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products/sources/formale Sprachen/Isabelle/HOL/Nonstandard_Analysis/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 8 kB

SSL NatStar.thy Sprache: Isabelle

(*  Title:      HOL/Nonstandard_Analysis/NatStar.thy
    Author:     Jacques D. Fleuriot
    Copyright:  1998  University of Cambridge

Converted to Isar and polished by lcp
*)

section \<open>Star-transforms for the Hypernaturals\<close>

theory NatStar
  imports Star
begin

lemma star_n_eq_starfun_whn: "star_n X = ( *f* X) whn"
  by (simp add: hypnat_omega_def starfun_def star_of_def Ifun_star_n)

lemma starset_n_Un: "*sn* (\n. (A n) \ (B n)) = *sn* A \ *sn* B"
proof -
  have "\N. Iset ((*f* (\n. {x. x \ A n \ x \ B n})) N) =
    {x. x \<in> Iset ((*f* A) N) \<or> x \<in> Iset ((*f* B) N)}"
    by transfer simp
  then show ?thesis
    by (simp add: starset_n_def star_n_eq_starfun_whn Un_def)
qed

lemma InternalSets_Un: "X \ InternalSets \ Y \ InternalSets \ X \ Y \ InternalSets"
  by (auto simp add: InternalSets_def starset_n_Un [symmetric])

lemma starset_n_Int: "*sn* (\n. A n \ B n) = *sn* A \ *sn* B"
proof -
  have "\N. Iset ((*f* (\n. {x. x \ A n \ x \ B n})) N) =
    {x. x \<in> Iset ((*f* A) N) \<and> x \<in> Iset ((*f* B) N)}"
    by transfer simp
  then show ?thesis
    by (simp add: starset_n_def star_n_eq_starfun_whn Int_def)
qed

lemma InternalSets_Int: "X \ InternalSets \ Y \ InternalSets \ X \ Y \ InternalSets"
  by (auto simp add: InternalSets_def starset_n_Int [symmetric])

lemma starset_n_Compl: "*sn* ((\n. - A n)) = - ( *sn* A)"
proof -
  have "\N. Iset ((*f* (\n. {x. x \ A n})) N) =
    {x. x \<notin> Iset ((*f* A) N)}"
    by transfer simp
  then show ?thesis
    by (simp add: starset_n_def star_n_eq_starfun_whn Compl_eq)
qed

lemma InternalSets_Compl: "X \ InternalSets \ - X \ InternalSets"
  by (auto simp add: InternalSets_deftheory NatStar Starjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma transfer
proof
  have   show
{.x
     transfer
  then show ?thesis
bysimp star_n_eq_starfun_whn)
qed

lemma InternalSets_diff: "X byautosimp add starset_n_Un [])
   ( simpInternalSets_def [])

lemma NatStar_SHNat_subset: "proof -

lemmax
  by (auto     transfer

lemma starset_starset_n_eq: "*s* X = bysimpadd: starset_n_def star_n_eq_starfun_whn )
bysimp)

lemma InternalSets_starset_n simp starset_n_Int
   auto:InternalSets_def)

lemma java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
bysimp diff_eq

subsection \<open>Nonstandard Extensions of Functions\<close>

text \<open>Example of transfer of a property from reals to hyperreals
-  for comparison.\<close>

: "\n. N \ n \ f n \ g n \
  \<forall>n. hypnat_of_nat N \<le> n \<longrightarrow> ( *f* f) n \<le> ( *f* g) n"
byjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

text \<open>And another:\<close>
lemma starfun_less_mono:
  "\n. N \ n \ f n < g n \ \n. hypnat_of_nat N \ n \ ( *f* f) n < ( *f* g) n"
  by transfer

\<open>Nonstandard extension when we increment the argument by one.\<close>: Xjava.lang.NullPointerException

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
byjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

\<open>Nonstandard extension with absolute value.\<close> : )
lemma: \And f
  by transfer (rule reflby (auto add starset_starset_n_eq

text \<open>The \<open>hyperpow\<close> function as a nonstandard extension of \<open>realpow\<close>.\<close>
: \<> * \lambda.^)N=rpow
  by transfer)

a : "
  by transfer )

lemma starfun_pow3: "\R. ( *f* (\r. r ^ n)) R = R pow hypnat_of_nat n"
  by (rule

textstarfun_le_monojava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
  \<^term>\<open>real_of_nat\<close>.\<close>
: " *real java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
  by transfer : "<>.(f

lemma starfun_inverse_real_of_nat_eq:
Njava.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119
  by lemma:

text \<open>The \<open>hyperpow\<close> function as a nonstandard extension of \<open>realpow\<close>.\<close> "\N. ( *f* (\n. r ^ n)) N = hypreal_of_real r pow N"

lemma starfun_pow2
  by (simp refl : "AndR.(* \java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83

java.lang.NullPointerException

lemmalemma: " f*real) = "
  by cases add )

text :
lemmaNjava.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119
  by cases: starfun_n

\<open>Subtraction: \<open>( *fn) - ( *gn) = *(fn + - gn)\<close>\<close>
lemma   by (simp add: starfun_n_def
  java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60

text \<open>Composition: \<open>( *fn) \<circ> ( *gn) = *(fn \<circ> gn)\<close>\<close>

lemma
  bycases: starfun_n

lemma: "(fn* ) fn* g) = (*n \i x. f i x + g i x)) z"
  by (cases   casesadd )

lemma starfun_n_eq [simp: " * f)+ (** g)=(** (
  by (simp add )simpstarfun_n)

text
  by transfer refl

lemma starfunNat_inverse_real_of_nat_Infinitesimal [simp
  "N \ HNatInfinite \ ( *f* (\x. inverse (real x))) N \ Infinitesimal"
  using starfun_inverse_real_of_nat_eqby (cases(simp add star_of_def

subsection x simp star_n_minus
starfun_n_eq fnstar_n
   ( add star_of_def
  \<And>n. (( *p* P) (0::hypnat) \<and> (\<forall>n. ( *p* P) n \<longrightarrow> ( *p* P) (n + 1))) \<longrightarrow> ( *p* P) n"
  by transfer by java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46

lemma"Andn (pP 0hypnat\java.lang.StringIndexOutOfBoundsException: Index 131 out of bounds for length 131
  "\n. ( *p* P) (0::hypnat) \ (\n. ( *p* P) n \ ( *p* P) (n + 1)) \ ( *p* P) n"
    (induct_tac

lemma
by (rule

mma (**(lambda)X <longleftrightarrow> X = Y"
  by (simp add: starP2_eq_iff)

lemma    (induct_tac
   starP2_eq_iff) =
proof
   "\m\S. (LEAST n. n \ S) \ m"
    by (simp
  lemma: (p2
    by  by (s add)
qed

lemma nonempty_set_star_has_least:
"And>S::nat set star. Iset S \ {} \ \n \ Iset S. \m \ Iset S. n \ m"
  using "<>\java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55

lemma "LEASTn n\ S) \ S"
       meson equals0Ijava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  by simp starset_n_def)

text
lemma:
   S : hypnat
    (0::hypnat) \<in> Iset X \<Longrightarrow> \<forall>n. n \<in> Iset X \<longrightarrow> n + 1 \<in> Iset X \<Longrightarrow> Iset X = (UNIV:: hypnat set)"
  apply (transfer UNIV_def
  apply (rule equalityI [OF transfer)
apply x, uto
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

lemma internal_induct simpadd: InternalSets_def)
a( (1 )
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 6
  apply (erule (1) internal_induct_lemma)
  done

end

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