--- a/src/HOL/Nonstandard_Analysis/Examples/NSPrimes.thy Sun Dec 18 16:13:20 2016 +0100
+++ b/src/HOL/Nonstandard_Analysis/Examples/NSPrimes.thy Sun Dec 18 20:01:24 2016 +0100
@@ -4,280 +4,287 @@
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)
-section\<open>The Nonstandard Primes as an Extension of the Prime Numbers\<close>
+section \<open>The Nonstandard Primes as an Extension of the Prime Numbers\<close>
theory NSPrimes
-imports "~~/src/HOL/Number_Theory/Primes" "../Hyperreal"
+ imports "~~/src/HOL/Number_Theory/Primes" "../Hyperreal"
begin
-text\<open>These can be used to derive an alternative proof of the infinitude of
+text \<open>These can be used to derive an alternative proof of the infinitude of
primes by considering a property of nonstandard sets.\<close>
-definition
- starprime :: "hypnat set" where
- [transfer_unfold]: "starprime = ( *s* {p. prime p})"
+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)"
+definition choicefun :: "'a set \<Rightarrow> 'a"
+ where "choicefun E = (SOME x. \<exists>X \<in> Pow E - {{}}. x \<in> X)"
-primrec injf_max :: "nat => ('a::{order} set) => 'a"
+primrec injf_max :: "nat \<Rightarrow> 'a::order set \<Rightarrow> '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})"
-
+| injf_max_Suc: "injf_max (Suc n) E = choicefun ({e. e \<in> E \<and> injf_max n E < e})"
-lemma dvd_by_all2:
- fixes M :: nat
- shows "\<exists>N>0. \<forall>m. 0 < m \<and> m \<le> M \<longrightarrow> m dvd N"
-apply (induct M)
-apply auto
-apply (rule_tac x = "N * (Suc M) " in exI)
-apply auto
-apply (metis dvdI dvd_add_times_triv_left_iff dvd_add_triv_right_iff dvd_refl dvd_trans le_Suc_eq mult_Suc_right)
-done
+lemma dvd_by_all2: "\<exists>N>0. \<forall>m. 0 < m \<and> m \<le> M \<longrightarrow> m dvd N"
+ for M :: nat
+ apply (induct M)
+ apply auto
+ apply (rule_tac x = "N * Suc M" in exI)
+ apply auto
+ apply (metis dvdI dvd_add_times_triv_left_iff dvd_add_triv_right_iff dvd_refl dvd_trans le_Suc_eq mult_Suc_right)
+ done
-lemma dvd_by_all:
- "\<forall>M::nat. \<exists>N>0. \<forall>m. 0 < m \<and> m \<le> M \<longrightarrow> m dvd N"
+lemma dvd_by_all: "\<forall>M::nat. \<exists>N>0. \<forall>m. 0 < m \<and> m \<le> M \<longrightarrow> 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)
+lemma hypnat_of_nat_le_zero_iff [simp]: "hypnat_of_nat n \<le> 0 \<longleftrightarrow> n = 0"
+ by transfer 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 dvd N)"
-by (transfer, rule dvd_by_all)
+text \<open>Goldblatt: Exercise 5.11(2) -- p. 57.\<close>
+lemma hdvd_by_all: "\<forall>M. \<exists>N. 0 < N \<and> (\<forall>m::hypnat. 0 < m \<and> m \<le> M \<longrightarrow> m dvd N)"
+ by transfer (rule dvd_by_all)
lemmas hdvd_by_all2 = hdvd_by_all [THEN spec]
-(* Goldblatt: Exercise 5.11(2) - p. 57 *)
+text \<open>Goldblatt: Exercise 5.11(2) -- p. 57.\<close>
lemma hypnat_dvd_all_hypnat_of_nat:
- "\<exists>(N::hypnat). 0 < N & (\<forall>n \<in> -{0::nat}. hypnat_of_nat(n) dvd 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)
-done
+ "\<exists>N::hypnat. 0 < N \<and> (\<forall>n \<in> - {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)
+ apply (auto simp add: linorder_not_less)
+ done
-text\<open>The nonstandard extension of the set prime numbers consists of precisely
-those hypernaturals exceeding 1 that have no nontrivial factors\<close>
+text \<open>The nonstandard extension of the set prime numbers consists of precisely
+ those hypernaturals exceeding 1 that have no nontrivial factors.\<close>
