# Theory NSPrimes

theory NSPrimes
imports Primes Hyperreal
```(*  Title       : NSPrimes.thy
Author      : Jacques D. Fleuriot
Copyright   : 2002 University of Edinburgh
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)

section ‹The Nonstandard Primes as an Extension of the Prime Numbers›

theory NSPrimes
imports "HOL-Computational_Algebra.Primes" "HOL-Nonstandard_Analysis.Hyperreal"
begin

text ‹These can be used to derive an alternative proof of the infinitude of
primes by considering a property of nonstandard sets.›

definition starprime :: "hypnat set"
where [transfer_unfold]: "starprime = *s* {p. prime p}"

definition choicefun :: "'a set ⇒ 'a"
where "choicefun E = (SOME x. ∃X ∈ Pow E - {{}}. x ∈ X)"

primrec injf_max :: "nat ⇒ 'a::order set ⇒ 'a"
where
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})"

lemma dvd_by_all2: "∃N>0. ∀m. 0 < m ∧ m ≤ M ⟶ m dvd N"
for M :: nat
apply (induct M)
apply auto
apply (rule_tac x = "N * Suc M" in exI)
apply auto
done

lemma dvd_by_all: "∀M::nat. ∃N>0. ∀m. 0 < m ∧ m ≤ M ⟶ m dvd N"
using dvd_by_all2 by blast

lemma hypnat_of_nat_le_zero_iff [simp]: "hypnat_of_nat n ≤ 0 ⟷ n = 0"
by transfer simp

text ‹Goldblatt: Exercise 5.11(2) -- p. 57.›
lemma hdvd_by_all: "∀M. ∃N. 0 < N ∧ (∀m::hypnat. 0 < m ∧ m ≤ M ⟶ m dvd N)"
by transfer (rule dvd_by_all)

lemmas hdvd_by_all2 = hdvd_by_all [THEN spec]

text ‹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 dvd N)"
apply (cut_tac hdvd_by_all)
apply (drule_tac x = whn in spec)
apply auto
apply (rule exI)
apply auto
apply (drule_tac x = "hypnat_of_nat n" in spec)
done

text ‹The nonstandard extension of the set prime numbers consists of precisely
those hypernaturals exceeding 1 that have no nontrivial factors.›

text ‹Goldblatt: Exercise 5.11(3a) -- p 57.›
lemma starprime: "starprime = {p. 1 < p ∧ (∀m. m dvd p ⟶ m = 1 ∨ m = p)}"
by transfer (auto simp add: prime_nat_iff)

text ‹Goldblatt Exercise 5.11(3b) -- p 57.›
lemma hyperprime_factor_exists: "⋀n. 1 < n ⟹ ∃k ∈ starprime. k dvd n"

text ‹Goldblatt Exercise 3.10(1) -- p. 29.›
lemma NatStar_hypnat_of_nat: "finite A ⟹ *s* A = hypnat_of_nat ` A"
by (rule starset_finite)

subsection ‹Another characterization of infinite set of natural numbers›

lemma finite_nat_set_bounded: "finite N ⟹ ∃n::nat. ∀i ∈ N. i < n"
apply (erule_tac F = N in finite_induct)
apply auto
apply (rule_tac x = "Suc n + x" in exI)
apply auto
done

lemma finite_nat_set_bounded_iff: "finite N ⟷ (∃n::nat. ∀i ∈ N. i < n)"
by (blast intro: finite_nat_set_bounded bounded_nat_set_is_finite)

lemma not_finite_nat_set_iff: "¬ finite N ⟷ (∀n::nat. ∃i ∈ N. n ≤ i)"
by (auto simp add: finite_nat_set_bounded_iff not_less)

lemma bounded_nat_set_is_finite2: "∀i::nat ∈ N. i ≤ n ⟹ finite N"
apply (rule finite_subset)
apply (rule_tac [2] finite_atMost)
apply auto
done

lemma finite_nat_set_bounded2: "finite N ⟹ ∃n::nat. ∀i ∈ N. i ≤ n"
apply (erule_tac F = N in finite_induct)
apply auto
apply (rule_tac x = "n + x" in exI)
apply auto
done

lemma finite_nat_set_bounded_iff2: "finite N ⟷ (∃n::nat. ∀i ∈ N. i ≤ n)"
by (blast intro: finite_nat_set_bounded2 bounded_nat_set_is_finite2)

lemma not_finite_nat_set_iff2: "¬ finite N ⟷ (∀n::nat. ∃i ∈ N. n < i)"
by (auto simp add: finite_nat_set_bounded_iff2 not_le)

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:
fixes f :: "nat ⇒ nat"
assumes inj_f: "inj f"
shows "starfun f whn ∉ Nats"
proof
from inj_f have inj_f': "inj (starfun f)"
by (transfer inj_on_def Ball_def UNIV_def)
assume "starfun f whn ∈ Nats"
then obtain N where N: "starfun f whn = hypnat_of_nat N"
by (auto simp: Nats_def)
then have "∃n. starfun f n = hypnat_of_nat N" ..
