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theory NSPrimes(* Title : NSPrimes.thy Author : Jacques D. Fleuriot Copyright : 2002 University of Edinburgh Conversion to Isar and new proofs by Lawrence C Paulson, 2004 *) header{*The Nonstandard Primes as an Extension of the Prime Numbers*} theory NSPrimes imports "~~/src/HOL/NumberTheory/Factorization" Complex_Main begin text{*These can be used to derive an alternative proof of the infinitude of primes by considering a property of nonstandard sets.*} constdefs hdvd :: "[hypnat, hypnat] => bool" (infixl "hdvd" 50) "(M::hypnat) hdvd N == ( *p2* (op dvd)) M N" declare hdvd_def [transfer_unfold] constdefs starprime :: "hypnat set" "starprime == ( *s* {p. prime p})" declare starprime_def [transfer_unfold] constdefs choicefun :: "'a set => 'a" "choicefun E == (@x. ∃X ∈ Pow(E) -{{}}. x : X)" consts injf_max :: "nat => ('a::{order} set) => 'a" primrec injf_max_zero: "injf_max 0 E = choicefun E" injf_max_Suc: "injf_max (Suc n) E = choicefun({e. e:E & injf_max n E < e})" text{*A "choice" theorem for ultrafilters, like almost everywhere quantification*} lemma UF_choice: "{n. ∃m. Q n m} : FreeUltrafilterNat ==> ∃f. {n. Q n (f n)} : FreeUltrafilterNat" apply (rule_tac x = "%n. (@x. Q n x) " in exI) apply (ultra, rule someI, auto) done lemma UF_if: "({n. P n} : FreeUltrafilterNat --> {n. Q n} : FreeUltrafilterNat) = ({n. P n --> Q n} : FreeUltrafilterNat)" apply auto apply ultra+ done lemma UF_conj: "({n. P n} : FreeUltrafilterNat & {n. Q n} : FreeUltrafilterNat) = ({n. P n & Q n} : FreeUltrafilterNat)" apply auto apply ultra+ done lemma UF_choice_ccontr: "(∀f. {n. Q n (f n)} : FreeUltrafilterNat) = ({n. ∀m. Q n m} : FreeUltrafilterNat)" apply auto prefer 2 apply ultra apply (rule ccontr) apply (rule contrapos_np) apply (erule_tac [2] asm_rl) apply (simp (no_asm) add: FreeUltrafilterNat_Compl_iff1 Collect_neg_eq [symmetric]) apply (rule UF_choice, ultra) done lemma dvd_by_all: "∀M. ∃N. 0 < N & (∀m. 0 < m & (m::nat) <= M --> m dvd N)" apply (rule allI) apply (induct_tac "M", auto) apply (rule_tac x = "N * (Suc n) " in exI) apply (safe, force) apply (drule le_imp_less_or_eq, erule disjE) apply (force intro!: dvd_mult2) apply (force intro!: dvd_mult) done lemmas dvd_by_all2 = dvd_by_all [THEN spec, standard] lemma lemma_hypnat_P_EX: "(∃(x::hypnat). P x) = (∃f. P (star_n f))" apply auto apply (rule_tac x = x in star_cases, auto) done lemma lemma_hypnat_P_ALL: "(∀(x::hypnat). P x) = (∀f. P (star_n f))" apply auto apply (rule_tac x = x in star_cases, auto) done lemma hdvd: "(star_n X hdvd star_n Y) = ({n. X n dvd Y n} : FreeUltrafilterNat)" by (simp add: hdvd_def starP2) lemma hypnat_of_nat_le_zero_iff: "(hypnat_of_nat n <= 0) = (n = 0)" by (transfer, simp) declare hypnat_of_nat_le_zero_iff [simp] (* Goldblatt: Exercise 5.11(2) - p. 57 *) lemma hdvd_by_all: "∀M. ∃N. 0 < N & (∀m. 0 < m & (m::hypnat) <= M --> m hdvd N)" by (transfer, rule dvd_by_all) lemmas hdvd_by_all2 = hdvd_by_all [THEN spec, standard] (* Goldblatt: Exercise 5.11(2) - p. 57 *) lemma hypnat_dvd_all_hypnat_of_nat: "∃(N::hypnat). 