-(* Goldblatt: Exercise 5.11(3a) - p 57 *)
-lemma starprime:
- "starprime = {p. 1 < p & (\<forall>m. m dvd p --> m = 1 | m = p)}"
-by (transfer, auto simp add: prime_nat_iff)
+text \<open>Goldblatt: Exercise 5.11(3a) -- p 57.\<close>
+lemma starprime: "starprime = {p. 1 < p \<and> (\<forall>m. m dvd p \<longrightarrow> m = 1 \<or> m = p)}"
+ by transfer (auto simp add: prime_nat_iff)
-(* Goldblatt Exercise 5.11(3b) - p 57 *)
-lemma hyperprime_factor_exists [rule_format]:
- "!!n. 1 < n ==> (\<exists>k \<in> starprime. k dvd n)"
-by (transfer, simp add: prime_factor_nat)
+text \<open>Goldblatt Exercise 5.11(3b) -- p 57.\<close>
+lemma hyperprime_factor_exists: "\<And>n. 1 < n \<Longrightarrow> \<exists>k \<in> starprime. k dvd n"
+ by transfer (simp add: prime_factor_nat)
-(* Goldblatt Exercise 3.10(1) - p. 29 *)
-lemma NatStar_hypnat_of_nat: "finite A ==> *s* A = hypnat_of_nat ` A"
-by (rule starset_finite)
+text \<open>Goldblatt Exercise 3.10(1) -- p. 29.\<close>
+lemma NatStar_hypnat_of_nat: "finite A \<Longrightarrow> *s* A = hypnat_of_nat ` A"
+ by (rule starset_finite)
-subsection\<open>Another characterization of infinite set of natural numbers\<close>
+subsection \<open>Another characterization of infinite set of natural numbers\<close>
-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: "finite N \<Longrightarrow> \<exists>n::nat. \<forall>i \<in> 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 = (\<exists>n. (\<forall>i \<in> N. i<(n::nat)))"
-by (blast intro: finite_nat_set_bounded bounded_nat_set_is_finite)
+lemma finite_nat_set_bounded_iff: "finite N \<longleftrightarrow> (\<exists>n::nat. \<forall>i \<in> N. i < n)"
+ 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 not_finite_nat_set_iff: "\<not> finite N \<longleftrightarrow> (\<forall>n::nat. \<exists>i \<in> N. n \<le> i)"
+ 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 bounded_nat_set_is_finite2: "\<forall>i::nat \<in> N. i \<le> n \<Longrightarrow> finite N"
+ apply (rule finite_subset)
+ apply (rule_tac [2] finite_atMost)
+ apply 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_bounded2: "finite N \<Longrightarrow> \<exists>n::nat. \<forall>i \<in> N. i \<le> 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 = (\<exists>n. (\<forall>i \<in> N. i<=(n::nat)))"
-by (blast intro: finite_nat_set_bounded2 bounded_nat_set_is_finite2)
+lemma finite_nat_set_bounded_iff2: "finite N \<longleftrightarrow> (\<exists>n::nat. \<forall>i \<in> N. i \<le> n)"
+ 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)
+lemma not_finite_nat_set_iff2: "\<not> finite N \<longleftrightarrow> (\<forall>n::nat. \<exists>i \<in> N. n < i)"
+ by (auto simp add: finite_nat_set_bounded_iff2 not_le)
-subsection\<open>An injective function cannot define an embedded natural number\<close>
+subsection \<open>An injective function cannot define an embedded natural number\<close>
-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 lemma_infinite_set_singleton:
+ "\<forall>m n. m \<noteq> n \<longrightarrow> f n \<noteq> f m \<Longrightarrow> {n. f n = N} = {} \<or> (\<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 \<longleftrightarrow> x = n")
+ apply auto
+ done
lemma inj_fun_not_hypnat_in_SHNat:
- assumes inj_f: "inj (f::nat=>nat)"
+ fixes f :: "nat \<Rightarrow> nat"
+ assumes inj_f: "inj f"
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 (auto simp: Nats_def)
+ then have "\<exists>n. starfun f n = hypnat_of_nat N" ..
