src/HOL/NSA/Examples/NSPrimes.thy

+(*  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" 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
+  hdvd  :: "[hypnat, hypnat] => bool"       (infixl "hdvd" 50) where
+  [transfer_unfold]: "(M::hypnat) hdvd N = ( *p2* (op dvd)) M N"
+
+definition
+  starprime :: "hypnat set" where
+  [transfer_unfold]: "starprime = ( *s* {p. prime p})"
+
+definition
+  choicefun :: "'a set => 'a" where
+  "choicefun E = (@x. \<exists>X \<in> 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})"
+
+
+lemma dvd_by_all: "\<forall>M. \<exists>N. 0 < N & (\<forall>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 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: "\<forall>M. \<exists>N. 0 < N & (\<forall>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:
+     "\<exists>(N::hypnat). 0 < N & (\<forall>n \<in> -{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 & (\<forall>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 --> (\<exists>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 ==> (\<exists>k \<in> 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"
+by (rule starset_finite)
+
+
+subsection{*Another characterization of infinite set of natural numbers*}
+
+lemma finite_nat_set_bounded: "finite N ==> \<exists>n. (\<forall>i \<in> 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 = (\<exists>n. (\<forall>i \<in> N. i<(n::nat)))"
+by (blast intro: finite_nat_set_bounded bounded_nat_set_is_finite)
+
+lemma not_finite_nat_set_iff: "(~ finite N) = (\<forall>n. \<exists>i \<in> N. n <= (i::nat))"
+by (auto simp add: finite_nat_set_bounded_iff not_less)
+
+lemma bounded_nat_set_is_finite2: "(\<forall>i \<in> 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 ==> \<exists>n. (\<forall>i \<in> 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 = (\<exists>n. (\<forall>i \<in> N. i<=(n::nat)))"
+by (blast intro: finite_nat_set_bounded2 bounded_nat_set_is_finite2)
+
+lemma not_finite_nat_set_iff2: "(~ finite N) = (\<forall>n. \<exists>i \<in> N. n < (i::nat))"
+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: "\<forall>m n. m \<noteq> n --> f n \<noteq> f m
+      ==>  {n. f n = N} = {} |  (\<exists>m. {n. f n = N} = {m})"
+apply auto
+apply (drule_tac x = x in spec, auto)
+apply (subgoal_tac "\<forall>n. (f n = f x) = (x = n) ")
+apply auto
+done
+
+lemma inj_fun_not_hypnat_in_SHNat:
+  assumes inj_f: "inj (f::nat=>nat)"
+  shows "starfun f whn \<notin> Nats"
+proof
+  from inj_f have inj_f': "inj (starfun f)"
+    by (transfer inj_on_def Ball_def UNIV_def)
+  assume "starfun f whn \<in> Nats"
+  then obtain N where N: "starfun f whn = hypnat_of_nat N"
+    by (auto simp add: Nats_def)
+  hence "\<exists>n. starfun f n = hypnat_of_nat N" ..
+  hence "\<exists>n. f n = N" by transfer
+  then obtain n where n: "f n = N" ..
+  hence "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)
+  thus "False"
+    by (simp add: whn_neq_hypnat_of_nat)
+qed
+
+lemma range_subset_mem_starsetNat:
+   "range f <= A ==> starfun f whn \<in> *s* A"
+apply (rule_tac x="whn" in spec)
+apply (transfer, 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 \<noteq> {} ==> \<exists>x. \<exists>X \<in> (Pow E - {{}}). x: X"
+by auto
+
+lemma choicefun_mem_set: "E \<noteq> {} ==> choicefun E \<in> E"
+apply (unfold choicefun_def)
+apply (rule lemmaPow3 [THEN someI2_ex], auto)
+done
+declare choicefun_mem_set [simp]
+
+lemma injf_max_mem_set: "[| E \<noteq>{}; \<forall>x. \<exists>y \<in> E. x < y |] ==> injf_max n E \<in> 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: "\<forall>x. \<exists>y \<in> 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: "\<forall>x. \<exists>y \<in> E. x < y
+      ==> \<forall>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: "\<forall>x. \<exists>y \<in> E. x < y ==> inj (%n. injf_max n E)"
+apply (rule inj_onI)
+apply (rule ccontr, auto)
+apply (drule injf_max_order_preserving2)
+apply (metis linorder_antisym_conv3 order_less_le)
+done
+
+lemma infinite_set_has_order_preserving_inj:
+     "[| (E::('a::{order} set)) \<noteq> {}; \<forall>x. \<exists>y \<in> E. x < y |]
+      ==> \<exists>f. range f <= E & inj (f::nat => 'a) & (\<forall>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 \<noteq> {}")
+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: "~ (\<forall>n \<in> - {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: "\<exists>n \<in> - {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 \<noteq> 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: "\<not> 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 \<notin> starprime"
+by (transfer, simp)
+declare hypnat_of_nat_zero_not_prime [simp]
+
+lemma hypnat_zero_not_prime:
+   "0 \<notin> starprime"
+by (cut_tac hypnat_of_nat_zero_not_prime, simp)
+declare hypnat_zero_not_prime [simp]
+
+lemma one_not_prime: "\<not> prime 1"
+apply safe
+apply (drule prime_g_one, auto)
+done
+declare one_not_prime [simp]
+
+lemma one_not_prime2: "\<not> 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 \<notin> starprime"
+by (transfer, simp)
+declare hypnat_of_nat_one_not_prime [simp]
+
+lemma hypnat_one_not_prime: "1 \<notin> 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
+apply (subgoal_tac "k \<notin> 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