src/HOL/Hyperreal/NatStar.thy

--- a/src/HOL/Hyperreal/NatStar.thy	Thu Apr 22 10:43:06 2004 +0200
+++ b/src/HOL/Hyperreal/NatStar.thy	Thu Apr 22 10:45:56 2004 +0200
@@ -533,6 +533,112 @@
 val starfunNat_inverse_real_of_nat_Infinitesimal = thm "starfunNat_inverse_real_of_nat_Infinitesimal";
 *}
 
+
+
+subsection{*Nonstandard Characterization of Induction*}
+
+constdefs
+
+  starPNat :: "(nat => bool) => hypnat => bool"  ("*pNat* _" [80] 80)
+  "*pNat* P == (%x. \<exists>X. (X \<in> Rep_hypnat(x) &
+                      {n. P(X n)} \<in> FreeUltrafilterNat))"
+
+
+  starPNat2 :: "(nat => nat => bool) => hypnat =>hypnat => bool"
+               ("*pNat2* _" [80] 80)
+  "*pNat2* P == (%x y. \<exists>X. \<exists>Y. (X \<in> Rep_hypnat(x) & Y \<in> Rep_hypnat(y) &
+                      {n. P(X n) (Y n)} \<in> FreeUltrafilterNat))"
+
+  hSuc :: "hypnat => hypnat"
+  "hSuc n == n + 1"
+
+
+lemma starPNat:
+     "(( *pNat* P) (Abs_hypnat(hypnatrel``{%n. X n}))) =
+      ({n. P (X n)} \<in> FreeUltrafilterNat)"
+by (auto simp add: starPNat_def, ultra)
+
+lemma starPNat_hypnat_of_nat [simp]: "( *pNat* P) (hypnat_of_nat n) = P n"
+by (auto simp add: starPNat hypnat_of_nat_eq)
+
+lemma hypnat_induct_obj:
+    "(( *pNat* P) 0 &
+            (\<forall>n. ( *pNat* P)(n) --> ( *pNat* P)(n + 1)))
+       --> ( *pNat* P)(n)"
+apply (cases n)
+apply (auto simp add: hypnat_zero_def hypnat_one_def starPNat, ultra)
+apply (erule nat_induct)
+apply (drule_tac x = "hypnat_of_nat n" in spec)
+apply (rule ccontr)
+apply (auto simp add: starPNat hypnat_of_nat_eq hypnat_add)
+done
+
+lemma hypnat_induct:
+  "[| ( *pNat* P) 0;
+      !!n. ( *pNat* P)(n) ==> ( *pNat* P)(n + 1)|]
+     ==> ( *pNat* P)(n)"
+by (insert hypnat_induct_obj [of P n], auto)
+
+lemma starPNat2:
+"(( *pNat2* P) (Abs_hypnat(hypnatrel``{%n. X n}))
+             (Abs_hypnat(hypnatrel``{%n. Y n}))) =
+      ({n. P (X n) (Y n)} \<in> FreeUltrafilterNat)"
+by (auto simp add: starPNat2_def, ultra)
+
+lemma starPNat2_eq_iff: "( *pNat2* (op =)) = (op =)"
+apply (simp add: starPNat2_def)
+apply (rule ext)
+apply (rule ext)
+apply (rule_tac z = x in eq_Abs_hypnat)
+apply (rule_tac z = y in eq_Abs_hypnat)
+apply (auto, ultra)
+done
+
+lemma starPNat2_eq_iff2: "( *pNat2* (%x y. x = y)) X Y = (X = Y)"
+by (simp add: starPNat2_eq_iff)
+
+lemma lemma_hyp: "(\<exists>h. P(h::hypnat)) = (\<exists>x. P(Abs_hypnat(hypnatrel `` {x})))"
+apply auto
+apply (rule_tac z = h in eq_Abs_hypnat, auto)
+done
+
+lemma hSuc_not_zero [iff]: "hSuc m \<noteq> 0"
+by (simp add: hSuc_def)
+
+lemmas zero_not_hSuc = hSuc_not_zero [THEN not_sym, standard, iff]
+
+lemma hSuc_hSuc_eq [iff]: "(hSuc m = hSuc n) = (m = n)"
+by (simp add: hSuc_def hypnat_one_def)
+
+lemma nonempty_nat_set_Least_mem: "c \<in> (S :: nat set) ==> (LEAST n. n  \<in> S)  \<in> S"
+by (erule LeastI)
+
+lemma nonempty_InternalNatSet_has_least:
+    "[| S \<in> InternalNatSets; S \<noteq> {} |] ==> \<exists>n \<in> S. \<forall>m \<in> S. n \<le> m"
+apply (auto simp add: InternalNatSets_def starsetNat_n_def lemma_hyp)
+apply (rule_tac z = x in eq_Abs_hypnat)
+apply (auto dest!: bspec simp add: hypnat_le)
+apply (drule_tac x = "%m. LEAST n. n \<in> As m" in spec, auto)
+apply (ultra, force dest: nonempty_nat_set_Least_mem)
+apply (drule_tac x = x in bspec, auto)
+apply (ultra, auto intro: Least_le)
+done
+
+text{* Goldblatt page 129 Thm 11.3.2*}
+lemma internal_induct:
+     "[| X \<in> InternalNatSets; 0 \<in> X; \<forall>n. n \<in> X --> n + 1 \<in> X |]
+      ==> X = (UNIV:: hypnat set)"
+apply (rule ccontr)
+apply (frule InternalNatSets_Compl)
+apply (drule_tac S = "- X" in nonempty_InternalNatSet_has_least, auto)
+apply (subgoal_tac "1 \<le> n")
+apply (drule_tac x = "n - 1" in bspec, safe)
+apply (drule_tac x = "n - 1" in spec)
+apply (drule_tac [2] c = 1 and a = n in add_right_mono)
+apply (auto simp add: linorder_not_less [symmetric] iff del: hypnat_neq0_conv)
+done
+
+
 end