+ then have "\<exists>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)
- thus "False"
+ then show 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
+lemma range_subset_mem_starsetNat: "range f \<subseteq> A \<Longrightarrow> starfun f whn \<in> *s* A"
+ apply (rule_tac x="whn" in spec)
+ apply transfer
+ apply auto
+ done
+
+text \<open>
+ Gleason Proposition 11-5.5. pg 149, pg 155 (ex. 3) and pg. 360.
-(*--------------------------------------------------------------------------------*)
-(* 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 *)
-(*--------------------------------------------------------------------------------*)
+ Let \<open>E\<close> be a nonvoid ordered set with no maximal elements (note: effectively an
+ infinite set if we take \<open>E = N\<close> (Nats)). Then there exists an order-preserving
+ injection from \<open>N\<close> to \<open>E\<close>. 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.
+\<close>
-lemma lemmaPow3: "E \<noteq> {} ==> \<exists>x. \<exists>X \<in> (Pow E - {{}}). x: X"
-by auto
+lemma lemmaPow3: "E \<noteq> {} \<Longrightarrow> \<exists>x. \<exists>X \<in> Pow E - {{}}. x \<in> X"
+ by auto
-lemma choicefun_mem_set [simp]: "E \<noteq> {} ==> choicefun E \<in> E"
-apply (unfold choicefun_def)
-apply (rule lemmaPow3 [THEN someI2_ex], auto)
-done
+lemma choicefun_mem_set [simp]: "E \<noteq> {} \<Longrightarrow> choicefun E \<in> E"
+ apply (unfold choicefun_def)
+ apply (rule lemmaPow3 [THEN someI2_ex], auto)
+ done
-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_mem_set: "E \<noteq>{} \<Longrightarrow> \<forall>x. \<exists>y \<in> E. x < y \<Longrightarrow> injf_max n E \<in> E"
+ apply (induct n)
+ apply force
+ apply (simp 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_preserving: "\<forall>x. \<exists>y \<in> E. x < y \<Longrightarrow> injf_max n E < injf_max (Suc n) E"
+ apply (simp add: choicefun_def)
+ apply (rule lemmaPow3 [THEN someI2_ex])
+ apply 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 injf_max_order_preserving2: "\<forall>x. \<exists>y \<in> E. x < y \<Longrightarrow> \<forall>n m. m < n \<longrightarrow> injf_max m E < injf_max n E"
+ apply (rule allI)
+ apply (induct_tac n)
+ apply auto
+ apply (simp 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)
+ apply auto
+ apply (drule_tac x = "injf_max m E" in order_less_trans)
+ apply 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 inj_injf_max: "\<forall>x. \<exists>y \<in> E. x < y \<Longrightarrow> inj (\<lambda>n. 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::('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
+ "E \<noteq> {} \<Longrightarrow> \<forall>x. \<exists>y \<in> E. x < y \<Longrightarrow> \<exists>f. range f \<subseteq> E \<and> inj f \<and> (\<forall>m. f m < f (Suc m))"
+ for E :: "'a::order set" and f :: "nat \<Rightarrow> 'a"
+ apply (rule_tac x = "\<lambda>n. 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\<open>Only need the existence of an injective function from N to A for proof\<close>
-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
+text \<open>Only need the existence of an injective function from \<open>N\<close> to \<open>A\<close> for proof.\<close>
-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 hypnat_infinite_has_nonstandard: "\<not> finite A \<Longrightarrow> 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])
+ apply auto
+ apply (drule inj_fun_not_hypnat_in_SHNat)
+ apply (drule range_subset_mem_starsetNat)
+ apply (auto simp add: SHNat_eq)
