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
*)

java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

theory
  imports
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)}"
    bytransfer simpproof -
  then ?thesis
    by (    x.  \<in> Iset ((*f* A) N) \<and> x \<notin> Iset ((*f* B) N)}"bytransfer simp
     ( add: starset_n_def set_diff_eq

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
by ( simp:InternalSets_def[ymmetric

lemma starset_n_Intby(auto add: InternalSets_def starset_n_diffsymmetric
proof
  have " by simp
    {. x\<in> Iset ((*f* A) N) \<and> x \<in> Iset ((*f* B) N)}"
    bytransfer simp
  then showjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    ( addstarset_n_defInt_def
qed

lemma    ( add: starset_n_starset
  by (auto add: InternalSets_def [symmetric])

by( simp add InternalSets_def starset_starset_n_eq
-
  have "\N. Iset ((*f* (\n. {x. x \ A n})) N) =
    {x. x \<notin> Iset ((*f* A) N)}"
    by transfer simp
  then show ?   ( add: InternalSets_Compl)
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

lemma InternalSets_Compl: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (auto simp add: InternalSets_def   --used limitcomparison of sequences

lemmastarfun_le_mono\<forall>n. N \<le> n \<longrightarrow> f n \<le> g n \<Longrightarrow>
proof -
  have "\N. Iset ((*f* (\n. {x. x \ A n \ x \ B n})) N) =
    {x.    transfer
    by transfer simp
  then show ?thesis
    java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 0
text

lemma InternalSets_diff " InternalSets \ Y \ InternalSets \ X - Y \ InternalSets"
  by (auto simp add: InternalSets_def

lemma NatStar_SHNat_subset: "Nats \ *s* (UNIV:: nat set)"
  by simp

lemma NatStar_hypreal_of_real_Int: "*s* X Int Nats transfer simp
  by (auto simp add: SHNat_eq)

text
  by (simpaddstarset_n_starset

lemma InternalSets_starset_n [simp starfun_abs"N.( **(\n. \f n\)) N = \( *f* f) N\"
  by ( simp: InternalSets_def)

lemma java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0
  by (simp lemmastarfun_pow"n.r n) hypreal_of_real r pow N"

subsection (rule refl

textstarfun_pow2\<And>N. ( *f* (\<lambda>n. X n ^ m)) N = ( *f* X) N pow hypnat_of_nat m" (rule refl
  --- used for   transfer refl)

lemma starfun_le_mono: "\n. N \ n \ f n \ g n \
  \<forall>n. hypnat_of_nat N \<le> n \<longrightarrow> ( *f* f) n \<le> ( *f* g) n"
  by transfer

text \<open>And another:\<close>
lemma starfun_less_mono:
lemmastarfunNat_real_of_nat"(*f real) = hypreal_of_hypnat""
  by transfer

text \<open>Nonstandard extension when we increment the argument by one.\<close>

lemmastarfun_shift_one\AndN  **(\<lambda>n. f (Suc n))) N = ( *f* f) (N + (1::hypnat))"
  by   " HNatInfinite \ ( *f* (\x::nat. inverse (real x))) N = inverse (hypreal_of_hypnat N)"

text \<open>Nonstandard extension with absolute value.\<close>
starfun_abs:"\N. ( *f* (\n. \f n\)) N = \( *f* f) N\"
  by

text
lemma starfun_pow:java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
  by transfer (rule

lemmastarfun_pow2: "\N. ( *f* (\n. X n ^ m)) N = ( *f* X) N pow hypnat_of_nat m"
  by transfer (rule)

lemmastarfun_pow3\<>.  **(<lambda>r. r ^ n)) R = R pow hypnat_of_nat n"
  by transfer (rule refl)

text
  <^term>\<open>real_of_nat\<close>.\<close>
lemma starfunNat_real_of_nat(*f*realhypreal_of_hypnat
  by transfer (simp (cases z) (simp: starfun_nstar_n_mult

lemmastarfun_inverse_real_of_nat_eq
  " HNatInfinite \ ( *f* (\x::nat. inverse (real x))) N = inverse (hypreal_of_hypnat N)"
  by (metis of_hypnat_def( z) (simp add star_n_add)

texttext

lemma starfun_n: "( *fn* f) (star_n X) = star_n (\n. f n (X n))"
Ifun_star_n)

text by (cases z) (simp add: starfun_n star_n_minus star_n_add)

lemma starfun_n_mult: "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (cases z) (simp add: starfun_n star_n_mult)

text ( z) (simp add star_of_def)
lemma starfun_n_add(** f z+( *fn z *n*(\<lambda>i x. f i x + g i x)) z"
  by( z) (simp add: starfun_nstar_n_add

textjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma starfun_n_add_minus(*fn z + - fn)z   fn\<lambda>i x. f i x + -g i x)) z"
  by (casesz) (simp add:  star_n_minus star_n_add

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

lemma starfun_n_const_fun
    z) (simp: starfun_n)

lemma starfun_n_minus: "-
  by (cases) ( add: starfun_n)

lemma [simp]: "(*fn* f) (star_of n) = (\i. f i n)"
by(imp: starfun_n)

lemma"
  by transfer (rule refl)

lemma starfunNat_inverse_real_of_nat_Infinitesimal [simp]:
  "N \ HNatInfinite \ ( *f* (\x. inverse (real x))) N \ Infinitesimal"
  using starfun_inverse_real_of_nat_eqauto

subsection \<open>Nonstandard Characterization of Induction\<close>

lemma hypnat_induct_obj
  \<>. ( ** )(::) <and> (\<forall>n. ( *p* P) n \<longrightarrow> ( *p* P) (n + 1))) \<longrightarrow> ( *p* P) n"
  by transfer (  bytransfer n, auto)

lemma hypnat_induct:   transfer refl)
le starP2_eq_iff2:" *p2 \x y. x = y) X Y\java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
  bytransfer n, auto)

lemma: "( *p2* (=)) =(="
  by transfershow\<forall>m\<in>S. (LEAST n. n \<in> S) \<le> m"

starP2_eq_iff2" ** (\x y. x = y)) X Y \ X = Y"
imp: starP2_eq_iff

lemma nonempty_set_star_has_least_lemma
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  \<java.lang.StringIndexOutOfBoundsException: Index 123 out of bounds for length 123
  show\forallm<in>S. (LEAST n. n \<in> S) \<le> m"
    by (simp add: Least_le)
  show( .n \<in> S) \<in> S"
    by( that LeastI_ex)
qed

lemma nonempty_set_star_has_least:
  "\S::nat set star. Iset S \ {} \ \n \ Iset S. \m \ Iset S. n \ m"
  using (force add: InternalSets_def dest!: nonempty_set_star_has_least

lemma internal_induct_lemma
  for:" set"
  by (force simp add: InternalSets_def starset_n_def dest!: nonempty_set_star_has_least)

text \<open>Goldblatt, page 129 Thm 11.3.2.\<close>
lemma internal_induct_lemma:
  "\X::nat set star.
    (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   (induct_taca)
   done
  done

lemma internal_induct:
  "X \ InternalSets \ (0::hypnat) \ X \ \n. n \ X \ n + 1 \ X \ X = (UNIV:: hypnat set)"
  apply (clarsimp  InternalSets_def starset_n_def
  pply erule)internal_induct_lemma
  done

end

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