author huffman Mon, 12 Sep 2005 23:13:46 +0200 changeset 17331 6b8b27894133 parent 17330 e007b307a09e child 17332 4910cf8c0cd2
updated to work with latest HOL-Complex
```--- a/src/HOL/Complex/ex/NSPrimes.thy	Mon Sep 12 23:07:58 2005 +0200
+++ b/src/HOL/Complex/ex/NSPrimes.thy	Mon Sep 12 23:13:46 2005 +0200
@@ -17,15 +17,15 @@
constdefs
hdvd  :: "[hypnat, hypnat] => bool"       (infixl "hdvd" 50)
"(M::hypnat) hdvd N == ( *p2* (op dvd)) M N"
-(*
-   "(M::hypnat) hdvd N ==
-	           \<exists>X Y. X: Rep_hypnat(M) & Y: Rep_hypnat(N) &
-                               {n::nat. X n dvd Y n} : FreeUltrafilterNat"
-*)
+
+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. \<exists>X \<in> Pow(E) -{{}}. x : X)"
@@ -102,7 +102,7 @@

(* 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 (unfold hdvd_def, transfer, rule dvd_by_all)
+by (transfer, rule dvd_by_all)

lemmas hdvd_by_all2 = hdvd_by_all [THEN spec, standard]

@@ -123,9 +123,7 @@
(* Goldblatt: Exercise 5.11(3a) - p 57  *)
lemma starprime:
"starprime = {p. 1 < p & (\<forall>m. m hdvd p --> m = 1 | m = p)}"
-apply (unfold starprime_def prime_def)
-apply (unfold hdvd_def, transfer, rule refl)
-done
+by (transfer, auto simp add: prime_def)

lemma prime_two:  "prime 2"
apply (unfold prime_def, auto)
@@ -149,10 +147,7 @@
(* Goldblatt Exercise 5.11(3b) - p 57  *)
lemma hyperprime_factor_exists [rule_format]:
"!!n. 1 < n ==> (\<exists>k \<in> starprime. k hdvd n)"
-apply (unfold starprime_def hdvd_def)
-apply (transfer prime_def)
-done

(* Goldblatt Exercise 3.10(1) - p. 29 *)
lemma NatStar_hypnat_of_nat: "finite A ==> *s* A = hypnat_of_nat ` A"
@@ -313,7 +308,7 @@
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 (unfold hdvd_def, transfer, auto)
+apply (transfer, auto)
done
declare lemma_not_dvd_hypnat_one [simp]

@@ -339,7 +334,7 @@
declare zero_not_prime [simp]

lemma hypnat_of_nat_zero_not_prime: "hypnat_of_nat 0 \<notin> starprime"
-by (unfold starprime_def, transfer prime_def, simp)
+by (transfer, simp)
declare hypnat_of_nat_zero_not_prime [simp]

lemma hypnat_zero_not_prime:
@@ -360,7 +355,7 @@
declare one_not_prime2 [simp]

lemma hypnat_of_nat_one_not_prime: "hypnat_of_nat 1 \<notin> starprime"
-by (unfold starprime_def, transfer prime_def, simp)
+by (transfer, simp)
declare hypnat_of_nat_one_not_prime [simp]

lemma hypnat_one_not_prime: "1 \<notin> starprime"
@@ -368,13 +363,13 @@
declare hypnat_one_not_prime [simp]

lemma hdvd_diff: "!!k m n. [| k hdvd m; k hdvd n |] ==> k hdvd (m - n)"
-by (unfold hdvd_def, transfer, rule dvd_diff)
+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 (unfold hdvd_def, transfer, rule dvd_one_eq_one)
+by (transfer, rule dvd_one_eq_one)

theorem not_finite_prime: "~ finite {p. prime p}"
apply (rule hypnat_infinite_has_nonstandard_iff [THEN iffD2])```