+ 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 starsetNat_eq_hypnat_of_nat_image_finite: "*s* A = hypnat_of_nat ` A \<Longrightarrow> finite A"
+ by (metis hypnat_infinite_has_nonstandard less_irrefl)
+
+lemma finite_starsetNat_iff: "*s* A = hypnat_of_nat ` A \<longleftrightarrow> 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
+lemma hypnat_infinite_has_nonstandard_iff: "\<not> finite A \<longleftrightarrow> 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\<open>Existence of Infinitely Many Primes: a Nonstandard Proof\<close>
-lemma lemma_not_dvd_hypnat_one [simp]: "~ (\<forall>n \<in> - {0}. hypnat_of_nat n dvd 1)"
-apply auto
-apply (rule_tac x = 2 in bexI)
-apply (transfer, auto)
-done
+subsection \<open>Existence of Infinitely Many Primes: a Nonstandard Proof\<close>
-lemma lemma_not_dvd_hypnat_one2 [simp]: "\<exists>n \<in> - {0}. ~ hypnat_of_nat n dvd 1"
-apply (cut_tac lemma_not_dvd_hypnat_one)
-apply (auto simp del: lemma_not_dvd_hypnat_one)
-done
+lemma lemma_not_dvd_hypnat_one [simp]: "\<not> (\<forall>n \<in> - {0}. hypnat_of_nat n dvd 1)"
+ apply auto
+ apply (rule_tac x = 2 in bexI)
+ apply transfer
+ apply auto
+ done
-lemma hypnat_add_one_gt_one:
- "!!N. 0 < N ==> 1 < (N::hypnat) + 1"
-by (transfer, simp)
+lemma lemma_not_dvd_hypnat_one2 [simp]: "\<exists>n \<in> - {0}. \<not> 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: "\<And>N::hypnat. 0 < N \<Longrightarrow> 1 < N + 1"
+ by transfer simp
lemma hypnat_of_nat_zero_not_prime [simp]: "hypnat_of_nat 0 \<notin> starprime"
-by (transfer, simp)
+ by transfer simp
-lemma hypnat_zero_not_prime [simp]:
- "0 \<notin> starprime"
-by (cut_tac hypnat_of_nat_zero_not_prime, simp)
+lemma hypnat_zero_not_prime [simp]: "0 \<notin> starprime"
+ using hypnat_of_nat_zero_not_prime by simp
lemma hypnat_of_nat_one_not_prime [simp]: "hypnat_of_nat 1 \<notin> starprime"
-by (transfer, simp)
+ by transfer simp
lemma hypnat_one_not_prime [simp]: "1 \<notin> starprime"
-by (cut_tac hypnat_of_nat_one_not_prime, simp)
+ 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_diff: "\<And>k m n :: hypnat. k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m - n)"
+ by transfer (rule dvd_diff_nat)
-lemma hdvd_one_eq_one:
- "\<And>x::hypnat. is_unit x \<Longrightarrow> x = 1"
+lemma hdvd_one_eq_one: "\<And>x::hypnat. is_unit x \<Longrightarrow> x = 1"
by transfer simp
-text\<open>Already proved as \<open>primes_infinite\<close>, but now using non-standard naturals.\<close>
-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
-apply (drule hypnat_add_one_gt_one)
-apply (drule hyperprime_factor_exists)
-apply clarify
-apply (subgoal_tac "k \<notin> hypnat_of_nat ` {p. prime p}")
-apply (force simp add: starprime_def)
- apply (metis Compl_iff add.commute dvd_add_left_iff empty_iff hdvd_one_eq_one hypnat_one_not_prime
- imageE insert_iff mem_Collect_eq not_prime_0)
-done
+text \<open>Already proved as \<open>primes_infinite\<close>, but now using non-standard naturals.\<close>
+theorem not_finite_prime: "\<not> finite {p::nat. prime p}"
+ apply (rule hypnat_infinite_has_nonstandard_iff [THEN iffD2])
+ using hypnat_dvd_all_hypnat_of_nat
+ apply clarify
+ apply (drule hypnat_add_one_gt_one)
+ apply (drule hyperprime_factor_exists)
+ apply clarify
+ apply (subgoal_tac "k \<notin> hypnat_of_nat ` {p. prime p}")
+ apply (force simp: starprime_def)
+ apply (metis Compl_iff add.commute dvd_add_left_iff empty_iff hdvd_one_eq_one hypnat_one_not_prime
+ imageE insert_iff mem_Collect_eq not_prime_0)
+ done
end