0 < N & (∀n ∈ -{0::nat}. hypnat_of_nat(n) hdvd N)" apply (cut_tac hdvd_by_all) apply (drule_tac x = whn in spec, auto) apply (rule exI, auto) apply (drule_tac x = "hypnat_of_nat n" in spec) apply (auto simp add: linorder_not_less star_of_eq_0) done text{*The nonstandard extension of the set prime numbers consists of precisely those hypernaturals exceeding 1 that have no nontrivial factors*} (* Goldblatt: Exercise 5.11(3a) - p 57 *) lemma starprime: "starprime = {p. 1 < p & (∀m. m hdvd p --> m = 1 | m = p)}" by (transfer, auto simp add: prime_def) lemma prime_two: "prime 2" apply (unfold prime_def, auto) apply (frule dvd_imp_le) apply (auto dest: dvd_0_left) apply (case_tac m, simp, arith) done declare prime_two [simp] (* proof uses course-of-value induction *) lemma prime_factor_exists [rule_format]: "Suc 0 < n --> (∃k. prime k & k dvd n)" apply (rule_tac n = n in nat_less_induct, auto) apply (case_tac "prime n") apply (rule_tac x = n in exI, auto) apply (drule conjI [THEN not_prime_ex_mk], auto) apply (drule_tac x = m in spec, auto) apply (rule_tac x = ka in exI) apply (auto intro: dvd_mult2) done (* Goldblatt Exercise 5.11(3b) - p 57 *) lemma hyperprime_factor_exists [rule_format]: "!!n. 1 < n ==> (∃k ∈ starprime. k hdvd n)" by (transfer, simp add: prime_factor_exists) (* Goldblatt Exercise 3.10(1) - p. 29 *) lemma NatStar_hypnat_of_nat: "finite A ==> *s* A = hypnat_of_nat ` A" apply (rule_tac P = "%x. *s* x = hypnat_of_nat ` x" in finite_induct) apply auto done (* proved elsewhere? *) lemma FreeUltrafilterNat_singleton_not_mem: "{x} ∉ FreeUltrafilterNat" by (auto intro!: FreeUltrafilterNat_finite) declare FreeUltrafilterNat_singleton_not_mem [simp] subsection{*Another characterization of infinite set of natural numbers*} lemma finite_nat_set_bounded: "finite N ==> ∃n. (∀i ∈ N. i<(n::nat))" apply (erule_tac F = N in finite_induct, auto) apply (rule_tac x = "Suc n + x" in exI, auto) done lemma finite_nat_set_bounded_iff: "finite N = (∃n. (∀i ∈ N. i<(n::nat)))" by (blast intro: finite_nat_set_bounded bounded_nat_set_is_finite) lemma not_finite_nat_set_iff: "(~ finite N) = (∀n. ∃i ∈ N. n <= (i::nat))" by (auto simp add: finite_nat_set_bounded_iff le_def) lemma bounded_nat_set_is_finite2: "(∀i ∈ N. i<=(n::nat)) ==> finite N" apply (rule finite_subset) apply (rule_tac [2] finite_atMost, auto) done lemma finite_nat_set_bounded2: "finite N ==> ∃n. (∀i ∈ N. i<=(n::nat))" apply (erule_tac F = N in finite_induct, auto) apply (rule_tac x = "n + x" in exI, auto) done lemma finite_nat_set_bounded_iff2: "finite N = (∃n. (∀i ∈ N. i<=(n::nat)))" by (blast intro: finite_nat_set_bounded2 bounded_nat_set_is_finite2) lemma not_finite_nat_set_iff2: "(~ finite N) = (∀n. ∃i ∈ N. n < (i::nat))" by (auto simp add: finite_nat_set_bounded_iff2 le_def) subsection{*An injective function cannot define an embedded natural number*} lemma lemma_infinite_set_singleton: "∀m n. m ≠ n --> f n ≠ f m ==> {n. f n = N} = {} | (∃m. {n. f n = N} = {m})" apply auto apply (drule_tac x = x in spec, auto) apply (subgoal_tac "∀n. (f n = f x) = (x = n) ") apply auto done lemma inj_fun_not_hypnat_in_SHNat: "inj (f::nat=>nat) ==> star_n f ∉ Nats" apply (auto simp add: SHNat_eq hypnat_of_nat_eq star_n_eq_iff) apply (subgoal_tac "∀m n. m ≠ n --> f n ≠ f m", auto) apply (drule_tac [2] injD) prefer 2 apply assumption apply (drule_tac N = N in lemma_infinite_set_singleton, auto) done lemma range_subset_mem_starsetNat: "range f <= A ==> star_n f ∈ *s* A" apply (simp add: starset_def star_of_def Iset_star_n) apply (subgoal_tac "∀n. f n ∈ A", auto) done (*--------------------------------------------------------------------------------*) (* Gleason Proposition 11-5.5. pg 149, pg 155 (ex. 3) and pg. 360 *) (* Let E be a nonvoid ordered set with no maximal elements (note: effectively an *) (* infinite set if we take E = N (Nats)). Then there exists an order-preserving *) (* injection from N to E. Of course, (as some doofus will undoubtedly point out! *) (* :-)) can use notion of least element in proof (i.e. no need for choice) if *) (* dealing with nats as we have well-ordering property *) (*--------------------------------------------------------------------------------*) lemma lemmaPow3: "E ≠ {} ==> ∃x. ∃X ∈ (Pow E - {{}}). x: X" by auto lemma choicefun_mem_set: "E ≠ {} ==> choicefun E ∈ E" apply (unfold choicefun_def) apply (rule lemmaPow3 [THEN someI2_ex], auto) done declare choicefun_mem_set [simp] lemma injf_max_mem_set: "[| E ≠{}; ∀x. ∃y ∈ E. x < y |] ==> injf_max n E ∈ E" apply (induct_tac "n", force) apply (simp (no_asm) add: choicefun_def) apply (rule lemmaPow3 [THEN someI2_ex], auto) done lemma injf_max_order_preserving: "∀x. ∃y ∈ E. x < y ==> injf_max n E < injf_max (Suc n) E" apply (simp (no_asm) add: choicefun_def) apply (rule lemmaPow3 [THEN someI2_ex], auto) done lemma injf_max_order_preserving2: "∀x. ∃y ∈ E. x < y ==> ∀n m. m < n --> injf_max m E < injf_max n E" apply (rule allI) apply (induct_tac "n", auto) apply (simp (no_asm) add: choicefun_def) apply (rule lemmaPow3 [THEN someI2_ex]) apply (auto simp add: less_Suc_eq) apply (drule_tac x = m in spec) apply (drule subsetD, auto) apply (drule_tac x = "injf_max m E" in order_less_trans, auto) done lemma inj_injf_max: "∀x. ∃y ∈ E. x < y ==> inj (%n. injf_max n E)" apply (rule inj_onI) apply (rule ccontr, auto) apply (drule injf_max_order_preserving2) apply (cut_tac m = x and n = y in less_linear, auto) apply (auto dest!: spec) done lemma infinite_set_has_order_preserving_inj: "[| (E::('a::{order} set)) ≠ {}; ∀x. ∃y ∈ E. x < y |] ==> ∃f. range f <= E & inj (f::nat => 'a) & (∀m. f m < f(Suc m))" apply (rule_tac x = "%n. injf_max n E" in exI, safe) apply (rule injf_max_mem_set) apply (rule_tac [3] inj_injf_max) apply (rule_tac [4] injf_max_order_preserving, auto) done text{*Only need the existence of an injective function from N to A for proof*} lemma hypnat_infinite_has_nonstandard: "~ finite A ==> hypnat_of_nat ` A < ( *s* A)" apply auto apply (subgoal_tac "A ≠ {}") prefer 2 apply force apply (drule infinite_set_has_order_preserving_inj) apply (erule not_finite_nat_set_iff2 [THEN iffD1], auto) apply (drule inj_fun_not_hypnat_in_SHNat) apply (drule range_subset_mem_starsetNat) apply (auto simp add: SHNat_eq) done lemma starsetNat_eq_hypnat_of_nat_image_finite: "*s* A = hypnat_of_nat ` A ==> finite A" apply (rule ccontr) apply (auto dest: