"
-proof -
- def z == "(PROD p: prime_factors (x::nat) Un prime_factors y.
- p ^ (max (multiplicity p x) (multiplicity p y)))"
- have [arith]: "z > 0"
- unfolding z_def by (rule setprod_pos_nat, auto)
- have aux: "!!p. prime p \<Longrightarrow> multiplicity p z =
- max (multiplicity p x) (multiplicity p y)"
- unfolding z_def
- apply (subst multiplicity_prod_prime_powers_nat)
- apply (auto simp add: multiplicity_not_factor_nat)
- done
- have "x dvd z"
- by (intro multiplicity_dvd'_nat, auto simp add: aux)
- moreover have "y dvd z"
- by (intro multiplicity_dvd'_nat, auto simp add: aux)
- moreover have "ALL w. x dvd w & y dvd w \<longrightarrow> z dvd w"
- apply auto
- apply (case_tac "w = 0", auto)
- apply (rule multiplicity_dvd'_nat)
- apply (auto intro: dvd_multiplicity_nat simp add: aux)
- done
- ultimately have "z = lcm x y"
- by (subst lcm_unique_nat [symmetric], blast)
- thus ?thesis
- unfolding z_def by auto
-qed
-
-lemma multiplicity_gcd_nat:
- assumes [arith]: "x > 0" "y > 0"
- shows "multiplicity (p::nat) (gcd x y) =
- min (multiplicity p x) (multiplicity p y)"
-
- apply (subst gcd_eq_nat)
- apply auto
- apply (subst multiplicity_prod_prime_powers_nat)
- apply auto
-done
-
-lemma multiplicity_lcm_nat:
- assumes [arith]: "x > 0" "y > 0"
- shows "multiplicity (p::nat) (lcm x y) =
- max (multiplicity p x) (multiplicity p y)"
-
- apply (subst lcm_eq_nat)
- apply auto
- apply (subst multiplicity_prod_prime_powers_nat)
- apply auto
-done
-
-lemma gcd_lcm_distrib_nat: "gcd (x::nat) (lcm y z) = lcm (gcd x y) (gcd x z)"
- apply (case_tac "x = 0 | y = 0 | z = 0")
- apply auto
- apply (rule multiplicity_eq_nat)
- apply (auto simp add: multiplicity_gcd_nat multiplicity_lcm_nat
- lcm_pos_nat)
-done
-
-lemma gcd_lcm_distrib_int: "gcd (x::int) (lcm y z) = lcm (gcd x y) (gcd x z)"
- apply (subst (1 2 3) gcd_abs_int)
- apply (subst lcm_abs_int)
- apply (subst (2) abs_of_nonneg)
- apply force
- apply (rule gcd_lcm_distrib_nat [transferred])
- apply auto
-done
-
-end