then have "∃n. f n = N" by transfer
then obtain n where "f n = N" ..
then have "starfun f (hypnat_of_nat n) = hypnat_of_nat N"
by transfer
with N have "starfun f whn = starfun f (hypnat_of_nat n)"
by simp
with inj_f' have "whn = hypnat_of_nat n"
by (rule injD)
then show False
qed

lemma range_subset_mem_starsetNat: "range f ⊆ A ⟹ starfun f whn ∈ *s* A"
apply (rule_tac x="whn" in spec)
apply transfer
apply auto
done

text ‹
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 [simp]: "E ≠ {} ⟹ choicefun E ∈ E"
apply (unfold choicefun_def)
apply (rule lemmaPow3 [THEN someI2_ex], auto)
done

lemma injf_max_mem_set: "E ≠{} ⟹ ∀x. ∃y ∈ E. x < y ⟹ injf_max n E ∈ E"
apply (induct n)
apply force
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 (rule lemmaPow3 [THEN someI2_ex])
apply 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)
apply auto
apply (rule lemmaPow3 [THEN someI2_ex])
apply (drule_tac x = m in spec)
apply (drule subsetD)
apply auto
apply (drule_tac x = "injf_max m E" in order_less_trans)
apply auto
done

lemma inj_injf_max: "∀x. ∃y ∈ E. x < y ⟹ inj (λn. injf_max n E)"
apply (rule inj_onI)
apply (rule ccontr)
apply auto
apply (drule injf_max_order_preserving2)
apply (metis linorder_antisym_conv3 order_less_le)
done

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))"
for E :: "'a::order set" and f :: "nat ⇒ 'a"
apply (rule_tac x = "λn. injf_max n E" in exI)
apply safe
apply (rule injf_max_mem_set)
apply (rule_tac [3] inj_injf_max)
apply (rule_tac [4] injf_max_order_preserving)
apply 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])
apply auto
apply (drule inj_fun_not_hypnat_in_SHNat)
apply (drule range_subset_mem_starsetNat)
done

lemma starsetNat_eq_hypnat_of_nat_image_finite: "*s* A =  hypnat_of_nat ` A ⟹ finite A"
by (metis hypnat_infinite_has_nonstandard less_irrefl)

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 [simp]: "¬ (∀n ∈ - {0}. hypnat_of_nat n dvd 1)"
apply auto
apply (rule_tac x = 2 in bexI)
apply transfer
apply auto
done

lemma lemma_not_dvd_hypnat_one2 [simp]: "∃n ∈ - {0}. ¬ hypnat_of_nat n dvd 1"
using lemma_not_dvd_hypnat_one by (auto simp del: lemma_not_dvd_hypnat_one)

lemma hypnat_add_one_gt_one: "⋀N::hypnat. 0 < N ⟹ 1 < N + 1"
by transfer simp

lemma hypnat_of_nat_zero_not_prime [simp]: "hypnat_of_nat 0 ∉ starprime"
by transfer simp

lemma hypnat_zero_not_prime [simp]: "0 ∉ starprime"
using hypnat_of_nat_zero_not_prime by simp

lemma hypnat_of_nat_one_not_prime [simp]: "hypnat_of_nat 1 ∉ starprime"
by transfer simp

lemma hypnat_one_not_prime [simp]: "1 ∉ starprime"
using hypnat_of_nat_one_not_prime by simp

lemma hdvd_diff: "⋀k m n :: hypnat. k dvd m ⟹ k dvd n ⟹ k dvd (m - n)"
by transfer (rule dvd_diff_nat)

lemma hdvd_one_eq_one: "⋀x::hypnat. is_unit x ⟹ x = 1"
by transfer simp

text ‹Already proved as ‹primes_infinite›, but now using non-standard naturals.›
theorem not_finite_prime: "¬ finite {p::nat. prime p}"
apply (rule hypnat_infinite_has_nonstandard_iff [THEN iffD2])
using hypnat_dvd_all_hypnat_of_nat
apply clarify