hypnat_infinite_has_nonstandard) done lemma finite_starsetNat_iff: "( *s* A = hypnat_of_nat ` A) = (finite A)" by (blast intro!: starsetNat_eq_hypnat_of_nat_image_finite NatStar_hypnat_of_nat) lemma hypnat_infinite_has_nonstandard_iff: "(~ finite A) = (hypnat_of_nat ` A < *s* A)" apply (rule iffI) apply (blast intro!: hypnat_infinite_has_nonstandard) apply (auto simp add: finite_starsetNat_iff [symmetric]) done subsection{*Existence of Infinitely Many Primes: a Nonstandard Proof*} lemma lemma_not_dvd_hypnat_one: "~ (∀n ∈ - {0}. hypnat_of_nat n hdvd 1)" apply auto apply (rule_tac x = 2 in bexI) apply (transfer, auto) done declare lemma_not_dvd_hypnat_one [simp] lemma lemma_not_dvd_hypnat_one2: "∃n ∈ - {0}. ~ hypnat_of_nat n hdvd 1" apply (cut_tac lemma_not_dvd_hypnat_one) apply (auto simp del: lemma_not_dvd_hypnat_one) done declare lemma_not_dvd_hypnat_one2 [simp] (* not needed here *) lemma hypnat_gt_zero_gt_one: "!!N. [| 0 < (N::hypnat); N ≠ 1 |] ==> 1 < N" by (transfer, simp) lemma hypnat_add_one_gt_one: "!!N. 0 < N ==> 1 < (N::hypnat) + 1" by (transfer, simp) lemma zero_not_prime: "¬ prime 0" apply safe apply (drule prime_g_zero, auto) done declare zero_not_prime [simp] lemma hypnat_of_nat_zero_not_prime: "hypnat_of_nat 0 ∉ starprime" by (transfer, simp) declare hypnat_of_nat_zero_not_prime [simp] lemma hypnat_zero_not_prime: "0 ∉ starprime" by (cut_tac hypnat_of_nat_zero_not_prime, simp) declare hypnat_zero_not_prime [simp] lemma one_not_prime: "¬ prime 1" apply safe apply (drule prime_g_one, auto) done declare one_not_prime [simp] lemma one_not_prime2: "¬ prime(Suc 0)" apply safe apply (drule prime_g_one, auto) done declare one_not_prime2 [simp] lemma hypnat_of_nat_one_not_prime: "hypnat_of_nat 1 ∉ starprime" by (transfer, simp) declare hypnat_of_nat_one_not_prime [simp] lemma hypnat_one_not_prime: "1 ∉ starprime" by (cut_tac hypnat_of_nat_one_not_prime, simp) declare hypnat_one_not_prime [simp] lemma hdvd_diff: "!!k m n. [| k hdvd m; k hdvd n |] ==> k hdvd (m - n)" by (transfer, rule dvd_diff) lemma dvd_one_eq_one: "x dvd (1::nat) ==> x = 1" by (unfold dvd_def, auto) lemma hdvd_one_eq_one: "!!x. x hdvd 1 ==> x = 1" by (transfer, rule dvd_one_eq_one) theorem not_finite_prime: "~ finite {p. prime p}" apply (rule hypnat_infinite_has_nonstandard_iff [THEN iffD2]) apply (cut_tac hypnat_dvd_all_hypnat_of_nat) apply (erule exE) apply (erule conjE) apply (subgoal_tac "1 < N + 1") prefer 2 apply (blast intro: hypnat_add_one_gt_one) apply (drule hyperprime_factor_exists) apply (auto intro: STAR_mem) apply (subgoal_tac "k ∉ hypnat_of_nat ` {p. prime p}") apply (force simp add: starprime_def, safe) apply (drule_tac x = x in bspec) apply (rule ccontr, simp) apply (drule hdvd_diff, assumption) apply (auto dest: hdvd_one_eq_one) done end
lemma UF_choice:
{n. ∃m. Q n m} ∈ \<U> ==> ∃f. {n. Q n (f n)} ∈ \<U>
lemma UF_if:
({n. P n} ∈ \<U> --> {n. Q n} ∈ \<U>) = ({n. P n --> Q n} ∈ \<U>)
lemma UF_conj:
({n. P n} ∈ \<U> ∧ {n. Q n} ∈ \<U>) = ({n. P n ∧ Q n} ∈ \<U>)
lemma UF_choice_ccontr:
(∀f. {n. Q n (f n)} ∈ \<U>) = ({n. ∀m. Q n m} ∈ \<U>)
lemma dvd_by_all:
∀M. ∃N>0. ∀m. 0 < m ∧ m ≤ M --> m dvd N
lemmas dvd_by_all2:
∃N>0. ∀m. 0 < m ∧ m ≤ x --> m dvd N
lemmas dvd_by_all2:
∃N>0. ∀m. 0 < m ∧ m ≤ x --> m dvd N
lemma lemma_hypnat_P_EX:
(∃x. P x) = (∃f. P (star_n f))
lemma lemma_hypnat_P_ALL:
(∀x. P x) = (∀f. P (star_n f))
lemma hdvd:
(star_n X hdvd star_n Y) = ({n. X n dvd Y n} ∈ \<U>)
lemma hypnat_of_nat_le_zero_iff:
(star_of n ≤ 0) = (n = 0)
lemma hdvd_by_all:
∀M. ∃N>0. ∀m. 0 < m ∧ m ≤ M --> m hdvd N
lemmas hdvd_by_all2:
∃N>0. ∀m. 0 < m ∧ m ≤ x --> m hdvd N
lemmas hdvd_by_all2:
∃N>0. ∀m. 0 < m ∧ m ≤ x --> m hdvd N
lemma hypnat_dvd_all_hypnat_of_nat:
∃N>0. ∀n∈- {0}. star_of n hdvd N
lemma starprime:
starprime = {p. 1 < p ∧ (∀m. m hdvd p --> m = 1 ∨ m = p)}
lemma prime_two:
prime 2
lemma prime_factor_exists:
Suc 0 < n ==> ∃k. prime k ∧ k dvd n
lemma hyperprime_factor_exists:
1 < n ==> ∃k∈starprime. k hdvd n
lemma NatStar_hypnat_of_nat:
finite A ==> *s* A = star_of ` A
lemma FreeUltrafilterNat_singleton_not_mem:
{x} ∉ \<U>
lemma finite_nat_set_bounded:
finite N ==> ∃n. ∀i∈N. i < n
lemma finite_nat_set_bounded_iff:
finite N = (∃n. ∀i∈N. i < n)
lemma not_finite_nat_set_iff:
infinite N = (∀n. ∃i∈N. n ≤ i)
lemma bounded_nat_set_is_finite2:
∀i∈N. i ≤ n ==> finite N
lemma finite_nat_set_bounded2:
finite N ==> ∃n. ∀i∈N. i ≤ n
lemma finite_nat_set_bounded_iff2:
finite N = (∃n. ∀i∈N. i ≤ n)
lemma not_finite_nat_set_iff2:
infinite N = (∀n. ∃i∈N. n < i)
lemma lemma_infinite_set_singleton:
∀m n. m ≠ n --> f n ≠ f m ==> {n. f n = N} = {} ∨ (∃m. {n. f n = N} = {m})
lemma inj_fun_not_hypnat_in_SHNat:
inj f ==> star_n f ∉ Nats
lemma range_subset_mem_starsetNat:
range f ⊆ A ==> star_n f ∈ *s* A
lemma lemmaPow3:
E ≠ {} ==> ∃x. ∃X∈Pow E - {{}}. x ∈ X
lemma choicefun_mem_set:
E ≠ {} ==> choicefun E ∈ E
lemma injf_max_mem_set:
[| E ≠ {}; ∀x. ∃y∈E. x < y |] ==> injf_max n E ∈ E
lemma injf_max_order_preserving:
∀x. ∃y∈E. x < y ==> injf_max n E < injf_max (Suc n) E
lemma injf_max_order_preserving2:
∀x. ∃y∈E. x < y ==> ∀n m. m < n --> injf_max m E < injf_max n E
lemma inj_injf_max:
∀x. ∃y∈E. x < y ==> inj (%n. injf_max n E)
lemma infinite_set_has_order_preserving_inj:
[| E ≠ {}; ∀x. ∃y∈E. x < y |] ==> ∃f. range f ⊆ E ∧ inj f ∧ (∀m. f m < f (Suc m))
lemma hypnat_infinite_has_nonstandard:
infinite A ==> star_of ` A ⊂ *s* A
lemma starsetNat_eq_hypnat_of_nat_image_finite:
*s* A = star_of ` A ==> finite A
lemma finite_starsetNat_iff:
(*s* A = star_of ` A) = finite A
lemma hypnat_infinite_has_nonstandard_iff:
infinite A = (star_of ` A ⊂ *s* A)
lemma lemma_not_dvd_hypnat_one:
¬ (∀n∈- {0}. star_of n hdvd 1)
lemma lemma_not_dvd_hypnat_one2:
∃n∈- {0}. ¬ star_of n hdvd 1
lemma hypnat_gt_zero_gt_one:
[| 0 < N; N ≠ 1 |] ==> 1 < N
lemma hypnat_add_one_gt_one:
0 < N ==> 1 < N + 1
lemma zero_not_prime:
¬ prime 0
lemma hypnat_of_nat_zero_not_prime:
star_of 0 ∉ starprime
lemma hypnat_zero_not_prime:
0 ∉ starprime
lemma one_not_prime:
¬ prime 1
lemma one_not_prime2:
¬ prime (Suc 0)
lemma hypnat_of_nat_one_not_prime:
star_of 1 ∉ starprime
lemma hypnat_one_not_prime:
1 ∉ starprime
lemma hdvd_diff:
[| k hdvd m; k hdvd n |] ==> k hdvd m - n
lemma dvd_one_eq_one:
x dvd 1 ==> x = 1
lemma hdvd_one_eq_one:
x hdvd 1 ==> x = 1
theorem not_finite_prime:
infinite {p. prime p}